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V . t”.v.’.r.!.v \‘c’nu‘ r ’10-:II’I'I t! . . £544.11 :4 519.239.. _ . {gait dilute 'niversity ‘ IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIOIIIIIII 3 1293 103859 This is to certify that the thesis entitled TWO-DIMENSIONAL POTENTIAL FLOW AND BOUNDARY LAYER ANALYSIS OF THE AIRFOIL OF A STOL WING PROPULSION SYSTEM presented by JAMES ARTHUR ALBERS has been accepted towards fulfillment of the requirements for P/70/ degreeinmcé’ [2?- Wfll’z Major professor Date f////7/ 0.7639 ABSTRACT TWO-DIMENSIONAL POTENTIAL FLOW AND BOUNDARY LAYER ANALYSIS ON THE AIRFOIL OF A STOL WING PROPULSEON SYSTEM By James Arthur Albers The analysis considers a two—dimensional wingufan system which consists of an airfoil with flap; the fans which have a distributed suction at their inlet and a jet at their exit; and a jet sheet leav- ing the flap trailing edge. The solution provides the incompressible potential flow for variable fan or engine mass flow coefficient, the thrust coefficient for the propulsion system exhaust, and the wing and flap angle of attack. This includes the approximate location of the free exhaust jet. This potential flow solution is used as an input to the boundary layer analysis which calculates both laminar and turbulent incompressible boundary layer parameters. In particular, the separa— tion point is determined on the airfoil of a blown flap wing propulsion system at various angles of attack. The calculated pressure distributions for a particular externally blown flap configuration indicated that the minimum pressure point is near the leading edge (less than 2 percent of chord) of the airfoil with severe adverse pressure gradients at high angles of attack (near 0 20 )- The results Of the boundary layer analysis indicated that the James Arthur Albers predicted turbulent separation point moved forward from the trailing edge as the angle of attack was increased. Trailing edge separation for the thick wing (t/c = 0.15) propulsion system combination consid- ered was verified by experimental data. TWO-DIMENSIONAL POTENTIAL FLOW AND BOUNDARY LAYER ANALYSIS OF THE AIRFOIL OF A STOL WING PROPULSION SYSTEM James Arthur Albers A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1971 ’ ”7 c7r”' ~ )/ -- 3/ PLEASE NOTE: Some Pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS To my Mother and Father, Theresa, and J. J. ii ACKNOWLEDGMENTS The author wishes to thank his advisor, Dr. Merle C. Potter, for his guidance and assistance throughout this study. His active par- ticipation in the project and his willingness to discuss problems as they arose made working with him a rewarding experience. The author also wishes to thank the other members of his guidance committee for their time and interest in this study: Dr. Mahlon C. Smith, Dr. James V. Beck, and Dr. Norman L. Hills. The author appreciates the assistance and guidance of Roger W. Luidens of NASA Lewis Research Center for his many constructive criti- cisms during the preparation of this dissertation. I would also like to acknowledge Newell D. Sanders of NASA Lewis Research Center for his continued support throughout the study. The financial support of NASA Lewis Research Center's Training Office made the continuation of graduate study possible. I would like to give special thanks to Gertrude R. Collins of the NASA Training Office for her assistance throughout the program. Special appreciation is extended to the author's wife, Theresa Ann, for her encouragement and understanding throughout the study. iii TABLE OF CONTENTS Page LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . vi S WRY O O O O O O O O l 0 O O O 0 O a O ‘1 u b 0 v 8 0 (v G O 0 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 3 POTENTIAL FLOW ANALYSIS . . . . . . . . . . . . . . . . . . . . 9 Representation of Wing Propulsion System. . . . . . . . . . . 9 Basic Equations and Boundary Conditions . . . . . . . . . . 11 Formulation of the Boundary Conditions as an Integral Equation. . . . . . . . . . . . . . . . . . . . . . . . . 12 Solution of Integral Equation . . . . . . . . . . . . . . . . 14 Basic Solutions . . . . . . . . . . . . . . . . . . . . . . . 16 Combination Solution. . . . . . . . . . . . . . . . . . . . 17 Location of Propulsion System Exhaust Jet . . . . . . . . . . 18 Potential Flow Computer Program . . . . . . . . . . . . . . . 20 DISCUSSION AND RESULTS OF POTENTIAL FLOW ANALYSIS . . . . . . . 21 Validity of Analysis. . . . . . . . . . . . . . . . . . . . . 21 Inlet air suction . . . . . . . . . . . . . . . . . . . . . 21 Exhaust jet shape . . . . . . . . . . . . . . . . . . . . . 22 Example Applications. . . . . . . . . . . . . . . . . . . . . 23 Flow field. . . . . . . . . . . . . . . . . . . . . . . . . 23 Pressure distribution . . . . . . . . . . . . . . . . . . . 24 Lift coefficients . . . . . . . . . . . . . . . . . . . . . 26 BOUNDARY LAYER ANALYSIS . . . . . . . . . . . . . . . . . . . . 29 Basic Equations of Motion . . . . . . . . . . . . . . . . . . 29 Transformed Equation of Motion. . . . . . . . . . . . . . . . 32 Effective Viscosity Hypothesis. . . . . . . . . . . . . . . . 33 Solution of the Boundary Layer Equation . . . . . . . . . . . 37 iv DISCUSSION OF RESULTS OF BOUNDARY LAYER ANALYSIS. . . . . . . . 4O Laminar Boundary Layer Growth . . . . . . . . . . . . . . . . 40 TranSition. O O O 0 0 O O O O I O O O O O O 0 O O O O O O O O 41 Turbulent Boundary Layer Growth and Separation. . . . . . . . 43 Boundary layer parameters . . . . . . . . . . . . . . . . . 43 velOCity PrOfileS o o o o o o c , c o o o o s o o o u o o o 44 Consequences of separation. . . _ . . . . . . . . . . . . . 45 CONCLUDING REMARKS. . . . . . . . . . . . . . . . . . . . . . . 48 APPENDICES A - SYMBOLS FOR POTENTIAL FLOW ANALYSIS. . . . . . . . . . . 51 B - VELOCITIES IN TERMS OF SOURCE DENSITIES. . . . . . . . . 54 C — COMPUTATION OF FLOW QUANTITIES FOR POTENTIAL FLOW ANALYSIS 0 O O O O O 0 O O O O I 0 O 0 0 o I i O O O 0 60 D - POTENTIAL FLOW COMPUTER PROGRAM. . . . . . . . . . . . . 62 E - SYMBOLS FOR BOUNDARY LAYER ANALYSIS. . . . . . . . . . . 92 F - DERIVATION OF TRANSFORMED BOUNDARY LAYER EQUATION OF MOTION. O O O I O O O I O 0 O O O 0 O O O O O O O O 95 REFERENCES 0 O O O O O O 0 O 0 O O O O C O 0 O 0 0 O 0 O O O O O 101 ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 106 Figure 10. LIST OF FIGURES Types of wing-flap systems considered . (a) Jet flap airfoil (b) Externally blown flap wing section Lewis wind tunnel model of multiple-fan blown flap wing propulsion system (a)‘Schematic of model (b) Two-dimensional representation of wing propulsion system Representation of boundary condition on body surface Finite element approximation to body surface Basic solutions of potential flow . (a) O0 uniform flow solution V0 (b) 900 uniform flow solution, V90 (c) Vbrtex solution, VV (Voo = o) (d) Suction solution, VS (vg— — 0) Basic solutions for inlet . . (a) 00 solution with duct closed; V1 (b) 00 solution with duct open, V2 1 (0) Cross flow solution with duct open, V5 Two-dimensional inlet configuration . Comparisons of theoretical velocity distributions with experimental data for two-dimensional inlet (a) Surface velocity distributions (b) Centerline velocity distributions Effect of jet shapes on the upper surface pressure distribution (flap angle,300; wing angle of attack, 00) (a) Illustration of jet shapes (b) Upper surface pressure distributions Comparison of theoretical nondimensional jet shapes (flap angle, BOO; thrust coefficient, 3) vi Page 106 107 109 110 111 112 115 114 115 116 Flow field for externally blown flap wing propulsion system (mass flow coefficient 0.58; thrust coefficient, 5). . . . . . . . . . . . . . . . 117 (a) Wing angle of attack, O0 3f1ap angle, 500 (b) Wing angle of attack, 200;f1ap angle, 500 (c) Wing angle of attack, 00; flap angle, 600 Calculated pressure distributions on upper surface for fan-wing combination for various angles of attack (mass flow coefficient, 0.580; thrust coefficient, 3; flap angle, 500) . . . . . . . . . . . . . 118 Effect of suction and jet on pressure distribution (flap angle, 300; wing angle of attack, 00) . . . . . . . . 119 Comparison of theoretical two-dimensional lift coefficient for blown flap (thrust coefficient, 5) . . . . 120 (a) Flap angle, 300 (b) Flap angle, 600 Comparison of calculated and experimental three- dimensional lift coefficients for blown flap (thrust coefficient, 5) . . . . . . . . . . . . . . . . . . 121 (a) Flap angle, 500 (b) Flap angle, 600 Illustration of notation for boundary layer analysis . . . 122 (a) Coordinate system (b) Description of velocity profile The turbulent effective viscosity hypothesis . . . . . . . 123 Calculated velocity distributions for fan-wing com- bination for various angles of attack (mass flow coefficient, 0.58; jet momentum coefficient, 3; flap angle, 500) . . . . . . . . . . . . . . . . . . . . . . . . 124 Laminar boundary layer parameters on the airfoil of a blown flap wing propulsion system (wing angle of attack, 150) . . . . . . . . . . . . . . . . . . . . . . 125 Turbulent boundary layer parameters on the airfoil of a blown flap wing propulsion system.. . . . . . . . . . . . 126 Velocity profiles at start of turbulent boundary layer growth . . . . . . . . . . . . . . . . . . . . . . . 127 Vll 22. 25. 24. 25. 26. 27. Turbulent foil of a angles of ) (b) (c ) (d) (e) The Angle Angle Angle Angle Angle boundary layer velocity profiles on the air- blown flap wing propulsion systems at various attack . . . . . . . . . . . . . . . . . . . of attack, 00 of attack, 15° of attack, 200 of attack, 250 of attack, 500 stalling characteristics of airfoils . . . . . . . . The stalling characteristics of blown flap wing propulsion system . . . . . . . . . . . . . . . (a) Experimental lift curve (b) Location of separation point for various angles of attack Notation for two-dimensional potential flows . . . . . . (a) Three-dimensional illustration (b) Two—dimensional cross section Element of body surface . . . . . . . . . . . . . . . . Schematic representation of computer program . viii 128 129 150 155 155 SUMMARY The analysis considers a tweedimensional combined wing and pro- pulsion system which consists of a flapped airfoil with fans located under the wing. The exhaust jet of the fans impacts the flap and is deflected downward. A numerical method is used which includes the effect of suction at the inlet of the propulsion system and treats the constant thick- ness exhaust jet as part of the solid body. This method includes the determination of the approximate exhaust jet location. The method provides the potential flow solution for any fan or engine mass flow coefficient, the thrust coefficient for the propulsion system exhaust, the flap deflection angle, and the wing angle of attack. Validity of the numerical solution for a case with suction (but with no jet) was indicated by application of the program to a two-dimensional inlet; excellent agreement was found with experimental results. The potential flow program was used to obtain the pressure dis— tribution, velocity field, and lift coefficient for a particular ex- ternally blown flap, high-lift configuration. The flow field for this configuration indicated high upwash angles (600 to 900) at the pro- pulsion system inlet and large jet penetrations at high angles of attack. A comparison of two—dimensional lift coefficients obtained by the method of this report with Spence's jet flap theory indicated that the method of this report yielded lift coefficients that were an average of 10.5 and 12.1 percent higher in the 30° and 60° flap angles, respectively. A comparison of three-dimensional lift coef— ficients with experimental data for the blown flap indicated good agreement for the 300 flap, with the predicted lift coefficient an average of 11.4 percent higher than experimental data. Calculated pressure distributions showed severe adverse pressure gradients over a large portion of the wing at angles of attack of 200 or greater. The surface velocity distribution obtained from the potential flow solution was used to determine the boundary layer growth and separation on the upper surface of the airfoil. The boundary layer solution was obtained by reduction of the partial differential equa— tions of motion to a set of ordinary differential equations at each x—location using finite differences for the x derivatives. By an iterative solution to the differential equations, the boundary layer parameters for both laminar and turbulent flow were found. The re— sults indicated that the predicted turbulent separation point moved forward from the trailing edge as the angle of attack is increased. Trailing edge separation for the thick wing (t/c = 0.15) propulsion system combination considered was verified by the experimental data. 1- _.._..J.. INTRODUCTION In recent years there has been much interest in short-takeoff- and-landing (STOL) aircraft for both civil and military applications. A STOL airplane must have the capability for both high lift at take- off and low drag at cruise. Past experimental work (refs. 1 to 3) demonstrated that the jet flap concept was capable of producing high lift. The jet flap airfoil injects high velocity air over the flap surface from a slot located at the trailing edge of the airfoil, as shown in Figure 1(a). (The jet flap airfoil (Fig. 1(a)) is sometimes referred to as a "jet augmented flap" in the literature.) One way to implement the jet flap concept is to use an externally blown flap. This may be accomplished by using high-bypass-ratio turbofan engines which exhaust into the wing flap system. A STOL concept under investigation at NASA Lewis Research Center is a multiple-fan externally blown flap (Fig. l(b)). Important in this design concept is passing some of the fan exhaust through the gaps in the flaps to control the boundary layer over the wing upper surface. Some of the aerodynamic problems associated with this con— cept are (1) airfoil design for takeoff, cruise, and landing; (2) fan location and orientation; (3) penetration of propulsion system exhaust jet; (4) the slot location and amount of blowing which are necessary for satisfactory boundary layer control. An analytical tool is needed to do detailed design studies of these aerodynamic problems. This tool should have the capability to handle both potential flow and boundary layer flow. The potential flow analysis is the first step in obtaining an analytical tool to design STOL wing propulsion systems. By shaping the airfoil geometry, the designer can modify the wing pressure dis- tributions to delay separation. Fan location and orientation can be improved by analysis of the velocity and pressure distributions and the flow field obtained from the potential flow solution of the com- bined wing and propulsion system. A method to handle the slot loca- tion and the amount of blowing is discussed in reference 4. This problem is not included in this study. From the potential flow solu- tion, we can determine the maximum attainable lift coefficient for the wing propulsion system. Using the surface velocity distribution as input to a boundary layer analysis, we can determine the separation point for a given engine—wing combination. The development of the potential flow solution was the first phase of this study. There are many approximate potential flow theories. Some approx- imate methods for calculating flow over two-dimensional bodies are discussed in references 5 to 7. Most approximate methods assume, for simplification, that the body is slender or that the perturbation velocities caused by the body are small. Another type of approximate solution utilizes a distribution of singularities on or interior to the body surface. Some of the methods, based on a distribution of vorticity over the body surface are discussed in references 8 to 10. The potential flow theory that is often used when considering high- lift wing systems is that of Spence's jet flap theory as discussed in references 11 and 12. This thin airfoil theory considers the effect of a highly idealized jet sheet leaving the trailing edge of the flap, and does not take into account the effect of the propul— sion system inlet and the thickness distribution of the lifting sys- tem. The most general and comprehensive two—dimensional incompress- ible potential flow method and program is the Douglas method as re- ported in references 13 to 15. This method utilizes a distribution of sources and sinks on the body surface, and does not require bodies to be slender and perturbation velocities to be small. This method has the potential for dealing with distributed suction over part of the surface, and hence can handle the propulsion system inlet air— flow. However, the program cannot handle problems for which the loca- tion of part of the boundary is unknown. For a combined wing and propulsion system, the shape and location of the jet exhaust of the fan or engine is not known a priori, hence, a method is developed to determine them. The second phase of this report includes a study of the boundary layer growth and separation on the airfoil of the STOL wing propulsion system. During takeoff and landing the wing operates at high—lift coefficients with adverse pressure gradients over a large portion of the wing. This adverse pressure gradient may cause either laminar and/or turbulent separation. In general, the designer employs the var— ious techniques at his disposal to avoid flow separation and to achieve the desired wing propulsion characteristics. Separation may be delayed by shaping the wing velocity distribution, by modification of airfoil geometry, by engine location and orientation, and with boundary layer control devices. By using the potential flow surface velocity distribution as input to the governing boundary layer equations, we may solve the boundary layer growth and separation characteristics on the airfoil of the wing propulsion system. Techniques for solving the boundary layer equations can be div— ided up into two general solution methods. The first includes the explicit—integral methods which require a procedure for solving ordi- nary differential equations for "integral" properties of the boundary layer. Some of the more commonly used integral methods are discussed in references 16 and 17. A discussion of a computer program based on the above methods is given in reference 18. These integral methods applied to the analysis of two—dimensional airfoils are discussed in reference 19. Integral methods are widely used at the present time for predicting the behavior of both laminar and turbulent boundary layers, but are not applicable for the strong adverse pressure grad- ients that are encountered on STOL wing propulsion systems at high angles of attack. The second method of solution of boundary layers is the finite difference methods which provide a procedure for solving the coupled partial differential equations of mass, momentum and energy. One accurate numerical procedure for solving partial differtial equations of the diffusion type was developed by Crank and Nicolson as discussed in reference 20. Numerical methods developed specially for hydrody— namic phenomena are given by Flfigge-Lotz and Bradshaw et al. in 7 references 21 and 22. The three finite difference boundary layer methods more commonly used are those of (1) Spalding and Patankar, (2) Cebeci and Smith, and (3) Mellor and Herring and are reported in references 23 to 26. Spalding and Patankar's compressible method obtains a finite—difference equation from the boundary layer partial differential equation by formulating each term in the partial differ- ential equation as an integrated average over a small control volume. Both of the other two methods are incompressible and transform and linearize the partial differential equations by using finite differ- ences for the x derivatives resulting in a series of ordinary dif— ferential equations. The ordinary differential equations are then integrated numerically across the boundary layer at each x-location. Numerical methods, used to solve the partial differential equa— tions for turbulent flow, require as input an empirically based ex— pression for the turbulent effective viscosity. The effective vis- cosity hypothesis used by Spalding and Patankar is based upon the mixing-length hypothesis of Prandtl and utilizes a Couette—flow re- lationship for the region close to the wall. Reference 27 indicates that the pressure gradient produced systematic deviations in the pre— dicted heat—transfer rate and that the mixing—length constants should depend on the pressure gradient. Then the mixing length should be in- creased for adverse and reduced for favorable pressure gradients. The method of Smith, et al., utilizes an eddy viscosity based on Prandtl's mixing—length theory in the inner region. In the outer region a con- stant eddy viscosity modified by an intermittency factor is used. Mellor and Herring in formulating their effective viscosity hypothesis 8 divide the boundary layer in terms of an inner layer and outer layer and an overlap layer. The value of each region is based on experi- mental data and is uniquely determined by values of a pressure gradi- ent parameter and displacement thickness Reynolds number. A more de- tailed discussion of this hypothesis is given in references 28 to 30. The method of Mellor and Herring was chosen for the prediction method to be used to calculate both laminar and turbulent boundary layer growth because of its accuracy, physical soundness, and adapta— bility to the particular application problem. Their effective vis— cosity hypothesis should be applicable to high adverse pressure grad- ient flows. Also an incompressible boundary layer analysis is suffi— cient, since here, we are only interested in studying the takeoff and landing flow characteristics of the wing-propulsion system. This cor— responds to a free stream Mach number of 0.12 or less. The purpose of this report is to develop an analysis to solve the potential flow and boundary layer growth of a STOL wing propulsion sys— tem. The potential flow solution was obtained by extending the two— dimensional Douglas analysis and computer program to include the effect of suction at the propulsion system inlet, and by the development of a technique for determining the approximate location of the exhaust jet of the propulsion system. The potential analysis was used to obtain the flow field including the surface velocity and pressure distribu—C tions, and the lift coefficient. The surface velocity distribution was used to obtain the boundary layer growth and separation on the airfoil of a particular externally blown flap, high-lift configuration. POTENTIAL FLOW ANALYSIS Representation of the Wing Propulsion System While the present development can be used for any tw0udimensional configuration, it is helpful in describing the analysis to consider a particular physical system. The high—lift wing propulsion system for STOL applications under investigation at Lewis is a multiple—fan exter— nally blown flap, as shown in Figure 2(a). The wind tunnel model is semispan with a NASA 4415 airfoil section, a 66 centimeter (26 in.) chord and a 165.1—centimeter (65—in.) span. The model has eight pro- pulsion units spaced spanwise with the inlets under the wing and the exhausts ahead of a double slotted flap. The 300 and 600 flap deflec— tions in Figure l(b) represent typical takeoff and landing configura- tions. Since the proposed STOL lifting system utilizes a large number of fans closely spaced spanwise on each wing, it is reasonable to approx- imate the actual flow with a two-dimensional flow. This approximation should be valid as long as there is a sufficient number of fans for blowing to be uniformly distributed along the wing trailing edge. The representation of the two-dimensional lifting system is shown in Fig— ure 2(b). The equivalent body surface over which the potential flow is calculated consists of the airfoil with flap; the fans, which have a distributed suction at their inlet and a jet at their exit; and the jet sheet leaving the flap trailing edge. 10 The wing propulsion system combination is idealized by consider— ing it to be one solid body with suction at the fan inlet. The jet stream, as it exits from the propulsion system is at a higher total pressure than the surrounding flow with free stream lines separating this jet from the remaining potential flow. In potential flow the total pressure is everywhere constant; hence, in this study the jet is considered to be part of the solid body. This assumes no mixing of the external free stream and the free jet. The equivalent two— dimensional propulsion system dimensions and jet sheet thickness were determined from the known mass flow rate and thrust of the Lewis pro— pulsion system (Fig. 2(a)). The method used to determine the location of the free jet is discussed in the section, Location of Propulsion System Exhaust Jet. The potential flow problem for a given wing propulsion system combination becomes one of calculating the velocities on and external to the body surface for any combination of the following variables: (1) free stream velocity V“, (2) fan or engine mass flow rate m per unit span (3) propulsion system thrust T per unit span, (4) flap angle 6, and (5) wing angle of attack a. The first three variables can be combined into two dimensionless parameters: the fan or engine mass flow coefficient CQ = m/meC and the thrust coefficient CT = T/(l/ZQVZC). The development of the theory to handle this calcu- lation is discussed in the following sections. All symbols used for the potential flow analysis are defined in appendix A. 11 Basic Equations and Boundary Conditions The basic potential flow equation is obtained from the incom- pressible continuity equation together with the condition of irrota— tionality which gives Laplace's equation 2 2 843,11“, (1, 2 3x 3y where ¢ is the velocity potential due to the presence of the body only. To ensure uniqueness of the solution, the regularity condition at infinity is specified as MI; 0 <2) The velocity field Y can be expressed as the sum of the two veloci— ties 17 = i; + 3 (3) m where Ya is the free stream velocity and N is the disturbance vel— ocity due to the presence of the body only. A general method of solving the potential flow for an arbitrary boundary is to use a large number of sources and sinks distributed on the surface of the body. This is the method presented in this report. The boundary condition, illustrated in Figure 3, specifies that the entire normal component of velocity of the fluid at any point p on the surface must be equal to the prescribed normal velocity on the surface. The contribution supplied by the source—sink distribution + + is v-n and that supplied by the free stream velocity is Ym°;. The prescribed normal velocity VN on the surface is due to suction or 12 blowing. Thus, the boundary condition becomes V 3| — m = v (4) m N P P + + Since V'n = ao/an, the boundary condition on ¢ is _ V (5) 3—31? = Vfil Equations (1), (2), and (5) form the classic Neumann problem of poten- tial theory which is the basic problem we wish to solve. The direcc problem as just defined, can be solved exactly by conformal transform- ation only for a limited class of boundary surfaces. By using a large number of sources and sinks distributed on the surface of the body, the boundary condition can be formulated into an integral equation. Formulation of the Boundary Condition as an Integral Equation A simple potential function which satisfies equation (1) is the potential due to a point source. The potential at a point P due to source at q is expressed as «mm = % (6) where o(q) is the local intensity per unit area of the source and r(P,q) is the distance between P and q. Since Laplace's equation is linear, the combined potential due to a distribution of sources is also a solution. By considering a continuous source distribution on the surface S, the potential at point P due to the entire body __ 0(9) ¢(P) — f r(P’q) d8 (7) S becomes 13 The potential as thus given satisfies equations (1) and (2), but it must also satisfy the boundary condition as given by equation (5). Applying the boundary condition requires evaluating the derivative ao/an at point p on the boundary surface. The derivative of l/r(p,q) becomes singular at p when p and q coincide so that the principal value of the integral must be extracted. The princi= pal value, according to reference 31, is —2no(p). This is the con— tribution to the normal velocity at p from the source at p. The contribution of the remainder of the sources to the normal velocity is given by the derivative of the integral of equation (7).eva1uated on the boundary. The normal derivative of ¢ becomes g—ii p = - 21ro(P) + f {EL—(131717] 0(q) d8 (8) 5 Applying the condition of equation (5) to equation (8) results in the integral equation for the source—intensity distribution o(p) 2......) - 1(4)...) ds = -20.; + v (9) an r(P,Q) N .S This equation is a Fredholm integral equation of the second kind whose solution is the central problem of the analysis. The quantity ~—8/8n[l/r(p,q)] is called the kernel of the inte- gral equation and depends only on the geometry of the surface. The first term of equation (9) is the normal velocity induced at p by a source at p. The second term is the combined effect of the sources at other points q on the surface of the body. The specific boundary conditions determine the right-hand side of equation (9). The first l4 term on the right is the normal component of the free-stream velocity at p. The second term on the right is the prescribed normal velocity on the boundary surface at p. The solution of this Fredholm integral equation then requires determining the unknown function 0 on the body surface. Solution of Integral Equation Since the boundary of the wing propulsion system is completely arbitrary, the integration of equation (9) with respect to S should be done numerically. The boundary is approximated by a large number of surface elements whose characteristic lengths are small compared to the body. It is assumed that the surface element is a flat segment as shown in Figure 4. As the number of elements increase, the assumed model approaches the shape of the body. The value of the source inten— sity is assumed to be constant over each surface element. By assuming this constant intensity over each element, the problem becomes one of finding a finite number of values of 0, one for each of the surface elements. This gives a number of linear equations equal to the number of unknown values of 0. On each element a control point (the midpoint of the element) is selected where equation (8) is required to hold. Rewriting equation (8) in summation form yields %3 = -2'rrO(P) + g :% [m] 0(q) AS (10) P ptq The right side of equation (10) now becomes a matrix consisting of the normal velocities induced by a source of intensity 0 at the control 15 point of all elements. The normal velocity at the control point of .th . the 1 element due to all surface elements is denoted as N N 3—4’ = A..0. + ) :A..o. = ) A..o. (11) an i 11 1 ij j ij j jzl j=l j 1 Thus a l A.. = — —— As 13 3n [drum] where i corresponds to p and j corresponds to q. The source densities of all the surface elements must be deter- mined in such a way that the normal velocity condition is satisfied at all control points. This results in N EAijoj = — V231 + VN’i (12) j=l This set of linear algebraic equations is an approximation to the in- tegral equation (9). Both Vw-h and the prescribed normal velocity VN’ in general, vary over the body surface. The linear equations are solved by a procedure of successive orthogonalization, as discussed in reference 32. Once the linear equations have been solved, flow velocities may be calculated for points on and off the body surface (see appendix B). The method just described is used to obtain the basic solutions of potential flow. 16 Basic Solutions The superposition of any solutions to the integral equation (9) is also a solution since Laplace's equation is linear. Hence, the flow about a body may be thought of as a linear combination of four basic flows illustrated in Figure 5: (1) Uniform flow at zero angle of attack (2) Uniform flow at 900 angle of attack (3) Vortex flow (4) Flow due to suction or blowing The uniform flow solutions are solutions due to free stream vel— ocity (rectilinear flow) past the body surface at 00 and 900, respec- tively. For these basic solutions, the boundary condition of zero normal velocity on the surface must be satisfied. Then the prescribed velocity normal to the surface must be zero for the basic uniform flow solutions. From equation (5) the boundary condition becomes (13) The solution for the body at any angle of attack may be obtained by a linear combination of the OO and 900 uniform flow solutions. For a lifting body the circulation is obtained by placing a vor- tex at any convenient location within the body. The boundary condi— tion of zero normal velocity on the surface still applies (eq. (5)) except that now Vm is replaced by the vortex velocity at any point. If Y; represents the velocity at any point p on the body caused by the vortex, the boundary condition for the basic vortex solution becomes 17 it an - (14) P The suction flow solution is obtained by specifying a prescribed normal velocity VN at the fan face with a zero free stream velocity. This gives the desired mass flow rate for the inlet of the propulsion system. From equation (5) the boundary condition for the basic suc- tion velocity solution becomes 3111-- 3n VN (15) P For each basic solution, the velocities on the body surface and at prescribed locations in the flow field may be obtained. From the basic solutions the total combined solution may be obtained. Combination Solution The total velocity tangent to the body surface can be obtained by adding the tangential velocities of the four basic solutions. V = V cos a + Vt t t,0 Sln a + T Vt + V (16) :90 ,V t,S where a is the angle of attack. The nondimensional circulation F is determined by satisfying the Kutta condition at the trailing edge of the body. This condition stip— ulates that the flow at the body trailing edge be smooth. Thus, the tangential velocities above and below the trailing edge must be equal in magnitude. If AV is defined as AV = V , the Kutta , - V . upper lower condition is satisfied if AV = O at the body trailing edge. Then 18 from equation (16) AV cos a + AVt’90 t,0 Sln a + AVt,s + P Avt,v = 0 (l7) Solving for P yields Athocos a + AVt’9031n a + AVt AV t,v '5 <18) Once the combined velocities on the body surface are known, the pressure coefficient, the lift coefficient, and the thrust coefficient can be obtained (see appendix C). For off-body points it is more convenient to combine the basic source intensities rather than the basic velocities. The equation for the combined source intensity is o = 00 cos a + 090 Sln a + F 0v + as (19) Then, the x and y components of velocity are calculated from the combined source intensities (see appendix B). This approach is the same as that used in reference 15, with the addition of the basic suc— tion source intensity being added to the other basic source intensi— ties. Location of Propulsion System Exhaust Jet The location of the propulsion system exhaust jet is determined by the following variables: (1) jet angle 6 at flap trailing edge, (2) jet penetration H, (3) jet angle 61 at trailing edge of jet and (4) total length of the free jet L The representation of these T. variables is shown in Figure 2(b). It was assumed that the free jet 19 leaves the flap trailing edge at the flap angle 6. The flap angle is defined as the angle between the free stream direction and lower flap surface. After the jet leaves the trailing edge, the free stream vel~ ocity turns the jet, which approaches a horizontal asymptote several chordlengths beyond the airfoil leading edge. For typical thrust coef- ficients CT corresponding to takeoff and landing conditions, this occurs at approximately four chord lengths from the airfoil leading edge (see ref. 10). For a reasonable approximation to the jet shape, the lift coeffi- cient is expected to depend principally on the vertical location of the jet asymptote. The problem then becomes one of finding the jet penetration H as shown in Figure 2(b). Initially, the penetration was assumed and a cubic equation used to approximate the shape of the jet sheet. (A cubic equation is the simplest expression that ade- quately approximates the jet shapes obtained from Spence's jet flap theory of ref. 12.) The correct distance H is that value for which the vertical component of thrust at the flap trailing edge balances the integrated vertical pressure forces on the free jet. Several val— ues of H were assumed until the correct value of H was obtained. Since the angle of the free jet is not exactly horizontal sev— eral chord lengths beyond the wing, a small angle 91 (50 or less) was assumed. The length of the jet LT was extended until the verti— cal component of force on the end of the jet (last 5 percent of the jet) was negligible for the chosen angle 61. Thus, if the jet is ex— tended beyond this length, it gives no significant contribution to the lift coefficient. Neglecting the vertical component of thrust at the 20 end of the jet for an angle of 50 results in only a 3 percent varia- tion in lift coefficient. Since the body was represented by the wing, flap, and jet, it was assumed to be closed at the jet trailing edge and the Kutta condition applied here. Potential Flow Computer Program A general description of the potential flow computer program along with the imputs and outputs of the program is given in appendix D. This appendix also includes a complete program listing. DISCUSSION AND RESULTS OF POTENTIAL FLOW ANALYSIS Validity of Analysis Inlet air suction To help ensure validity of the analysis, comparisons were made with known existing flow solutions. One existing solution solves the suction problem indirectly. It is also based on the Douglas method, but has application only to inlets and ducts (see ref. 33). This method utilizes three basic solutions, shown in Figure 6, to obtain a combined solution of physical interest. The flow about the inlet is obtained by considering the three basic solutions; V with inlet 1 2 with the duct open, and V3 the crossflow solution. With these three solutions any combination of free—stream ' + duct extension closed, V velocity and mass flow through the inlet can be obtained. The duct must be extended far downstream of the region of interest to obtain valid solutions. This method could not be used to get solutions for a wing—engine combination since the body must be closed to consider it a lifting body. To make a comparison between this existing flow solu- tion and the method presented in this report a two—dimensional inlet, shown in Figure 7, was considered. This inlet was chosen because ex— perimental data were available. In the present analysis, mass flow through the inlet was obtained by considering a distributed suction VN downstream of the inlet (see Fig. 7). Comparison of the 21 22 nondimensional surface velocities for the two methods are shown in Figures 8(a) and (b). Also shown is experimental data obtained.from reference 34. The reference velocity Vre was arbitrarily selected f as the average velocity at an x/L of 0.89. Agreement between the two predictions is excellent for both the surface and centerline velocities. Comparison of experimental data with the prediction is quite acceptable for the centerline velocities. There is a slight variation between the experimental and predicted surface velocities. One reason for this variation could be boundary layer effects. The preceding discussion indicates that the combined uniform flow and suction solution is valid. Exhaust jet.shape For a valid solution of a wing propulsion system there must be a reasonable approximation to the jet shape. The lift coefficients and pressure distributions for a given thrust coefficient depend princi— pally on the flap angle 6 and jet penetration H, as outlined in the analysis section, and not on the precise local shape of the jet. This is illustrated in Figure 9, which considers various free jet shapes for a 300 flap. For clarity the jet thickness is not shown. The as— sumed cubic equation is shown, along with representative upper and lower bounds for the jet shape. For the jet shapes. A and C shown, the solution results in only a 12.5 percent variation in lift coeffi- cient from the assumed cubic shape B. This percent variation is dis- tributed over the entire wing surface, as illustrated by the pressure distributions in Figure 9(b). Figure 9(b) shows only a 3 percent var— iation in pressure distribution for the jet shapes considered. Thus, 23 the lift coefficients and pressure distributions depend principally on the flap angle and jet penetration and not on the precise local shape of the jet for the present configuration. As a point of interest, a comparison of the jet shape based on the Spence's theory of reference 12 was made with the jet shape obtained from the present method. Spence of reference 12 assumes that all flow deflections from the free stream are not large and uses vortex distri— butions that are placed on the x—axis rather than on the airfoil or jet. Thus, a comparison could only be made for relatively small flap deflections (300 or less). A comparison of the nondimensional jet shape predicted from Spence's theory and from the method of this re- port is shown in Figure 10 for a30o flap deflection and a thrust coef- ficient of 3. The basic shapes of the two cases are the same close to the wing. The jet penetration of the present method is larger than Spence's theory at the greater distances, as would be expected. The present method, besides not assuming small angle approximations, in— cludes the wing thickness and camber effect which would increase the lift coefficient and would also result in a greater penetration. Example Applications Flow field Potential flow solutions are adequate representations of the flow around bodies if the surface boundary layers are thin and remain at— tached. It is assumed that the final design of a high-lift wing pro— pulsion system will be one in which boundary layer separation is 24 prevented at relatively high angles of attack and flap settings. Rep— resentative flow fields for an externally blown flap high—lift config— uration are shown in Figures 11(a) to (c). The flow fields were ob- tained by sketching streamlines tangent to the calculated velocity vectors at various points in the flow. The wing propulsion system is shown, along with the shape of the jet exhaust of the propulsion sys- tem. For the conditions shown, the upwash angles at the propulsion system inlet are quite large, varying from 600 to 900 depending on flap angles and wing angles of attack. Two stagnation points occur on the lifting body. One occurs ahead of the inlet below the leading edge of the wing, and the other occurs downstream of the inlet on the under surface of the fan. Both stagnation points move further aft as the flap angle and the wing angle are increased. By observation of the flow fields it is seen that the under surface of the wing is in a rel- atively stagnant region. The jet penetration increases with angle of attack (Figs. 11(a) and (b)). For a flap angle of 600 (Fig. ll(c)) the jet penetration distance is approximately three chord lengths at five chord lengths beyond the wing leading edge. This jet penetra- tion is also important when considering the effect of the ground on lift coefficient. Pressure distribution The predicted pressures on the surface of the airfoil are valid only if the boundary layer is very thin and attached to the surface. The potential flow pressure distributions can be used both to calcu- late the boundary layer growth on the surface of the airfoil and as a design aid for the combined wing and propulsion system. Pressure i 25 distributions on the wing upper surface with a 500 blown flap at vari— ous angles of attack are presented in Figure 12. The incompressible pressure coefficient, corresponding to the minimum pressure point, ranges from -7.5 to -51 for the OO and 200 angle of attack, respec- tively. These extremely high negative pressure coefficients corres- pond to the very high lift coefficients which are discussed in the following section. The minimum pressure point for all angles of at- tack occurs very near the leading edge of the airfoil, and severe ad- verse pressure gradients over a large portion of the wing result at the higher angles of attack. The stagnation point moves further under the h leading edge as angle of attack increases, resulting in high velocity gradients about the leading edge. To illustrate the effect of the inlet airflow of the propulsion system and the effect of the exhaust jet a comparison was made of the pres- sure distributions for (l) the wing alone, (2) the wing with jet but with- out inlet air suction, and (5) the wing with inlet air suction and jet. This comparison is presented in Figure 15 for a 300 flap. At the mini- mum pressure point for the wing alone there exists a pressure coefficient of -4.8 near the wing leading edge, followed by a mild adverse pressure gradient. The wing with jet but without inlet air suction would be re- presentative of a jet flap airfoil shown in Figure 1(a). Jet flap theory does not include the effect of the inlet airflow of the propulsion sys- tem. For the wing with jet (without suction) the pressure coefficient is about -18 at the minimum pressure point, and there is a severe ad- verse pressure gradient over a large portion of the wing upper surface. When the effect of the suction at the propulsion system inlet is 26 included, the magnitude of the pressures is reduced considerably over the wing upper surface, resulting in a minimum pressure coefficient of -7.6, followed by a much milder adverse pressure gradient. It may ap- pear from the upper surface pressure distributions of Figure 13 that the lift with the jet alone is much larger than the lift associated with the jet with suction; but this is not the case if both upper and lower sur— faces of the airfoil are considered. The change in pressure distribu- tion between the zero suction case and the suction case is a result of a shift in the stagnation point (Cp = 1.0) on the under surface of the wing. For the wing, without suction, one stagnation point occurs just ahead of the inlet of the propulsion system. For the wing with suction this stagnation point moves closer to the wing leading edge and another stagnation point occurs on the under surface of the fan (see Fig. 11(a». The corresponding shift in the pressure distributions on both the upper and lower surfaces presented in Figure 13 results in less than 5 percent decrease in lift when the effect of inlet suction is included for the selected inlet location. The preceding discussion indicates that the effect of suction resulting from a fan or inlet installed under the wing affects the pressure distribution on the wing upper surface favorably, with only a small effect on total lift coefficient. Lift coefficients In order to further indicate the applicability of the present analysis a comparison (Fig. 14) was made between Spence's theory (ref. 12) and the method of this report for two-dimensional lift coef- ficients for the blown flap configuration (Fig. l(b)). The lift coef- ficients predicted by the method of this report for the 300 flap range 27 from 6.5 to 13, while those for the 600 flap range from 15 to 21. The lift coefficients predicted by the present method generally range from 9.1 to 12 percent and from 10.6 to 13.6 percent higher than Spence's theory for the 300 and 600 flap, reSpectively. This difference exists since Spence's theory does not take into account the effects of the thickness and camber of the wing. The suction effect decreases the lift by approximately 5 percent, as discussed previously. The thick= ness and camber effect corresponds to approximately 15 percent increase in lift coefficient. The two-dimensional lift coefficients were used to determine three-dimensional lift coefficients to compare with experimental data of a semispan blown flap model (Fig. 2(a)). The three—dimensional lift coefficient is CL = fC1 (20) where f is a function of aspect ratio and thrust coefficient (assum- ing an elliptical lift distribution) and was obtained from reference 35 as 2c AR +-—41 f — ' " (21) AR + 2 + 0.604(CT)1/2 + 0.87CT Calculated three-dimensional lift coefficients along with experimental data obtained from the Lewis test program are presented in Figure 15. The aspect ratio was 5 for the blown flap model. The theoretical lift coefficients range from 4 to 7.5 and from 9 to 13 for the 300 and 600 flap, respectively. There is good agreement between theory and exper- iment for the 300 flap case, with theory ranging from 10.8 to 12 28 percent higher than the experimental data. The lift coefficients for the 600 flap range from 28.6 to 28.4 percent greater than the data. The calculated lift coefficient is the maximum attainable lift coef- ficient for each configuration corresponding to complete boundary layer control and negligible viscous effects. This may indicate that the 600 flap configuration did not have optimum boundary layer control and that improvements could be made in obtaining better experimental coeffi- cients. BOUNDARY LAYER ANALYSIS Basic Equations of Motion Consider the motion of a viscous incompressible fluid along a curved two-dimensional surface. Let x represent the distance meas— ured along the surface of the airfoil from the stagnation point and y represent the distance normal to the airfoil surface, as shown in Figure 16. The time average velocity components in the x and y directions are designated at E and V, respectively. The curvature of the surface is denoted by K, which is a continuous function of x. (All symbols used for the boundary layer analysis are defined in appen- dix E.) For steady turbulent motion, the Navier—Stokes equations may be written (see ref. 36) as: 1 E§+V£+ KEV =_ 1 E,” 1. 3217+321I 1 + Ky 3x 3y 1 + Ky o(l + Ky) 8x e (1 + Ky)2 3x2 8y ___z_. an, K a Kit + 2K a, V as (1+Ky)3 3X 3X' 1 + Ky 3y (1 + Ky)2 (l + Ky)2 3x (1+Ky)3 3X (22a) 29 1 -3; Vaw_I (fl) " X (383) T = —-1— ¢(Re *X) + can — x (38b) Reaig (5 where _ U6* Reo" ' T For laminar flow T = l/Re6* (39) 37 Solution of the Boundary Layer Equation The first phase of the solution of the boundary layer equation (51) which is a nonlinear partial differential equation, is the reduction to a set of ordinary differential equations using finite differences for the x derivatives. The x derivatives are represented by finite differences in the x—direction according to an adaptation of the Crank-Nicolson scheme (ref. 20). Equation (31) is written in terms of average functions at a point halfway between the x posi— tion of the known profile, Xi—l’ and that of the profile to be calcu- lated xi as follows: I — _ _. V .. _ __ _. [III] + Eh; + P)(n - f) - 7%]f" + [P(f'-2)]f' = E3 (1 - f')(f' — f ) + é:E‘Kf — f ) (40) Ax i i—l Ax i i-l where Then, using the relations _ l I = _ I I f 2 (fi + fi—l)’ etc. Equation (40) can be written in terms of functions at position x. as H l ' _ = _ II II I I (Tf )i Tb + cl(fi + fi—l) + c2(fi + fi— 1) ‘ °3(fi ' fi—I) ’ C4(fi ‘ fi—l) (41) 38 where -7—1 _ c — (5x + P)[h - 2 (fi + fi-lfl (vwi + Vwi—l)/(Ui + Ui-l) 1 (42a) _—l I I _ c2 — P [2 (fi + fi—l) 2] (42b) = * * _ 1_ , , _ c3 (Si + 5i—1)[I 2 (fi + fi-lfl /Ax (42c) ._ * * A; II II c4 — (61 + 61—1) 2 (fi + fi_l)/Ax (42d) I I__ II Tb — [Tf ]i—l (426) Finally, the form in which this equation is solved is [bf"]'=b+bf"+bf'+bf (43) 5 i 4 3 i 2 i l i where the coefficients are bl = _ c4 (44a) b2 = c2 - c3 (44b) b3 = cl (44c) I I, I b4 — - Tb + lei-l + (c2 + C3)fi—l + C4fi-l (44d) b5 = — Ti (44e) Since equations (43) is nonlinear, the solution is carried out itera- tively for each i value. The coefficients bl to b5 are evaluated using the results at the previous (i-l) step. The resulting linear equation is then solved for f' and f". 5* is adjusted so that f(m) = l to some specified accuracy. The parameters P and Q are 39 recalculated and the effective viscosity function, T, is recalculated. Then the cycle begins again and is continued until the desired accur— acy is obtained at a particular step. The second phase of the method is the solution of the ordinary differential equations. Equation (43) is rewritten as a set of first- order differential equations. The Runge—Kutta method is then used for solving the equations. See Hildebrand (ref. 44) and McCracken (ref. 45) for details of this method. DISCUSSION OF RESULTS OF BOUNDARY LAYER ANALYSIS Laminar Boundary Layer Growth By using the surface velocity distribution obtained from the po- tential flow analysis as input to the boundary layer analysis, we may obtain the boundary layer growth on the airfoil of the wing propulsion system. Velocity distributions used for the boundary layer analysis are shown in Figure 18. The incompressible velocity distributions are illustrated at the start of the stagnation point to the trailing edge of the airfoil. For high angles of attack (150 or greater) the flow becomes compressible over a small (10 percent) portion of airfoil sur- face from x/L of 0.05 to x/L of 0.15. For typical takeoff and landing conditions the free stream Mach number is 0.12 or less. Since the flow is incompressible for 90 percent of the airfoil surface, the incompressible velocity distributions were used as inputs to the in- compressible boundary layer analysis discussed in the previous section. For practical applications we are concerned with angles of attaCR, 150 or less. In order to compute a boundary layer solution,.it is necessary to prescribe the velocity profile in the boundary layer at the start of the calculation; namely, the stagnation point of the airfoil. The velocity profile resulting from a similarity wedge flow solution for 40 41 stagnation point flow over a circular cylinder was assumed (see ref. 38). This and other similar profiles could also be generated from a specialization of equation (31). This is described in refer- ence 26. The boundary layer is laminar in the region of velocity increase (i.e., roughly from the stagnation point to the point of maximum vel— ocity) and becomes turbulent in most cases from that point on and throughout the region of velocity decrease. The velocity profiles of Figure 18 indicate that laminar flow exists only on the first 5 per— cent of the leading edge of the airfoil surface for an angle of attack of 15°. Typical parameters for the laminar portion of the flow are shown in Figure 19. The strong accelerated flow results in a large rate of decrease of the local skin friction coefficient (CE = Tw/pUz/Z). The skin friction coefficient ranged from 0.035 near the stagnation point to 0.0025 at the point of maximum velocity. The shape factor remained a constant value of approximately 2.2 throughout the laminar region, as would be expected (see ref. 19). The displacement thickness Reyn— olds number increases linearly from the stagnation point to the point of maximum velocity. Transition The pressure distribution in the external flow exerts a decisive influence on the position of the transition point. In ranges of de- creasing pressure (accelerated flow) the boundary layer generally re- mains laminar, whereas even a very small pressure increase always 42 brings transition with it. The location of the transition point is generally determined by experiment but may also be predicted by empir— ical methods. From experimental data, Crabtree (ref. 46) established a curve of momentum thickness Reynolds number and a pressure gradient parameter at transition. When the curve obtained from predicted bound- ary layer calculations intersect the experimental curve, the location of transition is determined. Michel's method (ref. 47) established an experimental curve of Re and Rex at transition. Both of these 6 methods are for smooth surfaces with low turbulence. Granville (ref. 48) predicted a method for finding the distance between instability and transition points. Theoretical investigations into the process of transition from laminar to turbulent flow are based on the acceptance of Reynolds' hypothesis that transition occurs as a consequence of an instability developed by the laminar boundary layer. Thus, the position of the point of maximum velocity of the potential velocity distribution (point of minimum pressure) influences decisively the position of the point of instability and the region of transition. Usually, the chordwise distance over which the transition region extends is rela- tively small. Thus, the transition region may be considered to take place at a point. A rough guide for the location of the transition point of airfoil shapes is given by Schlichting (ref. 38). According to Schlighting's rule, the point of transition almost coincides with the point of minimum pressure or maximum velocity of the potential flow in the range of Rex from 106 to 107. At very large Rex, the transition may be a short distance ahead of the maximum velocity. At 43 small Rex, the transition may take place some distance after the max- imum velocity. In summary, we can establish the rule that the point of transition lies behind the point of minimum pressure but in front of the point of laminar separation, at all except very large Reynolds numbers. Taking into account the Reynolds number (6x106) the adverse pressure gradient, and the turbulence intensity usually associated with flow over STOL wing propulsion systems, transition was assumed to take place at the point of minimum pressure. Turbulent Boundary Layer Growth and Separation Boundary.layer.parameters It is important that the development of the turbulent boundary layer from the transition point be accurately determined to find out whether the turbulent boundary layer would separate and, if so, at what point on the airfoil. The laminar velocity profile at the tran- sition point is used for the initial turbulent boundary layer calcu— lation. It is necessary to know the shape factor, and skin friction coefficient which are indicative of separation. The turbulent bound— ary layer parameters on the airfoil of a blown flap wing propulsion system at various angles of attack are illustrated in Figure 20. The parameters are shown up to but just shy of the point of separation on thénairfoil. The displacement and momentum thicknesses are nondimen- sionalized by L, the distance along the airfoil from stagnation point to trailing edge of the airfoil. As angle of attack is increased the displacement thickness, momentum thickness, and displacement thickness Reynolds number increase at a faster rate. Thus, the separation point 44 occurs closer to the leading edge of the airfoil as the angle of attack is increased. . The point of separation is determined by the condition of zero wall shear stress which gives zero skin friction coefficient. Another condition of the point of separation is the increase in shape factor H as the separation point is approached. At zero angle of attack the shape factor remains a relatively constant value (1.45) with a slight increase at the trailing edge of the airfoil. The skin friction coef— ficient decreases to a value of 0.0018 at the trailing edge. The shape factor increases at a faster rate as the angle of attack is increased and reaches a value of 2.0 or greater at the point of separation. Likewise the skin friction coefficient approaches zero at a faster rate as angle of attack is increased. The above result is caused by the in- crease in the adverse pressure gradient as angle of attack is increased. For angles of attaCk 150 and greater, the skin friction coefficient was 0.0001 or smaller just shy of separation. Hence, this point was used as the separation point. Velocity.Profiles The effect of the starting velocity profile on the turbulent boundary layer separation point is now considered. Three velocity pro— files at the start of the turbulent boundary layer growth are illustra— ted in Figure 21. Curve B is obtained by assuming a similarity wedge flow solution for a circular cylinder at the stagnation point. Curves A and C are arbitrary selected profiles. Using the profiles in Fig— ure 21 and the surface velocity.distributions on the airfoil of the 45 wing propulsion system resulted in a negligible effect on the turbu- lent separation point. The boundary layer velocity profiles on the airfoil of a wing- propulsion system at various angles of attack are shown in Figure 22. At zero angle of attack there exists a very mild adverse pressure gradient along the surface of the airfoil (Fig. 18). This results in a small change in the velocity profile along the surface of the air- foil (Fig. 22(a)). For angles of attack 150 and greater, the velocity profiles are shown up to the point near separation (Figs. 22(b) to (e)). As the separation point is approached, the boundary layer thickens and results in an inflection point in the velocity profile. These profiles give approximately zero skin friction (Cf = 0.0001) and hence are in- dicative of the profile near separation. The change in the velocity profiles at the various locations along the surface resulted in a cross-over of the velocity profiles in the outer portion of the bound— ary layer. This occurred at a velocity ratio EYU of approximately 0.75. Schlichting (ref. 38, p. 630) reported this cross-over char- acteristic in velocity profiles for convergent and.divergent channels. Conseguences of separation The separation at high angles of attack as indicated in the pre— vious section results in a loss of lift and the airfoil stalls.. Air- foil stall refers to the angle of attack corresponding to the maximum lift coefficient. ,Typical lift curves illustrating the stall charac- teristics of airfoils in subsonic flows are shown in Figure 23 (see refs. 49 and 50). Three main classifications of stalling behavior 46 occur, depending on airfoil shape and Reynolds number: (1) trailing- edge stall, where there is a gradual loss of lift at high lift coefm ficient as the turbulent separation point moves forward from the trailing edge; (2) leading—edge stall, where there is an abrupt loss of lift, as the angle of attack for maximum lift is exceeded, with little or no rounding over of the lift curves; and (3) thinuairfoii stall, where there is a gradual loss of lift at low lift coefficients as the turbulent reattachment point moves rearward. Trailing edge stall is characteristic of most conventional thick airfoils (say t/c > 12%) at moderate to high Reynolds numbers. Leading‘edge stall is characteristic of moderate airfoils (t/c = 9%) and is caused by an abrupt separation of the flow near the nose without subsequent reat- tachment, i.e., short bubble "bursting". The process of laminar bound— ary layer separation, transition, turbulent reattachment is referred to as a "short bubble”. Thin-airfoil stall is characteristic of thin- airfoil sections (t/c = 6%) and is the result of laminar separation near the leading—edge and turbulent reattachment moving progressively rearward with increasing incidence, i.e., a long bubble. The process of laminar boundary layer separation just aft of the leading edge, transition to turbulence, but reattachment not so quickly established is referred to as a "long bubble". The above stall characteristics for airfoils can be used as an aid in the classification of the stall associated with STOL wing propulsion systems. The experimental lift curve of the blown flap wing propulsion system with airfoil t/c of 15 percent (see Fig. 2(a» is illustrated 47 in Figure 24(a). The lift curve increases almost linearly to an angle of attack of 200 and then there is a gradual loss in lift coefficient at high angles of attack. This is typical of the trailing edge stall of thick airfoils as shown in curve (a) of Figure 23. Having assumed that the transition point is near the point of minimum pressure re- sulted in the predicted turbulent separation point to move forward from the trailing edge as angle of attack is increased. This is illus— trated in Figure 24(b). At an angle of attack 150 the separation point is near the trailing edge and moves slightly forward at 200. At the stall angle of attack of 250 (angle of maximum lift coefficient) the separation point moves still further near the leading edge correspond— ing to an x/L of 0.57. At an angle of attack of 300 the separation point moves considerably forward as would be expected to an x/L of 0.34. The above experimental curve and indicated separation points validate the assumption of the transition point to occur near the point of minimum pressure for the given airfoil shape and fan location. How— ever, boundary layer characteristics could be quite different for other airfoil geometries and fan locations. CONCLUDING REMARKS A method was developed to determine the two-dimensional potential flow solution of STOL wing propulsion systems. The Douglas potential flow computer program was extended to include the effect of suction at the propulsion system inlet and to provide a technique for determining the approximate location of the exhaust jet of the propulsion system. The effect of suction was obtained by combining a basic suction solu- tion with the uniform flow solution for a lifting body. The jet ex- haust was considered as part of the solid body and its location was determined by balancing the vertical component of thrust at the flap with the integrated vertical pressure forces of the free jet. The applicability of the potential flow program is illustrated by considering a multiple-fan externally blown flap under high-lift con— ditions. The results indicated high upwash angles (600 to 900) at the fan inlet and large jet penetration at high angles of attack. (The predicted two-dimensional lift coefficients for the 300 flap ranged from 6.5 to 13 while for the 600 flap ranged from 15 to 21 (for angles of attack from O0 to 200). The predicted two-dimensional lift coeffi- cients for a 300 flap were an average of 10.5 percent higher than pre— dicted by Spence's jet flap theory which neglects thickness effects. 48 49 The predicted three-dimensional lift coefficients were an average 11.4 percent higher than experimental data for the 300 blown flap high=lift configuration. The calculated pressure distributions indicated that the minimum pressure point is near the leading edge (less than 2 per— cent of chord) of the airfoil, with severe adverse pressure gradients at high angles of attack. The pressure coefficient, corresponding to the minimum pressure point, ranged from —7.5 to -51 for the 00 and 200 angle of attack, respectively. The effect of suction due to a fan or inlet installed under the wing decreases the magnitude of the upper surface pressure distribution with only a small effect on total lift coefficient (obtained by integration of pressure distribution on both the upper and lower surfaces of the airfoil). The surface velocity distribution obtained from.the potential flow solution was used to determine the boundary layer growth and sep- aration on the upper surface of the airfoil. The boundary layer solu- tion was obtained by reduction of the partial differential equations of motion to a set of ordinary differential equations using finite dif- ferences for the x derivatives. By an iterative solution to the differential equations the boundary layer parameters for both laminar and turbulent flow were found. Near separation the shape factor was found to be 2.0 or greater. This corresponded to a skin friction coef— ficient of approximately 0.0001. The results indicated that the pre— dicted turbulent separation point moved forward from.the.trailing edge as the angle of attack is increased. Trailing edge separation for a thick wing (t/c = 0.15) propulsion system combination considered was 50 verified by the experimental lift curve. .However, this boundary layer characteristic could be quite different for other wing geometries and propulsion system locations. The ability to predict the potential flow and boundary layer sol— ution makes the analysis in this report extremely useful as a tool in the design of a STOL wing propulsion system. The.analysis can be used to design the airfoil and to determine the optimum location and orien— tation of the propulsion system. From the predicted surface velocity distributions and boundary layer calculations one may minimize the frictional drag for a given wing propulsion system. The potential flow solution can be used to determine the jet penetration - an impor— tant quantity when considering ground effect. The analysis has application not only to wing propulsion sys- tems, but to any lifting or nonlifting body where suction or blowing is applied. APPENDIX A SYMBOLS FOR POTENTIAL FLOW ANALYSIS ii A.. 1] Bij APPENDIX A SYMBOLS FOR POTENTIAL FLOW ANALYSIS normal velocity of element 1 caused by a unit source at element i normal velocity of element 1 caused by a unit source at element j aspect ratio tangential velocity of element 1 due to a unit source at element j chord length two-dimensional lift coefficient three-dimensional lift coefficient pressure coefficient mass flow coefficient thrust coefficient correction factor (eq. (21)) force in vertical direction jet penetration (see Fig. 2(b)) number of elements that describe the jet length total length of free jet (see Fig. 2(b)) fan or engine mass flow rate per unit span 51 52 number of elements that describe the body a normal to the body surface static pressure arbitrary point in the flow field off the surface distance between two points surface of body thrust per unit span disturbance velocity velocity complex velocity complex potential velocity in x direction at point j due to a unit source at element k Cartesian coordinate Cartesian coordinate velocity in y direction at point j due to a unit source at element k Cartesian coordinate angle of attack orientation of surface element nondimensional Circulation variable Cartesian coordinate (see Fig. (25)) turning efficiency of exhaust jet variable Cartesian coordinate (see Fig. (25)) flap angle (see Fig. 2(b)) 55 (25)) 91 jet angle at trailing edge of jet (see Fig. (2(b)) 0 surface source intensity per unit area p density C variable Cartesian coordinate (see Fig. ¢ velocity potential w stream function Subscripts 1 control point 0f ith element j control point of jth element N normal p arbitrary point on the surface q a surface point ref reference 5 refers to suction flow solution t tangential v vortex flow solution w free stream 0 flow solution at zero angle of attack 90 flow solution at 900 angle of attack Superscripts _s vector APPENDIX B VELOCITIES IN TERMS OF SOURCE DENSITIES APPENDIX B VELOCITIES IN TERMS OF SOURCE DENSITIES Consider the body illustrated in Figure 25(a), which extends over the range - m g z §_+ w. Any point on the body is described by p(x,y,z) the general point being considered. The point q(€,n,c) is the variable point for which integration is performed. The potential ¢p due to a point source at any point p in the region R bounded by z = — w, z = + w, S = So and S = S is _ an R It is evident from Figure 25(a) that if p is the plane 2 = 0 1/2 r= [(x-E)2+(y-n)2+cz] Hence, S1 m _ (S)d§dS ¢ - 2 . 0 —- (32) P S 0 [(x-£)2+(y-n)2+1:2]l/2 0 Here the upper limit for the g variable of integration signifies a large but finite value. The normal and tangential velocities 3¢/3n and a¢/as can be evaluated in terms of x and y derivatives of the potential from equation (Bl) 54 55 S do (it) = _ 2 ‘/ o(s> = ‘ o(S)(x - £2dS 8x P s (x - E)2 + (y - n)2 0 (B4) s (if) = L/p I 0(S)(z - n)dS 3y P s (x — E)2 + (y - n)2 o . The problem is reduced to one in a single plane, the plane 2 = 0. The continuous boundary curve S is approximated by a series of segments as shown in Figure 25(b). The front, middle, and rear points of a seg— ment are designated by S. S , and S. If the source density is j-l’ j j+l' assumed constant over each surface element, the preceding equations become N Sj+1 3x j 2 p ._ (x - E) + (y - n) j-l Sj-l (BS) N j+l 3}) =25 ‘ 4L§n“_—“‘)ds 2 P -= (x - ) + (y - ) j l Sj-l E n 56" A transformation of the E,n variables into S,rij variables yields (see Fig. 25(b)) S ai = N o j+1 [rij sin aj + (S - Sij) cos dj]dS 3x j 2 2 P s rij + (S — Sij) S N j+l . . fli = o [—rij cos aj + (S Sij) sin dilds 3y j 2 2 P - S r.. +‘ S — S.. j-l 1] ( 1]) (B6) The quantities in integrals of equation (B6) are functions only of the geometry of the body. If they are identified as X . and Y . re- 13 spectively, the x and y components of velocity become N V = 1?.) = U.X.. x 8x i Z : j ij j=1 (B7) N a E > V = -$- = o.Y , y 3y>i J 13 j=1 where i represents an arbitrary point on the body surface. The nor- mal and tangential velocities can be obtained by using the directional derivative formula 32. = _ ii - Bi '3n?1 3x i Sin ai + 3y 1 cos 01 (B8) 39 = 39 89 as 1 3x 1 cos 01 + 8y i sin “i 57 Then using equation (B7) N it = _ - 3n 1 oj( xij Sin ai + Yij cos a1) i=1 (B9) N ii = as i oj(Xij cos ai + Yij sin a1) j-l Letting the terms in the brackets be Aij and Bij respectively, the normal and tangential velocities due to the source contributions become N gi) = o.A . n i E > J 13 i=1 N 1 (B10) 3¢ _ as). ‘ 2°91,- j: The total velocities are made up of the contribution due to the source- The entire normal and sink distribution and the free stream velocity. tangential velocities on the body become =93; _ VN,i 3n . V0° sin mi 1 (Bll) he) Vt,i as ' + V” cos mi 1 The velocity at any point off the body can be obtained by N 07¢)k = i (iji + ijj)ok (312) i=1 58' . . . th where 0k is the combined source 1ntens1ty of the k element as given by equation (19) of the text, and ij and ij are the effects in the x and y direction, respectively at any point p due to the kth element. 4 The terms Aij’ Bij’ ij, and ij are called influence coeffi- cients and all represent velocities at some point that are resolved in a particular direction. These velocities are generated by the jth source element at point pi (on the body surface) or pk (off the body surface) and resolved normal and tangent to the body surface (Aij’ B ) or x ax1s (ij, ij). In terms of complex veloc1ties ij Wij and ij the influence coefficients for points on and off the body surface can be expressed as _ —iai B., + iA.. = W.. e 13 13 1J (313) ij + 1ka = ij where the bar indicates the conjugate and ai is the ith element angular orientation. The complex potential at 2k for a unit source located at C(S) is expressed as w= ¢+iw=-2—11;1n[Zk- r(sn (314) The complex velocity ij is the influence of element j at the point pk. Since W = dw/dz the influence coefficient becomes W = -£- -g-1n(z — C(S))dS (B15) kj 2n dzk k ’ o J elem 59 Referring to Figure 26, ds = e j d; (816) Replacing dS in equation (315) and evaluating the integral there results Cz- . -a J —1a. e j d e 3 (2k ' ‘11) ij — 2" 3;; 1n[zk — §(S)] dg — -§;—— In (2k _ Czj) C13- (317) APPENDIX C COMPUTATION OF FLOW QUANTITIES FOR POTENTIAL FLOW ANALYSIS APPENDIX C COMPUTATION OF FLOW QUANTITIES FOR POTENTIAL FLOW ANALYSIS Once the combined velocities on the body surface are calculated, the pressure distribution and the lift coefficient of the body can be found. The equation of motion for steady, incompressible, inviscid fluid can be expressed as -> -> 1 (V'V)V = —-; VP (Cl) For potential (irrotational) flow Bernoulli's equation results E + %-V2 = Constant (C2) and is applicable everywhere. The pressure coefficient CP is defined as C = —— (C3) c=1—V— (04) v The two—dimensional lift coefficient is defined as L c = —— (c5) 1 %pvic 60 61 This can be obtained by integration of the pressure distribution over the surface of the body. Since L =JP pi cos a1 dSi, S N C = l- C cos a AS (C6) 1 C p,i i 1 i=1 where CP i represents the pressure coefficient at the control point 9 of the ith element. The thrust coefficient is defined as C = _T__ T %DV.2.C (07) where T is the exit thrust of the propulsion system. The exit thrust is obtained from the vertical force on the jet, the jet deflection angle, and the experimental turning efficiency' n between the propul- sion system exhaust and the trailing edge of the flap. Then Fy = n sin 6 (C8) where n is the turning efficiency of the exhaust jet. The vertical force is calculated by integration of the pressures on the jet M F = .pi cos ai ASi (C9) where a is the angular orientation of the ith element. 1 APPENDIX D POTENTIAL FLOW COMPUTER PROGRAM APPENDIX D POTENTIAL FLOW COMPUTER PROGRAM Summary A schematic representation of the main subroutines of the com- puter program is illustrated in.Figure 27. The program is divided up into seven parts. These are called from the main program. Part 1 performs computations with the basic data input. It calculates angu- lar orientation of elements of the body, mid point of elements, ro- tates the body, etc. Subroutine 22YA generates the initial shape of the exhaust jet of the propulsion system.from the input data. Part 2 formulates the matrix including the complex velocity potential for points on and off the body surface. Part 4 solves the above matrix. The influence coefficients Aij’ Bij’ ij, and ij (see appendix B) are determined in this subroutine. The combination solu- tion is obtained in part 6. Part 7 determines if the jet exhaust is properly orientated. It integrates for the mass flux into the pro- pulsion system and calculates the forces on the exhaust jet. Input The inputs required by the program are as follows: 62 FLG02, FLG03, etc. MON CDIM VINF BETA DELT ALF2 KKK POSS CHORD THETA BDN X(I), x<1 + 1), etc. Y(I), Y(I + 1), etc. NN 65 control flags (see comment cards main program) controls amount of output desired (see comment cards main program) total x distance of body including ex= haust jet free stream velocity point on body at trailing edge of flap (start of jet) point of initial location of trailing edge of exhaust jet flap angle thickness of jet initial jet angle at trailing edge of jet number of elements on jet turning efficiency of exhaust jet chord length of airfoil number of points on body (first time thru D0 loop) rotation angle for airfoil one if on body points follow coordinates of points on the body surface coordinates of points off the body surface number of off body points (second time thru DO loop) 64 THETA - rotation angle for off body points BDN - zero if off body points follow X(I), X(I + 1), etc. - coordinate of points off the body surface Y(I), Y(I + 1), etc. - coordinate of points off the body surface NTYPE, XP, YR - equal zero for prescribed velocity normal to surface of body NUF(I), NUF(I + 1), etc. - prescribed velocity normal to surface of body (known value for inlet of propul= sion system, zero for rest of body) TUF(I), TUF(I + 1), etc. - tangential velocity on surface of body, input as zero Output The output consist of tangential velocities and.pressure coeffi= cients for each element on the body for the basic solutions and the combination solution. It also includes the mass flow rate into the propulsion system, the forces on the exhaust jet, the thrust coeffi- cient of the prOpulsion system, the lift coefficient of the body, and the basic input data. Also included are the x and y components of velocity and angular orientation of points off the body (in the flow field). The complete program listing follows: 65 Complete Program Listing SMAJUND NOPRINT SZERDIV SIBJOB NOMAP SIBFTC BZVI CDECK BZVI 82 0040 nannnnnnnn COMMON IM HHEDRnCASE RPI.RZPIoSPcCL'ALPHA.FALPHA.DALFA.CHORD'SUMDS.BZV00100 1 XMC.YMC. ADDYoFL602.FLGO3gFLGO4yFLGOSpFLGO6pFLGOT'FLGOG' 82110 2 FLGO9vFLGIOnFLGIIpFLGIZoND. NLFoNER. NT NB.NCFLG.FLGIS.FL616 COMMON SUMSIGyVINFnMONyBETAoCOIM'VREFuDELToFLGIBoFL614'ITER'ALFZ COMMON /SPACER/DUMMYIIOOOOI CCMMON lFORCUR/XCURV(ZOOI.YCURVIZOOIIKKK COMMON I NBSAVE I NBOLD 82Y00140 DIMENSION NDIIOI’ NLF(IO)v SUMDSIIO), XMCIBI. YMCIBI' ADDYIBI BZYOOOBG 1' HEDRIISIvCASEIZI 82Y00090 COMPLEX IM BZVOOOSO INTEGER FLGOZ.FLGOByFLGOthLGOS FLGObgFLGOT 82'00060 I. FLGOB. FLGO91 FLGIO. FLGII. FLGIZuFLGI3 FLGIkoFLGIS'FLGIb DATA KORE/IOOOO/ *##*** MON=0 IS REGULAR OUTPUT tttttt HON=1 IS MINIMUM OUTPUT t‘ttt FLGOI=1 ONE BODY t¥¥#t‘* FLGOZ=I DOES OFF BODY POINTS stat: FL503=1 ALPHA IS INPUT titttt FLGIIII DOES SUCTION ##3##: FL613=1 DOES THE TAIL totttt FL614=1 COMPUTES THE TAIL HITH A CUBIC astttt FL615=1 SKIPS INTEGRATING FOR THE MASS FLUX titttt FLGlb=1 CACULATES THE FORCE ON THE END OF THE TAIL ONLY NBOLD 3 0 BZY00160 ITER =0 CALL PARTI BZYOOIBO IF ( FLGOB .NE. 0 I GO TO 60 BZV00190 CALL PAR T2 BZYOOZOO CALL PA 82Y00220 CALL SDLVIT (DUMMY. NT. NCFLG' KORE. 1: 21 8. 3. 9100' BZY00230 CALL PA RT5 BZYOOZbO CALL PA BZYCOZTO IFIFLGI3I 680.80970 CALL PART GO TO 10 END BZYOO330 SIBMAP 22VZ ENTRY .UNIZ. oUNlZ. PZE UNI T12 UNITIZ FILE vUTloINOUTpBIN BLK=256 NOLIST ENTRY .UN13. oUN13. PlE UNIT13 UNITIB FILE pUT3.INOUT¢8INvBLK=256pNOLIST END 66 SORIGIN ALPHA SIBFTC BZYZ CDECK B2Y2 82Y00340 SLOROUTINE PARTl 82Y00350 COMMON IM.HEDR.CASE.RPI.RZPI.SP.CL.ALPHA.FALPHA.DALFA.CHORD.SUMDS.BZY00460 1 XMC.YMC.ADDYoFLGOZ.FLGO3.FLGOfi.FLGOS.FLGOb.FLG07.FLGOB. 52Y00470 2 ’ FLGO9.FLGIO.FLGII.FLGIZ.ND.NLF.NER.NT0NB.NCFLG.FL615gFL616 COMMON SUMSIG.VINF.MON.BETA.CDIM.VREF.DELT.FLG13.FLGIA.ITERoALFZ COMMON /FORCUR/XCURV(200I.YCURVIZOOI.KKK.POSS COMMON INBS/ KII COMMON/ NBSAVE I NBOLD 82Y00360 DIMENSION XIBOOI. YI300I. XMPIZ99I. YMPI299I. ALFAI299I. 82Y00420 1 RSDSI299I. SINAI299I. COSAI299I. DELSIZ99I. DALFI298I. lI299I. 82Y00430 ZQI3OOI.HEDRI15I.NDI10I.NLFIIOI.SUMDSI10I.XMCI8I.YMCI8I. BZV00460 3ADDYI8I.CASEI2I.NUFI299I.TUFI299I 32Y00450 DIMENSION RADCI250I COMPLEX IM. 1. Q BZY00370 REAL MX. MY. NUF BZY00380 INTEGER BDN. SUBKS. SE01. SEOZ ' 82Y00390 INTEGER FLGOZ. FLGO3. FLG04. FLGOS. FLGOb. FLGO7 BZYOOAOO 1. FLGOB. FLGO9. FLGIO. FLGII. FLG12.FLGl3.FLGI4.FLGlS.FL616 EQLIVALENCE INUF. XI. ITUF. YI 82Y00490 [M = (0.. 1.I BZYODSOO N = FLGIZ BZYOOSIO READ (5.4I HEDR. CASE. NB. FLGOZ. FLGOS. FLGOA. FLG05.FLGO6. BZYOOSZO 1 FLGO7. FLGOB. FLGO9. FLGIO. FLGll. FLG12.FLG13.FL614. 2 FLGI§.FL616.SEOI. 4 FORMA TI 15A“. 2X. N2A4/18I1I lFI NBOLD .EQ. 0 I N8 OLD I N8 BZY00540 IFI NIN .E0. 0 I NIN 5 82V00550 READ (5.6) MON.CDIM.VINF.X1.Y1.X2.Y2.BETA. DELT.VREF.ALF2 READ I5.7I KKK. POSS 6 FORMAT IIS.6FIO.5.3F5.Z/1F10o5I 7 FORMAT II5.F10.4I READ I5. BI SP. CL.I ALPHA. FALPHA. DALFA. CHORD. SEOZ BZYOOSTO 8 FORMAT I6F10.0. 16X 82Y00580 IF I SE02 .GE. SE01 I4 'GO TO 80 82Y00590 60 NRITE I 6.9 I BZYOOOOO 9 FORMAT I SOHODATA OUT OF SEQUENCE. SORT DATA ON 77-80. RELOAD. I 82Y00610 STOP BZY00620 80 5501 = SE02 BZY00630 82 IF I CHORD .E0. 0. I CHORD = 1.82Y00640 WRITE I6. IZI HEDR. CASE. NB. FLGOZ. FLGO3. FLGOQ. FLGOS. FLG069 82Y00650 1 FLGO7. FLGOB. FLGO9. FLGIO. FLGI1. FLG12.FLGI3.FLGI“.FLGI§.FLGI6I 2 MON.VINFI 2 SP. CL. ALPHA. FALPHAI 32'00660 2 DALFA. CHORD. NIN 82V00670 12 FORMAT I1H1 25X ZbHDOUGLAS AIRCRAFT COMPANY / 28X 21HLONG BEACH32Y00680 1 DIVISION III 6X 26HPROGRAM BZYC-' Z-D CASCADE // 11X 29H#‘*‘* CA62Y00690 ZSE CONTROL DATA ‘**‘*.///6X.15A4.4X. 9HCASE NO. .2A4.I/6X.9HBODIESBZYOO700 3 =I3.20X 9HFLAG 2 :I3/ 6X 9HFLAG 3 =I3.ZOX 9HFLAG 4 8 I3] 6X 82Y00710 4 9HFLAG 5 = I3.20X 9HFLAG 6 = I3/ 6X 9HFLAG 7 = I3.20X 9HFLAG 882Y00720 5 = I3/ 6X 9HFLAG 9 = I3.20X 9HFLAG 13 = I3] 6X 9HFLAG 11 = I3. 82Y00730 620x 9HFLAG 12 =I3/6X 9HFLAG 13 =I3.ZOX 9HFLAG 14 =I3I6X 9HFLAG 15 A:IZ.20X 9HFLAG 16 = I3.20X 5HMON = I3.2)X 6HVINF = FB-ZII B 11X 10H SPACING : FI3.8/ 16X 5H CL 8 F13.8/ 82Y00740 7 13X Q1 ALPHA = F13.8/ 7X 14H INLET ALPHA = F13.8/ 7X 14H DELTA AL82Y00750 OPHA = Fl3.8/ 13X 8H CHORD = F13.8/13X.58HINPUT TAPE N0. FOR COORDIBZYOO760 9NATES AND NON-UNIFORM FLOH ONLY 8. I5 I 82Y00770 IF IFLGOB .EQ. OI GO TO 119 82Y00780 122 16 141 139 142 67 IF I FLGOZ .NE. K0 I K2 = N8 + 1 READ (5. 15I NN. MX. MY. THETA. ADDX. ADDYILI. SE02 FORMAT (5X (5. 5F10.0. 16X I4 I IF I SE02 .LT. SE01 I GO TC 60 5601 = SE02 READ (5.16I BDN.SUBKS.NLFF.XMCILI.YMCILI.ELPSTH.SE02 FORMAT I3I5X I5I.3F10.0.16X.I4I IF I SE02 .LT. SE01 I GO TC 60 SE01 = SE 02 NDILIxNN JJ=NDI1 II3NDI1I-KKK+1 NT = NT 0 NN IFI NLFF .E0. 0 .AND. BOA .NE. 0 I NCFLG = NCFLG 9 1 IF I SUBKS .E0. 0 I GO TO 140 IFI L .NE. K2 I GO TO 145 NTIMES = NBOLD - NB IFI NTIMES .LE. 0 I GC TO 145 GO TO 145 ELPSTH = ABSIELPSTHI IF (ELPSTH .LE. 0.0I GO TO 139 DANGLE = 6.2831853E0 / FLOAT (MI ANGLE = 0.0 XIII = 1.0 X(NNI = 1.0 Y(II = 0.0 YINNI = 0.0 N I M - 1 DO 141 I = 1.N ANGLE = ANGLE - DANGLE X(IO1I = COS (ANGLE) YII+1I = ELPSTH * SIN IANGLEI DD 142 I = 1.NN.6 READ ININ.20I XIII. X(I+1I. X(IOZI. X(IO3I. X(IO4I. XIIOSI. SE02 FORMAT (6F10.0. 16X 14) IF I SE02 .LT. SE01 I GO TO b0 DO 144 I = 1. NN. READ (NIN.20I Y(II. Y(I+1I. YII‘ZI. Y(I+3I. YII*4I. Y(I+5I. SE02 IF ( SE02 .LT. SE01 I GO T060 SE01 8 SE02 TFETA-THETA/57o2957795E0 82Y00790 BZYOOBOO 82Y00810 BZY00820 BZY00830 BZY00840 BZY00850 82Y00860 82Y00890 82Y00900 BZY00910 BZY00920 BZY0093O BZY00940 82Y00950 BZY00960 BZY0097O 82Y00980 BZY00990 BZYOIOIO 82Y01020 82Y01030 82Y01050 BZY01060 BZY01070 BZYOIOBO BZY01090 82Y01100 BZY01110 BZY01120 BZYOIIBO BZY01190 82Y01200 82Y01210 82Y01220 BZYOIZ3O 82Y01240 82Y01250 BZY01260 82Y01270 BZYOIZBO 82Y01290 82Y01300 82Y01310 82Y01330 82Y01340 82Y01350 BZY01360 BZY01370 82Y01380 82Y01390 82Y01400 BZY01410 IF ( THETA .E0. 0. I GO TO 300 CSTHT COSI THETA I SNTHT SINI THETA I DO 290 I = 1. NN T1 = XIII XIII = T1*CSTHT + YIII‘SNTHT 290 Y(I) = YIII*CSTHT - T1*SNTHT 300 CONTINUE 145 IF (FLG12 .GT. OI GO TO 150 82Y01440 IFIFLGI4 .50. 0) GO TO 148 IF IBDN.NE.1I GO TO 148 T1=X1 . X1=T1*COSITHETAI+Y1*SINITHETAI Y1=Y1ECOSITHETAI-TI‘SINITHETAI T2=X2 X2=T2*COSITHETAI9Y2*SIN(THETAI Y2=Y2*COSITHETAI-T2*SINITHETAI X3=X2+9.72 Y 3=Y2'10 32 TFETA=THETA‘57.Z957795E0 ALF=8ETA9THETA CALL SUBCURIX1.Y1.X2.Y2.X3.Y3.ALF.ALFZ.DELTI MMM=KKK NNN=1 143 DC 147 I=NNN.MMM XIII=XCURVIII 147 YIII=YCURVIII IF INNN-III 146.148.148 146 NNN=II MMM=JJ GO TO 143 148 WRITE I13) IXIII.I=1.NNI HRITE (13) (YIII.I=1.NNI 82Y01460 150 IF I BDN .E0. 0 I GO TO 200 82Y01470 IF (FLGlZ .GT. OI GO TO 163 82Y01480 DO 160 I = 1. M 82Y01490 XMPIII = I XII01I * XIII I l 2. 82Y01500 160 YMPII) = ( YII+1I + YIII I / 2. BZY01510 HRITE I13) I XMPIII. I = 1. M I BZY01520 WRITE (13) I YMPIII. I E 1. M I BZY01530 GO TO 200 BZY01540 163 SUMS = 0.0 BZY01550 DO 164 I = 1.M BZY01560 XMPIII = I X(I'1I+XIII I/Z. 82Y01570 YMPIII = I YII*1I+YIII I/Z. 82Y01580 TI 3 XIIflI-XIII 82Y01590 T2 I YIIOlI-YIII 82Y01600 DELSIII = SQRTI T1*T1+T2*T2I BZY01610 SUMS = SUMS + DELSIII 32Y01620 RSDSIII = SUMS 82Y01630 164 ALFAIII = ATANZIT2yT1I 82Y0164O MM = NN-Z 82Y01650 DO 165 I = 1. MM BZY01660 165 DALFII) = IALFAIIOII-ALFAIIII#57.2957795E0 BZY01670 200 HRITE I6. 24) HEDR. NN. NLFILI. MX. MY. THETA. ADDX. ADDYILI. 82Y01680 1 XMCIL). YMCILI 82Y01690 24 FORMAT I1H 25X 26HDOUGLAS AIRCRAFT COMPANY I 28X 21HLONG BEACHBZYOlTOO 1 DIVISION.///5X.15A4.// 5X.4HNN =.I4.4X.5HNLF =.I4.5X.4HMX =. 82Y01710 2 F13.8. 4X 4HMY = F13.8 I 5X 7HTHETA = F13.8. 4X 6HADDX = F13.8. BZYC1720 3 2X 6HADDY = F13.8 / 7X 5HXMC = F13.8. 5X 5HYMC = F13.8 I 82Y01730 69 IF (MON-1) 27.240.240 27 IF I BDN .E0. 0 I GO TO 220 IF (FLG12.LE.0I WRITE (6.25) IF (FLGIZ.GT.0I WRITE . 25 FORMAT I1H0 4X 33HON-BODY CCORDINATES (TRANSFORMEDII 26 FORMAT (1H0 4X 35HON-BODY CCDRDINATES IJNTRANSFORMEDI I WRITE (6.28) D 28 FORMATI9H BODY NO.I3//12X1HX13X1HY11X7HDELTA S 7X 5HSUMOS 8X 1 7HD ALPHA //I GO TO 230 220 IF (FLG12. LE.0I WRITE (6.31) 31 FORMAT (1H0 4X 34HOFF- BODY COORDINATES (TRANSFORMEDI // 10X 1 5HX-OFF 9X 5HY-OFF2 II IF IFLGIZ.GT.OI NRITE (6.3 32 FORMAT (1H0 4X 36HOFF-BODY2 COORDINATES IUNTRANSFORMEDI // 10X 1 5HX-OFF 9X 5HY-OFF ) 230 (F (FL512.LE.0I GO TO 240 IF IBDN.LE.0I GO TO 235 WRITE (6.36) XIII.YI1I.XMPI1I.YMPIII.DELSI1I.RSDS(1I WRITE (6.40) ( I. XIII. Y(I). DALFII-1I. XMPIII. YMPIII. 1 DELSIII. RSDSIII. I = 2. WRITE (6.44) NN. XINNI. YINNI GO TO 0 235 WRITE (6.48I (I. XIII. Y(I). I = 1. MN) 240 IF I MX .E0. 0. I GO 0 26 DO 250 I = 1. NM 250 X(I) = XIII * MX XMCILI = XMCILI * MX 260 IF I MY .E0. 0. ) GO TO 280 OD 270 I = 1. NN 270 Y(I) = Y(I) * MY YMCILI = YMCILI * MY 280 IF I ADDX .E0. 0. I GO TO 320 00 310 I = I. N 310 XIII = XIII o AODX XMCILI = XMCILI O ADDX 320 T1 = ADDYILI IF I T1 .E0. 0. I GO TO 340 00 330 I = 1. NN 330 Y(I) = Y(I) + T1 YMCILI = YMCILI 6 T1 340 [F I CHORD .E0. 1. I GO TO 360 00 350 I = 1. NM X(I) = XIII/CHORD 350 Y(I) = YIII/CHORD XMCILI = XMCILI/CHORD YMCILI = YMCILI/CHORD 360 IF I BDN .E0. 0 I GO TO 500 a. I) N & DO 400 I = 1. T1 = XII+1I - XIII T2 = Y(IFII ' Y(I) XMPII) = (XI101I + X(III / 2. YMPIII = IYII+1I 0 Y(III / 2. TOS = SORT I T1*T1 0 T2*T2 I COSAII) = T1 / TDS SINAIII = T2 I TDS 82Y01740 82Y01750 BZY01760 BZYOITTO 82Y01780 BZY01790 82Y01800 BZY01810 82Y01820 BZY01830 BZY01840 32Y01850 82Y01860 BZYO1870 BZY01880 BZY01890 BZY01900 BZY01910 BZY01920 32Y01930 BZY01940 BZY01950 82Y01960 82Y01970 BZY01980 BZY01990 82Y02000 BZYOZOIO BZYOZOZO BZY02030 82Y02040 82Y02170 BZY02180 BZY02190 BZYOZZOO BZY02210 BZYOZZZO 82Y02230 BZY02240 BZY02250 BZY02260 82Y02270 BZYOZZBO 82Y02290 BZY02300 82Y02310 82Y02320 82Y02330 BZY02340 BZY02350 BZY02360 82Y02370 BZY02380 BZY02390 82Y02400 BZY02410 82Y02420 82Y02430 '70 ALFAIII = ATAN2 (T2. T1) BZY02440 ZIII = CMPLX I XMPIII. YMPIII I 82Y02450 400 0(1) 8 CMPLX I X(I). Y(I I I 82Y02460 QINNI = CMPLX I XINNI. YINNI I BZY02470 SUMDSILI = RSDSIMI BZY02480 WRITE (12) I XMPIII. I = 1. M I BZY02490 WRITE (12) I YMPIII. I = 1. M I BZYOZSOO WRITE (12) I DELSIII. I = 1. M I BZY02510 IF (FL612.GT.OI GO TO 450 . BZY02520 M = NN - 2 82Y02530 DO 420 I = 1. M 62Y02540 DALFIII = I ALFAIIilI-ALFAIII I P 57.295779 IF (ABSIDALFIIII.GT.270.I DALFIII= 360. )- ABSIOALFIIII 420 CONTINUE IF (MON. GE.1I GO TO 600 WRITE (6. 36) X(1I. Y(1I. XMPIII. YMPI1I. DELSIII. RSDSI1I BZY02560 36 FORMAT (1H 3H 1 2F14.B. / 4X 4F14.3 I BZY02570 M = NN - 1 BZYOZSBO WRITE (6. 40) ( I. XIII. Y(I). DALFII-l). XMPIII. YMPIII. DELSIII BZY02590 1 . RSDSIII. I = 2. M I BZY02600 4O FORMAT (1H I3. 2F14.8. 28X F14.8 / 4X 4F14.8 I BZY02610 WRITE (6. 44) NN. XINNI. YINNI BZY02620 44 FORMAT (1H I3. 2F14.B I BZY02630 GO TO 600 BZY02640 450 WRITE (13) (XIII. I =1. NNI 82Y02650 WRITE (13) (VIII .I=1.NNI 82Y02660 M = NN-1 BZY02670 WRITE I13) IXMPII I. I=1.M I BZY02680 WRITE I13) IYMPII I. I=1.M I BZY02690 GO TO 600 82Y02700 500 IF (MON.GE.1I GO TO 501 IF IFLGIZ .LE. 0) WRITE (6.48) (I. XIII. Y(I). I 8 1. NNI BZY02710 48 FORMAT (1H I3. ZF14.8 ) BZY02720 501 M=NN IF (FLGIZ .LE. 0) GO TO 530 82Y02740 WRITE I13) (XIII. I31.NNI B2Y02750 WRITE (13) (VIII. I=1.NNI BZY02760 530 D0 550 I = 1. MN 82Y02770 550 III) = CMPLXI XIII. Y(I) I BZY02780 600 WRITE I9) I III). I 8 1. NN I 82Y02790 IF I BDN .E0. 0 I GO TO 2000 BZY02800 M=NN-1 WRITE (9) I SINAIII. I I 1. M ) BZYOZBIO WRITE I4) I SINAIII. I = 1. M I BZY02820 WRITE (9) I COSAIII . I = 1. M I BZY02830 WRITE (4) I COSAIII . I 8 1. M I BZYOZB40 WRITE I9) I QIII. I = 1. NN I BZY02850 2000 CONTINUE 82Y02860 NT = NT - NB - NDINBOII BZYOZBTO NT = TOTAL NO. OF ELEMENTS BZY02880 IF (FL611aE0.0I RETURN 82Y02890 REWIND 12 B2Y02900 REWIND 4 82Y02910 M I 1 BZY02920 N = NDI1I-1 BZY02930 DO 2050 J 3 1. NB BZY02940 READ I12) (XMPIII.I=M.NI BZY02950 READ (4) (SINAIII.I=M.NI 82Y02960 READ I12) IYMPIII.I=M.NI 82Y02970 READ I4) ICOSAIII.I=M.NI 82Y02980 71 READ (12) 82Y0299O M 8 N+1 BZY03000 2050 N = N+NDIJ61I-1 82Y03010 IF IFLGII.LE.8I GO TO 2100 82Y03020 WRITE (6.56) 82Y03030 56 FORMAT (1H1 5X 37HNUM8ER OF NON-UNIFORM FLOWS EXCEEDS 8 // 82Y03040 1 6X 18HPROGRAM TERMINATED I 82Y03050 STOP 82Y03060 2100 NCFLG = NCFLG f FLG11 DO 4000 K = 1. FLGll 82Y03OBO READ (5.64) NTYPE.XR.YR.SEQZ 82Y03090 64 FORMAT ((1. 9X 2F10.0. 46X I4) 82Y03100 IF ISE02.LT.SE01I GO TO 60 BZY03110 SE01 = SE02 82Y03120 IF (NTYPEoGT. 1) NGO TD 2400 BZY03130 DO 2200 I = 1. 6 82Y03140 1READININ.20INUFIII.NUFII01I.NUFIIOZI.NUFII03I.NUFII+4I.NUFI1+5). BZY03150 SE02 BZY03160 22001 SE01 3 SE02 BZYG3180 2160 DO 2300 I = 1. NT REACININ.20ITUFIII.TUFII#1I.TUFII’ZI.TUFI103I.TUFIIO4I.TUFII95I. BZY03200 1 SE02 BZY03210 2300 SE01 = SE02 B2Y03230 2260 IF (NTYPEI 3000.3000.2800 2400 IF INTYPE. E0.3I GO TO 2600 82Y03250 DO 2500 I = 1. NT 82Y03260 T1 = (XMPIII- XRI**2 0 IYMPII)-YRI*‘2 82Y03270 NUFIII = (YMPIII-YRI/Tl 82Y03280 2500 TLFIII = IXR-XMPIIII/TI 82YO3290 GO TO 2800 BZY03300 2600 DO 2700 I = 1. NT 82Y03310 NUFIII = YMPIII- YR 82Y03320 2700 TUFIII = XR- XMPIII BZY03330 2800 DO 2900 I = 1. NT BZY03340 T1 = NUFIII 82Y03350 NUFIII =-T1‘SINAIII+TUFIII*COSAII) 82Y03360 2900 TUFIII = T1*COSAIII+TUFIII‘SINAIII 82Y03370 3000 WRITE I4) INUFIII.I=1.NT) 82YO33BO WRITE I4) ITUFIII.I=1.NTI 82Y03390 WRITE (6.68) HEDR. K. II.NUFIII.NUFII41I.NUFII02).NUFI143I. 1 NUF(Ii4I.NUF(I*5I.I=1.NT.6 68 FORMATI1H1.6X.15A4.//7X.ZOHNON-UNIFORM FLOW NO..I3.//12X.2HNG. BZY03410 1 11X ZHTG // I1X I5. 6F13.2I 4000 CONTINUE 82Y03430 RETURN BZY03440 END 82Y03450 SIBFTC ZZYAA SCBROUTINE SUBCURIX1.Y1.XZ.Y2.X3.Y3.ALF.ALF2.DELTI COMMON /FORCUR/XCURVI200).YCURVIZDO).KKK COMMON INBSI KII COMMON XDI50I.YDI50I COMMON IFNC/ 8. C. DIMENSION XSI50I. DXI50I. Y(50I. XLI50I. YLI50I.NDI10I DATA RAD/. 01745329/ 72 DATA TOL/1-E‘6/ EXTERNAL FUNC WRITE (6.23) XZ.Y2.X3.Y3.ALF.DELT .ALF2.X1.Y1 23 FORMAT (9F10.4I .— NH ALF2=ALF2*RAD X38X2§9.72‘COSIALF2I-DELT/2.0*SINIALF2) Y3=Y2-DELT/Z.0*COSIALF2I-9.72#SINIALFZI ALF2=-ALF2 ALF a -ALF * RAD X2MX1= (X2-X1I X2PX1= X2*X1 XITXZ: X1 ‘ X2 Y2MY1= Y2-Y1 X153 X1 *X1 X25a X2‘X2 DET = -1.0 # IX2MX1I ‘*3 ALF = SINIALFIICOSIALFI ALF2=SINIALF2IICDSIALF2I A=IXZS*IX2MX1#XI‘ALF §Y1*I3.0*X1-X2IIOXIS‘IYZ‘IX1-3oo‘XZIOALF2*X2‘ 1X2MX1IIIDET 8=(X2‘I6.0*X1¥Y2MY1*ALF‘(2.0‘XIS-X1TX2~XZSIIOALFZ‘X1*IXISQX1TX2-2. 10*XZSII/DET C=I-3.0*Y2MY1*X2PX193.0‘ALF‘IXZS-X1$I+(ALF2-ALFI‘IX25*XITX2-2.0*X1 ISII/DET .— u D D'I2.0'Y2MY1-2.0*X2MX1¢ALF-(ALF2-ALFI‘XZMXII/DET SC 3 SIMPSIIX1.X2.FUNC.KI S= SC 0 DSQRTI IX3-X2I**2 * (Y3-Y2I**2I JJ=KKK+1 KK=KKK-1 DS‘S/FLDATIKKI DO I (=1.KKK AY a 1-1 XSIII= AY *DS XG=IX2MX1IIFLOATIKKI+X1 XI1I= X1 SIMPSIIX1.XG.FUNC.KI + X6 # FUNCIXGI I IF (REL .GT. TOLI GO TO 3 IFIXG .GE. X2) GO TO 4 Y(I)z A + X5 *(BTXG‘IC*D‘XGII YP ' 8 P XII) *(2.0tc+3.0¥ D ‘XIIII YP ' -1.0/YP TI'ET 3 ATANIYPI IFIYP .LT. 0.0) 50 TU 10 DY "DELT ‘ SINITHETI Dxi‘DELT ‘ COSITHETI GO TO 11 DY ’ DELT * SINITHETI DX ‘ DELT * COSITHET) XLIII' XIII 0 OX YLIII' Y(I) O DY SLOP = (Y3-Y2I/IX3-X2) THET I ATANISLOPI SLOPL= IY3-IY2-DELTII/IX3-X2I X6 = XSIIII - SC DY = XG * SINITHETI OX 8 XG * CDSITHETI XIII) = X2 + OX YIIlI= Y2 + DY IFIALFZ .NE. 0.0) GO TO 20 YLII1I= SLOPL * (XIIII-XZI * (Y2 - DELT) XLII1)= XIII) GO TO 21 20 DELTG = (1.0 - XG/(S-SCII * DELT SLOPL = '1.0/ALF2 SLOPL = ATANISLOPLI DYL = -DELTG * SINISLOPLI DYX = -DELTG # COSISLOPLI YLII1I = YIIII 6 DYL XLII1) = XIIlI 9 DXL 21 I1 = II * 1 IFII1 .EQ. KKK) GO TO 5 GO TO 6 X(KKKI=X3 XLIKKKI=X3 Y(KKKI=Y3 YLIKKKI=Y3 YP = -1.0/ALF THET = ATANIYPI DY a -DELT ‘ SINITHETI OX 2 -DELT * CDSITHETI XLI1I= X1 4 0X YLIII= Y1 f DY DO 7 I=1.KKK J=JJ-I XCLRVIII= XLIJI YCURVIII= YLIJI K=KII-KKK+I XCLRVIKI= XIII YCLRVIKI= Y(I) RETURN END VI .5 SIBFTC FXYZ FLNCTIDN FUNCIXI COMMON /FNC/ B.C.D FUNC= $0RTI1.0 F (8+2.U * C‘X03.0‘D‘X‘XI 1*‘2I RETURN END SORIGIN ALPHA SIBFTC 82Y3 CDECK BZY3 82Y03460 SLBROUTINE PARTZ BZY03470 74 C ** MATRIX FORMATION SUBROUTINE *2 COMPLEX I l. 0. H1. H2. TF. T2. T1. CLUG. CSINH INTEGER FLGOZ. FLGD3. FLGO4. FLGO5. FLGOO. FLGDT 1. FLGOB. FLGO9. FLGlO. FLG11. FLGIZ DIMENSION l(Z99I. QI300I. SINAI299I. CDSAI300I. NDI10II 1 VNSI400 I. VTSI400 I. AIZ99I. 3(299). SUMDSIIDI. NLFI10) DIMENSION VNSTI10).HEDRI15I.CASEIZI.XMCIBI.YMCI8I.ADDYI8I COMMON IM.HEDR.CASE.RPI.RZPI.SP.CL.ALPHA.FALPHA.DALFA.CHORD.SUMDS. 1 XMC.YMC.ADDY.FLGOZ.FLGO3.FLGD4.FLGO5.FL506.FLGOT.FLGOB. FLGO9.FLG10.FLGI1.FLG12.ND.NLF.NER.NT.NB.NCFLG RPI = 0.31830989E0 RZPI =‘Oo15915494E0 REWIND 9 REHIND 10 REWIND B M E 1 N = NDI1I - 1 M = 1 N1 = NDI1I DU 100 L g 1. B READ I9) (III). I = M. N) READ I9) ISINAIII. I = M. N) READ I9) (COSAIII. I = M. N) READ I9) IQIII. I = M1. N1) M: f N = N 9 NDIL91I ' 1 M1 = N * 1 100 N1 = N1 0 NDIL‘II K = NB * NT 00 200 I = 1. K VNSIII = 0. 200 VTSII) = 0. ASSIGN 850 T0 N50 ASSIGN 1050 TD N51 IF (FL611.LE.0I 50 T0 400 REWIND 4 C C M1 = ZtNB DO 250 K = 1. M1 250 READ (4) DO 300 K = 1. FLGlI IVNSIII. I = M1. N1) 400 NPFLG = O A230. 0 CMPLXI COSAIJ). 200 I = 1. N8 -SINAIJI 82Y03480 82Y03490 82Y03500 82YC3510 BZY03520 82Y03530 BZYO3540 82Y03550 82Y03560 82Y0357O 82Y03580 82Y03590 82Y03600 82Y03610 82Y03630 BZY03640 82Y03650 BZY03660 82Y03670 BZY03680 BZY03690 82Y03700 82Y03710 BZYO3720 82Y03730 BZY03740 BZY03750 B2Y03760 BZY03770 BZY03780 82Y03790 BZY03800 82Y03810 BZY03820 82Y03830 82Y03860 82Y03870 82Y03880 82Y03910 82Y03920 82Y03930 BZY03940 BZY03950 82Y03960 BZY03970 82Y03980 BZY03990 82Y04000 82Y04010 B2Y04020 82Y04030 82Y04040 BZY04050 B2Y04060 82Y04070 700 720 800 850 900 1000 1050 1100 1200 1250 1275 U) 10 1300 1500 C 1800 2000 15 20 3000 1 IF (SP ONE. 000) TF = 00 1000 K = M1. N1 J2 = J2 * 1 CALL FORMI I J. K. J2. Z. T IF I NPFLG .NE. 0 ) T2 = CONJGIHI) ‘ T1 A1 3 AIMAGT T2 ) [F (J .EO. J2) A1 = 31 8 REALT T2 ) GO TO 800 A1 a - AIMAGI H1 ) B! I REALI H1 ) GO TO N50.I850.900) VNSTJ1) = VNSIJ1) - 81 0 82 VTSIJl) = VTSTJI) 0 A1 - A2 ATJZ) = A1 * A2 BTJZ) = Bl O 82 GO TO N51.(1050.1100) VNSIJl) = VNSlJI) I SUMDSTJ4) VTSIJI) = VTSIJI) / SUMDSIJé) M1 = N1 * 2 N1 = N1 * NDTIOI) VNSTTI) = - SINAIJ) VNSTIZ) = COSAIJ) IFI FLGIl .LT. 1 ) KL = J - L 00 1250 I = 3. NCFLG VNSTII) = VNSIKL) WRITE (10) I AII). I NRITE (10) I 8(1). I IF I FLGOT .E0. 0 ) WRITE I6. 5) J. (All). FORMAT (1H0 12H AJK HRITE l6. 10)J. IBII). I FORMAT (1H0 12H BJK WRITE ( 8) I All). I HRITE ( 8) ( BII). I CONTINUE M 3 1 N = L 00 2000 J = 1. N8 IF ( NLFIJ) .NE. 0 WRITE (4) (VNSTI). HRITE (4) (VTSTI). M = N + 1 N = N + L IF ( FLGOT .E0. 0 ) N 3 NB * L ) I I WRITE (6. 15) I VNSTI). HRITE (6. 20) I VTSII). FORMAT (1H0/10X 3HVNS ll/ 75 CSINHT3.14159265EJ*(Z(J)-Q(M1))ISP) Q. SINA. COSA. H1 ) 0 750 ABST A1 ) GO TO 1275 3 1. NT). T VNSTIKL). KL NT) 4 // (6F15o8) ) 14 // (6F15.8) ) GO TO 1800 p 60 TO 3000 l. N ) NI - I (6F15o8) ) FORMAT (1H0 I 10X 3HVTS /// (6F15.8) ) IF I FLGOZ .EQ. 0 .OR. NPFLG = 1 L I NDTNB*1) READ (9) ( III). I I .NE. 0 ) RETURN 82Y04080 82Y04090 BZY04100 82Y04110 82V00120 BZY04140 BZV04260 82Y06270 82Y04280 82V04290 82Y06300 BZV04310 BZY04320 BZV04330 82Y04340 82Y04350 BZV04360 82Y04370 82Y04380 82Y04390 82Y04400 BZY04410 BZY04420 82Y04430 BZV04440 BZV04450 82Y04460 BZY04470 82Y04480 82Y04490 82V04500 BZY04510 BZV04520 82Y04530 82Y04540 82Y04550 BZY04560 82V04570 82Y04580 82Y04600 82Y04620 BZV04630 82Y04640 BZY04650 82Y04660 82Y04670 82Y04680 82V04690 82V04700 32V04710 BZY04720 82Y04730 BZY04740 BZY04750 82Y04760 BZY04770 BZYOkTBO 82Y04790 K = N8 ' L BZY04800 DD 3100 I = 1. K BZYC4810 VNSTI) = 0. 82Y04820 3100 VTS(I) = 0. 82Y04830 ASSIGN 850 T0 N50 82Y04840 ASSIGN 1050 TO N51 82V04850 GO TO 500 BZY04860 END BZY04870 SIBFTC 82Y4 coecx 02v4 32.04800 SLBROUTINE FORMl I J. K. 02. z. o. SINA. COSA. H I ezv04890 COMPLEX IM. 1. O. H. CLOG Ozvc4900 COMMON IM.HEDR.CASE.RPI.R2PI.SP.CL.ALPHA.FALPHA.NER.NT.NB.NCFLG 32v04910 DIMENSION zI299I. 0I300I. SINAI299I. COSAI299I 32vo4920 DIMENSION HEDRTIS).CASE(2) azv04930 u =CLUG I (l(JI-QTKI) / IZIJI-DIK+1II I 32.04940 N . I NCOSATJZ) - IM*SINA(J2) I . RZPI I N 02.04950 REI Ozvo49so ENDUR Dzv04910 sDRIcIN ALPHA sIDEIc BZY9 CDECK azvq BZY06110 StBROUTINE RARI4 BZV06120 CCMPLEX IM BZY06130 INTEGER FLGCZ. FLGO3. FLGD4. FLGOS. FLGOb. FLGO7 32v0614o I. FLGOB. FLGO9. FLGIO. FLGll. FL012 BZYCélSO DIMENSION A(300 I. RI300. 5I. NDIIDI. NLFTIO) BZY06160 DIMENSION MEDRIISI.CASEI2I.SUMDSIIDI.XMcIaI.VMCIeI.ADDVI 8) 82Y06110 CCMMON IM.MEDR. CASE.RPI. R2PI.SP.CL.ALPHA. FALPHA.DALFA. CHORD.SUMDS.BZY06180 1 xMc.VMc.ADDv.FLDoz.FL003.EL004.FL005.FL006.ELDDI.FLOOD. 82V06190 2 FLGO9.FLGIO.FLGII.FLGlZ.ND.NLF.NER.NT.NB.NCFLG BZVObZOO REHIND 1 BZV06210 REuIND 3 82V06220 RENIND 4 azvoozao REhIND 10 azv0624o M s 1 82V06250 N . NDIII - 1 I 82Y06260 DO 100 K = 1. N8 BZY06270 READ I4I I RII.II. I = M. N I BZYObZBO READ I4I I RII.2I. I = M. N I BZY06290 M = N . 1 ezvooaoo 100 N = N . NDIK+1I - 1 BZY06310 c PRECEDING READS IN SINES. CCSINES. DNSEI FLOHS NEXT (IF ANvI. BZY06320 [F I NCFLG .LE. 2 I GO ID 180 Dzvooaao DD 150 J = 3. NCFLG 02v06340 READ I4) I RII.JI. I = 1. NI I BZY06350 150 READ I4I Dzvoeaeo 180 OD 200 J = 2. NCFLG 32v00370 D0 200 I = I. NI 82Y06380 200 RII. J) = -RII. J) 82V06390 250 D0 300 I - 1. NT Dzvob4oo READ (10) I AI J I. J = 1. NI I azv06410 READ T10) 32v06420 300 leTEIl) (ATJ). 4:1. NII.IRII. J). 0:1. NCFLG) 02v06430 77 END FILE 1 82Y06440 REHIND 1 82V06450 RETURN 82Y06460 END 82Y06470 SORIGIN ALPHA $IBFTC C20X9 C20X9 BZY06480 SLBROUTINE SOLVIT (A. ND. ND. KD. NI. MM. NO. NH. *) 62Y06490 DIMENSION A ( KD ) BZY06620 C BZV06630 LOGICAL LAST BZY06640 C 82Y06650 N = ND 82Y06680 M = MD BZY06690 KORE = KD 82Y06700 NPM = N + M _ 82Y06710 IF (MAXOI3 * NPM. M t N) .GT. KORE) RETURN 1 BZY06720 MT 8 MM BZY06730 REHIND MT BZV06740 NIN = NI BZY06750 REhIND NIN 82Y06760 NOUT 3 NO 82Y06770 REHIND NOUT 82Y06780 MP1 = M O 1 82Y06790 NN = N 82Y06800 NEL = NPM 82Y06810 C 82Y06820 C - - CALCULATE THE MAXIMUM NO. OF ROHS. 'K' BZY06830 C BZY06840 10 K = (KORE - NEL) / NEL BZY06850 C 82V06860 C - - TEST TO SEE IF THE REST OF THE MATRIX HILL FIT IN CORE BZYC5870 C BZY06880 LAST 3 K .GE. NN BZY06890 IF (LAST) K = NN BZY06900 C 82Y06910 C - - READ 'K' ROHS OF THE AUGMENTED 'A' MATRIX BZY06920 C ' 82V06930 30 NT BZY06940 DO 20° 18 311. K BZV06950 NS = NT + 82Y06960 NT = NT + NEL 82V06970 40 READ (NIN) (AIIUI. ID = NS. NT) 82Y06980 C 2V06990 C - - CFECK TO SEE IF HE HERE UNLUCKV ENOUGH TO END UP HITH ONLY ONE ROHBZYOTOOO C 2Y07010 IF (K .EQ. 1T GO TO 90 BZYOTOZO C BZY07030 C - - ‘K' IS GREATER THAN '1' SD hE CAN START THE TRIANGULARIZATION BZY07040 C BZY07050 NELP1 3 NEL 6 1 82Y07060 NS = - NEL BZVOTOTO NELPZ = NELP1 + 1 82Y07080 C BZY07090 C - - FORM THE 'TRAPEZOIDAL' ARRAY (SI C 060 non COO DD 50 [B = 2. K NP = NELPZ - 18 NS = NS 0 NELP1 NT 5 NS DO 50 IO = NIB. K NT t NT 0 MN = NT N8 N5 AINT) = I- AINT)) / AINS) DO 50 NF = 2. MN = MN I 1 NB = N8 + 1 AIMN) = AIMN) f AINTI * AINB) IF (LAST) GO TO 90 HRITE THE 'TRAPElOIDAL' MATRIX ON TAPE NT = 0 NP = NEL NS = ' NEL DO 60 ID = 1. K NS = NS + NELP1 N = NT NEL HRITE (MTI 1NP. IAIIBI. I8 = NS. NT) NP NP NP = NP NS = KORE - NEL O 1 READ ANOTHER ROH DO 80 IO = I. READ (NIN) (AIIB). I3 = NS. KORE) MODIFY THIS ROH BY THE 'TRAPEZOIDAL' ARRAY NF = MN 0 AIMN) = I- AIMN)) / AINTI DO 65 NN =1NF. RE AINNI = AINN) * AIMN) * AINB) MN — NT : NT 4 NELP1 HRITE THE MODIFIED RDH ON TAPE HRITE INOUTI (AINTI. NT F MN. KORE) REHIND NOUT REHIND NIN ShITCH THE TAPES NT = NIN NIN = NOUT NOUT = NT BZY07100 82Y07110 82Y07120 82Y07130 82Y07140 82Y07150 BZY07160 82Y07170 82Y07180 82Y07190 82Y07200 82Y07210 BZY07220 BZY07230 BZY07240 BZY07250 BZY07260 82Y07270 BZYOTZBO 82Y07290 32Y07300 BZY07310 BZY01320 82Y07330 82Y07340 BZY07350 BZY07360 BZY07370 82Y07380 BZY01390 BZY07400 82Y07410 BZY07420 82Y01430 82Y07440 82Y07450 BZY07460 82Y07470 82Y07480 82Y07490 82Y07500 BZY07510 32Y07520 82Y07530 BZY07540 BZY07550 BZY07560 BZY07570 BZY07580 82Y07590 62Y07600 BZY07610 82Y07620 BZY07630 BZY07640 BZY07650 82Y07660 BZY07670 82Y07660 82Y07690 non fifififi nan 100 105 110 125 130 '79 RE-CALCULATE ROH LENGTH AND LOOP BACK NEL = NEL - K NN = NEL - M GO ID 10 REHIND ALL TAPES REHIND MI REHIND NIN REhIND NOUT CONDENSE THE MATRIX NN = NEL NL = NELP1 IF IK .E0. 1) GO TO 105 NS = 1 NT = NEL DD 100 I8 = 2. K NS = NS 4 NELP1 NT = NT 0 NEL DO 100 ID = NS. NT AINL) = AIIOI NL = NL O 1 N1 = KORE - K * M f 1 THERE. NOH HE CAN START THE BACK-SOLUTION NDTE..THE FIRST AVAILABLE LCCATION FOR THE SOLUTIONS IS AIN1I NREM = N NEL = NPM LAST = K .EQ. N NPASS = 0 SOLVE FOR THE ANSHERS CORRESPONDING TO 'K' ROHS KMI = K - 1 KP1=K01 NS = NL - MP1 NPASS = NPASS 9 1 00 130 MN = 1. M NF = NS + MN AINF) = AINF) / AINS) NT NS IF IKM1 .E0. 0) GO TO 130 00 125 I8 = 1. KM1 NF = NF - IB - M NT 3 NT - MP1 - IB SLM = 0.0 NP = NF N2 = MP1 + ID DO 120 ID : 1. I8 NN = NI + ID NP : NP + N2 - ID 120 SUM = SUM * AINNI # AINP) AINFI = IAINF) - SUM) / AINU CONTINUE BZYOTTOO BZY07710 BZYOTTZO 82Y07730 BZY07740 82Y07750 BZY07760 BZY07770 BZYOTTBO BZY07790 82Y07800 BZYOTBIO BZYO7820 BZY07830 82Y07840 BZY07850 BZY07860 BZYCTBTO BZY07880 82Y07890 32Y07900 82Y07910 BZY07920 BZY07930 82Y07940 BZY07950 82Y07960 BZY07970 BZY07980 82Y07990 BZYCBOOO 82Y08010 BZY08020 62Y08030 BZY08040 BZYOBOSO 82Y08060 82Y08070 82Y08080 BZY08090 BZY08100 82Y08110 82Y08120 BZY08130 BZY08140 BZYOBISO BZY08160 62Y08170 32Y08180 BZY08190 82Y08200 BZY08210 82Y08220 82Y08230 BZYOBZAO BZY08250 82Y08260 BZYOBZTO BZYC8280 82Y08290 GOO GOO fihnhfififi non ("TOO 80 - - MOVE THE SOLUTIONS TO CONTIGUOUS LOCATIONS STARTING AT AINI) N1 = KORE DO 140 NN DD 135 MN NL 3 NL - N1 = N1 - 0 1 =1IK a1.M 1 135 AINI) = AINL) 140 NL I NL - NN - - HRITE THE SOLUTIONS ON TAPE N 145 HRI .— # HRITE (NIN NS = N1 - DO 145 MN I = NS 0 TE I NI IK 1 ‘1.M MN N ) IAIIOI. IO 8 NT. KORE. MI - TEST IF THIS IS THE LAST PASS IF (LAST) - HE MUST NOH MODIFY THE TRIAKGULAR MATRIX TO REFLECT THE EFFECT OF 2 GO TO 200 THE SOLUTIONS OBTAINED SC FAR ‘ NOTE..LOCATIONS AIlI TO AIN1-1I ARE NOH FREE TO USE - CALCULATE THE NEXT VALUES OF 'NEL' AND 'NREM' NELOLD = NEL KOLD NEL = NEL - K NREM = NREM - K - NOH APPLY K:I-l.* THE INCREDIBLE FORMULA FOR THE NEH 'K' M - 1) / 2 + IFIXISDRTIO.25 P FLOATII4 * M I 2) * M * 1 2 ‘ (KORE - NELOLD)I)) NROH = NRE IF (K .LT. LAST = .TRUE NROh I 1 K = NREM 150 NS M - K O 1 NREM) GO TO 150 O 3 1 NT 8 NELOLD O 1 - - READ IN THE ROHS TO BE MODIFIED DO 190 [B NT ' NT 8 NT + 160 READ I MT NP 3 N1 - NF 1 NT - NN = MN - DO 170 MN N2 = NF NA = NP 9 NB 1 NA = 1. NREM - Nf - IF ([8 .LE. NROH) GO TO 160 S * NN NN I NN. IAIIOI. I0 3 NS. NT) M - KMI KOLO ‘1." MN BZY08300 BZY08310 82Y08320 82Y08330 BZY08340 BZY08350 BZY08360 BZY08370 BZYDB380 82Y08390 BZYCBQOO BZY08410 BZY08620 BZY08430 82Y08640 82Y08450 BZY08460 82Y08470 BZY08480 BZY08490 BZY08500 BZY08510 82Y08520 BZY08530 82Y08540 BZY08550 82Y08560 BZY08570 82Y08580 BZY08590 BZY08600 82Y08610 BZY08620 82Y08630 BZY08640 82Y08650 BZY08660 BZY08670 32Y08680 BZY08690 BZY08700 BZY08710 BZY08720 BZY08730 32Y08740 BZY08750 BZY08760 BZYOBTTO BZYOBTBO BZY08790 BZYOBBOO 82Y08810 BZYOBBZO 82Y08830 BZY08840 BZY08850 BZY08660 BZY08870 BZY08880 82Y08890 GOO nan COO 175 180 190 210 220 D0 165 ID = 1. KOLD SLM.= SUM 9 AIN2) * AINA) I MN - 1 AINZ) = AINZI - SUM HRITE THE MODIFIED RDH ON TAPE DR CONDENSE THE ROH NL = NT - M + 1 IF (IB .GE. NROHI GO TO 175 NF = NL - KP1 HRITE (NOUT) NN. IAIIO). IO = NS. NF). (AIIO). IO = NL. NT) GO TO 190 NF = NL - KOLD DO 180 MN = NL. NT AINF) = AIMNI NF = NF 5 1 CCNTINUE REHIND MT REHIND NOUT ShITCH THE TAPES NT = MT MT : NOUT NDLT = NT LOOP BACK THRU THE SOLUTION ML 3 NF GO TO 110 START TD HRAP IT UP REHIND NIN N2 = NOTEo. AT THIS POINT ALL LOCATIONS AI1) THRU AIKORE) ARE FREE DO 220 I8 1 1. NPASS READ (NIN) K N1 = N2 - K 6 1 NS = N1 NT = N2 READ IN THE SOLUTIONS DO 210 ID = 1. M READ (NIN) IAINN). NN = NS. NT) NT = NT 0 N NS = NS 0 N N2 3 N1 ‘ 1 HRITE THE SOLUTIONS ON TAPE NT 3 0 DO 230 ID = 1. M NS 3 NT * 1 BZY08900 BZY08910 82Y08920 BZY08930 BZY08940 82Y08950 82Y08960 BZY08970 BZY08980 BZY08990 82Y09000 BZY09010 82Y09020 BZY09030 82Y09040 BZY09050 BZY09060 82Y09070 82Y09080 82Y09090 BZY09100 BZY09110 82Y09120 BZY09130 BZY09140 BZY09150 BZY09160 62Y09170 BZY09180 BZY09190 82Y09200 BZY09210 BZY09220 BZY09230 82Y09240 82Y09250 BZY09260 BZY09270 82Y09280 BZY09290 BZY09300 BZY09310 82Y09320 82Y09330 BZY09340 82Y09350 82Y09360 BZY09370 82Y09380 BZY09390 82Y09400 82Y09410 82Y09420 BZY09430 82Y09460 BZY09450 BZY09660 BZY09470 BZY09480 BZY09490 NT 2 NT 0 N 230 HRITE INHI IAINNI. NN = NS. NT) C RETLRN END SORIGIN ALPHA SIBFTC BZYE CDECK BZYE SUBRDUTINE PARTS INTEGER FLGOZ. FLG03. FLGO4. FLGOS. FLGOb. FLGOT 82 1. FLGOB. FLG09. FLGlO. FLG11. FLGlZ DIMENSION 3(299). VTI299. 5). SIGI299. 5). TIZ99. 5|. 1. NDIIOI. NLFI10I. XI300I. YI300I. XMPIZ99). CPI299.‘6I YMPI299I.CASEIZI 2. SUMDSIIDI. XMCIBI. YMCIBI. ADDYIBI.HEDRI15I. VELIDIIOI.VIDI2I COMMON IM.HEDR.CASE.RPI.RZPI.SP.CL.ALPHA.FALPHA.DALFA.CIORD.SUMDS.BZY10820 XMC.YMC.ADDY.FLGOZ.FLGO3.FLGO4.FLGOS.FLGDG.FLGOT.FLGOB. 2 FLG09.FLGIO.FL611.FLGIZ.ND.NLF.NER.NT.NB.NCFLG.FL015.FL616 COMMON SUMSIG.VINF.MON.BETA.CDIM.VREF.DELT.FLG13.FL614.ITER.ALF2 EQUIVALENCE I T. CP I DATA VELID I 3H V0. 4H V90. 3H V1. 3H V2. 1 2H V6. 3H v7. 3H V8 /.BLANK/1H I REhIND 3 REhIND 4 REhIND 10 REHIND 8 REHIND 12 REHIND 13 H = 1 N = NDII) - 1 DO 100 K = 1. NB READ (4) I TII.ZI. I = M. N I READ I4) I TII.1). I = M. N I C READS IN SINES AND COSINES M = N I 100 N = N + NDIK+1I - 1 IF ( NCFLG .LE. 2 I GO TO 200 OD 150 J = 3. NCFLG READ (4) 150 READ (4) TII.JI. I = 1. NT I I 200 00 250 J = 1. NCFLG 250 READ I3I I SIGII.JI. I = 1. NT ) DO 400 I = 1. NT READ (10) READ I10) I BIL). L = I. NTI DO 400 J z 1. NCFLG PR 3 00 DO 300 L ’ 1. NT 300 PR = PR + BIL)*SIGIL.JI VTII.J) = PR 0 TII.J) 400 CPII.JI = 10 - VTII.JI**2 DO 500 J = 1. NCFLG 500 HRITE I 8) I VTII.J). I = 1. NT I IF (MON-1) 510.520.520 3H V3. 3H V4. 3H V5. 82Y09500 82Y09510 BZY09520 82Y09580 BZY09590 82Y10740 82Y10750 82Y10760 82Y10770 82Y10780 BZY10790 BZY10800 BZY10810 BZY10830 BZY10850 BZY10860 BZY10870 BZY10880 BZY10890 BZY10900 BZY10920 82Y10930 BZY10940 BZY10950 BZY10960 BZY10970 BZY10980 BZY10990 82Y11000 BZY11010 BZY11020 82Y11030 82Y11040 BZY11050 BZY11060 BZY11070 BZY11080 62Y11090 BZY11100 82Y11110 82Y11120 82Y11130 82Y11140 BZY11150 82Y11160 32Y11170 510 M I 1 N i NDI1) M1 3 1 N1 1 NDI1) - 1 DD 700 J 8 1. NB READ I13) I XII). I I M. N I READ I13) I YIII. I 8 M. N I READ I13) I XMPIII. I I M1. N1 I READ I13I I YMPIII. I 8 M1. N1 I M I N O 1 N 8 N 9 NDIJ’1I M1 = N1 0 1 700 N1 = N1 * NDIJOII - 1 DD 2500 L = 1. NCFLG L ' 2 IF IFLGIO .LT. 2) GO TO 1000 1000 HRITE I6. 1100) HEDR. CASE 1100 FORMAT I1H1 25X Z6HOOUGLAS AIRCRAFT COMPANY I 23X 21HLONG 1 DIVISION ///I 5X.15A4.// 6H CASE I2A4I IFI K I 1150. 1300. 1500 1150 HRITE I6. 1200) 1200 FORMAT IIH 19HSTREAMFLOH SOLUTION I GO TO 1700 1300 WRITE I6. 14°0I 1400 FORMAT I1H 23H90'OEGREE FLO“ SOLUTION I GO TO 1700 1500 WRITE I6. 1600) K 1600 FORMAT I1H 35HNON- UNIFORM ONSET FLOH SOLUTION NO. I3 I 1700 IF IFL512.LE.0I HRITE I6 .1800I 1800 FORMAT I1H Z5HUNTRANSFORMEO COORDINATES // I IF IFLG12.GT.0I HRITE I6. 900) 1900 FORMAT I1“ 23HTRANSFORMEO1 COORDINATES // I WRITE (6.1950) 1950 FORMAT Ilzx 1HX 13X 1HY 14X IHV 12X ZHCP 11X 5H$IGMA // I 2000 HRITE I6. 2100) I. XIII. VIII. XMPIJI. VMPIJI. VTIJ.LI . CPIJ.LI. SIGIJvLI 2100 FORMAT I1H I3. ZF14.8 / 4X 5F14oe I I 3 I + J = J 0 1 IF I I .EQ. N ) GO TO 2200 IF I I .LE. LCTRI GO TO 2000 LCTR = LCTR o 22 RITE I6. 2300) I. XIII. VIII 2300 FORMAT I1H I3. 2F14.8 // I I I 1 IF. I J .65. NT I GO TO 2500 GO TO 2000 2500 CONTINUE 520 RETURN END 82Y11190 82V11200 82Y11210 82Y11220 BZY11230 82Y11240 82V11250 82V11260 BZY11270 32'11280 BZY11290 32V11300 82Y11310 52Y11320 82'11330 82Y11340 82Y11350 82V11360 62Y11380 82V11390 11440 8EACHBZY11450 11460 52Y11470 82V11480 82V11490 82V11500 82Y11510 82Y11520 82V11530 BZV11540 82'11550 BZV11560 BZY11570 BZY11580 82Y11590 32Y11600 82Y11610 BZV11620 82Y11630 82Y11640 BZY11650 82V11660 82Y11670 BZY11680 82Y11690 82Y11700 82Y11710 82Y11720 82Y11730 82Y11740 82Y11750 82Y11760 82Y11770 82Y11780 82V11190 82V11800 ‘lu... 84 SORIGIN ALPHA SIBFTC BZYF CDECK BZYF SLBROUTINE PARTb COMPLEX IM 30 40 VI 0 INTEGER FLGOZ. FLGO3. FLGO4. FLGOS. FLGOb. FLGOT 1. FLGOB. FL609. FLGIO. FLGII. FLGlZ DIMENSION SIGMAI299. 4). SUMAI300. 4). SUM8I300. 4) DIMENSION THETVI299) DIMENSION VCI299I. XI300). YI300). XMPI299I. YMPI299I. XMI299I 1. YMI299). SINAI299). CDSAI299). CPI299). DELSI299). NDIIOI 2. NLFI10I. SUMDSIIOI. GAMI9I. XMCIB). YMCIB). HEDRI15I.ADDVI8) 4. YTEMPIISOO). VXLI299I. VYLI299I. XIJI299). YIJI299I. DVAI9.8I 3. DVTI9.10).DVI9.9I.OELSUTI299I.SIGTIZ99). XTEMPII500I DVXI9.10I. ZTEMPI300).VCIDIZI.VXIDIZI.VYIDI2I.GAMTI8).CASEI2I DIMENSION PRE$I299I.XNI299I.YNI299) DIMENSION XIDIZI. YIDIZI. XIDDFFIZI. YIDDFFIZ) BZY11810 82Y11820 BZY11830 82Y11840 82Y11850 82Y11810 82Y11880 82Y1189 82Y11910 BZY11900 82Y11920 BZV11930 COMMON IM.HEDR.CASE.RPI.RZPI.SP.CL.ALPHA.FALPHA.DALFA.CHORD.SUMDS.BZY11940 XMC.YMC.ADDY.FLGOZ.FLGO3.FLGO4.FLG05.FLGO6.FL607.FLGOB. FLGO9.FLGIO.FL611.FLGIZ.ND.NLF.NER.NT.NB.NCFLG.FLGI§.FL616 COMMON SUMSIG.VINF.MONIBETA.COIM1VREFIDELT0FLGI3IFLGIQ.[TERvALFZ EOUIVALENCE IGAM. DVT). IDV. DVTI10II. IOVA. OVTI19II 1. IVXL. XMP). IVVL. YMPI. ISIGT. XMI’ IXIJ. SINAIV IV 1J1 COSA) 2v IZTEMP. XTEMFI300I) DATA VCID.VXID. VYIO/4H V.1HC.4H V.1HX.4H V. IHY/ DATA XID /6H XMS .1HP/.YID/6H YM .IH P/ DATA XIDOFF [4“ X0 v3HFFB/vYIDOFF/4H YO.3HFFB/ SP? 3 SP It .577 .58. 0.0) SPP 3 1. 0 E6 IF IFLGOO .50. 0 00R. FLGOZ .60. 0) GO TO 40 REMINO 10 K 8 NCFLG - 2 REHIND REhIND 3 REhIND 4 REHIND 8 RENIND 12 REHIND 13 DD 50 J = 1. 10 DO 50 I = 1. 9 OVXIIoJ) = 0. DVTII.J) = 0. IF I FLGO4 .NE. 0 I ALPHA = DALFA IF I FLGOé .NE. 0 I ALPHA = 0. ALPHA I ALPHA I 57.295779550 CSALF = COSIALPHAI SNALF = SINIALPHA) READ I 8) I XTEMPIII. I = 1. NT I READ I 8) I YTEMPII). I a 1. NT I M = 1 q 82Y11950 BZY11970 BZY11960 82Y12000 BZY12010 82Y12020 BZY12030 6 Y 040 BZY12050 BZY12060 BZYIZOTO 82V12080 BZY12090 62Y12100 BZY12110 82V12120 82Y12130 BZY12140 BZY12150 82Y12170 BZYIZIBO BZY12190 BZY12200 BZYIZZIO BZY12220 BZY12240 32Y12250 82Y12260 82Y12210 BZV12280 BZY12310 It 150 160 5040 5070 5100 5125 5150 5200 5250 5210 5220 85 N = NDI1) - 1 32V12320 DO 100 I = 1. NB 82Y12330 GAMII) = IXTEMPIMI * XTEMPINI)*CSALF 82Y12340 GAMIII = - I GAMIII 9 IVTEMPIMI+YTEMPINII ’SNALF I BZY12350 M = N+1 82V12360 N = N f NDIIIII - 1 82Y12370 IF I K .E0. 0 I GO TO 160 82Y12380 DD 150 J 1. K BZY12390 READ I 8) I ZTEMPIII. I = 1. NT I M = 1 82Y12410 N = NDI1) - 1 BZY12420 DO 150 I = 1. NB 82Y12430 DVAII.JI = ZTEMPIM) 0 ZTEMPINI 82Y12440 DVXII.J) = DVAII.JI BZY12450 REthD 8 GAMI1I=IGAMI1I-DVAI1.1II IDVAI1.2I READ I 8) I PII = SNALF = SINI xALPHA ) 32713870 READ I 8) I YTEMPII). I = 1. NT I DO 5070 I = 1. NT 82Y13890 VCII) = XTEMPIII‘CSALF + VTEMPIII‘SNALF BZY13900 READ I 8) I XTEMPIII. I = 1 DO 5100 I = 1. NT 82Y13940 g VCIII = VCIlI 0 XTEMPIII IFINLFI1I.NE.OI GO TO 5150 , READIS) IXTEMPII).I=1.NTI DO 5125 I=I.NT VCIII=VCIII+XTEMPII)*GAMI1I DD 5200 I = 1. NT BZY13960 CPII) = 1. - VCII)*VCII) 82Y13970 M = 1 82Y13980 N = NDI1) - 1 82Y13990 M1 = 82V14000 N1 = NDI1) BZY14010 DO 5250 J = 1. 82Y14020 READ I4) I SINAIII. I = M. N I 82Y14030 READ I13) I XIII. I = M1. N1 I BZY14040 I READ I4) I COSAIII. I = M. N I 82Y14050 READ I13) I VIII. 1 = M1. N1 I BZV14060 READ I13) I XMPIII. I = M. N I 82Y14070 READ I13) I YMPIII. I = M. h ) 82V14080 READ I12) I XMII). I = M. N I 82Y14090 READ I12) I YMII). I = M. N ) 82Y14100 READ I12) I DELSIII. I = M. N ) 82V14110 M1 = N1 6 1 82V14120 N1 = N1 0 NDIJOII 82V14130 M = N + 1 BZY14140 N = N * NDIJ#1I - 1 BZY14150 M=1 N=NDI1)-1 DO 5210 I=M.N XNII01I=IXIII+XIII1I)/2.O YNII+1I=IYIII0YII+1II/2.0 PRESII*1I=CPIII WRITE I7) IPRESII+1I.I=M.NI WRITE I7) ICP II).I=M.NI NRITE I7) IXNI101I.I=M.N) WRITE I7) IYNIIFII.I=M.N) F (MON-2) 5220.5230.5230 PLNCH 5551. L.ALPHA 5551 FORMAT I15.F10.5) CALL BCDUMP IXMPIMI.XMPINII CALL BCDUMP (YMPIM).YMPINII CALL BCDUMP IVC (MI.VC (NI) 5230 IF (FLGIZ .LE. 0) GO TO 5252 O 5251 I = 1.NT 5251 DELSUTIII = DELSIII GO TO 5301 5252 DD 5300 I = 5300 DELSUTII) = 5301 GT = 0.0 DO 5350 I = I. 1 5350 GT = GT * GAMII) 1. SQRTI IXII+1I-X(I))‘*2 f (VII+1)-YIIII‘#2 I T = .5 * GT / SP IF I FLGO5 .E0. 0 I FALPHA= ATANZ ISNALF+T . CSALF) ALFEX = ATANZ (SNALF - T. CSALF I ALFEX = ALFEX # 57.2957795E0 FALPHA = FALPHA * 57.2957795E0 ALPHA = ALPHA * 57.2957795E0 IFN I FLG04 .E0. 0 I DALFA = FALPHA - ALFEX = SORT I 1. + 2.'SNALF‘T + T*T I EX = SORT I 1. - 2.*SNALF*T 0 T‘T I 5375 1 DO 5400 I = M. N T = CPIII * DELSIII CL = CL - T*COSA(II CD = CD 9 T‘SINAII) 5400 CM = CM 0 T‘ICOSAIII*IXMII)-XMCILII * SINAIII‘IYMIII-YMCILII I AAA=CHORDICDIM CL=CL*AAA I = 1 K2 3 NDILI 5500 WRITE I6. 5550) HEDR. SP. ALPHA. DALFA. FALPHA. VIN. XMCILI. 1 ALFEX. VEX. YMCILI. CASE 82V14170 BZY14180 BZV14190 62Y14200 82V14210 BZV14220 BZY14240 BZY14250 82V14260 82V14270 BZY14280 82Y14290 82V14300 82Y14310 82Y14320 BZY14330 BZY14340 82Y14350 82Y14360 82Y14370 82Y14380 BZY14400 82Y14410 82Y14420 BZY14430 82V14440 BZY14450 82Y14460 BZY14470 82Y14480 BZY14490 82Y14500 82Y14510 5550 FORMAT (1H1 25X 26HDOUGLAS AIRCRAFT COMPANY / 28X 21HLONG BEACHBZY14520 1 DIVISION //I 5X 15A4I/ 5X 9HSPACING = F13.8. 5X THALPHA = F13.8 82Y14530 2. 5X 13HDELTA ALPHA = F13.8 // 14H INLET ALPHA = F13.8. 3X 82Y14540 3 GHV INLET = F13o8. 13X 5HXMC = F13.8 // 2X 12HEXIT ALPHA = F13.B.62Y14550 4 4X 8HV EXIT = F13.8. 13X 5HYMC = F13.8 // 6H CASE 2A4.22H COMBIBZV1456O SNED VELOCITIES I IF (FLGIZ.LE.0) HRITE (6.5560) L 5560 FORMAT I10H BODY NU.I3.27H URTRANSFORMED COORDINATES // I IF IFLGIZ.GT.0) WRITE (6.5570) 5570 FORMAT (10H BODY NO. [3. 25H TRANSFORMED COORDINATES // I WRITE (6.5580) 5580 FORMAT (11X IHX 13X 1HY 13X ZHVC 12X ZHCP 10X 7HDELTA S // I 5600 HRITE (6. 5650) I. X(J). VIJI. XMPIKI). YMPIK1I. VCIK1I. CPIKII 1 .DELSUTIKI) 5650 FORMAT (1H 13. 2F14.8 / 4X 5F14.8 I I = I * 1 J = J 9 1 BZY14570 BZY14580 BZV14590 BZY14600 82V14610 82V14620 BZY14630 BZV14640 82Y14650 BZY14660 BZY14670 82Y14680 5700 5750 5800 5830 5840 5850 6100 C 6110 6120 6155 = K1 0 1 IF I I .EQ. K2 I GO TO 5700 IF I I .LE. LCTR I GO TO 5600 LCTR = LCTR + 19 GO TO 5500 WRITE (6. 5650) I. X(J). VIJ) * 1 HRITE (6. 5750) CL FORMAT (1H0 I 5X 4HCY = F13.8) M = N I 1 N = N * NDILOII - 1 K = NCFLG-2 = NUMBER OF GAMMAS CL=2.*GT#AAA IF (FLGIO .LT. 2) GO TO 5830 HRITE (6. 5840) CL FORMAT I1HO 4X 4HCL = F13.8 I IF IFLGOZ .E0. 0 .DR. FLGlO .E0. 1 .OR. N = NDINB+1I K2 = K + 2 DO 6100 J=1.K2 READI3I ISIGMAII.JI.I=1.NT) CONTINUE NOH ALL SIGMAS ARE IN CORE. ORDER DO 6110 I=1.NT SIGTIII= SIGMAII.1I*CSALF I SIGMAII.2I‘SNALF ISIGMAII.3) IFIK.EQ.0I GO TO 6150 DO 6120 J = 1.1 READI4I READI4I DO 6120 I = 1 NT I SIGTII) = SIGTII) I SIGMAII.J*3I*GAMIJI M = 1 M1 = N DO 6130 J ‘ 87 ‘ I IFI J .LE. FLGlI I GO TO 6122 READI4I IXTEMPIII.I=M.M1) READI4I (YTEMPIII.I=M.M1I GO TO 6125 DO 6123 I = M. M1 XTEMPII) YTEMPII) .0 IFINLFII).NE.O) GO TO 6125 READ (4) 0.0 READ (4) M = 1 M1 = N READI13IIXIII.I=1.NI READIIBIIYIII.I=1.NI IF (MON-1) 6152.6155.6155 CONTINUE HRITE I 6.6151 I HEDR. SP. ALPHA FORMATI1H1.25X.26HDOUGLAS AIRCRAFT 1 DIVISION ///5X.15A4//5X.9HSPACING 2///39H OFF-BODY POINT COMPONENT VELOCITIES. 4TOUT IS STREAMFLOH.90- DEGREE FLCH. NON‘UNIFORM FLOH 1. 82Y14690 BZV14700 82Y14710 82Y14720 82Y14730 82Y14740 82Y14750 82V14780 82Y14790 82Y14800 82Y14820 82Y14890 82V14900 TD 6700 82Y14910 82Y14920 82Y14930 82Y14940 82Y14950 82V14960 82V14970 82V14980 82Y15000 82Y15020 82Y15030 82Y15040 B 82Y15060 82Y15070 82Y15080 82Y15090 BZY15100 82V15110 BZY15120 82Y15130 82Y15140 82V15150 82Yl5180 82Y15190 COMPANY I 28X.21HLONG BEACH 82V15210 .F13.8.4X.7HALPHA =.F13.8 82V15220 ORDER OF PRIN82Y15230 ETC. // BZY15240 513X.1HX.20X.1HY.18X.3HVXL.17X.3HVVL.17X.3HVXL.17X.3HVYL III 82Y15250 DO 6400 J = 1.N READIlOI IYIJII).I =1.NT) READIIOI (XIJIII.I=1.NT) SUMI =0. 82Y15260 82Y15270 82Y15280 82Y15290 SLMZ =0. 6210 SUMDIJ.II SLMI I SUM1 9 T‘XIJII) 6200 SUM2 I SUM2 0 TIYIJII) IFIK.EQ.0) GO TO 6300 SUM1 I SUM1 I TIYTEMPINII SUM2 I SUM2 f T‘XTEMPINII 6250 N1 = N1 +N 6300 VXLIJ) = SUM1 * CSALF VYLIJI = SUM2 * SNALF IFIMON-l) 6305.6400.6400 6305 DO 6370 I = 1.K2 DO 6310 LSD = 1.NT T = SIGMAILSD.II SUMAIJ.II = SUMAIJ.II f T‘XIJILSDI 6310 SUMBIJ.II I SUMBIJ.II 9 TIYIJILSD) IF (I.LE.2 I GO TO 6355 N1 = J DO 6350 LSDI1.1 IFILSD*2.NE.I) GO TO 6350 SLMAIJ.II = SUMAIJ.II 0 YTEMPINII SUMBIJ.II = SUMBIJ.I) 0 XTEMPINI) 6350 N1 = N1 4 N GO TO 6370 6355 IF ( I.EQ.2 I GO TO 6360 C‘*# *‘*I = 1 MEANS AXISYMMETRIC FLOW SUMAIJ.II = SUMAIJ.II f 1.0 GO TO 6370 C#*‘ #*#I I 2 MEANS 90 DEGREE FLOW 6360 SUMBIJ.II = SUMBIJ.I) O 1.0 6370 CONTINUE WRITEI 6.6371 I J. XIJI. YIJI. (SUMAIJ.II.SUMBIJ.II.I=1.K2) 6371 FORMATIIH .I3.6F20.8 I I44X.4F20.8II 6400 THETVIJIIATANZIVVLIJI.VXLIJII*57.2957795 WRITE (7) IVXLII).I=1.NI WRITE (7) (VYLIII.I=1.NI IF (MON-2) 6420.6430.6430 6420 CALL 8CDUMP (XIII.XINII CALL 833UMPIYI1).YIN)I CALL BCDUMP (VXLII).VXLINI) CALL BCDUMP IVYLI1I.VYLINII 6430 LCTR = 45 I IF (FLGlO .LT. 2) GO TO 6500 6500 WRITE I6. 6550) HEDR. SP. ALPHA 82Y15300 82Y15310 82Y15320 82Y15330 82Y15340 82Y15350 82Y15360 82Y15370 82Y15380 82Y15390 82Y15410 82Y15420 82Y15430 82Y15440 82Y15450 82Y15460 82Y15470 82Y15480 82Y15490 BZY15500 82Y15510 82Y15520 82Y15530 82Y15550 82Y15560 82Y15570 82Y15580 82Y15590 82Y15600 82Y15610 82Y15620 82Y15630 BZY15640 82Y15650 82Y15660 82Y15670 82Y15680 82Y15700 82Y15710 82Y15720 82Y15860 6550 FORMAT (1H1 25X 26HDOUGLAS AIRCRAFT COMPANY / 28X 21HLONG 8EACH82Y15870 1 DIVISION ///5X 15A4 ll 5X 9HSPACING = F13.8. 4X 7HALPHA = F13.8 82Y15880 2 I/l28H OFF-BODY POINT VELOCITIES // 11X IHX 13X 1HY 12X 3HVXL 3 11X 3HVYL 10X 5HTHETA/l) 6600 WRITE (6. 6650) I. X(I). Y(I). VXLII). VYLIII . THETVIII 6650 FORMAT (1H I3. 5F14.8 I I 1 IF (I .GT. N) GO TO 7000 82Y15890 82Y15930 82Y15940 IF ( I .LE. LCTR I GO TO 6600 82Y15950 LCTR = LCTR # 45 BZY15960 GO TO 6500 82Y15970 6700 IF (FLGlO .E0. 0 .OR. FLGIO .E0. 3) GO TO 7000 82Y15980 FL 10 = 3 ‘ FLG 10 - 3 BZY15990 IF ( FLGOZ oNE. 0 I HRITE (6' 6750) 82V16000 6750 FORMAT (32H1FLAS 10 IS NON-ZERO - OFF- BODY / 82Y16010 30H VELOCITIES CANNOT BE COMPUTED I BZY16020 FLGOZ = 0 BZY16030 READ (5. 6800) (XTEMP(II. I = 1. NT) 82Y16040 6800 FORMAT (6F10.0I 62V16050 DO 6900 I = 1. NT 82Y16060 VC(II = VCIII‘XTEMP(II BZY16070 6900 CP(I) = 1. - VCII)‘VC(I) 82Y16080 GO TO 5375 82Y16090 7000 DO 7100 I = IoNB BZY16100 7100 GAMT(II = GAMIII 82Y16110 RETURN 82Y16120 END BZV16130 l SORIGIN ALPHA ‘ $IBFTC 22V? SLBROUTINE PART 7 C C THIS SLBROUTINE INTEGRATES FOR THE MASS FLUX AND IS USED TO C DETERMINE IF THE JET STREAM IS PROPERLY ORIENTED C COMMON IMpHEDRpCASEcRPI.RZPI.SP.CL.ALPHA.FALPHA.DALFA.CHORD.SUMDS. XMCnYMCIADDVnFLGOZ'FLGO3pFLGO4vFLGOSoFL606cFLGOTIFLGOBI FLGO99FLGIOyFLGIIoFLGlZoNDvNLFvNERvNTpNB'NCFLG'FLGI5cFLGl6 COMMON SUMSIGIVINFoMONuBETAoCDIMuVREFcDELToFL613vFLGI4oITERnALFZ COMMON IFORCUR/XCURVIZOOInYCURVIZOOIvKKKcPOSS ~ COMPLEX IM INTEGER FLGI5'FLG16IFLGOZ DIMENSION X(300). Y(300I' HEDR(15I.CASE(2IIADDY(8)v 2 ND(10)o NLFIIO): SUMDS(10I. XMC(8)v YMC(8) C 1 RSDS(499). SINA(499), COSA(499I. DELS(499). DALF(498). ((499). 7 DIMENSION RADCIZSOI'PRESIZSOIIVXLIZSOIoVYLIZSOIvXNIZSOIvYNIZSOII 1 CP(250I.DPRESI50IyV(5OIuPTOT(50IgRADCCL(50IoXCLI50IcVCL(50) DIMENSION VSPEC(4I cX8(5OOIpYB(500I REhIND 13 REkIND 7 M8NDI1I READ (13) (XB(I)oI=1pMI READ (IBIIYBIIIII=IIMI READ (13) READ (13) M=NT M1=NT-3 N2NDIZI KK=KKK-1 JJ=ND(1I II=ND(1I-KKK+1 READ (I3I (X(IIoI=1oN) READ (13) (Y(I),I=1'NI 90 READ(7I'(PRES(I+1).I=1.MI READ (7) (CP(II.I=1.MI READ (7) (XN(I+1).I=1.M) READ (1. (YNII91I.I=1.MI READ (7) (VXL(II.I=1.NI READ (1) (VYL(I).I=1.NI Do 30 I=1.N 30 PRESIII=CP([ IMAss=o.o M=l T I‘0.00119*VINF##2 N=36 DO 100 I=M.N VXL(II=(VXL(I)+VXL(I+1II/2.0 VYLIII=IVYLIII’VYL(I¢1)I/2o0 DELY=YII+1I-Y(II DELX=X(I61I-XIII DMASS=-VXL(II*DELY DMASSY=DELX#VYLII) 100 TMASS= TMASS+DMASS+DMASSV VAVG=TMASSl3o61 TMASS=TMASS*0.00238‘VINF*32.2/12.0*4o5 TMASS IS THE MASS FLOW IATO THE FANS FOUND BY INTEGRATION WRITE (6.1000) TMASS.VINF.VAVG 1000 FORMAT (1H1/23H THE INTEGRATED MASS =9F10o497H LB/SEC/ZBH FREE ST 1REAM VELOCITY =.F10.4.4H FPS/.23H THE AVERAGE VELOCITY =.F10.4//) IF (VINF—300.) 200.200.999 200 WRITE (6.1006) 1004 FORMAT (1H .2F10.4.6F15.8////I 1006 FORMAT (1H1//8X1HX.9X.1HY.8X.8HPRESSURE) 645 M=Z N=KKK MM=II+Z 650 00 700 I=M.N 700 WRITE (6.1004) XN(II.YNIII.PRESII'1I IF (N-MMI 750.300.800 750 M=II N3JJ GO TO 650 800 TFY=0.0 TFX=OIO IF (FLGIéoEQ-l) GO TO 840 N31 M=KK GO TO 850 840 N31 M810 JJJ=1 850 OD 900 I=N.M T1=XB(I*II-XBII) T2=YB(I+1I-YB(II TDS=SQRT(T1‘T1*T2*T2I COSAL=T1ITDS SINAL=T2ITDS IF (FL816.EQ.1I GO TO 860 DFX!PRES(II tTDS*SINAL*4.5/12.0 DFY=-PRES(II *TDS‘CDSAL‘4.5/12o0 GO TO 870 860 SLOPx-T2/T1 91 AL1=ATAN(SLDPI ALZMAL=-ALF2-AL1 CJJJ= JJJ DFY= PRE5(II‘TDS‘COS(ALZMALI‘CJJJ TFYITFV+DFY GO TO 900 870 TFX=TFXODFX TFYzTFYfOFV 900 CONTINUE IF (N-II) 950.975.975 950 IF (FL616.EQ.1I GO TO 960 NSII M=JJ-1 GO TO 850 960 N=ND(1)-10 M=ND(1I-1 JJJ=-1 GO TO 850 975 WRITE(6.1008I TFX.TFY 1008 FORMAT ( 1H .//.48H THE TOTAL FORCE ON THE JET IN THE X-DIRECTIDN 1=.F10.4/48H THE TOTAL FORCE ON THE JET IN THE Y-DIRECTION =.F10.4) IF (FLG16.EQ.0I GO TO 980 FLGIé‘D 800 980 BETA=8ETA#3.14159/180. THRUST =TFY/(SINIBETAI‘POSSI CMU=THRUST‘12.I(0.00119*4.5#CDIM‘VINF'iz) WRITE (6.1010) THRUST.CMU 1010 FORMAT (1H .//.9M THRUST =.F10.4.3H LEI/23H CMU FOR THE JET FLAP = 1.FIC.4I TPA=THRUST*12.0/(4.5*DELTI PRATIO=((TPA-40.I*0.2/760.+1.0I‘2116./(2116.0.00119*VINF“2I WRITE (6.1020) PRATIO 1020 FORMAT(1H .9H PRATIO =.F10.4I 999 RETURN APPENDIX E SYMBOLS FOR BOUNDARY LAYER ANALYSIS t—] 5| three-dimensional lift coefficient velocity defect variable, (U - E)/U shape factor, 6*/9 curvature distance along airfoil surface from stagnation point to trailing edge static pressure parameter in equation (31) parameter in equation (31) Reynolds number based on x, xU/v Reynolds number based on 6*, 6*U/v Reynolds number based on e, BU/v thickness of airfoil non-dimensional effective viscosity (see eq. (38)) time average velocities in the x and y directions, respectively 92 95 U velocity at the outer edge of the boundary layer (potential flow velocity) u v Reynolds stress V; wall transpiration velocity x streamwise coordinate (see Fig. (16)) y coordinate normal to wall (see Fig. (16)) a angle of attack 8 the Clauser equilibrium pressure gradient, 6*(dp/dx)/'rw 6 boundary layer thickness 6* displacement thickness, U/p (U - E)/U dy 0 n non-dimensional coordinate normal to wall, y/6* 6 momentum thickness, d/1 fi(U — fi)/U2 dy 0 K von Karman constant in the effective viscosity function (taken here to be 0.41) v molecular kinematic viscosity ve effective kinematic viscosity 0 density T local shear stress 1% non-dimensional shear stress gradient (see eq. (42)) ¢,¢ wall and defect effective kinematic viscosity functions x,X wall and defect layer variables for the effective kinematic viscosity function Subscripts: 1 index of variable in the x direction w evaluated at wall x differentiation with respect to x w evaluated at edge of the boundary layer r... 94 Superscripts: ( ) used with functions of x only, denotes average value, [( )1+1 + ( )11/2 I ( ) differentiation with respect to n APPCI'TSIII F DERIVATION OF THE TRANSFORMED BOUNDARY LAYER EQUATION OF MOTION APPENDIX F DERIVATION OF THE TRANSFORMED BOUNDARY LAYER EQUATION OF MOTION The transformed boundary layer equation of motion is obtained by making a coordinate transformation to the standard continuity and mom- entum equations for two-dimensional, incompressible flow. tum equation as given by equation (25) of the text is: aau+v§2=ud£+i(v Lu) 3; 3y dx 3y e 8y or — —— \) — 2§E+lfl£=dl+i ifl U 3x U 8y dx 8y U 8y Transform variables from (x,y) to (£,n) where E = x and and utilizing Mellor's transformation, = l - f' or E'= (l - f')U Cilfil where f' = 3f/8n. Then E- _.d_U_£ 3x - (l f ) dx U 2k Here f' = f'(€.n) 95 The momen— (F1) (F2) * n = y/6 , (F3) (F4) (F5) Then E -E'LE ELL” (w 3x ’31: ax an ax L” The following conventional notation will be used: I Tn =f" (F7) It follows that 5* a _ 35* d6* m==_3(y)__aafix_--li¥_ (F8, 8x 8x 5 (6*)2 6* Hence, using equation (F7) and (F8), equation (F6) becomes db* af' _ af' f” n E , (3x ) _ 3E 6* (F9) Substituting equation (F9) into (F4) yields §=(1-£')%-u%+¥1§—M (F10) Also 3% = U-§% (1 - f') = - u-ggl =-u%3—3—=-Uf"%(f£) (mm or 93: _%" (F12) 97 The continuity equation 33' 37 _ __= , (1.1. UK By 0 \E 3 is used to obtain V, and BVYBX which are inserted into the momentum equation. Integrating each term of the continuity equation yields: y V(X,y) y E T — '—T_ - — __ a“; ' .‘-_ / 3y.dy - dv - v vw — 8x dy (P14) 0 V(x,0) 0_ Let Y' = n+n (F22) Substituting equations (F3), (F10), (F12), and (F22) into the momentum equation (F2) gives: , (1 a f' )f"n ——-U V' n (1 - fvfism- (1 - fv)u2£.+_2_.___.-_wvf dx 35 6* U 6* 3f 6* —— Uf" - P 1* - f' Uf" 3 6* 6 = 3:5 + — (- TUf") (F23) where * T — ve/UG Using gg_ * dU 6* 2 dx 3; 2 (1 - f') ——-—- — -———— = P(1 — 2f' + f' - l) = P(f' — 2)f' U U (F24) and flan LG" (1 — f') ——:—E—x—= (Q — Pm — f')f"n (F25) The transformed boundary layer equation of motion becomes (Tf")‘+ [Q(n - f)- Vw/Ulf" + P(f' - 2)f' f =* _v2£L *n3_ 5(1 f)a +6f3x (F26) 100 where _d_ * _ dx (6 U) _ 6*dU/dx Ve Q — U a P — U 9 T = UO" and x has been Substituted for E. The independent variablés are then x and n. REFERENCES ._. REFERENCES Koenig, David G.; Corsiglia, Victor R.; and Morelli, Joseph P.: Aerodynamic Characteristics of a Large—Scale Model with an Unswept Wing and Augmented Jet Flap. NASA TN D-4610, 1968. . Campbell, John P.; and Johnson, Joseph L., Jr.: Wind Tunnel Investigation of an External-Flow Jet Augmented Slotted Flap Suitable for Application to Airplanes with Pod-Mounted Jet Engines. NACA TN 3898. 1956. Parlett, Lysle P.; and Freeman, Delma C., Jr.; and Smith, Charles C., Jr.: Wind Tunnel Investigation of a Jet Transport Airplane Configuration with High Thrust-Weight Ratio and an External- Flow Jet Flap. NASA TN D-6058, 1970. Gartshore, I. 8.: Prediction of the Blowing Required to Suppress Separation from High—Lift Aerofoils. Paper no. 70-872, AIAA, July 1970. . Thwaites, Bryan, ed.: Incompressible Aerodynamics. Clarendon Press, Oxford, 1960. Weber, J.: The Calculation of the Pressure Distribution over the Surface of Two-Dimensional and Swept Wings with Symmetrical Aerofoil Sections. Rep. R & M 2918, Aeronautical Research Council, Gt. Britain, 1956. . Goldstein, 8.: Approximate Two-Dimensional Airfoil Theory. Parts I—VI. Aeronautical Research Council, Gt. Britain, Reps. CP-68 to CP-73, 1952. . Anon: A Direct Iteration Method for the Calculation of the Vel- ocity Distribution of Bodies of Revolution and Symmetrical Pro- files. Rep. ARL/RI/G/HY/lZ/Z, Admiralty Research Laboratory, Gt. Britain, August 1951. . Jacob, K.: Some Programs for lncompressible Aerodynamic Flow Ca1- culations. Tech. Rep. 122, Calif. Inst. Technology Computing Center, 1960. 101 10 11 12 13 14 15. 16 17 18. 19 20. 21 102 Davenport, F. J.: Singularity Solutions to General Potential- Flow Airfoil Problems. Rep. D6-7202, Rev., Boeing Co., 1963. Spence, D. A.: The Lift Chefficient of 3 Thin, Jet—Flapped Wing. Proc. Roy Soc. (London), Ser. A, vol. 238, December 4, 1956, pp. 46—68. Spence, D. A.: The Lift on a_Thin Aerofoil with a Jet-Augmented Flap. Aeron. Quart., vol. 9, August 1958, pp. 287—299. Smith, A. M. 0.; and Pierce, J.: Exact Solution of the Neumann Problem. Calculation of Noncirculatory Plane and Axially Sym- metric Flows about or Within Arbitrary Boundaries. Rep. ES—26988-V, Douglas Aircraft, Apr. 25, 1968. Giesing, Joseph P.: Extension of the Doublas Neumann Program to Problems of Lifting Infinite Cascades. Rep. LB-31653, Douglas Aircraft Co., July 2, 1964. (Available from DDC as AD-605207.) Giesing, Joseph P.: Potential Flow About Two-Dimensional Airfoils. Part I: A Summary of Two—Dimensional Airfoil Methods. Part II: Solution of the Flow Field About One or More Airfoils of Arbi— trary Share in Uniform or Non-uniform Flows by the Douglas Neu- mann Method. Rep. LB-31946, Douglas Aircraft Co., Dec. 1, 1965. Cohen, Clarence 3.; and Reshotko, Eli: The Compressible Laminar Boundary Layer with Heat Transfer and Arbitrary Pressure Gradi- ent. NACA TR 1294, 1956. Sasman, Philip K.; and Cresci, Robert J.: Compressible Turbulent Boundary Layer with Pressure Gradient and Heat Transfer. AIAA J., vol. 4, no. 1, Jan. 1966, pp. 19-25. McNally, William D.: Fortran Program for Calculating Compressible Laminar and Turbulent Boundary Layers in Arbitrary Pressure Grad— ients. NASA TN D—568l, May 1970. Bennett, J. A.; and Goradia, S. H.: Methods for Analysis of TWO- Dimensional Airfoils with Subsonic and Transonic Applications. Rep. ER—8591, Lockheed—Georgia Co., July 1966. Crank, J.; and Nicolson, P.: A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type. Proc. Cambridge Phil. Soc., vol. 43, 1947, pp. 50-67. Flfigge—Lotz, 1.: The Computation of the Laminar Compressible. Boundary Layer. Dept. Mech. Eng., Stanfordeniv., Rep. R9352— 30-7, 1954, or Office of Scientific Research, Tech. Note 54—144, 1954. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 105 Bradshaw, P.; Ferriss, D. H.; and Atwell, N. P.: Calculation of Boundary Layer Development Using the Turbulent Energy Equation. J. Fluid Mech., Vol. 28, pt. 3, 1967, pp. 953-616. Patankar, S. V.; Spalding, D. B.; and Ng, K. B.: The Hydrodynamic Turbulent Boundhry Layer on a Smooth Wall, Calculated by a Finite Difference Method. Proc. Conf. on Computation of Turbulent Bound= ary Layer Prediction, Stanford University, 1968, pp. 356-365. Cebeci, T.; and Smith, A. M. 0.: A FinitesDifference Solution of the Incompressible Turbulent Boundary Layer Equations by an Eddy— Viscosity Concept. Proc. Conf. on Computation of Turbulent. Boundary Layer Prediction, Stanford University, 1968, pp. 346-355. Smith, A. M. 0.; and Cebeci, T.: Numerical Solution of the Turbu= lent Boundary Layer Equations. Rep. DAG 33725, Douglas Aircraft Co., 1967. Herring, H. J.; and Mellor, G. L.: Computer Program to Calculate Incompressible Laminar and Turbulent Boundary Layer Development. NASA CR-1564, 1970. Patankar, S. V.; and Spalding, D. B.: Heat and Mass Transfer in Boundary Layers. C. R. C. Press, 1968. Mellor, G. L.; and Gibson, D. M.: Equilibrium Turbulent Boundary Layers. J. Fluid Mec., v. 24, pt. 1, 1966, pp. 225—253. Mellor, G. L.: The Effects of Pressure Gradients on Turbulent Flow Near a Smooth Wall. J. Fluid Mech., v. 24, pt. 2, 1966, pp. 254- 274. Mellor, G. L.; and Herring, H. J.; Two Methods of Calculating Tur- bulent Boundary Layer Behavior Based on Numerical Solutions of the Equations of Motion. Proc. Conf. on Computation of Turbulent Boundary Layer Prediction, Stanford University, 1968, pp. 331-545. Kellogg, Oliver D.: Foundations of Potential Theory. Dover Publi., Inc., 1953. Purcell, E. W.: The Vector Method of Solving Simultaneous Linear Equations. Math. Phys., vol. 32, 1953, pp. 180-183. Stockman, Norbert 0.: Potential Flow Solutions for Inlets of VTOL Lift Fans and Engines. Analytic Methods in Aircraft Aerodynam— ics, NASA SP-228, 1970, pp. 659—681. Emslie, K.: Wind Tunnel Test on Models of V. T. 0. Aircraft. Aircraft Eng., vol. 38, no. 6, June 1966, pp. 8-13. 35 36 37 38 39 40 41. 42 43 44 45 46. 47. 48 104 Williams, J.; Butler, F. J.; and Wood, M. N.: The Aerodynamics of Jet Flaps. Rep. R & M 3304, Aeronautical Research Council, Gt. Britain, 1963. Goldstein, 8.: Modern Developments in Fluids Dynamics, vol. 1, Dover Pub., 1965, p. 119. Massey, B. S.; and Clayton, B. R.: Some Properties of Laminar Boundary Layers on Curved Surfaces. Journal of Basic Engineerw ing, June 1968, pp. 301-312. Schlichting, H.: Boundary Layer Theory. McGraw Hill Book Co., 1968. Hartree, D. R.; and Womersley, J. R.: A Method for the Numerical or Mechanical Solution of Certain Types of Partial Differential Equations. Proc. Roy. So. (London), Series A, vol. 161, no. 906, Aug. 1937, p. 353—366. Clauser, F. B.: The Turbulent Boundary Layer. Advances in Applied Mechanics, vol. IV, Academic Press, Inc., 1956, pp. 2-51. Mellor, G. L.; Turbulent Boundary Layers with Arbitrary Pressure Gradients on Convergent Cross Flows. AIAA Journal, 1967, pp. 1570-1578. Laufer, J.: The Structure of Turbulence in Fully Developed Pipe Flow. NACA TR 1174, 1954. Van Dyke, M.: Perturbation Methods in Fluid Mechanics. Academic Press, New York, 1964. Hildebrand, F. B.: Advanced Calculus for Engineers. Prentice— Hall, 1958. McCracken, D. D.; and Dorn, W. 5.: Numerical Methods and Fortran Programming. John Wiley and Sons, Inc., 1965. Crabtree, L. P.: Prediction of Transition in the Boundary Layer. on an Aerofoil. Royal Aircraft Establishment, London, Techni— cal Note Aero. 2491, Jan. 1957. Michel, R.: Determination of Transition Point and Calculation of Drag of Airfoils in Incompressible Flow. Publication no. 58, 0. N. E. R. A. (France), 1952. Granville, Paul S.: The Calculation of Viscous Drag of Bodies of Revolution Navy Department. Rep. no. 849, David Taylor Model Basin, 1953. 105 49. Hancock, G. J.: Problems of Aircraft Behavior at High Angles of Attack. AGARD 136, April 1969. 50. Chappell, P. D.: Flow Separation and Stall Characteristics of. Plane, Constant—Section Wings in Subcritical Flow. The Aero— nautical Journal of the Royal Aeronautical Society. Vol. 72, Jan. 1968, pp. 82-90. ILLUSTRATIONS 106 (a) let flap airfoil. \ ‘ I ll <5 (b-ll 30° flap deflection. (b-2) 60° flap deflection. (bl Externally blown flap wing section. Figure 1. - Types of wing flap systems considered. .U l A... 9.0 21m U© @ @ @bi.‘ \ ‘\ \\ \Tunnel floor-I’ 10'7 [- Wing lower surface ,’ 66. (26) ll ) 165.1 ) (65) :> ‘ " ‘ ' ' ‘ ' ) CD-10819 -01 7 z z / f, r” I \‘~ Fan drive air supply and model support-’” (a) Schematic of model. Figure 2. — Lewis wind tunnel model of multiple-fan, blown flap, wing propulsion system. (Dimensions are in cm (in. l.) 108 \ Tu rbofan A Fan exhaust J Free jet —/ (bl Two-dimensional representation at wing propulsion system. Figure 2. - Concluded. 109 Figure 3. - Representation of boundary condition on body surface. 110 Figure 4. - Finite-element approximation to body surface. 111 :/—\\___ (F lllll (al 0° uniform flow solution, V0. (b) 90° uniform flow solution, V90. \R <3 _._v,,- I K I Trailing edge -J (cl Vortex solution, Vv- (Vw= 0.) (d) Suction solution, Vs. (Voo= 0.) W Figure 5. - Basic solutions of potential flow. 112 @ v. _, J (3) 0° solution with duct closed, v1. —. @111, 11111 v,o _.' (b) 0° solution with duct open, 172. @11 l. , llvll em ~ , (c) Crossflow solution with duct open, 173. Figure 6. - Basic solutions for inlet. 113 VN H 2'L le Figure 7. — Two-dimensional inlet configuration. Velocity ratlo, V/Vref 114 — — Predicted from ref. 33 —-— Predicted from this report --—— Experimental data from ref. 34 _,,-\ / \ ‘ o l ‘\~-4\. I (a) Surface velocity distributions. | l l I .50 Dimensionless length, x/L lb) Centerline velocity distributions. Figure 8. - Comparison of theoretical velocity distributions with experimental data for two-dimensional inlet 115 y/C P Pressure coefficient, C L c———J 0 ————— \\ .4 — Jet shape /—A / we (assumed cubic) H . 8 — / //,_c 1.2 — 1 6 l l l l l 0 1 2 3 4 5 xlc (3) Illustration of jet shapes. 10 //_A ///-B (assumed cubic) 5 \\f r0 '\. e l l i \‘j -. 2 0 .2 4 .6 .8 1. 0 )(IC (bl Upper-su rtace pressure distributions. Figure 9. - Effect of jet shapes on upper-surface pressure distribution. Flap angle, 30°; wing angle of attack, 0°. 116 '.4 LC—u <3 y/C .4— ,~Spence's theory (ref. 12) Method of this report—/ I 1 2 3 4 XIC Figure 10. - Comparison of theoretical nondimensional jet shapes. Flap angle, 30°; thrust coefficient, 3. 117 (a) Wlng angle of attack, 0°; flap angle, 30°, (bl Wing angle of attack, 20°; flap angle, 50°. \ (c) Wing angle of attack, 0°; flap angle, 60°. Figure 11. - Flow field for externally blown flap, wlng propulsion system. Mass flow coefficient, 0.38; thrust coefficient, 3. Pressure coefficient, Cp 118 '60 F— -50 — -40 _ -30 —- Angle of attack, a! -20 —— deg //-10 '10 _ / / f0 9 l l I fl -. 1 0 l 2 3 4 5 .6 7 8 9 L 0 x/C Figure 12. - Calculated pressure distributions on upper surface for fan-wing combination for various angles of attack. Flap angle, 30°; mass flow coefficient, 0.38; thrust coefficient, 3. Pressure coefficient, Cp 119 Wing with suction and jet —-— Wing with jet L ———- Wing alone (without suction or jet) Upper surface )\ \ K \‘~ “‘\T_}—' x \ ‘\\ _l *Lower surface I l l l 0 .2 .4 .6 .8 1.0 x/C Figure 13. - Effect of suction and jet on pressure distribution. Flap angle, 30°; angle of a Lift coefficient, Cl 120 Method of this report —— Spence'stheory (ref. 12) I l (a) Flap angle, 30°. 10 -‘ 5 r— i i I I J 0 20 25 10 15 Angle of attack, 0, deg (bl Flap angle, 60°. Figure 14. - Comparison of theoretical two-dimensional lift coefficients for blown flap. Thrust coefficient, 3. Lift coefficient, CL 121 10 —0— Experimental Predicted (a) Flap angle, 30°. l l l | l 0 5 10 15 20 25 Angle of attack, (1, deg (b) Flap angle, 60°. Figure 15. - Comparison of calculated and experimental th ree-dimensional lift coefficients for blown flap. Thrust coefficient, 3. 122 Stagnation / point (x = 0)-/ (a) Coordinate system. i 6 bounda layer thickness \ x \ \6° displacement thickness (bl Description of velocity profile. Figure 16. - Illustration of notation for boundary layer analysis. 123 —m . Ky I xiv“) 010 015 . 020 025 .020 o- 0.016 0 .015 obi, U6 . 010 oq+ £+mw l l l o 5 m u x . EX(231/2 Figure 17. - The turbulent effective viscosity hypothesis. Nondimensional velocity, V/Vno 124 Angle of attack 0. —' deg /F 25 _ , / /’//I— 20 / oI/rlf; I,— l l l | l 2 .4 .6 .8 1.0 Distance along airfoil surface/total length, xIL Figure 18. - Calculated velocity distributions for fan-wing combination for various angles of attack. Flap angle, 30°; mass flow coefficient, 0.38; thrust coefficient, 3. Skin friction coefficient, Cf Shape factor H Reynolds number, Rey 0 125 l l l 06 .02 .04 . Distance along airfoil surface/total length, le Figure 19. - Laminar boundary layer parameters on the airfoil of a blown flap wing propulsion system. Wing angle of attack, 15°. Shape factor, H Skin friction coefficient, Cf Reynolds number, Robo Angleofattack, ah “I deg 2.. 30 25 20 15 o 1..— e I l I 14 6x10'3 Angle of attack, a. v. -3 d v. 810 99 $3 20 ea “‘ 3 0 E3 0 §§ o E -2 810‘ :- L6K10 E 15 6— - 12— 20 :3 25 a 4— =3: .8— ‘g o E 30 2— E .4»— Q) 5’ a l 14 a l l l o .2 .6 6 8 Lo .8 1.0 0 .2 .4 Distance along airfoil surface/total length, x/L Figure 20 - Turbulent boundary layer parameters on the airfoil of a blown flap wing propulsion system at various angles of attack. 127 Parameter normal to the wall, y/(Jc o‘ .2 .4 .6 .8 1.0 Velocity ratio, U/le) Figure 21. - Velocity profiles at start of tu rbu- lent boundary layer growth. Parameter normal to the wall, yl6' 128 6 —— _ Ratio of distance along airfoil surface to 5 — total length, — x/L 4 —- _ ct = 0. 00005 3 —— — (near separationlw 2 — _ 1 — _ 9 I 1 1 l l (a) Angle of attack, 0°. ' (b) Angle of attack, 15°. 6 —- _ S —— ._. 4 — _ Cf “ 0. W1 3— (near separatioan\ — f = 0. 0002 (near separationi«\ 2— _ 1— ._ l l | l o .2 4 6 1. 0 0 4 .6 .8 1. 0 Velocity ratio, fi/lel lcl Angle of attack, 20°. (d) Angle of attack, 25°. 6 _ _ 5 _ 4 _ 3 _. . 1 (near separation)«\ Parameter normal to the wall, ylb‘ 2 .d .6 .8 Velocity ratio, Ullel (e) Angle of attack, 30°. Figure 22. - Turbulent boundary layer velocity profiles on the airfoil of a blown flap wing propulsion system at various angles of attack. Lift coefficient, CL 129 (a) trailing edge stall (b) leading edge stall (c) thin airfoil stall Angle of attack, a Figure 23. - The stalling characteristics of airfoils. 130 10 —- 8 .— _l U I 6 r .2 E 8 4C :3 2 _ l l l l l l 0 5 10 15 20 25 30 Angle of attack, 0, deg (a) Emerimental lift curve. Stagnation point location 300 for angle of attack, 0, deg 0 lb) Location of separation point for various angles of attack. Figure 24. - The stalling characteristics of a blown flap wing propulsion system. 131 \— p(x, y, 0) /— S \ 2‘; \— Body contour s 4 (a) Three-dimensional illustration. l,— Approximated body contour (b) Two-dimensional cross section. Figure 25. - Notation for two-dimensional potential flows. 132 Mi Body surface? / / [C2] Figure 26. - Element of body surface. 133 Subroutine - BZYl Main Program Calls other parts of program . Subroutine - 22YA Subroatlr: 1') BZYZ Generates initial shape a of exhaust jet of pro— Basic data pulsion system I Subroutines - BZY3, BZY4 (Part 2) Matrix formation Subroutines - BZY9, CZOX9 (Part 4) Solves matrix I Subroutine - BZYE (Part 5) Basic solutions I Subroutine - BZYF (Part 6) Combination solution Subroutine - 22YP (Part 7) Determines jet exhaust orientation Figure 27. - Schematic representation of computer program. nnnnn "IiILIIIIIIjIIIIIIIIIIIIIIIIIIIII 3