KNEW-FEES} .GE GaéffhET 2' GFTEEEZEEEGR .4 EE‘sSEi E E, 35033. CGMPEESS {EB ROTOR “tests EM EEIG Deg?“ 0E pk. D. MECEEEGAN STATE UI‘EEVERSEEY Pauf ChasEesG; Emce E97E ,r‘fi‘u 1‘ LIBRA R 1/ Michigan State University “ - ‘- I 'I-ul .0- This is to certify that the thesis entitled FW-PASSAGE-GEOIVETRY OPTIMIZATION INSIDE A MODEL COMPRESSOR ROTOR presented by Paul Charles Glance has been accepted towards fulfillment of the requirements for 31,1). degree inMechanical Engineering flz V‘ MW’é’ém/éh ' Majotflprofeuor Date December 28, 1971 ABSTRACT F ww - PASSAGE -GEOMETRY OPTIMIZATION INSIDE A MODEL COMPRESSOR ROTOR BY Paul Charles Glance The goal of this work is to determine optimal internal flow passages for compressor rotors. This work should be regarded as a first approach to the problem of designing optimal internal flow passages. A great many other phenomena such as shocks, boundary- layers, etc. need to be considered if a realistic compressor rotor is to be designed. Steady isentropic flow of a fluid, which obeys the ideal gas relations, is assumed throughout this work. The equations developed in this work may be applied to axial-flow, mix-flow, and radial-flow rotors. The problem of describing the motion of the fluid continuum is formulated as a minimum problem of Variational Calculus, and the equation which results from this formulation is called "the fluid particle minimum principle". Basically, this minimum principle states that of all the possible motions the fluid will travel along the one family of pathlines (or streamlines) which causes the kinetic energy minus the potential and enthalpy energies of each fluid particle to be a minimum. A fluid particle is defined as an infinitesimal volume of fluid whose surface is impervious to the flow of matter. Paul Charles Glance The "optimal flow passage geometry" is defined as the geometry for which the entire flow region satisfies the fluid particle minimum principle, continuity equation, boundary condi- tions, and various "optimal" constraints. An optimal constraint is any side condition which is imposed on the problem in an effort to produce desirable or optimal results. Constraints are imposed in order to control the pressure and energy increase of the fluid inside a rotor and they often simplify the problem. The flow problem is treated as a boundary value problem. One of the boundary conditions is that the pathlines of the flow region must coincide with the walls of the passage. When the flow is rotational, the following procedure is employed to determine the optimal flow passage geometry. A family of pathlines is determined which satisfy all the equations and boundary conditions except the above mentioned boundary condition. The passage geometry is then selected to coincide with any set of pathlines, belonging to the given family of pathlines, and the boundary value problem is thus completely determined. When the flow is irrotational, the boundary value problem is determined by the velocity potential function. In general, a rotational flow, inside a rotor, can satisfy the fluid particle minimum principle at only one set of operating conditions. Whereas, an irrotational flow will satisfy the fluid particle minimum principle over a wide range of operating conditions. In order to demonstrate the optimization procedure, a "maximum kinetic energy increase" centrifugal rotor is investigated. Application of the optimization procedure to axial-flow rotors is also discussed. FIOW-PASSAGE-GEOMETRY OPTIMIZATION INSIDE A MODEL COMPRESSOR ROTOR BY Paul Charles Glance 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 ACKNOWLEDGEMENTS The author is deeply indebted to his major professor, Dr. Maria Z.v. Krzywoblocki, for his guidance and assistance throughout the course of this study. The author also wishes to thank the other members of his guidance committee for their interest and suggestions: Dr. George E. Mase, Dr. Gerald D. Ludden, and Professor George H. Martin. Thanks are due to the Mechanical Engineering Department for financial support during graduate study and research. To his wife Joanne, daughter Michele, and parents Edmond and Juanita, the author dedicates this work for their encouragement and understanding throughout this study. ii TABLE OF CONTENTS Page ACKNOW LEDGEMENTS i i LIST OF FIGURES iv LIST OF APPENDICES v NOMENCLATURE vi INTRODUCTION 1 Section 1. KINEMATICS . 4 2. LAGRANGIAN FORM OF THE MOMENTUM AND CONTINUITY mUATIONS 0....0....OOOOIOCCCOICOOOCCCOOOOO..00....0 5 3. VARIATIONAL CALCULUS ............................... 8 4. FUJID PARTICLE MINIMUM PRINCIPLE .. 12 5. SOLUTION OF SOME CLASSICAL EXAMPLES ................ 20 6. ROTATING FILM PASSAGE .............................. 31 7. IRROTATIONALFLOW .................................. 4O 8. ROTATIONAL FUN‘.................................... 41 9. OPTIMAL (DNSTRAINTS ................................ 45 10. A "MAXIMUM KINETIC ENERGY INCREASE" CENTRIFUGAL ROTOR PASSAGE 0......OO...OOOOOOOOOOOOOOOOCOOOOOOOO. 52 11. AXIAL‘FW ROTORS o.oooooo0.0000000000000000...ooooo 64 12. CONCLUDING REWS OOOOOOOOOOOOOOOOOIOOOOOOO00...... 67 REFERENCES O...00....OOOOOOOOOOCOOOOOOOOOOOOOOOOOOOO 76 iii Figure 2.1 Figure 5.1 Figure 5.2 Figure 6.1 Figure 10.1 Figure 10.2 Figure 10.3 Figure 10.4 LIST OF FIGURES CUrVili-near curves ......OOOOOOOOOOOOO Conjugate POIDtS 000.000.00.00... ooooooooooo Radial Pathlines Enclosing a Pathtube ...... Rotating Passage ........................... An Optimal Centrifugal Rotor Passage ....... Flow Net ... Area Incremnt .....OOOOOOOOOOOOOOOOOO Flow Chart . iv 22 27 32 53 55 57 63 LIST OF APPENDICES Page Appendix A. First Law of Thermodynamics 70 Appendix 8' Fields Of a Functional ......OOOOOOOOOOOOOOO 72 h = C T kv2+h :11 Ill V°() dt agzp aY Y 5() A(H) = H - H ”l NOMENCLATURE entropy pressure temperature density enthalpy of ideal gas total enthalpy Specific heat at constant pressure specific heat at constant volume ratio of Specific heats ideal gas constant total relative enthalpy curl operator gradient operator divergence operator time derivative partial derivative variational derivative operator difference Operator volume cross sectional area arc length position vector vi W]. O ('7 Ian I! :nv II <1 I stun. 1 ...: n H n I ”b .... II 81 2"“! min. "4 O O 8 det F Elsi), 1:20:52?) V1 g yl’yz ’y3 *<1- «4. 1'92’5’3 u1,u2,u3 x,y,z r,a,z 1,3,k lr,la,lz 11’31’E‘1 Er’le’i 2 position vector at t = 0 time velocity position vector drawn from the relative reference frame relative velocity body forces per unit mass non-conservative body forces conservative body forces force potential function (VG 3 -f) velocity potential function (Vg = V) functional integrand of functional functions of the coordinates of the end points of the pathline space variables time derivative of space variables orthogonal curvilinear coordinates Cartesian coordinates cylindrical coordinates Cartesian unit vectors cylindrical unit vectors Cartesian unit vectors of rotating reference frame cylindrical unit vectors of rotating reference frame vii W ['11 Ill Subscripts * a cp/cv % mV angular Speed of rotor (w is constant) absolute angular Speed of a fluid particle angular Speed of a fluid particle with reapect to rotor Speed of sound of an ideal gas Mach number internal energy work kinetic energy 111888 denotes sonic conditions denotes stagnation conditions denotes intake point denotes discharge point ratio of Specific heats viii INTRODUCTION The Lagrangian method of describing the motion of a con- tinuum is employed in this work. The Eulerian description of motion is the traditional method used to describe the motion of a fluid continuum. However, the Lagrangian description was chosen because the methods of Variational Calculus are considerably less complicated for functions of one independent variable. The problem of describing the motion of the fluid continuum is formulated as a minimum problem of Variational Calculus, and the equation which results from this formulation is called "the fluid particle minimum principle". Basically, the fluid particle minimum prin- ciple states that of all the possible motions the fluid will travel along the one family of pathlines which causes the kinetic energy minus the potential and enthalpy energies of each fluid particle to be a minimum. The energy equation is of primary importance in the present formulation, while the momentum equation is always satisfied. The energy equation is the first integral of the momentum equation for the case of isentrOpic flow. This approach differs from the traditional Eulerian method, wherein the momentum equation is the equation primarily operated upon. Steady isentropic flow of a fluid, which obeys the ideal gas relations, is assumed throughout this work. The path of a fluid particle is called a pathline. For the case of steady flow pathlines and streamlines coincide. In section-1 the Lagrangian dcscript ion of motion of a fluid continuum is explained and a fluid particle is defined. In section-2 the Lagrangian form of the momentum and continuity equations are listed. A special form of the Lagrangian continuity equation is derived for the case of steady flow. Some of the results of Variational Calculus, that are employed in later sections, are listed in section-3. In section-4, the problem of describing the motion of a fluid con- tinuum is formulated as a minimum problem of Variational Calculus. The fluid particle ndninuniprinciple is then developed. In section-5 some classical fluid problems are investigated using the previously developed theory. All (irrotational) potential flow problems are Shown to satisfy the fluid particle minimum principle. The con- ditions under which (one-dimensional) isentropic compressible flow satisfies the fluid particle minimum principle are also investigated. In section-6 the fluid particle minimum principle of section-4 is adapted to the flow inside a rotating reference frame. The energy equation is then derived. In section-7 it is shown that there exists only one trivial case for which the flow inside a rotating passage is irrotational. Thus this case is excluded from the following investigations. In section-8 the continuity equation and energy equation are combined and the boundary value problem is described for the case of rotational flow. In section-9 the optimal constraints are selected. The System of equations and boundary conditions, which are employed to determine the optimal compressor passages, are summarized at the end of section-9. Section-10 contains a demonstration of how an Optimal flow passage may be determined. A "maximum kinetic energy increase" radial blade centrifugal rotor is considered in this section. In section-11 flows which are irrotational in the relative reference frame are investigated. Section-12 contains some concluding remarks. 1 . KINEMATI CS The Lagrangian method of describing the motion of a con- tinuum will be employed in this thesis. In the Lagrangian des- cription of motion, the path of each particle is described by the locus of points traced out by the end point of a position vector, R[Ro, x(t),y(t),z(t)], with respect to a fixed (Newtonian) reference frame [8]. The reference position of each particle is given by the constant position vector, R6, which is the position of the particle at time, t B O. The coordinates of the particle at t = O are known as the material coordinates of the particle. A fluid particle is defined as a differential volume of fluid which.umy change shape, volume, and density but must always contain the same molecules of the fluid [10]. A fluid particle is an infinitesimal closed system since no mass may cross its boundary. When the Lagrangian description of motion is employed to describe the motion of a fluid continuum, the trajectory (or pathline) of each fluid particle is described by the locus of points traced out by the end point of the position vector, R[Ro, x(t),y(t),z(t)]. An infinite number of position vectors is needed to describe the motion of the fluid continuum. For the present time we assume that the pathlines traced out by the posi- tion vectors do not intersect in the flow region under considera- tion. For the case of steady flow, pathlines and streamlines coincide [10]. Only steady flow is considered in this work. 4 2. LAGRANGIAN FORM OF THE MOMENTUM AND CONTINUITY EQUATIONS The Lagrangian form of the momentum equation, for a non- viscous fluid, that will be employed in this work is 2-4 £1—%+YR-'t’x=o, (2.1) p dt where p is the density of the fluid, p. is the pressure, f represents the body forces per unit mass acting on the fluid particle, and E is the position vector of the fluid particle [10] . The Lagrangian form of the continuity equation is often written in the below form pdV = pldvl = constant. (2.2) A second form is gi.1_1en=_ififl (23, V'dt dv dt pdt’ ° where t denotes time, p is density, V is the volume of the fluid, and R is the position vector of the fluid particle [10]. We now seek a more convenient form of the continuity equa- tion for the case of steady flow. We select the material co- ordinates to be a set of orthogonal curvilinear coordinates, (u1,u2,u3),and the u1 curve is selected to coincide with the pathlines of the fluid particles. That is, at time t = O the 5 fluid continuum is described by the curves: u1(x,y,z) = C1 = constant along a pathline, (2.4a) u2(x,y,z) = C2 = constant, (2.4b) u3(x,y,z) = C3 = constant. (2.4c) For the case of steady flow, the path of the fluid particles will remain coincident with the u1 curves. A volume element, dV, about any point, P for a moving orthogonal curvilinear co- B, ordinate system (u1,u2,u3) is defined as [12] . a3. . 35. 313. dv ‘aul dul (3‘12 duz x 8‘13 du3)‘ , (2.5) where E(ul,u2,u3) is a position vector drawn from the origin to the fluid particle at Pb, see figure 2.1. Figure 2.1 Curv 1 linear Curves The cross product in (2.5) may be interpreted as the change in cross sectional area, dA, normal to the curve. And the term ”1 h§_ du is the change in arc length, ds , along the u curve. aul 1 1 1 Thus (2.5) may be rewritten as dv = dsldA , (2.6a) =33. a3. = where dA - Bug du2 X Bu3 du3 - dszds3 , (2.6b) and dividing (2.6a) by dt yields dV ds1 d? = d—t_ dA . (2.7) Since the ratio of the change in arc length to the change in time is a measure of the Speed of a fluid particle along the u1 curve, equation (2.7) can be rewritten as d“ a a a gE‘FE'dA=v°dA’ (2-3) Dividing equation (2.2) by dt and then substituting (2.8) into the resulting equation yields pV.dA‘=p1V1-dK1 , (2.9a) or __ Vi’dAi VldAl 29b 9 ‘:—-_.—=—VTA—- <-) 1 V-dA Equation (2.9) is the Lagrangian form of the continuity equation that is employed in this work. It is only valid for the flow of a fluid particle along a time independent pathline. 3. VARIATIONAL CALCULUS We shall be concerned with the following problem from The Calculus of Variations. Consider the variable end point problem for the functional t=b t=a F(§’;)dt + E1(8 93:) + E2039?) 3 (3.1) where E1 and E2 are functions of the coordinates of the end points of the path along which the functional is considered, 9 = y1,y2,y3 represents the space variables, and y = yl,§2,93 represents the components of velocity. Calculating the variation of the functional, (3.1), and setting the result equal to zero we obtain the well known Euler-Lagrange equations F - d—F = o (i = 1,2,3) , (3.2) 3'1 dt 3'1 and the boundary conditions; (F§ - E1 )| = o (1 = 1,2,3) , (3.3a) F. - E = 0 i = 1 2,3 3.3b (y >|t___b < . > . < > where the subscripts y1 and yi denote partial differentiation, i.e., §%_ 5 Fy [2]. The solution to the Euler-Lagrange equations, . i i (3.2), is called an extremum or an extremal curve. The family of curves satisfying, (3.2), is called a family of extremal curves. We shall be concerned with minimizing the functional, (3.1). Consider the functional, 1‘: F<§J>dt . (3.4) where the end points of each extremal curve is specified. Equa- tion (3.4) will be called the fixed end point functional. Calculat- ing the variation of (3.4) and setting the result equal to zero yields the Euler-Lagrange equations, (3.2). Thus the problem of minimizing the variable end point functional, (3.1), is equivalent to minimizing the fixed end point functional, (3.4), subject to the boundary conditions (or side conditions), (3.3). From Variational Calculus it is known that (3.4) is a minimum when the following conditions are satiSfied, see Gelfand [2], pg. 146-148. I. The Euler-Lagrange equations, (3.2), are satisfied. II. The matrix “FEiEJH is positive definite along the extremal. III. The interval [a,b] contains no conjugate points. A conjugate point is a point of intersection of the neighboring extremals. IV. The value of the functional, (3.4), is independent of the path of integration. Or, more precisely, the Weierstrass E-function is 2 0 along the extremal curve. And for the functional, (3.1), we also impose the following addi- tional condition. V. The boundary conditions, (3.3), must be satisfied. 10 When conditions I-V are satisfied, the minimum is said to be a "Strong minimum". When all the conditions except IV are satisfied, the minimum is said to be a "weak minimum". For the case of a weak minimum, the family of extermals always possess conjugate points and the functional is a minimum only in local regions which are free of conjugate points. The family of extremal curves is obtained by integrating the Euler-Lagrange equations, (3.2). The Euler-Lagrange equations, (3.2), may be integrated in the below manner. Multiplying (3.2) by ii and then adding and subtracting the term, F; vi, yields 1 F ° +-F " - F Y - ° 9"F = O (i = 1 2 3) (3 5) yyi 9Y1 5:1 yidty ’ ”° ' i i i i Adding the above equations yields 3 d z F°.+F.“ - . +‘—-F. =0. 3.6 mu yiy1 yiyp eyiy, yi dt Y1” < ) Employing the chain-rule of Calculus, we observe that d 3 3;F(y1.y2.y3.y1.y2.y3) = .2 [Fy.yi +F5,’yi] - (3-7) 1=1 1 1 Substituting(3.7)into(3.6)yeilds dF 3 d —- z(r.°)=.+§.-—F.)=o, (3.8a) dt i=1 yi 1 1 dt yi or d 3 . a [F - .2 yiF).,.] = O . (3.8b) 1=l 1 Integrating (3.8b) yields 3 O F - 2 y F. = -H = constant along each extremal (3.9) i=1 i yi curve . 11 Equation (3.9) is called the first integral of the Euler-Lagrange equations, (3.2). Since the first integral of a system of dif- ferential equations is a function which has a constant value along each integral curve of the system, we see that the function, H, is a constant along each integral curve determined by (3.2). 4. FLUID PARTICLE MINIMUM PRINCIPLE In this section, the problem of describing the motion of a fluid continuum will be formulated as a minimum problem of The Calculus of Variation. Consider one fluid particle moving along one pathline during time t = a to t = b. The motion of the fluid particle is described by the locus of points traced out by the end of the position vector, RERo,yl(t),y2(t),y3(t)]. As explained in section-1, the motion of the fluid continuum is described by an infinite number of position vectors which trace out a family of pathlines. The motion of each fluid particle must obey the Lagrangian form of the momentum equation, (2.1). To derive the fluid particle minimum principle, the dot product of the momentum equation, (2.1), times the variation of the position vector, GR, is taken and the result is integrated with reSpect to time from t = a to t 3 b, which yields t=b 3 ' —o —0 —-¢ t=a[R - 6R +-§£-- 6R - i - 6R]dt = o , (4.1) 2* where R 5 fl_% . The first term of (4.1) may be integrated by parts dt in the below manner b a a _ a b b s . L fa R - 6R dt — R . ija - fa R 6R dt (4.2) ° ~ b b 1 L 2 R aRja - fa 2 5[(R) ]dt 12 13 When the end points of the pathline are Specified, the boundary conditions are 5Rja = o, and 6R]b = 0 ; (4.3) and (4.2) becomes b 3 a b l L 2 [a R - 5R dt = - a -2- 6[(R) ]dt . (4-4) The second term of (4.1) may be expanded in the below manner 3 1 —. a2 From Thermodynamics it is known that for the isentropic flow of fluid obeying the perfect gas laws that TdS = 0 = c dT - EB , (4.6a) P P or p P where T is temperature, 8 is entropy, CI) is the Specific heat of the fluid at constant pressure, p is pressure, and p is density [10]. Replacing the total derivatives in (4.6b) by varia- tional derivatives and equating the resulting equation with (4.5) yields 2P-'-£>'1'i=r=92=c w. @J) p p 9 When the body forces, f, acting on the fluid particle are con- servative, the third term of (4.1) may be replaced by a force potential, i.e. VG = -fc, and 14 66 = ~fc - 6R (when I is conservative, V x f = 0). (4.8) Substituting (4.8), (4.7), and (4.4) into (4.1) yields b 1 L 2 ja[- 2 6[(R) 1+ cpa'r + OGJdt — o . (4.9) Factoring out the variational derivative Operator, 5, and multiplying by a minus one yields b 1 A 2 sja [2(a) - CpT - G]dt 0 . (4.10) In words, equation (4.10) states that the isentropic flow of an ideal fluid particle moves between two Specified points in a con- servative force field in such a way that the functional, (4.10), is a minimum. A result which is equivalent to (4.10), but written in a more general form, was published by Nantanson in a series of papers from 1896 to 1902 [6]. In most fluid flow problems, the end points of the pathlines are unknown. We therefore consider the variable end point functional E: [21'6“ ' CpTdi) ‘ C(33)“ + E10351) + E2(b.§) . (4.11) where E1 and E2 are known functions of the coordinates of the end points of the pathline along which the functional is considered. Calculating the variation of (4.11), and remembering that boundary condition (4.3) no longer applies, yields b 1 L 2 a a bja [2(R) - CPT(R) - G(R)]dt + [(VEI - i) - 6K], +EKVE2 - K) . 6R]b = 0 , (4.12) where T(R) and G(R) are functions of the Space variables, i.e. 15 a L 2 .2 .2 .2 T(R) E T(y1,y2,y3), also (R) = y1 +y2 +y3 . In order to describe the motion of a fluid continuum, the functional must be solved along each pathline of the flow region. However, the functions VEl and sz will be chosen so that they Specify the velocity, E, along every pathline at the cross sections, 1 and 2. Then equation (4.12) will apply to every pathline in the flow region. And the solution of (4.12) will be a family of pathlines which describe the motion of the fluid continuum. Equation (4.12) will be called the "fluid particle minimum principle". In words, (4.12) states that the isentropic flow of a fluid, obeying the ideal gas laws in a conservative force field, will travel along the one family of pathlines which causes the variable end point func- tional, (4.12), to be a minimum. The functional, (4.12), is a minimum*when the five condi- tions, (I-V), of section-3 are satisfied. We will now discuss these conditions for the special case of (4.11). The Euler-Lagrange equation of (4.11) is identical to the momentum equation, (2.1). This statement is easily verified by substituting the integrand, F, of (4.11) into (3.2) which yields .. T a§_ +c3——+ =0 1=1,23 , 4.13 3'1 p ayi 5’1 ( .) ( ) or in vector form i + cpvr + vc = 0 . {4-14) and substituting (-f a VG) and (%2'= CPVT) into (4.14) yields :R'+%P--?=o, (4.15) 16 which is identical to (2.1). The first integral of the Euler- Lagrange equation of (4.11) is obtained by substituting the integrand, F, of (4.11) into (3.9) which yields 1-12 3 —(R) -cr-G-z:§§=-H; (4-178) 2 p i=1 i i and Since (R)2 = 2(§i)2,(4.17a) becomes 1 L 2 2(R) + CPT +'G = H = constant along each pathline. (4.17b) Equation (4.17b) is equivalent to Bernoulli's equation. We shall call (4.17b) the energy equation, and it is shown in appendix-A that the First law of Thermodynamics agrees with (4.17b). Since the energy equation, (4.17b), is derived by integrating the Euler- Lagrange (or momentum) equatiOn, we conclude that condition I of section-3 is satisfied when (4.17b) is satisfied. Condition II of section-3 is always satisfied for the functional (4.11) since 1 0 O “F9191“ = g 3 2 (4.18) is always positive. Let us now consider condition IV. In appendix- B it is shown that condition IV is satisfied when the flow is irrotational. When the flow is irrotational there exists a velocity potential function, g, such that ° gas. 1=123; 4.19s) yll 6% ( ..) < or in vector form R‘ = Vg . (4.19s) 17 It is also Shown in appendix-B that the integrand, F, reduces to F = fig, (4.20) for the case of irrotational flow. Let g = E = E1 = E2 and sub- stituting (4.20) into (4.11) yields b 23 a dt dt + 3(a) + go). (4.21) From (4.21) we conclude that the known functions, E1 and E2, and the functional, (4.11), are completely determined by the potential function, g, for the case of irrotational flow. Since the value of (4.21) does not depend on the path of integration, condition IV is satisfied. We now list the conditions for which the fluid particle minimum principle, (4.12), is satisfied. We consider two cases, rotational and irrotational flow.. A. Rotational flow (weak minimum) f'4.A.1 The energy equation, (4.17b), is satisfied. 4.A.2 The boundary conditions, (3.3), or; < (VEl - R')‘a = o , (4.22a) 4.A _.. (V132 - R)‘b - o (4.221;) are satisfied. 4.A.3 The pathlines do not intersect in the flow region, i.e., there are no conjugate points. The following procedure from.Gelfand [2] pg. 130 may be employed to test for conjugate points. Let y - y(t,a,a) be a general solution of(3.2)depending on two parameters, a and B- When the ratio 231% , (4.22c) 18 is the same at two points, the points are conjugate. B. Irrotational flow (strong minimum) r 4.3.1 The energy equation, (4.17b), is satisfied. 4.8.2 The potential function, g, satisfies the boundary conditions; 4.3 é (Vg - fi)‘a = 0 , (4.23a) (Vs - {Mb - 0 . (4.23b) t4-3°3 The pathlines do not intersect in the flow region. The fluid particle minimum principle, (4.12), is said to be satisfied when conditions (4.A) or (4.3) are satisfied. Whenever the fluid particle minimum principle is satisfied both the momentum equation, (4.15), and the energy equation, (4.17b), are satisfied. In fact, they are equivalent equationsfor the case of isentrOpic flow. In addition to the fluid particle minimum conditions, (4.A) or (4.3), the flow must also satisfy the continuity equation and the condition that the pathlines of the flow region coincide with the walls of the passage. The fluid particle minimum principle may be extended to include forces, fN, function, G. Letting f = fC +fN in equation (4.1) yields which are not derivable from a potential force b I... -o 12 —o -o -o b . —o a La [3 6R + 9 6R - fc - 6R]dt - j‘a IN 6R dt 0 , (4.24) where fc represents the conservative forces. Repeating the pre- vious steps of this section up to equation (4.10) yields 61‘: [El-(3)2 - cp'r - G]dt - J”: IN - 5i dt = o . (4.25) 19 Calculating the variation of the first term of (4.25), (4.25) becomes I: [i +'C VT +-VG] ° 6E dt - fb f - 6g dt - 0 . (4.26) P a N Since 6g is arbitrary, (4.26) reduces to g + CPVT +'VG - ffi = 0 ; (4.27) which is the momentum equation, see equation (2.1) (with prT = Vp/p and VG - ffi ‘ f). The variable end point form of (4.25) is b 1 ' 2 b , 6L! [2(3) - CpT G]dt fa IN 5i dt + [(VE1 - R) o 51218 +[(VE2 - R) . 5R]b - o , (4.28) which is similar to (4.12). The Euler-Lagrange equation of (4.28) is again the momentum equation, (2.1). The boundary conditions of (4.28) are the same as the boundary conditions for (4.12), i.e., the boundary conditions are given by (4.22). It should be pointed out that (4.28) is, in general, difficult to employ because the second integral in (4.28) cannot be evaluated, in practice, without additional information. Fortunately, for the case, which we shall consider, ffi is normal to bi and thus the second integral in (4.28) reduces to zero. 5. SOLUTION OF SOME CLASSICAL EXAMPLES In this section, some classical example problems are in- vestigated using the previously developed equations. One of the purposes of this section is to demonstrate that the Lagrangian description of motion may be employed to solve fluid problems, which are traditionally solved by the Eulerian method. Two examples will be considered, incompressible potential flow and isentrOpic compressible flow. A. Incompressible Potential Flow Consider the isentropic flow of a fluid in a region where the body forces, VG - 4?, may be neglected. We assume that the flow is irrotational and that the velocity potential function, g, is known. The velocity, V, of the fluid is then determined- from the gradient of the potential function, i.e., V E R = Vg . (5.1) Substituting (5.1) into the energy equation (4.17b), with G E 0, yields 2 SE§Z_.+-C§T = H = constant along each pathline. (5.2) For the case of isentropic flow, an a in - cpd'r ; (5.3) p 20 21 and since the density p is constant, integration of (5.3) yields CpT = §'+ constant . (5.4) Substituting (5.4) into (5.2) yields the incompressible form of the energy equation, $231—-+-§ = constant along each pathline . (5~5) Since the flow is incompressible, from (2.2) we observe that the change in volume is constant, i.e., dV = constant . (5.6) Substituting (5.6) into (2.3) yields v-Vao. . (5.7) Substituting (5.1) into (5.7) yields V ° (V3) = V23 3 0 , (5.8) which is the well known Laplace equation. Once a potential function, g, is selected which satisfies the boundary conditions and Laplace's equation, the family of path- lines is uniquely determined by the potential function, g. The pressure distribution is then determined by the energy equation, (5.5). Of course, this result is exactly the same as the results of "Classical Incompressible Potential Flow Theory" [4]. The only difference is that the classical results are derived from the Eulerian point of view, whereas, the present derivation is from the Lagrangian point of view. Since for the case of steady flow 22 pathlines and streamlines coincide, the same equations result regardless of our point of view. Let us now consider the fluid particle minimum principle. Condition (4.3.1) is satisfied by (5.5). Condition (4.3.2) is satisfied by (5.1) for all values of time t I a and t = b. Con- dition (4.3.3) is satisfied in regions which do not contain con- jugate points. Thus we conclude that all incompressible potential flow prdblems satisfy the fluid particle minimum principle in flow regions which exclude conjugate points. For the case of flow around a two-dimensional airfoil (with circulation), conjugate points occur at the stagnation point and trailing edge of the air-foil as shown in figure 5.1. Thus, these two points are excluded from.the flow region. The fluid particle minimum principle does not predict the nature of the flow in the neighborhood of these two points. The stagnation stream- line divides the flow into two regions and the flow in these two regions, excluding the stagnation streamline, satisfy the fluid particle minimum principle at all points. streamline stagnation conjugate points Figure 5.1 Conjugate Points 23 3. Isentropic Compressible Flow Consider the isentropic flow of a fluid obeying the ideal Assume that the flow is in a region where the body gas relations. forces, VG I -f, may be neglected. The energy equation as given by (4.17b), with G E 0, is 1 2 E-V +-C T = constant along each pathline, (5.9) where V a R. The constant in (5.9) may be evaluated at the stagnation condition (denoted by the subscript o), i.e., (5.10) 1'V2 +' T ' C T 2 OF P o ' Or the constant can be evaluated at sonic conditions (denoted by the subscript *), i.e., l 2 l 2 v + cp'r cp'r* + 2 v* . (5.11) 2 The definitions of the Mach number, M, and the speed of sound in an ideal gas, a, are M = V/a , (5.12) M* = V/V* a v/a* , (5.13) (5.14) 2 a B KRCT CPGK-1)T , where Rc is the ideal gas constant and K E CP/Cv is the ratio From Thermodynamics we recall the isentropic of specific heats. relations fl _ . .. [0—]1‘4 , (5.15) 91 2. T1 p1 24 Dividing each term of (5.10) by (5.14) yields T +-L=—1—-1-:—°- . (5.16) Substituting (5.12) into (5.16) and solving for TO/T yields '1‘ o K-1 2 '—— = 1 +~——— Dividing each term of (5.11) by £5.14) with T = T*] yields 2 2 V 2 a2 + x-1 1'*1<-1 2 2 ' (5'18) * a* Substituting V* E and M* E V/a* into (5.18) and solving 3* 2 for M“, yields 2 -2 T K+1 Mk " ETI[F' ' ‘2‘]- (5°19) * pO p0 Substituting (5.15) into (5.17) and solving for E-' and 'E-' yields p __‘.‘_ .52 - (1 + 5&1 M2)K'1 , and (5.20) p —1f 33 = (1 + Eg—l M2)K' . (5.21) At sonic conditions, M = l and (5.17), (5.20), and (5.21) reduce to: T F:- , KT+1 , (5.22) p K 33 - (531-5171. (5.23) p 1 3:1- (IS—:1)“. (5.24) All of the above equations, (5.10-5.24), agree with the classical one-dimensional isentropic flow relations [11]. This agreement is 25 expected since all the equations and definitions used so far are exactly the same. The only difference is that the above equations give the changes in fluid properties along a pathline, whereas in the classical one-dimensional method the above equations give the changes in properties along a "one-dimensional" streamline. How- ever, we now introduce the Lagrangian continuity equation, (2.9), which differs from the continuity equation employed in the classical method. The Lagrangian continuity equation, (2.9a), is p V dA = p*v* dA* . (5.25) The Eulerian continuity equation employed in classical one-dimensional gas dynamics is p V A = p*V*A* = constant . (5.26) We will now use the Lagrangian form of the continuity equation, (5.25), to obtain dA in terms of the Mach number, M. This result will then be compared to the similar equation obtained when the Eulerian continuity equation, (5.26), is employed. Substituting (5.25) into (5.15) yields T E_.K-1 V*dA* K—l dA 1-K ... = = d = M* a——- , (5.278) T* PfJ . V A A* or __1_ 1-K dA _ -1 T_. dA — M* [T J . (5.27b) * * Substituting (5.19) into (5.27b) yields dA -2 1' K+l '3 T 1'K cl——a= — -—-—-— — . (5.28) 11* -1 T* 2 2* 26 Substituting (5.22) into (5.28) yields .1. T -% l-K dA 2 T l: o] [T --- — --- — . (5.29) «111* K-Tl T 2* Substituting (5.17) into (5.29) yields .1. 9L=."___[1-1 K__1.2"MZ 35151-1( dA* 1(- 2* -§K+l) (5.30) ..l. 1;. 20‘ 1’ M T* ' Dividing (5 22) by (5 17) yields 1...:ch [1+L21M 2 (531) 2* 2*1'0 1(—+1 which when substituted into (5.30) yields K+1 2(K- l) §é_.=_ 1 K- 1 M2 (121* —[K-+——l [1 +—— M3] . (5.32) We now compare the above equation to the similar relation, (5.33), from classical one-dimensional gas dynamics which is given below [11] _1_<___+1 2(x-1) A .1 x_-_21M2 A* M[K+-_2-1 [1 +— M3] . (533) Notice that the right hand side of (5.32) and (5.33) are identical. We will now show that for a hypothetical case of flow bounded by radial pathlines (or streamlines), the left hand side of (5.32) and (5.33) are equivalent. Consider a family of radial pathlines as shown in figure 5.2. Let r* locate the cross section at which sonic conditions occur. 27 Figure 5.2 Radial Pathlines Enclosing a Pathtube The cross sectional area normal to the pathtube (or streamtube) is given by ~ 2 A - n(rAe) , (5.34) and the area ratio is 2 2 A. a n_i_)__rA9 .- L2. , (5.35) A* n(r*A9)2 r* The rate of change of the cross sectional area is 2 . dA = r Sin 9 d9 d¢ ; (5.36) and since d9 and do are constant along each pathline the dA ratio is N dA dA* r. . (5.37) 2 'k r 28 which is the same as (5.35). Thus we can conclude that for the case of radial pathline flow, (5.32) reduces to the classical result, (5.33). If we would analyze the flow through a straight- radial wall nozzle by the present equations and by the one- dimensional method we would obtain exactly the same results (all the equations are identical). However, when we analyze the flow through a nozzle with curved walls, the two methods would not agree because the ratio of increment of areas in (5.32) is not equal to the ratio of areas in (5.33). The greater the curvature of the walls, the greater will be the disagreement. Let us now consider the fluid particle minimum principle. Consider the case of flow through an internal flow passage where the pressure at the intake and discharge of the passage is known. The geometry of the passage is also known. A family of pathlines is then selected so that the outer pathlines of the flow region coincide with the walls of the passage. For example, for the case of flow through a conical passage, as shown in figure 5.2, a radial family of pathlines is selected. The ratio dA/dA* can then be calculated. We now seek the function V3 in terms of dA/dA*. Substituting [F-]1/K - 2—- (from 5.15) into (5.27s) yields * 9* [L]1/K = V_* 353:. (5 38) 9* v dA ° ' Solving for V yields 1/K dA 9* * Dividing (5.23) by (5.20) yields 29 .K_. p K-l L=L_g=[ K+1 2] ; (5.40) p* pop* 2(1+KT’1M) which when substituted into (5.39) yields ....1... K-l dA K+l * V = - —- V E'Vgl, (5.41) [2(1+—K1M2)] dA * 2 where we have defined the right hand side of (5.41) to be equal to lVg'. The Mach number M is determined implicitly in terms of dA/dA* by (5.32). Since dA/dA* is a function of the Space variables, the function Vg, defined by (5.41), is also determined as a function of the Space variables. Substituting the known pressures, pd and pi, into (5.39) determines the velocity at both ends of the flow passage, i.e., ’p*‘1/K dA* Vd = REC—U 3151—]d V* , and (5.428) Fpfll/K dA*] Vi = L;:‘ 52—, i V* . (5.42b) Condition (4.3.1) is satisfied by (5.9). Condition (4.3.2) is satisfied when (5.42) is satisfied or when (5.41) is satisfied. Since (5.41) is not valid across a shock, condition (4.3.2) is satisfied in flow regions which exclude shocks. Condition (4.3.3) is satisfied in regions free of conjugate points. Thus we can conclude that the fluid particle minimum principle is satisfied for the case of flow through a passage when (5.42) is satisfied in a region free of shocks and conjugate points. It should be pointed out that it is difficult to select a family of pathlines that coincide with the walls of some given 30 flow passage. An iteration procedure may be necessary in order to determine this proper family of pathlines. The inverse pro- cedure is much easier. That is, given a family of pathlines we may choose any set of pathlines to be the outer pathlines of the flow region and thus determine the geometry of the flow passage. This inverse procedure will be employed in later sections to determine optimal geometries of compressor rotor passages. It should also be pointed out that all compressible (irrotational) potential flows satisfy the fluid particle minimum principle in regions free of shocks and conjugate points. That is, once a velocity potential function, g, is selected which satisfies the boundary conditions and the continuity equation (2.9), the family of pathlines is uniquely determined, see (5.41). The boundary conditions are (5.42) plus the condition that the family of pathlines coincide with the walls of the flow passage. For the case of rotational flows, there does not exist a potential function and the fluid particle minimum principle is often not satisfied. 6. ROTATING Flow PASSAGE In this section, the fluid particle minimum principles, (4.12) and (4.28), are adapted to the flow inside a rotating passage. Consider a fluid particle that is moving along a path- line which lies inside a rotating passage. The fluid particle is rotating at an angular speed, é, relative to the passage as shown in figure 6.1. The wall of the passage is rotating at a constant angular speed, m. The total angular speed of the particle, &, is dt=w+é. (6.1) The absolute position vector, R, of the fluid particle may be expressed with respect to the fixed reference frame (with unit vectors 1,3,k) in the below manner 3 = r cos a i +'r sin a 3 +-z k . (6.2) Where (r,a,z) are the cylindrical coordinates related to the Cartesian coordinates (x,y,z) by: x = r cos a , y = r sin a , z . z . (6.3) The first time derivative of (6.2) is d "9 o o A :§'= V = (r cos a - a r sin a)i + (6.4) (i sin a +-& r cos a)3 +.é k ; 31 32 i y - fixed reference frame - («ital hig Ii'felxniec! irann‘ Y1 relative pathline Jaw Figure 6.1 Rotating Passage and the kinetic energy is . .2 .2 .2 %(§')2 5%v2 =_)__(r2 +9325L+5§L. (6-5) Since the Euler-Lagrange equations and the functional, (3.1), are invariant under coordinate transformation, the introduction of cylindrical coordinates does not change any of the equations developed in the previous sections [2]. In order to apply the result of the previous sections, we simply replace the terms 33 R, R, é-Vz by equations (6.2), (6.4), (6.5) respectively. For exam- 1 4 2 pie, replacing 3(R) by (6.5) in the fluid particle minimum principle, [24.12) (with G = 0)], yields b . 2 . 2 . 2 6 [[11+5—r—012—+£z—)— - CTjdt + a 2 2 2 p [(VEl - R) - 6R]a +[(vE2 - R) . 6R]b = 0 . (6.6) The Euler-Lagrange equations of the above functionalanmaobtained by substituting the integrand of (6.6) into (3.2) (with y1 = r, y2 = 0, Y3 = 2) which yields df .2 ah _ dgrzéz ah . dt +IBQ = O , (6.7b) dz h __ + L = - dt 82 0 , (6 7c) where h E CPT. Multiplying (6.7b) by 1/r and substituting vh='V-R=l[a£i +133} +923] (6.8) P P at r r 80 a az 2 into (6.7) yields . 1 f - ra2 + "a2'= 0 , (6.98) P 5r .. 1 r&+2ra+”la£=0, (6.9b) P r 80 .. 132 z +'— = 0 ; (6.9c) P 62 which is the Lagrangian form of the momentum equation written in terms of cylindrical coordinates for the case of isentropic (frictionless) flow, see Owezarek [10] page 93. The functional 34 (6.6) depends on time, t, because the total energy of a fluid particle is increased while passing through a rotating passage. When the functional depends on time, the first integral of the Euler-Lagrange eqaution (i.e., the energy equation) is no longer given by (3.9) but is given by the below equation, see Gelfand [2] page 70 33 dB d V2 at=az=az- “'10) We now choose a reference frame which is attached to the passage and thus is rotating at a constant angular speed, w, see figure 6.1. This rotating reference frame will be called the relative reference frame. The position vector drawn from the relative reference frame to the fluid particle is called the relative position vector, K , which is written in terms of the relative 1 unit vectors, (i1,31,k1) as a c R = r cos 9 1 1 + r sin e jl + 2 k1 . (6.11) 1 The relative velocity, W, is the first time derivative of (6.11), i.e., RI 5 W’= (f cos 9 - é r sin 9)11 + (f sin e + 6 r cos 9)}1 + 2 k1. (6.12) where i1,]1,k1 are the unit vectors of the rotating reference frame. The relative kinetic energy is 1 5 2 _ 2 2(R1) = w = [£2 + (ré)2 + £2] . (6.13) NIH Nh—I 35 We now seek an expression for the acceleration of the particle along the rotating pathline. Drawing the absolute posi- tion vector, R, from the fixed reference frame to the particle, we may differentiate ‘R with reSpect to time in order to obtain the absolute velocity and acceleration of the fluid particle. The absolute position vector, R, and the relative position vector, RI, coincide, i.e., R = R1 . (6.14) Differentiating (6.14) yields a 6R =—-='——+ =w Q 0 . v dt dt (1) x R1 +u)r 19 (6 15) Notice that the relative position vector, RI, 18 Changing its magnitude and direction. The term, a X R in (6.15), arises 1, because 31 is rotating. In general, derivatives taken in the rotating reference frame are related to derivatives taken in the inertial reference frame according to the operator [10], 91.1 d .] dt ROTATING dt] +63 x (). (6.16) INERTIAL The absolute acceleration is obtained by differentiating (6.15) and employing (6.16) which yields R = R1 + 23 x R1 + 3 x (6 x R1) , (6.17a) 01' 4 ° 3 2 Q v=fi+2wxW-wrir, (6.17b) where %%EV and ---‘-'='W. 36 We will now formulate a fluid particle minitmm principle in terms of quantities measured from the relative reference frame, i.e., in terms of R1 and W: The absolute acceleration, V, is given in terms of Quantities measured in the relative reference frame by the right hand side of (6.17). Substituting (6.17b) into the momentum equation, (2.1), (with f E 0) yields .2 _. _. 2 . W+2wXW-wrir+%2=0; (6.18) which is the momentum equation as viewed from the rotating reference frame. We observe that Newton's second law of motion does not retain its form (W ='§E-+-f) in the rotating reference frame [3]. Taking the dot product of (6.18) times 6R and then multiplying 1 by dt and integrating yields b . —‘. 2 A [[W-oR +213. .612 +2(&ixfi) - 6R’ -wr i - 6?? 3d: =0 . (6.19) a 1 p 1 1 r 1 When the inertia accelerations are treated as forces which act on the relative reference frame, i.e., when we let l Ha I! *2 8 x W’, and (6.203) (6.20b) Hal II E '1 H then (6.19) is of the same form as the general functional, (4.24), and as shown in section-4 the general form of the fluid particle minimum principle is given by (4.28). We now seek an expression for the "potential energy", G, where we define C so that 66 = -fC ' 63 First, observe that 1. 2 A the centrifugal acceleration, w r ir, produces a conservative force 37 f ie 1d 8 ince l1 1 la r r 9 r k 7- ~. g a. a. a. .. V X (m r 1r) Br 89 82 0 (6.20c) r w2 0 0 Thus the potential energy of a fluid particle in the rotating reference frame is 2 A —o 2 (0er 66 = -u) r it ° 6R1 = -0) 1761‘ = -5 T . (6.21) Substituting (6.21) and (6.20a) into the general form of the minimum principle, (4.28), yields b w2 w2r2 . b _ __ _ , -o . -o d 6 [[2 + 2 h]dt +£(2m x W) 6R1 t + [(vs1 - W) . 6R118+ [(vs2 - W) - 5R1]b = o , (6.22) where h a CPT and fN = 2&3 X W. The second term of (6.22) is zero since the Coriolis acceleration, 2; X‘W, is normal to 6R1. However this term will be retained since its omission will alter the Euler-Lagrange equation. The Euler-Lagrange equation of 2 2 (6.22) is given by (4.27) (with c = ‘er , Z’N =23 x if , and K =‘W), i.e., W + CPVT - wzr it + 263 x W = 0 , (6.23a) or W+%E-w2rir+2axfi=0. (6.23b) Substituting (6.17b) into (6.23b) yields \ .lilji 14’ I .l 38 V+%E=O; (6-24) which is the momentum equation as viewed from the fixed reference frame. Expanding the above vector equation into its three scalar equations, in terms of cylindrical coordinates, yields (6.9). Thus we can conclude that the Euler-Lagrange equations of the functionals (6.6) and (6.22) are identical. It is known from Variational Calculus that two functionals are equivalent when their respective Euler-Lagrange equations are identical. The two fluid particle minimum principles, (6.6) and (6.22), are therefore equi- valent and thus we may operate in the relative reference frame using therminimum principle given by (6.22). The first integral of_the Euler-Lagrange equation may be found by two methods. In the first method we form the dot product of (6.23b) times dfil and then integrate. In the second method we substitute the integrand of (6.22) into (3.9). In both methods, we observe that (ZE’X‘W) - dR B 0. Employing one of the above 1 methods yields 1 2 2 2 §“W - wzr +-h = HR I constant along each relative pathline; (6.258) or 1 2 2 2 w2r2 ' ° ° .— = . 6.25b flit +(r9) +2] 2 +h HR ( ) The relative pathline is the path of the fluid particle as viewed from the relative reference frame. Substituting 6 = & - m from (6.1) into (6.25b) yields 2 NII—I . . .2 . [r + (r6)2 + z 1 - 1‘sz + h = HR. (6.26) 39 Substituting (6.5) into (6.26) yields 1 2 2. E'V - r aw + h = HR = constant , (6.273) or v2 2 §*'+-h = r &m + HR = H # constant ; (6.27b) V2 where H E §-+ h, and H is called the total absolute enthalpy. The symbol HR is called the total relative enthalpy. Taking the time derivative of (6.27b) yields 2 d V dH d 2. Equation (6.28) is called Euler's turbine equation which is valid for steady isentropic flow, see Owczarek [10] page 95. The right hand side of (6.28) represents the rate at which work is added to a fluid particle along an absolute streamline or an absolute path- line in this derivation. Comparing (6.28) and (6.10) we see that they are equivalent equations. Thus we can conclude that the same absolute energy equation results regardless of the minimum principle that is employed to describe the motion. In later sections, the fluid particle minimum principle, (6.22), will be employed to determine the family of pathlines which describe the motion of the fluid continuum. The "relative energy equation", (6.25), is the energy equation corresponding to (6.22) which must be satisfied in order that condition (4.A.l) of section- 4 is satisfied. 7. IRROTATIONAL FLOW In this section we seek the condition at which the flow inside the rotating passage is irrotational, i.e., we seek the condition, V X V = 0. Substituting (6.15) into V X V = 0 yields v x (17+wr i9) =0 , or (7.1a) a 2 A 2 VXW=-i 951L1+1 9m=-izw. (7.16) z r ar r r 32 2 Since the curl of a velocity vector is equal to twice the angular speed of the fluid particle, the angular Speed of the relative velocity vector is equal to -w when the fluid is irrotational, i.e., the irrotational condition is va=0 IFF é=-w. (7.2) This type of flow is called a "free-vortex" flow in turbomachinery literature [13]. When (7.2) is Substituted into Euler's turbine equation, (6.28),we see that the energy level, H, of a free- vortex flow remains constant, i.e., Substituting 5 = w + 8 = w - w = 0 into (6.28) yields 3 = 0. Thus we conclude that the strong minimum energy State of the system corresponds to a zero change in energy along an absolute pathline. Since the principle function of a compressor is to add energy to the fluid, the irrotational case, (7.2), is dismissed as a trivial case. 40 8. ROTATIONAL FLOW In the previous section, it was shown that there exists only one trivial case for which the flow inside a rotating passage is irrotational. In this section, we investigate the conditions for which the fluid particle minimum principle, (6.22), and the continuity equation are satisfied in a rotational flow. In section-6 the relative energy equation, (6.25), was derived, which is the first integral of the Euler-Lagrange equation corresponding to the fluid particle minimum principle, (6.22). As shown in section-6, the (6.22) is identical to the and momentum equations are minimum principle, (6.22), Euler-Lagrange equation, (6.24), of momentum equation. Thus the energy satisfied whenever the fluid particle is satisfied. The continuity equation is now combined with the relative energy equation and an expression for the change in cross sectional area, dA, of a differential pathtube is obtained in terms of (w,r,W,h) as shown below. A precise definition of dA was given in section-2, see equation (2.6b). in the second example of section-5. We proceed in a manner similar to the procedure employed The relative energy equation, (6.25), as derived in section-6 is 12. (621-2 2 2 + CpT = H = constant along each relative pathline . (8.1) 41. 42 Dividing each term of (8.1) by the Speed of sound squared, 0 a; = Cp(k-1)T*, where * denotes sonic conditions, yields 2 2 2 w r 28: __1.'1_'_ K-1 1* + (8.2) NIp—I a: It! 31- N feign“ Substituting the continuity equation, given by (5.27s with W = V) into (8.2) yields .1.“ _w2r2+i LilLl-Kgljfl (83) 2 2‘2 2 2 R-1 * dA* 2 ° ° a* a* a* Solving for dA/dA* yields _.1._ dA a* HR u)2r2 w2 l-K —. — (K-l) ——+—- - -— . (8.4) dA* W a2 2a2 2a2 * * * Equation (8.4) can be written for any two points along a pathline in the below manner 1 l 2 2 ‘1 2 l-K dAZ.dA2/dA*.h HR+2wr2-2W2 (85) dA dA 7dA W l 2 2 l 2 ' ' 1 1 * 2 HR+§mr1-EW1 Multiplying (8.5) by W2/W1 and using the fact that the relative velocity vector, W, is perpendicular to the change in area vector, dX, yields 1 s s 1. 2 2 2 2 2 'ffii w2 cm2 _w2dA2 _ hi 2&2 W1 ‘” (r2 r1)] 8 6) -° dK WldA1 hl ’ ( ° w1 1 — . l. 2 2 .l 2 where h1 = CPT HR +2 w r1 - 2W1 (see 8.1). The changes in the thermodynamic properties of the fluid can be found by employ- ing the below equations after (8.6) has been evaluated. Employing the continuity equation, (2.9b) with W B V, yields 43 _= 1 1 . (8.7) Substituting (8.7) into the isentropic gas relation, (5.15), yields :2 = [El-JK-l =[WldAl]I(-l . (8.8) T1 91 1.723112 Substituting (8.7) and (8.8) into the ideal gas equation of state yields I: (8'9) P1 9 1RcT1 w2““‘2 :2 pZRcT2 =[W1dA1]K When equation (8.5) or (8.6) is satisfied along a relative pathline, the momentum, energy, and continuity equations are satisfied. If the flow is in‘a region which does not contain conjugate points and if the boundary conditions, (4.22), are satisfied, then the fluid particle minimum principle is satisfied. That is, when conditions (4.A) of section-4 are satisfied, the fluid particle minimum principle is satisfied” We can now formulate a boundary value problem. Equation (8.5) must be satisfied along every pathline inside the flow region. At the ends of the flow region, boundary conditions (4.22) must hold. Along the walls of the flow passage the pathlines must coincide with the walls of the passage.. The following "inverse" procedure is used to de- termine the optimal geometry of the flow passage. We impose "optimal" constraints on the problem. A family of pathlines is then determined which satisfy (8.4), (4.22), and the optimal con- straints. The passage geometry is then selected to coincide with 44 the pathlines, and thus the boundary value problem is completely determined. The optimal constraints are developed in the next section. 9. OPTIMAL CONSTRAINTS In this section a few optimal constraints are selected which, in the author's opinion, are expected to produce optimal performance of a compressor rotor. The choice of these optimal constraints is supported by intuitive arguments. The author knows of no rigorous procedure for selecting the optimal constraints. Optimal constraints are imposed in order to produce desirable operating conditions. However, it is not always possible to solve the boundary value problem subject to several constraints. When this occurs, it may be necessary to remove one or more con- straints. A rotor usually has a uniform inlet pressure and back pressure imposed on the intake and discharge cross sections (see Vavra [13] page 212). However, the intake pressure, p1, and the discharge pressure, pd, inside the rotor is, in general, not uniform. This type of situation may produce secondary flows eSpecially if the pressure distribution is highly non-uniform. It is therefore reasonable to constrain pi and pd to be approximately uniform. Examination of the momentum equation, (6.9), yields the following steady flow case in which the intake pressure, pi, is uniform over the (r,e)—plane of the intake section . . . 1 u r = constant, a, =1» - 61 = 0: 2 = ..32., f = a = 0 . (9-1) 45 46 We Shall call (9.1) the "free-vortex intake condition". Notice that this condition can hold only at one given rotor speed, w. Given a uniform pressure (or approximately uniform pressure), pi, at the intake section, the discharge pressure, pd, will be uniform (or approximately uniform) only if the pressure increases by the same amount along every pathline between the intake and discharge sections. We will also constain the pressure to mono- tonically increase along each pathline. A local region of rapid pressure change could cause the boundary-layer to separate, which is undesirable. Upon examining (8.6) and (8.9), we conclude that the above "uniform pressure increase constraint" is met when -R 1 2 2 2 2 2 TIE L_WdA'K_hi 2[w 'WL’w“ ’11)] pi WidAi bi = L(r.6.2) . (9-2) where L(r,e,z) is a monotonic increasing function which has the same value when evaluated between the intake and discharge points of each pathline. A Special case of (9.2) is III H O J-IIK (9.3) l‘fififi—i = [L(r.e.2) From (8.7 - 8.9) we observe that the thermodynamic variables, (p,T,p), remain constant along the pathlines when constraint (9.3) is satisfied. Constraint (9.3) will be called the "maximum kinetic energy increase constraint" because all the energy being added to the fluid is being converted into kinetic energy, while the enthalpy, h, of the fluid remains constant. 47 Solving (8.6) for dA/dAi yields 1 2 2 dA .21 hi-ZEW -wi “11 w hi _1_.. - 6.26:2 - 1%)] 1-x (9.4) The above equation determines the ratio dA/dA1 along each path- line in the flow region. For a given set of design conditions, (h1,W1,w, etc.), equation (9.4) is employed to determine a family of pathlines. The walls of the flow passage are then selected to coincide with the pathlines of the flow region, and thus the geo- metry of the flow passage is also determined from (9.4). When one of the design conditions, (hi’wi’w’ etc.), is changed, the pathlines of the flow region will, in general, no longer coincide with the walls of the flow passage. There is only one special case of (9.4) in which the pathlines coincide with the passage walls at all rotor Speeds, w, and initial conditions, (h1,W1). That is, if we impose the constraints ‘1—1-1 and §—=1, (9.5a) a (9.5b) which is independent of the parameters (h1,Wi,w). The above equations, (9.5), will be called the "maximum speed range" con- straint. It is common practice to design a rotor so that the same amount of energy is added to each fluid particle which passes through the rotor [13]. The "uniform energy increase constraint" 48 is imposed in order to reduce mixing losses after the flow leaves the rotor. Thus the change in energy, AH, is constrained to be a monotonic increasing function, L(r,9,z), which has the same value when evaluated between the intake and discharge points of each pathline. The change in energy, AH, along a pathline is determined by integrating (6.28) with respect to time from t = t to t = t. i Substituting & 8 6 +-w into the resulting equation yields 2 2 2 2 . 2 . H - H1 = w (r - r1) +’r we - riwei 8 L(r,e,z) . (9.7) The equation of a pathline can be expressed in the following para- metric form; r = r(t). z = 2(t). e - 6(t) - (9-8) For the case of steady flow, it is possible to eliminate time, t, from one of the above equations and express the equation of the pathline in terms of the other two Space variables, i.e., we may write 9 = 8(r.2) . (9.9) Differentiating the above equation and employing the chain rule of Calculus yields '.i&_aaae 6322. 9 dt 52 dt +~Br dt . (9.10) Substituting (9.10) into (9.7) yields 2 2 2 2 a§_ 2 .33 H - H = - 2 - z + 1 w (r r1) +'r w 82 rim 1[52]1 2 . 32. 2 . .33 r wr 3r - riwri[3r]i . (9.11) 49 Defining the angles y, g so that tan y a r ha and tan 5 a r BE. (9.12) dz 82 equation (9.11) becomes 2 2 2 . . H - Hi - w (r - ri) +-rwz tan y - riwzi tan Yi + rwf tan g - r wr tan g1 . (9.13) i i We now list the system of equations and boundary conditions which will be employed to determine optimal internal flow passages. I. The uniform pressure increase constraint, (9.2), plus the continuity, momemtum, and energy equation, (8.6), requires that -K 6.. -.. [m-iin-Wi-wsz-rb] T7“ [TEX-J = ‘ h ’ =L(r.e.2). (9-14) 1 i L 1 where L(r,e,z) is a monotonic increasing function which has the same value when evaluated between the intake and discharge points of each pathline. When the 9free-vortex intake condition", (9.1), is satisfied, the intake pressure, p1, and discharge pressure, pd, are uniform over their reSpective cross sections. Two Special cases of (9.14) are listed below. A. "Maximum Kinetic Energy Increase" Constraint 1) W2 - W3 - (1)2(r2 - ri) , (9.15s) 2) L§fi=1=h=1 and Lal. (9.156) "1 i 91 hi 3. "Maximum Speed Range" Constraint r W dA 1) —-1,—=1,——=1 (9.16a) ‘1 W1 “1 2) gfi=1=L=1 and L=1 (9.166) i i 1”i hi II. Boundary Conditions A. The initial or intake conditions, (pi’ W1, hi’ etc.) are known for each pathline. The intake pressure, pi’ is uniform over the intake section when the "free-vortex intake condition", (9.1), is satisfied (i.e., when u) - «'51). B. The discharge pressure, pa, is known for each pathline. It is constrained, by equation (9.14), to be uniform over the discharge section when pi is uniform (i.e., when w = -éi). C. The boundary condition, (4.22b), for the variable end point D. functional is determined in terms of the pressure boundary con- ditions by equation (8.9), i.e., dAi pi 1/K = — —- =-.: . .1 wd w, “a Pd] vs2 (9 7) The pathlines of the flow region mst coincide with the walls of the flow passage. Since the walls of the flow passage are determined from (9.14), this boundary condition is, in general, satisfied throughout the entire flow region only at one set of design conditions, (w, h Wi, etc.). However, when is the maximm speed range constraint, (9.16), is satisfied, this boundary condition is satisfied for all rotor Speeds, w, and intake conditions, (hi’ W , etc.). i When the above equations and boundary conditions are satisfied in regions which exclude conjugate points and shocks, the fluid particle minimum principle, (6.22), is satisfied. That is, 51 conditions (4.A) of section-4 are satisfied for equation (6.22). Shocks must be excluded from the flow region since equation (9.14) does not hold across a shock. The following two con- straints may also be imposed in order to reduce mixing losses after the flow leaves the rotor. III. The "uniform energy increase constraint" requires that H _ 2 2 2) + . . t + - H1 - w (r - ri rwz tan v - rimz1 an v1 rwf tan g - riwf1 tan gi = K(r,e,z) , (9.18) where K(r,e,z) is a monotonic increasing function which has the same value when evaluated between the intake and discharge points of each pathline. IV. The "uniform discharge velocity constraint" requires that 2 2 2 2 2 2 2 Vd 2(Hd - Hi) +Vi - (rd - ri)w +~Wd - Wi (9.19) have a uniform value over the discharge section. ‘Wherev(9.19) was obtained by integrating (6.28) with respect to time from t - i to t 8 d and then substituting h and h 1 d’ which are evaluated from (6.25s), into the resulting equation. 10. A "MAXIMUM KINETIC ENERGY INCREASE" CENTRIFUGAL ROTOR PASSAGE In this section, we seek the geometry of the flow passage of a centrifugal (mix-flow) rotor which will satisfy the "fluid particle minimum principle" and the "maximum kinetic energy in- crease constraint" of section-9. This example is intended to serve only as an academic demonstration of how an optimal rotor passage may be determined. We consider only centrifugal rotors having radial blades in this section. We constrain each pathline to lie on a (r,z)- plane (radial plane) as shown in figure 10.1. All the initial conditions of each pathline inside the rotor, (P1, f1, etc.), must be known. The pressure distribution over the intake section is then determined from the momentum equa- tion, (6.9). It is interesting to observe that when the flow satisfies the "free-vortex intake condition", (9.1), (i.e., when we assume 61 I -w), then the present example reduces to the trivial irrotational case discussed in section-7. That is, when condition (9.1) is satisfied, the pressure distribution is uniform over the intake section. And since the pressure is constrained by (9.15) to be constant along each pathline, the pressure is constant throughout the flow region. The momentum equation, (6.9), then reduces to f I constant, 2 I constant, and é = -w through- out the flow region. And Euler's turbine equation, (6.28), then 52 owmmmmm scuom Homemauuomo Hmawuao c< a.oa Shaman .i. ll ... L QOH UQUOH HO m.“ Kw J I. Innnmoofiasuma 54 reduces to H a 0, which is a trivial case. For the purpose of demonstration, we shall assume that the fluid enters the rotor with zero velocity in the tangential direction, i.e., we assume éi = O. The other intake conditions, (£1, £1, Pi’ etc.), are assumed to be known but will not be assigned specific values. The discharge pressure, pd, is constrained to equal the intake pressure, pi’ (see equation 9.15). Since the pathlines lie in the (r,z)-p1ane, 6 = 6 = o . (10.1) In order to satisfy the "maximum kinetic energy constraint", (9.15), we let N a) W2 - W. = w2Q(z) , (10.2a) 2 b) r -r =Q(z) . (10.2b) H-NH Substituting (10.1) and (10.2) into (9.14) and (9.7) it is easily verified that W dA = 1 , (10.3) widAi and H - Hi 3 w2Q(z) , or (10.4a) = 2 2 1 Hd -Hi wQ(zd) -(1)Q(zi) , (0.4b) where we require that Q(z) be a monotonic increasing function which has the same value when evaluated between the intake and discharge points of each pathline. Equation (10.4b) then satisfies 55 the "uniform energy increase constraint", (9.18). Substituting (10.4b) and (10.2) into the "uniform discharge velocity constraint", (9.19), yields Vi = 2w2[Q(zd) - Q(zi)] +-V: . (10.5) From (10.5) we observe that V3 is uniform when Vi is uniform 2 over the intake section. Expanding V 1 yields 2 .2 .2 2 2 = + . . Vi r1 zi +-ri(n (10 6) or i =W2 - 22 - r2 6.2 . (10.7) i l i i Letting v: = constant, the intake angle, 61’ of each pathline is W2_é2_r2w2 I tan.1 1 3 i i i 1 N1 H- Bi 5 tan- , (10.8) and the "uniform discharge velocity constraint", (9.19), is then satisfied at the design conditions, Gn,‘V , etc.). i We now seek an expression for the change in area, dA, (see equation 2.6b for a definition of dA). We assume that the flow may be represented by a family of pathlines, t, and orthogonal curves, o, as shown in figure 10.2. O = C5 Figure 10.2 Flow Net 56 The equation of the pathlines is determined from (10.2b), i.e., 2 2 w(r,z) = r - Q(z) = r1 = constant . (10.9) The slape of the pathlines is obtained by solving (10.9) for r and differentiating with reSpect to z, i.e., <1: .9. /2 (dzw dz[ ri+Q(Z)] = 1/: dQ/dz =gérgzz . (10.10) /ri +-Q Since the V and m curves are orthogonal, dr = _(d3 = -2r dr V Ql ’ ( d_z.. Q5 (10.11) where the subscripts o and V denote the curve along which the differentiation is performed. The equation of the ¢ curves is found by integrating (10.11), i.e., 2r d r = -‘r -Q-'- Z " ¢ ’ (10.12) where p is the constant of integration. Substituting (10.12) into (10.9) yields 2 V = E+IQEEZ) dz] - Q(z) . (10.13) Differentiating (10.13) with respect to z, and holding m constant yields 9i i ‘ 2r. 22.- (62),, 2L¢+fq,d;] q, 0'. (10.14) Substituting (10.12) into (10.14) yields 2 ($0,) = 33-1.— - Q' . (10.15) The differential change in area, dA, is equal to the differential change in arc length, dsl, of the ¢(r,z) I constant curve times the change in arc length, dsz, in the r-e plane as shown in figure 10.3. O. h “ i=0 ds / 1 Figure 10.3 Area Increment dr 2 dA == dslds2 = iVI + (a?)¢ dz r (11] , or (10.16a) (M 9.22 d_2 37 . _\/1 + (dz)¢ (“)0 r an . (10.166) Substituting (10.15) and (10.11) into the minus value of equation (10.16b) yields 95 _ - i/1;+ 4r2/(Q')2 r dn .. 7—13—71’ 6 ' , (10 17) d1 - [Q' + 4r2/Q'] (Q') 4’ 4r Substituting (10.2b) into (10.17) yields dA 5;-= r dn {[0']2 +-4rf -+-4<2(z>}"i . <10 18> 58 Substituting the derivative of (10.2b) with respect to 2 into (10.17) yields 2 - g? = r an (4.- ($92 + 4.2) ’5 . (10.19) Substituting tan 5 a g; into (10.19) yields dA 2 2 - d 3;- = r 61} [4r (tan B + 1)] 2 2:35 . (10.20) Evaluating (10.20) at the intake point (i) and then dividing it into (10.18) yields dA . 2 - dAi = 2" sec 51 {[Q (2)] + 4r? + 42(2)} J5 , (10.21) where dv and dn cancels with dli and dni reSpectively because they are constants along each pathline. Substituting (10.2a) into (10.3) and solving for dA/dAi yields .dA. . ii; __w_.i__ (10 22) dA w ' ‘ 1 VWi +(u32 Equating (10.22) and (10.21) yields Wi 2r sec Bi Substituting r = Vr +-Q from (10.2b) into (10.23), then squaring both sides and solving for Q' yields 2 ’2 4 SEC 8, 2 3% . __2_1. (if +w2Q)(ri +0) - 41f - 40 . (10.24) W. 1 59 Multiplying (10.24) by 1/ri, and then separating variables and integrating yields z = %-.r 49* + L , (10.258) 1 \LQZ +-bQ +-a 2 2 where; a = 4(sec Bi - 1) = 4 tan Bi , (10.256) 2 b=48ecai 2.122 5‘— 1025 22 mi 11“")‘2’ (-c) ri Wi r. 1 4w sec Bi c - , (10.25d) 2 w2 rii L = constant of integration . (10.25e) Performing the integration of (10.25a) yields z: 1 ing/6624.66... + 676+ b +1. (10.26a) r C JZVC; i where c > 0; or 1 1 M +4, , (10.266) 2 = sinh- ‘ r. ,7: J4... - b2 1 where 4ac - b2 > 0. Only (10.26a) will be considered in detail. Substituting (10.2b) into (10.268) yields Ln \/C(r2-ri)2 + b(r2-ri) + a + (rZ-ri)‘,c- + + L, r1 ° _ 2 (I? (10.27) 28 Evaluating the above equation at (ri,zi) and (rd,zd) respectively yields 60 zi = 1 Ln[\/a-+ b ]+L , and (10.28) ' ri J:. 2 V;— 1 '2 6 zd= Ln[c(Qd) +de+a+Qdc+ ]+t. (10-29) :3_VG? 2\/E where Qd I r: - r: = A%'= constant. Equation (10.29) determines w the end point of each pathline so that the "uniform energy increase constraint", (9.18), is satisfied. The constant of integration, L, is arbitrary and may be set equal to zero. Equation (10.27) determines the equation of each pathline in the r-z plane. Notice that the constants, (a,b,c), vary from pathline to pathline. In order to satisfy boundary condition D of section-9, we must select the hub and shroud profiles of the centrifugal rotor to coincide with equation (10.27), see figure 10.1. Since the constants, b and c, in (10.27) depend on the design conditions, (w and W1), boundary condition D is, in general, satiSfied only at one set of design conditions. This set of design conditions is the only operating point at which the fluid particle minimum principle is satisfied throughout the entire flow region. Boundary condition C of section-9 is satisfied, when the discharge pressure, pd, equals the intake pressure, Pi’ i.e., when pd I pi then I K W dA p d d g widAi [pi] ’ which is required by condition (10.3). In most flow situations the discharge pressure, pd, will equal the pressure of the chamber 61 into which the flow is being emitted. In these situations the pressure of the chamber must be controlled so that it is approx- imately equal to the discharge pressure, pd. See Sharpiro [11] page 91 for a discussion of the effect of back pressure on flow through nozzles. Although it is theoretically possible to satisfy all the constraints mentioned in this section, employment of all these con- straints may result in an impractical rotor. If the above situation arises, the "uniform discharge velocity constraint", (10.8), may be omitted, or the condition that Qd possess exactly the same value when evaluated between the intake and discharge points of each pathline may be relaxed. In conclusion, we observe that the "maximum kinetic energy increase constraint", (9.15), and the "fluid particle minimum principle" are satisfied when the following five conditions are satisfied: 1. The rotor operates at the given design conditions, 0», AH, Bi’ Wi’ etc.). 2. The hub and shroud profiles of the rotor conform to equation (10.27). 3. The discharge pressure, pd, and intake pressure, pi, of each pathline are equal. 4. Shocks are excluded from the flow region. 5. A flow region is selected from the family of pathlines, (10.27),which is free of conjugate points. We also observe that it is possible to satisfy the "uniform energy increase constraint", (9.18), and the "uniform discharge velocity 62 constraint", (9.19). The initial and end points of each pathline are partly determined by the employment of these constraints, (9.18) and (9.19). The flow chart in figure 10.4 outlines the procedure for determining the optimal geometry of the internal flow passage for the centrifugal rotor discussed in this section. Figure 10.1 shows the geometry of the rotor which is determined by the set of design conditions listed below: 5 . 2 2 w = 2000 RAD/SEC, vi = 500 FT/SEC, 91 = 0, AH = 4.16 x 10 FT lssc . The initial points of some representative pathlines are: 1 r1 = 1.00" Bi - 28° 2 r1 = 1.166" 91 - 29° 3 r1 = 1.333" Bi - 30° 4 r1 = 1.50" . 31 - 31° Because the parameters, b and c, depend on the operating conditions, m and W1, the family of pathlines, (10.27), will coincide with the hub and shroud profiles of the flow passage only at the design point, (w,wi). However, if the flow can be controlled so that w - constant W1, then the parameters, b and c, no longer depend on w and W The family of pathlines, (10.27), is then 1. independent of all operating conditions, (p1, T1, W1, w), and the fluid particle minimum principle is satisfied at all operating points , (0.), W1) . C 63 START I READ w, AH, Vi I F——_"C READ ri, 51’ (let L I: 0) i D D initial data for each pathline CALCULATE w? =v’: 1 1 see see see 2 see i 2 see d r2 w2 i equation (10.25b) equation (10.25c) equation (10.25d) equation (10.28) Qd = Ali/w2 equation (10.29) z 8 21 + Az CALCULATE r(z) see equation (10.27) C PRINT r(z), z END Figure 10.4 Flow Chart 3 output used to plot each pathline 11. AXIAL-FILM ROTORS In this section it is shown that it is possible to design "special axial-flow" rotors which satisfy the fluid particle min- imum principle over a wide range of operating conditions. We shall impose the constraint, r/r = 1 , (11.1) on the flows discussed in this section. This constraint requires each pathline to lie on a right cylindrical surface. The Coriolis acceleration is then normal to the flow as shown below 11 A 1. Far 19 Fix 26xfi=2 0 o w =-2rwéir- (ll-2) 0 r26 5 Thus both the Coriolis acceleration and the centrifugal acceleration, finzr it, are normal to the cylindrical surface containing the flow. We shall assume that the inertia forces and pressure gradient are in stable equilibrium in the r-direction. That is, we assume that the radial component of the momentum.equation, (6.9a), is satisfied throughout the flow region. This condition is called the "radial equilibrium condition". The flow in each (re,z)—cylindrical plane can then be treated as a two-dimensional flow which is independent 65 of the radial component of the momentum.equation, (6.9a). The fluid particle minimum principle, (6.22), then reduces to b 2 a a b where W2 = (r6)2 +’é2 and r = constant. The Euler-Lagrange equations corresponding to (11.3) are 9 (11.48) "'3‘. $33; II 0 . (11.4b) N + ‘OIo—I ‘Oh—I °éPé I 0 And the "radial equilibrium condition" is ° 2 13.2 _ + +-— :0 , 11.5 r(e w) p at ( ) Substituting r = constant, & = 6 +’m, and id E i9 into the momentum equation, (6.9), yields (11.4) and (11.5). Thus we con- clude that the Euler-Lagrange equations of the functional (11.3) plus the "radial equilibrium condition" are identical to the Euler- Lagrange equations of the functional (6.22). It is known from Variational Calculus that two functionals are equivalent when their respective Euler-Lagrange equations are identical. The fluid particle minimum principle (11.3) plus the constraint (11.5) is therefore equivalent to the fluid particle minimum principle (6.6). Thus we may operate in the relative reference frame using the fluid particle minimum.principle,.(1l.3), and the constraints, (11.5) and (11.1). Since the inertia forces do not act in the (re,z)cy1indrical plane (they act normal to the plane), the flow may be irrotational in each (r9,z)-p1ane. That is, the flow is 66 irrotational when 11 ’i la r 1' e r 2 . 2. VXfi: 5. a.— a— cli [&-M]BO; (11.6) ar 89 52 1’ 1‘ 89 52 0 r20 2 or the flow is irrotational in each (re,z)-plane when 39 dz . (11.7) However, the flow is still rotational as viewed from the fixed reference frame. When the flow is irrotational, the fluid particle minimum principle, (11.3), reduces to the irrotational flow (strong minimum) problem (4.B) discussed in sections 4 and S. In this case, it is possible to satisfy the fluid particle minimum principle over a wide range of operating conditions. The traditional method of designing blades for axial-flow rotors is to first assume that the flow is two-dimensional on each (re,z)-cylindrica1 surface. Next, the "radial equilibrium condition" is satisfied. Then the classical incompressible potential flow theory (or some other method) is employed to map the flow through a cascade of blades, see Vavra [13] page 312. Thus we can conclude that the assumptions employed in the tradi- tional method are often equivalent to the constraints employed in the optimization procedure. 12. CONCLUDING REMARKS The present optimization procedure predicts the "minimal energy configuration" of the flow field, inside a rotating passage, only when the fluid particle minimum principle is satisfied. When the optimization procedure is employed, the intake conditions of the fluid, inside the rotor, must be accurately known. The intake conditions may be difficult to determine in practice. The present work does not discuss how the intake conditions may be determined. In order to design a practical rotor, the present optimiza- tion procedure as demonstrated in section-10, must be employed in conjunction with "other analytical methods". The author suggests the following iteration procedure for incorporating the present optimization procedure into a design program. The design conditions (W1, w, r1, etc.) may be treated as unknown parameters each of which is restricted to lie within a specified range. Employment of the optimization procedure then yields the equation of a family of path- lines, which depends on the parameters, (W1, m, r1, etc.). For example, the constants, (AH, w, V1, Bi’ r1), in the problem dis- cussed in section-10 could have been treated as unknown parameters. Then, the design problem is to determine a passage geometry which coincides with one of the set of pathlines determined by the optimization procedure and which also appears to be a reasonable geometry based on the "other analytical methods" (i.e., based on 67 68 the boundary-layer analysis, off-design analysis, etc.). This would involve an iteration procedure in which the "other analytical methods" are employed for each set of parameters, (AH, w, V1, 31’ r1)° This work, in general, agrees with the well known equations and assumptions traditionally employed in turbomachinery design work. However, we will now discuss a few points in the present work which deviate from the traditional procedures. Often in turbomachinery design procedures, the one-dimensional compressible flow theory, as described in reference [11], is employed to investigate the flow inside the rotor. As mentioned in section-5, the one-dimensional theory becomes increasingly inaccurate as the curvature of the streamlines inerease. Since the absolute stream- lines inside a rotor are usually curved lines, this procedure may yield inaccurate results. A more accurate procedure is to employ the three-dimensional equations of section-8 in the investigation of the flow inside the rotor. The employment of "optimal constraints" in the present work also deviates from the traditional procedures employed in turbomachinery design work. Any constraint may be imposed on the flow as long as the fluid particle minimum principle is satisfied. Thus if we impose the constraint, that the flow is two-dimensional, and then discover that the fluid particle minimum principle is not satisfied, we must conclude that the flow will not be two- dimensional. That is, we can only force (or constrain) a flow to be two-dimensiona1.when the fluid particle minimum principle is satisfied. The traditional methods employed in turbomachinery 69 work often contain simplifying assumptions, such as the assump- tion, that the flow is approximately two-dimensional. In the present optimization procedure the simplifying assumptions are often replaced by the "optimal constraints". It was shown in sections 4 and 5 that all (irrotational) potential flow problems satisfy the fluid particle minimum prin- ciple. However, for the case of flow inside a rotating passage, the fluid particle mininum principle is seldom satisfied. In general, the flow inside a (rigid geometry) rotating passage will satisfy the fluid particle minimum principle at no more than one set of design conditions. The "maximum speed range" constraint, (9.16), and the special-rotors discussed in sections 10 and 11 represent cases in which it is, theoretically, possible to satisfy the fluid particle minimum principle over a wide range of operat- ing conditions. APPENDIX-A In this section it will be shown that the energy equation, (4.17b), is equivalent to the First Law of Thermodynamics for the case of steady isentropic flow. Consider a fluid particle, i.e., an infinitesimal closed system, moving at velocity, V E E , along a pathline. The First Law of Thermodynamics for a moving closed system and for steady isentrOpic conditions is * _ 91.1. EUR-E.) _dG sis 0-dt+dt +dt +dt ’ (A.1) where: U = mu internal energy of system, 1 2 K.E. = m E'V kinetic energy of system, * G = mG potential energy of system, w = work injected into system, m - mass of system. Substituting the definition of the mass of the fluid particle, m E I pdv , (A.2) into(A.l)yields dw d V2 =— — . A.3 0 dt+_dtvp[u+2 +GJdV ( ) The thermodynamic reversible compression work for a closed system is [7] w =f pdV . (A.4) v 70 71 Substituting (A.4) into (A.3) and rearranging terms yields 0=9— £+u+fi+c 0v (A5) at; p 2 p ‘ ' Since the size of the volume, V, is arbitrary, the integrand of (A.5) must vanish everywhere in the flow region. Observing that the element of mass, pdv, is a constant, (A.5) becomes 2 d_ 2 E. dt[p+u+2 +6] 0. (A.6) Multiplying (A.6) by dt and integrating with reapect to time yields 2 v_2 p+u+2 +G=H, (A-7a) or fl 2 CPT + $—%— + G B H = constant along each pathline , (A.7b) where CpT E h E u +“E and V E H. The constant of integration, H, is a constant along each pathline because equation (A.l) applies to one fluid particle which is traveling along one pathline. Equation (A.7b) is identical to the energy equation,(4.17b). APPENDIX B In this section a brief review of a "field of a functional" is presented. All the definitions and theorems listed in this section are taken from reference [2] chapters 5 and 6. Consider a system of second order differential equations (such as the Euler- Lagrange equations) 3’1 = fiiyiu). y2(t). y3(t)]. (1 = 1.2.3) . (3.1) In order to single out a definite solution of this system, we have to specify six boundary conditions of the form 511 = witylm. yzm. yam]. (1 -- 1.2.3) (3.2) for two values of time, t. The family of boundary conditions, (3.2), is called a field (of directions) for the given system (3.1) when equation (3.2) holds at all values of time, t. A necessary condition for (8.2) to be a field of a functional is that (3.2) must first be a field of the system of the Euler- Lagrange equations of the functional. TheoremrB-l. A necessary and sufficient condition for the family of directions, (3.2), to be a field of the functional, t2 . [1 PG. §>dt . (3.3) 72 73 is that the self-adjointness conditions (this is the irrotational condition in our application), 0P 0P = __l 8 __E (1 1,2,3) ’ (B 4) ayk ayi (k = 1,2,3) and the consistency conditions (this is the momentum or Euler- Lagrange equation in our application), P. i.e.: 35—. (1 = 1,2,3) , (8.5) at ay1 be satisfied at every point t in [t1,t2], where P1 is the "momenta" (it is the velocity in our application) defined as Pi = F§. , (8.6) l and H is the Hamiltonian function (it is the total enthalpy plus a constant in our application) is defined as 3 H = 2 P 9 - Fi-constant . (3.7) 1 i i=1 TheoremrB-Z. The expression 3P 3P ' = ...i -...E (1 1’2'3> (3.8) Byk 5V1 (k - 1,2,3) (this is the vorticity in our application) has a constant value along each extremal (i.e., along each pathline). The self-adjointness condition (or irrotational condition), (3.4), implies that there exists a potential function (a velocity potential function in our application), g, such that 5.8—3? . (3.9) ayi i 74 TheoremrB-3. The boundary conditions (3.2) defined by (3.9) are a field of the Euler-Lagrange equations if and only if the potential function, g, satisfies the Hamilton-Jacobi equation (this is the energy equation in our application) g§-+'H(§, Vg) = 0 . (3.10) We observe that the Hamilton-Jacobi equation, (3.10), and the self-adjointness conditions, (3.4), (i.e., the energy equation and irrotational condition) require that the integrand, th, have an exact differential, dg. That is, since at ayi i ( ) then dCB-dt'i' Pd III‘afidt'l' a's—d =d . .12 F a z iyi at 15153'1 yi s (B ) The above equation forms the basis for Hilbert's Invariant Theorem which is formally stated in the below manner. TheoremrB-4. Given a field of directions (3.2) of the Euler-Lagrange equation, the directions 08.2) define a field of the functional J‘tl F dt (13.13) if the Hilbert integral 3 3 £[ i=1 1 Y1 1-1 3’1 1 depends only on the end points of the curve along which it is taken and not on the curve itself. If the curve c along which 75 the integral, (3.14), is evaluated is one of the extremals (path- lines) of the field, then dy = #1 dt (B.15) 1 along c, and hence (3.14) reduces to 1‘ F dt . (3.16) c When the conditions (3.10) and (3.4) hold (i.e., when the flow is irrotational and the energy equation holds), then (3.12) may be substituted into (3.16) which yields In? t dt = g2 - g1 , (3.17) D.- Ith=£ c which is independent of the path of integration. When the potential function, g, is known, it may be used as the boundary conditions of the variable end point functional, (3.1). That is, equating (3.9) and (3.6) yields g = F. , (Bel-8) Y1 3'1 which when substituted into the boundary conditions, (3.3), yields a—-’ E - = 0 a‘-' E - = 0 or E]. = E2 = g 0 (3°19) at all values of time, t. 10. 11. 12. 13. REFERENCES Denn, M.M., Optimization by Variational Methods. McGraw- Hill Co., Inc., New‘York, 1969. Gelfand, I.M. and Fomin, S.V., Calculus of Variations. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963. Goodman, L.E. and Warner, W.H., Dynamics. Wadsworth Publishing Co., Inc., Belmont, California, 1963. Hansen, A.G., Fluid Mechanics. John Wiley and Sons, Inc., New York, 1967, pg. 358. Johnsen, I.A. and Bullock, R.O., Aerodynamic Design of Axial-Flow Compressors. U.S. Government Printing Office, Washington, D.C. , 1956. Krzywoblocki, M.Z.v., On the Variational Principles in Fluid Dynamics. 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