AN ANALYTECAL AND EXPERIMENTAL STUDY OF HEAT AND MOMENTUM TRANSFER {N WRSULEN‘F $EFARATED FLOW PAS? A RECYANGULAR CA‘V’ETY Thesis for the Degree a? pk. D. MICHIGAN STATE UNWERSETY Rona-Ed L. Haugen 1966 "1P 1 Q‘ 2' ‘ “55.5 __ LIB RA R Q5, , \ :5 :‘vfichigan in C L .'ILIIIHHUMMEHUWM“"WWI ; ’1 ”mm“? ABSTRACT AN ANALYTICAL AND EXPERIMENTAL STUDY OF HEAT AND MOMENTUM TRANSFER IN TURBULENT SEPARATED FLOW PAST A RECTANGULAR CAVITY by Ronald L. Haugen The study given here presents results of an analytical and experimental investigation aimed at describing the turbulent heat and momentum transfer mechanism in the separated flow region of a trans- verse rectangular cavity facing an oncoming boundary layer. A flow model of the mixing region in the slot postulated on the basis of eddy diffusion gives values of velocity, temperature, drag, and heat transfer in good agreement with experimental measurements. In each case, experiments were conducted with air at Reynolds num- bers up to 1.3 x 106, cavity height to width ratios from O. 2 to 4. 5, and with aspect ratios exceeding 10. The results further point to the sig— nificant effects exerted by the oncoming boundary layer on transfer rates from the slot. It was found that the average Stanton number is represented by the semi -theoretical correlation equation St = 0.0365 (6/b)-0'1367i 15% for 6/b (ratio of boundary layer thickness to slot width) ranging from 0.1 to 0.8. AN ANALYTICAL AND EXPERIMENTAL STUDY OF HEAT AND MOMENTUM TRANSFER IN TURBULENT SEPARATED FLOW PAST A RECTANGULAR CAVITY BY ._ c Ronald L“. Haugen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1966 ACKNOWLEDGMENTS The author wishes to express his sincere appreciation and thanks to his major professor, Dr. Amritlal M. Dhanak, for his generous help and encouragement which made this thesis possible. Special thanks are also extended to Dr. Joachim E. Lay (Mechanical Engineering), Dr. Edward A. Nordhaus (Mathematics), and Dr. Terry Triffet (Metalurgy, Mechanics, and Material Science) for serving on the guidance committee. Finally, the author acknowledges that this thesis, indeed his entire graduate program, would not have been possible without the patience, understanding, and consistent encouragement of his wife, Mary. ii TAB LE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . ii LIST OF FIGURES . . . . . . . . . . . . . . . . . iv NOMENCLATURE................. Vii 1. INTRODUCTION . . . . . . . . . . . . . . . . l 2. EXPERIMENTAL STUDIES . . . . . . . . . . . . 6 2.1 Description of Test Equipment . . . . . . . . 6 2. 2 Instrumentation and Measurements . . . . . . 11 3. ANALYTICAL STUDIES . . . . . . . . . . . . . 19 3.1 Analytical Flow Model . . . . . . . . . . . 19 3.2 Basic Equations . . . . . . . . . . . 20 3. 3 Solution of Momentum Equation . . . . . . . 21 3. 4 Solution of Energy Equation . . . . . . . . . 33 4. ANALYTICAL AND EXPERIMENTAL RESULTS . . . . 41 4. 1 Flow Patterns . . . . . . . . . . . . . . 41 4.2 Pressure . . . . . . . . . . . . . . . . 41 4.3 Velocity. . . . . . . . . . . . ». . . . . 48 4.4 Turbulence . . . . . . . . . . . . . . . 52 4.5 Temperature . . . . . . . . . . . . . . . 55 4.6 Heat Transfer . . . . . . . . . . . . . . 62 5. CONCLUSIONS . . . . . . . . . . . . . . . . . 72 REFERENCES . . . . . . . . . . . . . . . . . . 74 APPENDIX . . . . . . . . . . . . . . . . . . . . 76 iii Figure 10. ll. 12. l3. l4. 15. LIST OF FIGURES Physical Flow Model Test Equipment . Flow Channel and Traversing Mechanism . Schematic Diagram of Test Rig Used for Measuring Heat Transfer Rates . Test Equipment . Page 12 13 Typical Calibration Curve for the Hot -wire Anemometer 16 Water Flow Model . A Typical Distribution of the Quantities J u"2 lil- and u‘v' / £2 (h/b = 1.0) Constants k , k , andk of Equation (3.22) l 2 3 Constants C1 and C2 of Equation (3. 29) Velocity Distribution for x/b = . 15 . Velocity Distribution for x/b = . 50 . Velocity Distribution for x/b = . 85 . Typical Distribution of Em for x/b = 0. 50 . Factor K of Equation (3. 38) iv 18 24 27 29 30 31 32 38 39 Figure Page 16. Pictures of Flow Patterns in Cavity with . . . 9 . 42 (a) h/b = 1.0 (b) h/b = 1.5 (C) h/b = 1.75 17. Pictures of Flow Patterns in Cavity with . . . . 43 (a) h/b = 2.0 (b) h/b = 2.5 (c) h/b = 3.0 18. Measured Pressure Distributions Along Cavity Wall Showing Influences of Height to Width Ratio . . . 45 19. Typical Longitudinal Pressure Distribution in the Shear Layer (at y = 0) . . . . . . . . 46 20. Predicted Effect of the Oncoming Boundary Layer Thickness of the Turbulent Shear Stress and Com- parison with Experiment . . . . . . . . 47 21. Comparison of Experimental and Analytical Velocity Profiles (x/b = 0.15) . . . . . . . . . 49 22. Comparison of Experimental and Analytical Velocity Profiles(s/b = 0.50) . . . . . .. . . . 50 23. Comparison of Experimental and Analytical Velocity Profiles (s/b = 0.85) . . . . . . . . . 51 24. Plot of Cavity Streamlines Calculated from the Analysis (h/b = 1/0, 6/b = 0.1) . . . . . 53 25. Turbulence Intensity and Shear Distributions (h/b=1.0)............ 54 26. Temperature Profiles for h/b = 1.0 and 6/b = .28 56 27. Temperature Profiles for h/b = 1.0 and 6/b = . 58 57 28. Temperature Profiles for h/b = 1. 5 and 6/b = . 56 58 29. Temperature Profiles for h/b = l. 5 and 6/b = . 79 59 .Figure Page 30. Temperature Profiles for h/b = 2.0 and 6/b = .38 60 31. Temperature Profiles for h/b = 2. O and 6/b = . 76 61 32_. Predicted Cavity Bulk Temperature and Comparison with Experimental Data . . . . . . . . 63 33. Predicted and Measured Temperature Profiles (h/b =0.5, 6/b = 0.21) ‘. . . . . .. . 64 34. Predicted and Measured Temperature Profiles , (h/b = 1.0, 6/b = 0.28) . . . . . . . 65 35. Predicted and Measured Temperature Profiles (h/b =l.0, 6/b = 0.58) . . . . . . . 66 36. Predicted and Measured Temperature Profiles (h/b 2 1.0, 6/b = 0.76) . . . . . . . 67 37. Local Cavity Heat Transfer Rates (h/b = 1.0) . 69 38. Average Heat Transfer from a Heated Cavity (Data Based on Average Heat Transfer Rates) . 70 39. Average Heat Transfer from a Heated Cavity (Data Based on Local Heat Transfer Rates). . I 71 vi rJ‘l .OI St H l ‘3 I NOMENCLAT URE constant defined in equation (3. 33) cavity width, ft. specific heat, BTU/lbf. oF pressure coefficient cavity depth, ft. average film coefficient, BTU/hr). ft. 2 0F. thermal conductivity, BTU/hr. ft. OF . 2 static pressure, lbf./ft . average heat transfer, BTU/ft. 2 hr. Stanton number . - ' 0 fluctuating component of temperature, F . 0 time ~mean temperature , F x-component of time mean velocity, ft. /sec. fluctuating x-componeint of velocity, ft. /sec. y-component of time mean velocity, ft. /sec. fluctuating y-component of velocity, ft. /‘sec. vii coordinate along dividing streamline, ft. coordinate normal to dividing streamline, ft. thickness of boundary layer at the leading edge, ft. . . . 2 eddy difquIVIty of heat, It /hr . eddy diffusivity of momentum, ftzhr, transformed coordinate transformed coordinate stream function . . 3 fluid denSIty, 1b /ft , m 2 shear stress, lbf/ft . SUBSCRIPTS wall conditions average inviscid core conditions free stream condition viii 1. INTRODUCTION Separated flows occur, for example, where indentations or proturbances are present on flow surfaces. In view of the impor- tance of such related flows to the design of aerodynamic surfaces, considerable attention has been drawn to the mechanism of flow separation. The boundary layer separation is characterized by the formation of reverse flows and vortices. Fox (1) showed that a shear layer like that of a free jet boundary forms over the cavity and borders the external flow. It widens from the front to the back of the cavity and part of this free shear layer is deflected into the notch at the back edge, giving rise to the flow within the cavity. In particular, the boundary layer (laminar or turbulent) separating at the leading edge of a cavity subsequently reattaches itself either at the recom- pression corner or at the base. This, as discovered by Charwat (2 ), depends largely upon the depth to width ratio, the oncoming boundary layer thickness, and the relative heights of the forward and rear steps. In the case of a fairly narrow slot, evidence suggests that the boundary layer "bridges" the notch. Charwat's data reveal that the oncoming boundary layer thickness and the free stream Mach number are im- portant parameters, primarily in the case of the cavity closure. The pressure and thermal measurements presented by Charwat were for l the bottom of rectangular cavities with height to length ratios (h/b) smaller than four. Pressures in the turbulent subsonic flow were shown to rise above the free stream pressure in the downstream end of the cavity and to be little affected in the upstream end. Larson (4) measured the average heat transfer coefficients for both laminar and turbulent flow in cavities substantially rectangular in shape and having height to length ratios of 0. 208 and less. For laminar flow, the average heat transfer coefficient was found to be about 56% of the average coefficient that was measured on equivalent models with a straight heated portion replacing the cavity. For turbulent flow, the average coefficient was proportional to (poo um)o° 6. However, no simple ratio (as the 56%.for laminar flows) exists between this flow and that of the straight heated portion. Some additional quantitative results are reported in papers by Wieghardt (5) and by Tillman (6 ) . Their studies of flow past cavities consisted of measuring the overall drag coefficients, these being obtained by subtracting the drag values with and without a sur- face cutout. Flow measurements in a rectangular cutout were also made by Roshko (7) . His results consist of pressure, velocity, and skin friction measurements on the cavity walls with emphasis on the effects of varying the cavity depth-breadth ratio. Roshko's work, completely experimental in nature, gives rise to a number of tentative conclusions regarding the prominent aspects of the flow in slots. Most importantly: l. The pressures and forces due to the flow may be ex- pected to depend on the state of the boundary layer ahead of the cavity. 2. The drag increment due to a cavity is almost entirely accounted for by the pressures on the walls. Tani (8) also presents the results of an experimental investigation of flow separation related to a step or a groove. As Roshko, Tani found that the shear stress developed in the mixing or shear zone is balanced, primarily by pressure forces exerted on the solid surface of the slot. Additionally, the base pressure is essen- tially the same for different values of step height and boundarylayer thickness and the pressure rise by flow reattachment increases as the step height is increased or the boundary layer thickness is reduced. It is Tani's viewpoint that the turbulent shear stress is necessarily set up in the mixing region in such a way that the forces acting on the fluid form a system of equilibrium, and that the most essential and intriquing part of the problem is concerned with the mixing process between the dissipative cavity flow and the non-dissipative main flow. Seban, §_t_3_1. (9), Seban and Fox (10), and Fox (1) pre- sented measurements of surface pressure, recovery factors, and heat transfer coefficients on the bottom of two rectangular notches. Here, the heat transfer coefficient was proportional to (poo um) 0' 8, with the coefficient being greatest at the rear or downstream portion of the cavity. Fox (1) made additional measurements for a number of narrow notches having height to length ratios (h/b) from 4 to 4/7 and again found the heat transfer coefficient to be proportional to (p00 uoo) 0' 8. Turbulent flow was ascertained in the notch region adjacent to the subsonic free stream, and vortex flow was observed within the cavity. Concerning the analytical study of flow in a cavity, a number of distinctly different models are advanced. Chapman's model (11) treats the cavity as an isothermal low-velocity “dead-air" sink to which heat and momentum is trans fered through the shear layer. In essence, it is a solution for an infinitely thick laminar shear layer with the boundary conditions at infinity represented by the conditions at the base of the slot. The shear layer is postulated to be the sole transfer—rate controlling mechanism disregarding the resistance between the inner wall and the fluid cavity. For laminar flow, these results compare very favorably with the experimental results of Larson (4). Charwat (3) proposes a pulsating shear layer type model in which the fluid moves periodically in and out of the cavity, thus governing a transport of heat and momentum. Charwat's model depends on the unsteadiness of the flow at the exclusion of the other mechanisms. The theory predicts the 0. 6 power dependence of the heat transfer coefficient upon the mass velocity poo u00 that was measured by Larson in turbulent flow, and the corresponding 0. 5 power in laminar flow. In this model, the cavity shape enters but incidentally with an allusion to a vortex located in the downstream corner. Burggraf (12) however takes the cavity shape fully into account. In his model, Batchelor's (15) proposal of constant vorticity is assumed within the cavity. The vortex flow within the cavity is then coupled to the (laminar) external flow. However, for turbulent flow the velocity distribution along the separating streamline must be previously found by some other theory before this model can be utilized. None of these analyses is adequate to explain the results in turbulent flow, particularly when the external flow is boundary layer in nature. Thus, the present analytical and experimental study was undertaken and is aimed at delineating the turbulent momentum and heat transfer mechanism in the mixing zone of a cavity. A flow model of this zone is postulated on the basis of the well known concepts of mixing length and eddy diffusion yielding a rather satisfactory corre- lation with the experimental data. 2.. EXPERIMENTAL STUDIES 2.1 Description of Test Equipment A schematic diagram and photograph of the test rig employed; for these tests are presented in Figures 1 and 2 respectively. The unit consists of a fan driven by a 3 horse-power variable speed motor, adjustable length flow channel, and a rectangular cavity. The channel was 2. 5 in. wide and had an aspect ratio of 10 assuring substantially two-dimensional flow. The cavity width was varied from 1. O to 2. 5 inches and its depth was varied in 0. 5 in. increments up to 4'. 5 inches. The flow channel (Figure 3) was varied in length from 13 to. 29 inches. thus assuring a wide range of boundary layer thicknesses. The flow channel was connected to the fan by a second rectangular channel 3. 5 in. wide x 21 in. high. Flow straighteners were placed within this ductyOork approximately 85 inches upstream from the flow channel. The straighteners consisted of 3 inch long sections of 24 ga. sheet metal formed into a mesh having 1/2 in. squares. Turbulence screens were placed immediately upstream from the flow channel. These screens, which consisted of wire in various diameters and. mesh sizes, effectively controlled the scale and intensity of the free stream turbulence . FREE PLANE JET (0.) ZONE I EDGES OF NIXING ZONE 2 X b CAVITY (b) FIGURE I PHYSICAL FLOW MODEL mam umwe .N musmfim - “—4.:- "" "" — .5 ~"""-. ~- _ Figure 3. Flow Channel and Traversing Mechanism 10 The cavity was heated by means of electric Calrod heaters placed within the copper cavity walls . The quantity of heat required to keep the walls at constant temperature was measured by means of a watt meter (Model 432 by Weston; Electrical Instrument Corporation), placed in the circuit of the heating rods. The heat losses were measured by two separate methods. First, thermo- couples were buried within the cavity walls and the surrounding insulation. The heat losses were then estimated from local conduc- tion losses. Radiation heat transfer was neglected due to the small temperature differences associated with the various tests. Secondly, the heat losses were measured directly for various wall temperatures using the wattmeter while the free stream velocity was held at zero. An estimate of the free convection losses within the cavity show them to be quite small (~' 5%) compared to the total heat loss. The two methods compare quite closely and remained within 8% for all tests. Local heat transfer rates were also measured. To do this, a. second cavity was employed (Figure 5) and which had 36 evenly spaced 0.005 x l/8 inch series-connected nichrome ribbons. These ribbons were then attached over a 2 inch thick phenolic cavity. The heat rate was deduced from the measured power which was used to heat the cavity walls while the local temperature was measured with thermocouples attached to the underside of each ribbon. Further, the error in the temperature measurements was estimated to be within 0.4%. 11 2.2 Instrumentation and Measurements Shown in Figures 2 and 5 are the various instruments used in this study. The probes used for sensing velocity, turbulence, and temperatures were placed on a traverse mechanism that could be moved longitudinally, parallel to the mean flow, and transversely across the mixing region. A micrometer, accurate to :l: O. OOTI inches, was used for actuating the movement and measuring the transverse probe positions. The longitudinal probe positions were controlled by locating blocks as shown in Figure 3. The temporal -mean velocity and the turbulent intensities were measured by means of a constant-current hot -wire anemometer (Model HWB ser. no. 216 by Flow Corporation, Arlington, Mass. ) , in accordance with the Flow Corporation hot -wire manual. In measuring the turbulent shear, a single hot -wire probe was placed in two angular positions for each probe location. The tur- bulent shear was then calculated as described by Hinze (13) . Figure 2 shows the various instruments used in conjunction with the hot -wire anemometer. The signal from the hot -wire anemome'ter amplifier was fed through a 7KC low pass filter to a true-root-mean- square voltmeter (Model No. 320 by Ballantine Laboratories, Boonton, N. J. ), using a response time of 2.5 seconds. I. '3 THERMOCOUPLE LOCATIONS . f . . 17 . 0 &\ Q I) . \.\ '\ T \\ Q CALROD GP; HEATERS A. SCHEMATIC OF CAVITY USED FOR MEASURING AVERAGE HEAT TRANSFER RATES ' ° J I I l ' ' ,, I I . A I ]A I L 1’ I ___________ I __ I I A 0 SECTION AA 36 (0.005 x we") NICHROME RIBBONS B. SCHEMATIC OF CAVITY USED FOR MEASURING LOCAL HEAT TRANSFER RATES FIGURE 4 13 3m 33. .m wusmwm 14 The general procedure used was first to balance the bridge cold, using the bridge null variable resistor. The value of the bridge balance was recorded and the current turned on. Again the bridge was balanced, and the value of the bridge balance was recorded. The wire current "I" was measured by means of the galvanometer and meter balance resistor and recorded in units of four times the current in milliamperes. The square wave was then turned on and the compensation frequency adjusted until the oscilloscope pattern was a perfect square wave. The square-wave and wire current were then switched off and the r. m. s. voltage measured. The r. m. 3. value corresponding to this reading was Mn, which is a measure of the noise level. The current was then turned on again and the r. m. 5. value again recorded using the designation Mn + v. The square wave was turned on again and the r. m. 5. reading repeated thereby ob- taining Mn + v + 5. With this procedure and the appropriate equation suggested by the Flow Corporation (1 6), the turbulent measurements were repeated for each location of the hot-wire probe. The resis- tance ratio was kept constant at l. 6 throughout the tests. The hot-wire anemometer was also used for measuring the temporal-mean velocity. The 0.00035 in. diameter tungsten hot- wire with 1/8 inch diameter probe stem was calibrated against a Prandtl pitot tube in a free stream directed normally to the hot wire 2 12 and probe stem. For each filament used, a calibration curve I vs U/ 15 was obtained. To obtain a velocity profile, the hot -wire was located at a number of positions and the filament currents recorded. From these known values of current at various locations the corresponding velocities were obtained through the most recent calibration curve for the particular filament in use. A typical calibration curve for the hot -wire anemometer probe is shown in Figure 6. The air stream temperature was sensed by means of a copper -constantan (24 B ¢ S gauge) thermocouple located on the traverse mechanism. The EMF from the thermocouple was measured with a millivolt potentiometer (No. 8686 by Leeds and Northrup Corp. of Philadelphia, Pa.) with an expected error of t 0.05% . The milli- volts measured were then converted to temperature by means of tables available with the instrument. The thermocouple was first calibrated with an accurate mercury in glass thermometer. The static pressure was sensed by static holes, 0.030 inches in diameter, drilled through the cavity walls, and was measured by means of a micromanometer (Model MM3 by Flow Corporation) which had n -butyl alcohol as its manometer fluid and a nominal accuracy of :I: 0. 0002 inches of manometer fluid. Since this manometer had a slow response, only temporal mean values of total head could be measured. Additionally. static pressure measurements were made within the shear layer (y = 0).. In view of the uncertainty in the direction of flow within the shear layer, the accuracy of the static I4 I2 2 4 6 8 IO I2 I/2 I/2 U , fps FIGURE 6 TYPICAL CALIBRATION CURVE FOR THE HOT-WIRE ANEMOMETER 1? pressure distribution at y =0 is questionable. But, since the flow is essentially two dimensional, the static pressure probe, held normally to the xy plane, is expected to yield reasonably accurate results. In an attempt to balance any flow component present which was normal to the xy plane, pressure distributions were taken with the static pressure tap held in both directions of the xy planes normal. The differences in pressure readings were negligible in this case and the associated measurements were considered satisfactory. For flow visualization studies, it was found convenient to construct another cavity model. The model (shown in Figure 7) ’Was subjected to flow of water approximately simulating the dynamic con- ditions ,in terms of the flow Reynolds number and the relative boundary layer thickness (6/b) . The streamlines Were made visible by strew- ing aluminum powder over the water surface illuminated by light. Figures 16 and l7,represent the photographs of the flow patterns. Hove: 33m .333 .5 spam: ..’;-<4f‘*-"‘°‘ ' ‘ \ “A ._. 3. ANALYTICAL STUDIES 3.1 Analytical Flow Model Many studies have been made of turbulent free plane jets, and in each case the flow was considered to consist of a uniform semi- infinite, plane-parallel jet issuing from a. wall with a velocity of uoo and merging with a motionless fluid (Figure 1).. Under such conditions the oncoming fluid becomes completely turbulent within a very short distance from the wall. Because of this, turbulent mixing occurs between the jet and the surrounding fluid at rest. Particles of fluid from the surroundings are entrained by the jet so that the mass flow increases, while the total momentum remains constant. Further, when dealing with such problems of turbulent jets it is assumed that this mixing is confined to a "shear zone, " the width of this zone being proportional to the longitudinal distance from the point where the jet begins. Outside this zone, the flow is con- sidered to be completely unaffected by the mixing process, while within the zone the flow is considered to be boundary layer in nature. That is, the solution being sought does not extend far in the transverse direction and only the transverse gradient of shear predominates. In our description of the cavity flow model, we have treated the shear l9 20 layer portion of the flow as a two-dimensional plane jet and accord- ingly applied the boundary layer equations of momentum and continuity. The turbulent shear stress and heat transfer terms are approximated by various phenomenological theories, e. g., mixing length, eddy diffusion, etc. With this, satisfactory solutions have been found. 3. 2 Basic Equations The equation of motion for a two dimensional flow can be written as: 2 2 Bu Bu Bu 8P Bu au P (B‘s-i" ua—X+v_3;- - - ax+p( 2+ 2) (3.1) 3v 2 2 8v 8v 8v 8P 3 v 3v P(——+u—+v—)= -—+p(""—+——Z') (3.2) 89 6x 8y 8y 8x 3y Making estimates of the order of magnitude of the various terms, it is found that for a steady state turbulent flow in a boundary layer, Prandtl's approximation of Reynold's equation applies, namely: ‘ —33 —33' dP 31- —+ — = -—+— 3.3 p(u3X v y) dx 8y I I p and 9— : o (3.4) 3y where T = NEE -p u'v' (3.5) 8y and where the bars over the symbols indicate temporal mean values. 21 The equation of energy can be written as: Z 2 pc (gmggwgl) . 41;... 9...} I...) p y ,k 3y which simplifies, with the same approximations as before to read as follows: — 3T — 3T 1 so u 8x v 8y pC By (3.7) P 3T . where Q = k-—- - C v'T' (3.8) 3y p p In addition to the equations of motion and energy, the equation of continuity must apply: 8:39qu + 8(3):” = 0 (3.9) 3. 3 Solution of Momentum Equation According to the experimental evidence (Figure 19) , the pressure was found to remain relatively constant along a substantial portion of the shear layer. Thus, the pressure term in equation (3. 3) is neglected, giving: 2£.+;;§i.=.L.EL (3Jm 3x Y 9 3y ‘3 | Upon assuming constant density, the continuity equation becomes: a; a? __ + —-—— : (3.11) 8x 8y 0 22 Boussinesq (17) suggested the coefficient e (eddy diffu- m _ sivity of momentum) for momentum transfer by turbulence. Using this theory, the turbulent shear stress may be expressed as: ‘3' I 3 -r :96 — (3.12) m < Assuming the turbulent effects are considerably larger than the molecular effects, equation (3.10) becomes: Q30? :45: - — Bu 1 8 Bu u v 3y p 8y (p 8y) (3.13) The overall transfer of momentum from the cavity to the free stream takes place through a mixing region as depicted in Figure l (b). For the purpose of this analysis, the flow region over the cavity was divided into three different zones. In the first zone [Figure l (b)] the flow remains unaffected by the mixing process within the shear layer. Thus, for the case of an oncoming turbulent boundary layer of thickness 6, the flow above the shear zone is assumed to obey the usual 1/7 power law. Within the second zone, the shear stress term is postu- lated to obey the identical relationship as for the free plane jet as discussed by Abramovich (14). The shear zone is postulated to be symmetrical about y = 0 with its growth rate proportional to x. The equation used for the boundaries of this zone was y = i 0. 08829 x. This value was chosen as it corresponds to the line which describes 23 the growth rate of a free plane jet. Also,the plot of u'v'/32 (Figure 8) is seen to justify this choice. Thus: 3 _ _ a 2 Bu Bu where "a" is an empirical constant characterizing the structure of the flow of a jet. Its value for free jets is 0.09 (14) . However, for the shear layer of the cavity a value of 0.12 yields a better correlation with the experimental data. The momentum eddy diffusivity is ex— pressed by: _l 328u Em — 2 a X By (3.15) Then equation (3.13) becomes: _ _ _ 2_ 1-1- Bu + ; Bu _ a3x2 8n 8 u 2 8x 8y 8y 8y (3.16) Choosing the coordinate system (x, n) as in the free jet case (14) where: n ___y_ (3.1?) and letting the stream function ([1 2 an xf(n) (3.18) 00 equation (3.16) becomes f"'+f = 0 (3.19) ‘ 0.I0 \ 0-05 -———P—0r \ 0 A IN.{ ~— L _ _ _ A I h “ ~. -0.05 .— ‘3‘! 0 ”I .- — ‘0. I0 A -0.I5 '0.20 0 0.05 0.I0 0.l5 - "I ”/62 @- = 0.I0 0 b 0.I38 I 0.225 A x —8 0.50 b 0.l2 0.08 K \\ 0.04 \-\c Y . l-o _ \ \ b O kit—k “0.04 ‘1% “ h -0.00 / : for“ '1" ‘— K/ A r L - me i _ 0 0| 0.2 0.3 0.4 0.5 p 0.0 \/u’/fi FIGURE 8 A TYPICAL: DISTRIBUTION OF THZE QUANTITIES u'z/U AND u'v'/ U ( I‘I / b = LG I 25 The solution of equation (3.19) is (14) . _u_ : f'(n) :k e_n+k en/Zcos( u l 2. 3 11/2 3 2 n) +k3e n) sin( 00 (3. 20) To evaluate the three constants k , k , and k 1 2 3 the boundar y conditions utilized were: 1. The velocity is continuous from zone 1 to zone 2. 2. The velocity gradient is continuous from zone 1 to zone 2. 3. The turbulent shear stress has a maximum at y = 0. The latter condition is justified by experimental evidence (see Figure 8 and Reference 1). For a zero oncoming boundary layer thickness, the velo- city in zone 1 is uniform. Thus, at the edge of zones 1 and 2 f‘ = l and f" = O. In this case, equation (3.20) becomes: {fi— = 0.0684e_n+0. 79415en/2cos J: T) +0.28854 en/Zsin [3:77) 00 (3.21) However, for a finite oncoming boundary layer thickness, the zone 1 velocity obeys the usual 1/7 power law. Boundary conditions I and 2 will then vary with x. and the similarity of solutions implied by equation (3.19) no longer exists. 26 Proceeding with these varying boundary conditions, while acknowledging that the assumption of similarity must be violated some- where, we obtained the following approximate but satisfactory solution. W II 0.06840 g(x) + 1.1610 g' (x) 1 k2 = 0. 79415 g(x) - 0.47552g' (x) (3.22) k3 = 0.28854 g(x) + 0.71076 g' (x) Where, g(x) denotes the local velocity ratio at the edge of zones 1 and 2 and g'(x) its gradient. . k , andk must, For similarity to hold, the values of k1 2 3 of course, be constant. The extent of this violation is shown in Figure 9, where these k's are plotted versus x/b. Then, since g(x) and g'(x) come directly from the usual 1/7 th power law, the entire solution may be expressed in terms of a constant term and the Reynolds number which describes the turbulent boundary layer thickness . Finally, for the third zone the flow velocity and shear stress cannot be taken as zero as for the free plane jet. However, experimental evidence indicates that within this third zone the quantity ”1‘; remains approximately uniform (independent of y) . u E?" Thus, ___,2 = constant. u __ . Then, since T~ u'v' (bar indicates time -averaged quantity) we have T~ u / ._ = 0.I 0.6 b 0.5 \\ \I C\ \\I I 0.4 0.3 0.2 _ J I—-—-"'""‘ .3 .—-— _______ .5 3 jQ/fdr/ // -| 0.03 0.I r 3- g #4 5.. 0.02 k. .———-—— .9 7 0.0I 0 0 0 2 0 4 0.6 0 8 I 0 .1‘. b I FIGURE 9 CONSTANTS k,,k2,s k3 0F EQUATION (3.22) 28 , 1 -2 Hence we write T = 7 P ku , where k = constant, and the expression for the eddy diffusivity in this zone is - _1_ .2 3i 6m - 2 ku /(8y) (3.23) Equation (3.13), applied to this zone becOmes: —-a§ -aE — a; — + = k —— u 8x v 3y u 8y (3°24) Again letting W = k umeW) (3.25) where ¢ =—L- we obtain, kx F'(¢)+F()=0 (3.26) with the solution as: -C -4, 27) F() = C16 = Cle (3.27) where as before: n = —L (3.28) The boundary conditions employed here for obtaining Cl and C2 were: 1. The velocity is continuous from zone 2 to zone 3. 2. The velocity gradient is continuous from zone 2 to zone 3. The velocities in this zone are readily calculated from 5 "C2" = F' (4)) = ~C1e (3.29) 00 Typical values for C and C2 are shown in Figure 10. l 0.34 . \ 0.32 0.30 7I / ., / / I...” ° / C =-I.6827 ALL §-\/ 0.26 2 b ' C /// 0.20 0 0.2 0.4 0.6 0.8 I.0 crI" FIGURE I0 CONSTANTS CI AND 0 OF EOUATIONI3.29) 2 UI~< 3O L O/b-OB 0.5 I. _ 0.5\ " 0.3\ I. O I- ' 0.I -05 l L L I 0 0.2 0.4 0.6 0.8 I.0 U / U... FIGURE II . VELOCITY DISTRIBUTION FOR X/b=0.l5 |.0 0.5 crI~< " 0.5 8/b=09 \ . \ 0.I fl \ r- \ _ 1/ r 0 I )- l I _l- l 0 02 04 0.6 08 IO U/ u... FIGURE l2 VELOCITY DISTRIBUTIONS FOR X/b=0.50 UI~< I.0 0.5 ‘ 0.5 sxbeos I 05 HGURE l3 VELOCITY DISTRIBUTIONS FOR X/b=0.85 33 It must be noted that at the edge of zone 2 and 3, the velocity and its gradient are forced to be continuous by choice of an appropriate number of constants, whereas the assumed shear distri- bution [in view of equations (3.15) and (3. 23)] is discontinuous . 3. 4 Solution of Energy Equation Similarly, the energy equation becomes: — 3T — 3T 3 3T __ + __ : —_ e —_ u x v y 8y ( h y) (3. 30) - v'T' d 2 —--=-——— : d ' ' ' . where eh ( BT/ 8y) e y diffuswity of heat . 1 Assuming the turbulent Prandtl number to be constant, B- E l prt : ELI: ___ _ h Then, for the shear zone: 3 2 35 = C —— 3. 31 6h a X By ( ) Equation (3.30) , applied to this zone, becomes — 3T —8T 323— BZ'T 3T 32; u a +v r: Cax LSE— 7+3— 2) (3.32) X Y Y 3Y Y 8y Measured temperature distributions within the cavity (Figures 26-31) show that the temperature gradient is very large across the shear zone. But, outside this zone and within the inviscid 34 core, the temperature remains uniform. For deeper cavities, the temperature increases again for locations deeper than y/b = -1, thus again indicating secondary vortices. Burggraf has shown similar behavior, with a nearly con- stant temperature over the inviscid core. Although his analysis does not consider secondary vortices, his results for predicting this core temperature agree quite well with experiment. _Burggraf gives: T -Too 1 _____° = ____ (3.33) Tw-Too (’(l+-)A h Where, for laminar flow A = 2 and To is the average core tempera- ture. With this temperature, we are now in a position to solve the energy equation ( 3. 32).. T-Tm let 9 = TIT-T?— o co and TI ___ _L . ax assuming 9 ‘—' 9(1)) equation (3. 32) becomes: Cf"9"+C0'f"'+f0'=0 (3.34) But, from the momentum equation we have: f"'+f =0 (3.35) 35 Therefore, Cflll eI+CfII 9H _fIII 9'30 (3.36) _0_'_: _ l—C f'” 3 3 9' C f” I ° 7) Integrating l-C C 9' = KM") (3.38) Where K is an arbitrary constant. Similarly, for the third zone the energy equation becomes: — 3T — 3T 3 3T __ __ z __ E _— u x + v y 8y (hay) (3.39) Once again assuming Eh = C 6m, we obtain for the third zone, 6 _ CkEZ h ‘ 2 _ag (3.40) Y — 8T — 3T 3 CkE aT/ay __._+ _ = -— _ .41 Thus, u x v 8y 8y( 2 8u/8y (3 ) T-T A ain let 9 = ————93— (3 42) g ’ T -T ’ o OD and, 4: = l— (3.43) kx Equation (3.41) reduces to: 2 C d f' 0' - I =_ _ f0 2 d4) f" (3.44) 36 However, from the momentum analysis, we have: f' = ~f (3.45) Thus, (3.44) becomes: C d lel : __ __ IeI f 2 d¢ (f ) (3.46) 12.2 . . C Integrating f' 0' = Boe (3.47) But, from equation (3. 29) f' = Cle‘¢ (3.43) C-2 C Therefore, 0' = Ble (3.49) Integrating, and combining constants yields B317 9(n) = Be +13 (3.50) 2 4 where B2 and B4 are arbitrary constants, while B3 = C C2 = 0.72116 Equation (3.38) was now integrated numerically, giving values for 0( 77) over the shear zone. These values were then used to calculate the constants B2 and B4, with the assumption that 0( n) and its gradient 0'(n) are continuous through zones 2 and 3. The constant K was next readjusted from initial "guesses" so as to 37 satisfy equation (3. 33). Based upon best agreement with experimental results, the constant C was taken to be 1.4. For this value K2: 0. 5 for all x/b and 6/b although it increases slightly with increasing x/b and decreases with increasing O/b. Representative values of K are plotted in Figure 15. The total heat flux was given as: 3T : 6 C -— . q hp pay (3 51) To-Too dB or q = C6 p C m p ax (17') Substituting for E and 9—9- for zone 2 gives: m (11') NC : - ll Cl Cap uOO(T0 Tm)K[f (11)] (3.52) or at n = Owhere kI J3 II : : _ _— f (n 0) 2 k2 + k3 k _ 1 L3 l/C and q — CapCp uoo(TO-TOO)K(Z+ 2 k2+k3) (3.53) Defining an average heat flux as: _ “C x q: 1qd(—)=apCp Cum (To-COT )OK(—+—kz+k3) d(b-) i: b (3. 54) UI~< 0.04 0.0 " 0.04 "’ 0.08 '0.I2 L 8/D=0J ,0.50 09/ )— 1 I A 0.5 I.0 2.0 3.0 6... '0-3 — X 9 90,, FIGURE l4 TYPICAL DISTRIBUTION 0F 6... FOR x I b . 0.50 05| 050 049 048 047 046 045 044 39 6/b=0J O3 0 . .0 «I 0| 0.2 0.4 0.6 0.8 x / b HGUREIS FACTOR IQ OF EQUATION (3.38) 40 The average Stanton number becomes: _ - a _ St - pC u (T -T) - Ca; (3.55) p 00 o oo 1 k where I, = [K(—;—+ 3743- k2+k3)1/Cd()bi) (3-56) 0 It should also be noted that the Stanton number, as de- fined in equation (3.55), is independent of cavity depth and follows the equation: -0.l367 St = 0.0365(-E) (3.57) Ifhowever the heat transfer is desired in terms of the more conventional (Tw-TOO) , equation (3. 33) may be used. Defining _ q( (3.58) Then, _ ‘Caé (3.59) 4. ANALYTICAL AND EXPERIMENTAL RESULTS 4.1 Flow Patterns The flow patterns protrayed in Figures 16 and 1? reveal interesting effects of the height to width ratio of the slot on the structure of the vortices inside the cavity. For h/b = 1, there is a single vortex and it is. stable, resembling almost a solid-body rotation. Around h/b of l. 75 there appear secondary vortices in transition and at a value of h/b = 2 one observes a rather well-defined double -vortex structure (presumably stable) . Transition again seems to take place around h/b = 2. 5 when the number of vortices oscillates between two and three. Finally, for h/b of approximately 3, we see three cells with vortices stacked on top of each other. The vortices were observed in a state of counter-rotation to each other. 4. 2 Pressure Pressure is represented for the notch surfaces as a pressure coefficient C that is defined by P-Poo CP:_1_ uz 2poooo where P is the reference surface pressure which was measured 00 41 42 h/b a LG h,b.I.5 h/bans FIGURE I6 PICTURES OF CAVITY FLOW PATTERNS 43 h/b=25 h/b83.0 FIGURE I7 PICTURES OF CAVITY FLOW PATTERNS 44 0. 25 inches ahead of the cavity. p00 is the density as given for air at the free stream temperature and pressure. Figures 18 and 19 represent the measured pressure dis- tribution along the cavity walls and along the shear layer. As pointed out by Roshko (7) , lower pressures near the center of the walls and bottom and high pressures at the corners are typical of a single, stable vortex. Near the top surface of the downstream edge, the pressure suddenly rises. This rather abrupt change is apparently caused by the boundary layer separating at the front edge and imping- ing on the downstream wall. It is noteworthy that the pressure dis- tribution at the walls is rather sensitive to changes in depth. Also, the tendency to form a single vortex in the cavity with h/b = 1 is seemingly unaffected by the state of the boundary layer ahead of the cavity. However, the pressures and forces due to the flow do indeed depend on the state of the boundary layer. The pressure distribution longitudinally along the shear layer evidently remains uniform except near the recompression corner. As remarked before, use was made of this fact in neglecting the pressure gradient in the momentum equation. Roshko (7) has shown the shear stress acting on the cavity walls to be of two orders of magnitude lower than the pressure coefficients and the pressure- drag coefficient. The difference between the integrated mean-value of C over the upstream and downstream walls, and made dimension- 2 less by the dynamic pressure -2!-p um , is plotted in Figure 20. 0.2 45 UPSTREAM WALL BOTTOM DOWNSTREAM wALL 0.2 .5. a 0.40 b 0.I c" \\ 0 \§ :7 I I I l 1 . UPSTREAM WALL FIGURE MEASURED PRESSURE DISTRIBUTION ALONG CAVITY WALL SHOWING INFLUENCES 0F HEIGHT T0 WIDTH RATIO 8 0TT 0M 0 II/b = 05 I h/b= (.0 A h/b = L5 I8 DOWN STRE AM WALL 46 I ‘ 'I/b = I825 I 0.05 -.... / - 0.I FIGURE I9 TYPICAL LONGITUDINAL PRESSURE DISTRIBUTION IN THE SHEAR LAYER (at y=0) .030 PREDICTED .025 / I .020 ° _. 0 9 o Th0 0 .L 2 0 233 use A A .OI5 K D 4 cI‘o\A “II *3 I‘ .0I0 0 ROSHKO _005 A TANI D PRESENT DATA 0.5gII/bg2 0 0 0. I O. 2 0.3 04 0.5 0. 6 E. b FIGURE 20 PREDICTED EFFECT OF THE ONCOMING BOUNDARY LAYER THICKNESS ON THE TURBULENT SHEAR STRESS AND COMPARISON WITH EXPERIMENT 48 The cavity flow is maintained chiefly by the turbulent shear stress which is set up within the mixing region. Thus, the turbulent shear stress developed in the mixing or shear zone is balanced, primarily by the pressure forces exerted on the cavity walls. The compari- son is made between this measured value and the integrated average shear stress (at the mean dividing streamline, i. e. , y =' 0) from the analysis. It is clear that the influence of the relative oncoming boundary layer thickness (O/b) is quite substantial on the drag due to the slot. The data points are for different h/b values ranging between 0.5 and 2.0. 4. 3 Velocity The velocity data are presented in a form of a ratio of the local velocity to the free—stream velocity measured at mid-channel immediately upstream of the cavity. The ratio is presented as IT/ uoo where no attempt is made to correct for temperature effects, because of the small temperature differences. In Figures 21, 22, and 23 the data from the velocity measurements are compared with the results of the equation: _3_ u (I) = [0.06840 g(x) + 1.1610 g'(x)]e’" +[0.79415 g(x) 047552 g'(x)]cos(_J;_,3—n erI/z +[0.28854 g(x) + 0.71076 g'(x)]sin( 23 n)en/2 (4.1) Ul~< O'l~< l n 0.2 I :- 0.I 0 A—Q—IK - u u... 0 A I as |.O / -0.I0 -o.20 4 0.4 -0.30 0.3 l 5% 20.3 b -0.40 m 0-2 m 0.I -0.50' w % 0 if” U/u... ° (0 E 05 I.0 -0.IO -0.20 -0.30 , -040‘ -0.50 1% 0 0 1:) Y -O.I0 -0.20 -0.30 D h/b = LO 4 O h/b 3’ LS -0. 0 A 11/1: . 2.0 -0.50 FREDICTED FIGURE 2| COMPARISON OF EXPERIMENTAL AND ANALYTICAL VELOCITY PROFILES (XIbBOJSI 5 O 6 5 l . 0.21 — = O.I-‘-\] .9. 0.I ‘3 . V -0.I0 -0.20 0.4 - 0.30 0.3 l .2. = 0.3~ b -0.40 b 0.2 - 050 0.I 05 IO - 1'. -0.I0 0.5 1'. -0.20 0.4 b -030 0.3 -% =0.5- - 0.40 0.2 -050 OJ / 11/11.... 05 I0 0 h/b = LO 0 h/b = Is A Ii/b = 2.0 PREDICTED FIGURE 22 COMPARISON OF EXPERIMENTAL AND ANALYTICAL VELOCITY PROFILES (X/b=0.50) Ul< O’I‘< ‘0.l0 - 0.20 - 0.30 - 0.40 - 0.50 -0.|0 -0.20 - 0.30 - 0.40 - 0.50 - 0.I0 -0.20 - 0.30 -0.40 - 0.50 E. ‘ 1 b 0.I b —u U,” 0.5 I.O / I 0.4 N, 0.3 _Y_ out 2:03 b b 0.2 0.I u/um 0.5 Lo 05 Y 04 g 0.3 g .0. —— 0.2 0.I u/uc, 0.5 (.0 D h/b = LO 0 h/b: IS A h/b =2.0 —— PREDICTED FIGURE 23 COMPARISON OF EXPERIMENTAL AND ANALYTICAL VELOCITY PROFILES (X/b=0.85) 52 This function contains the constant "a" which was chosen to be 0.12. Velocity profiles from the tests, each at three distances from the cavity origin (x/b = .15, . 50, and .85) , are presented as data points in Figures 21, 22, and 23. The solid line represents equation (4.1) . This line is seen to correlate the data reasonably well. Furthermore, streamlines calculated from these equations are shown in Figure 24, and again appear to be reasonable when compared with the observed pattern of Figure 16. 4. 4 Turbulence Figure 25 shows the transverse distributions of the tur- bulent intensity and the shear stress as measured by the hot-wire anemometer. The solid line in Figure 25 (b) represents the theoretical shear stress and is seen to correlate the experimental data adequately. From the figure it is quite apparent that both the local shear stress and the turbulence are maximum at a point coinciding with the dividing streamline (y = 0); and they are quite sensitive to changes in the relative thickness of the approaching boundary layer (6/b) . In Figure 8, transverse distributions of the quantities —— -2 /— 2 — . . - u'v' / u and u' /u are shown. The relative Size of the turbulent boundary layer once more appears to have significant effects on the . —- —2 . . latter quantity whereas the former u'v' /u remains relatively un- . — —2 . affected. It must also be noted that the quantity ' u'v' / u remains 53 i \\\ \\\\\ \\ FIGURE 24 PLOT 0F CAVITY STREAMLINES CALCULATED FROM THE ANALYSIS (h/ b=|.0, 6/1380.” U|~< 0. I2 0.08 0.04 ' 0.04 " 0.08 ‘ 0.I2 Ul~< \\ 0 5‘3. 4 \ T "K A“; _+‘ l / /A I z: 42’ I I 0 0.04 0.06 0.12 0.16 0.20 0.24 u' us €- =0.I0 . =0.138 I 20.225 A % =0.50 0.06 0.04 *\ - ‘3‘ b ‘ \A \n \\64 0.02 \ \ "~ \ \ ‘1 0 . / - 0.02 - 0.04 - 0°06 /1 0.225 - 0.06 ‘1' / PREDICTED FROM - °-'° CALCULATED SHEAR STRE - 0.14 ' , — 0 .004 .008 .012 .016 .020 2. FIT/111. FIGURE 25 TURBULENCE INTENSITY AND SHEAR DISTRIBUTIONS (h/ b= I.OI 55 fairly constant for y/b below -0.1 in the cavity. The cavity depth appears to have insignificant effects upon these four quantities when they are compared for similar values of 6/b. 4.5 Temperature Temperatures that were measured at x/b = 0.15, 0. 50, and 0.85 from the frontside of the notch are shown in Figures 26. through 31. The temperatures are presented in the dimensionless ratio (T - TOO)/(TW- Too) . The temperatures TW and Too were established as the average cavity wall temperature and the mid-channel free stream temperature respectively. A major temperature increase occurs across the shear layer. The temperature then remains nearly constant over the inviscid core with a second major temperature increase occuring near the surface. For deeper cavities, the in- fluence of secondary eddies becomes noticeable. For the case of h/b = 1.5, the core temperature remains approximately uniform until reaching y/b = ~— 1 . O, at which time the temperature begins increasing. Because of the cavity size, this increase is hidden by the thermal boundary layer formed along the cavity floor. However, when the cavity depth is increased to h/b = 2.0, the sudden increase in tempera- ture is easily seen. Again it is seen to occur at y/b= -l. 0 indicating the formation of flow cells within the cavity. Burggraf, neglecting the influences of secondary eddies (with Pr = 1 and constant pressure) crl'< 0.4 ' 0.2 '02 ‘0.4 ‘0.6 “0.8 ' LG 56 TEMPERATURE PROFILES FOR h/b cm a 8 /b=0.2e l 0 I 5 " a . i fl I + F * / * A 0 I A. l u A . x/ b =O.I5 A x/b . 0.50 . Vb . 0.05 o 0-I 0.2 0.3 0.6 0.7 T -T... Tw- T“ FIGURE 26 O'|‘< 0.4 0.2 '0.2 '0.4 ' 0.6 “'03 “LG 57 Y 4 5 - €14 / I u- / 5 " 0A. ':*b*$_I_ —-.‘—1‘— l‘ . ///a/ .A 0 A. . x/b =0.I5 A x/b 30.50 I x/b =0.05 IA. IA. IL. IA. FIGURE 27 TEMPERATURE PROFILES FOR h/b=l.0 a. 6/b=0.50 0.7 U'I~< 58 0.6 i | 0'3 4313*)! 11/ / / , 5 n I /-- b *2 { l / 0' ///// I A O b A o “0.3 '0.6 ’09 . x/b =0.25 ‘ x/b 80.50 I x/b =0.75 -|.2 -|.5 O 0.I 0.2 0.6 0.7 FIGURE 28 TEMPERATURE PROFILES FOR h/b=|.5 s 8/b=0.56 U'I‘< 0.6 0.3 -O.s '|.2 "1.5 59‘ TEMPERATURE PROFILES FOR h/b=l.5 s 6/b=0.79 I 7; I5 x I / I . ’ / h / b / I /"" */ I ///;—i I 0 I A 0 4* I A O I 0 A 0 A 0 ,0 V223? IA 0 ‘ X/b ' ‘0. I x/b30.75 I 0 A I 0.1 0.2 0.3 0.4 05 05 0-7 T -T.. 11,711,a FIGURE 29 UI~< 60 0.8 0.4 41' A \\\\ F‘— i ° ~—I—Lk / I A IA I ‘ A ‘0.4 a '0.8 -|°2 . x/ b a 0.25 A x/b =0.50 I x/ b =0.75 +A0 .I'6 I A I I A. - {- '2.0 O 0.I 0.2 0.3 0.4 05 0.6 0.7 T -T.. T: T” FIGURE 30 TEMPERATURE PROFILES FOR h/b =2.0 e 5 /b=0.38 0.8 0.4 "O.4 crl‘< ‘O.8 . X/b’0.25 I x/b I050 I x/b -O.75 'I.6 " 2.0 0.I 0.2 FIGURE 3| TEMPERATURE PROFILES FOR h/b=2.0 a 5/b=0.76 62 gives for the average inviscid core temperature T , o To-Too 1 Tw " Too J2(1 + b/h) This value has been shown (analytically) to agree very well for laminar flow and h/b = 1. 0 (12) . For the present case (turbulent flow and variable h/b) a comparison between measured and predicted values is shown in Figure 32. Although the predicted values are consistently high, the correlation is adequate. However, as shown in Figure 32 a closer correlation is found using the equation: TO-T 1 J3(1+b/h) 8 T-T w oo Using this expression for To’ equations (3. 38) and (3. 50) were solved. The resulting expressions for 0(n) are shown in Figures 33 through 36 where they are seen to correlate closely with the experi- mental data . 4. 6 Heat Transfer The results from the measurements of the average heat transfer from a heated cavity are shown in Figure 38. The form of the heat-transfer coefficient was taken as 4.7 =).| b 3 -I/2 T-T 0 00: + T; T” [Ml b/h)] l l I . L 0.5 I.O |.5 2.0 2.5 h/ b FIGURE 32 PREDICTED CAVITY BULK TEMPERATURE AND COMPARISON WITH EXPERIMENTAL DATA o X/b=0.l5 I 050 ‘ _. ' I A 0.85 . ” A-Z _ I ' A-3 1 l L I l 0 0| 02 0.3 04 05 T - T” Tw- 7.. FIGURE 33 PREDICTED AND MEASURED TEMPERATURE PROFILES (h/ b = 0.50 , 5/b = O.2|I 6‘3 I xlb=0.l5 _ I 0.50 A 0.85 ~A=2 [— - r-‘A=3 1 J I l I 0 Ol 02 0.3 04 05 T- T... I; 1', FIGURE 34 PREDICTED AND MEASURED TEMPERATURE PROFILES (h/b=I.O, 6/b=0.28) '6 \ 'IO 'IZ ’I6 'I8 OI) I A I ' I X/b= 0.I5 ‘ A - n 0.50 0 A 0.85 F - ~A=2 O L. I l l L l 0 0| 0.2 03 0.4 05 T _ T... Tw- To, FIGURE 35 PREDICTED AND MEASURED TEMPERATURE PROFILES (h/b=I.O, 5/b=O.58) 6 7 "IZ F A 0.85 _A=2 I -I4 I. -I6 .- LIBS .I8 I- J l l J I 0 Ol 02 0.3 04 05 T - Tm Tw- T00 FIGURE 36 PREDICTED AND MEASURED TEMPERATURE PROFILES (h/ b =I.O,6/b = 0.76) 68 To correlate the data, reference is made to equation (3. 55) where the semi-theoretical expression describing the heat transfer is derived. Since the Stanton number is defined in terms of the tempera- ture difference (TO - TOO) rather than (TW - Too) , the results are independent of cavity depth. This result is confirmed reasonably well by the experimental data (Figure 38) . Additional heat transfer data is presented in Figures 37 and 39. Here local heat transfer rates measured along the cavity walls are given with the heat transfer coefficient taken as: q pC u (T -T) poo w oo St = q is the heat liberated per unit time and per unit exposed surface area of the ribbon heating elements. The heat rate was deduced from the measured power which was used to heat the series -connected ribbons. The heat transfer rates are greatest at the top of the down- stream side, most probably due to the impinging external flow. The heat transfer rates decrease rapidly through this zone remaining somewhat uniform along the remainder of the downstream wall. Secondary peaks appear along the bottom and upstream walls, with, these rates then decreasing slightly along the direction of flow. The relative size of the oncoming boundary layer also influences the local heat transfer rates and is most pronounced in the impingement region of the downstream wall. 69 09.1.): mwk<0 oukdm... 4 10¢“. mumm14m... :3... mad on 9.50.... 97.. . 0.. 0.0 0.0 (.0 N0 0.. x 0...... I «0.0 0.. I who 4 0.0 a n 2 .0. 7, 4. a. A no.0 x .... x I x I 4 .m I girl? / #00 I II I I I «4 a. g 4‘ I . I 000 :n.n..0u 00.0 .00 0.06 0.05 - 0.04 -- 0.03 PREDICTED (E0. 3.57) / I F I I 0.I 0.2 0.3 0.4 5/ 0 FIGURE 39 AVERAGE HEAT TRANSFER FROMA HEATED CAVITY (DATA BASED ON LOCAL 'HEAT TRANSFER RATES) (h/b=l.0) 5 . C ONCLUSIONS 1. The expression for the predicted velocity distribution as given by equation (3. 20) correlates the data reasonably well with a maximum deviation of 20 percent within a substantial portion of the flow. The coefficient V"a" was taken as 0.12. For a plane free‘jet, a value for "a" of 0.09 has been established (14) . Evidently, this increase in "a" is due to the additional shear layer‘ turbulence caused by the cellular flow within the cavity. 2. The experimental distribution of percent turbulence and turbulent shear stress (Figure 24) have maximum values at a point coinciding with the dividing streamline (y =._ 0) and are both quite sensitive to changes in the relative size of the approaching ' boundary layer. The predicted distribution for the turbulent shear distri- bution (Figure 24) is compared with the hot- wire measurements well within :1: 20 percent. . 3. The time -averaged temperature distribution described by equatiOns (3. 38) and (3. 50) correlates the data reasonably well with the coefficient C taken as 1.4 giving the best correlation with experiment. The average inviscid core temperature T is seen to be 0 72 73 adequately determined by Burggraf's equation: o-oo l w oo JAI1+ b/h) However, the present study indicates that for turbulent flow a value of 3.0 for the constant A gives better agreement with experiment. With this temperature difference (To -- Too) used in this definition, the average Stanton number is seen to be independent of the relative cavity depth. The average heat transfer in the cavity (Equation 3. 56) was confirmed within :1: 15 percent by experiment and is closely approximated by the correlation: _ -0.1367 St = 0.0365 (g) REFERENCES Fox, Jay, "Surface Pressure and Turbulent Airflow in Transverse Rectangular Notches, " NASA Technical Note D-2501, 1964. Charwat, A. F. , et al, "An Investigation of Separated Flows-- Part I: The Pressure Field, " Journal of the Aero /Space Sciences, Vol. 28, p. 457, 1961. Charwat, A. F. , et al, "An Investigation of Separated Flows-- Part 11: Flow in the Cavity and Heat Transfer, " Journal of the Aero/Space Sciences, Vol. 28, pp. 513-527, 1961. Larson, H. K., "Heat Transfer in Separated Flows, ” Journal of the Aero/Space Sciences, Vol. 26, No.11, pp. 731-38, November, 1959. Wieghardt, K., "Erhohung des turbulenten Reibungswiderstandes durch Oberflachenstorungen. " Forschungshefte fur Schiffstechnik, Heft 2, pp. 65-81, 1953. Tillmann, W. , "Neue Widerstandsmessungen an Oberflachen- storungen in der turbulenten Reibungschicht. Forschung- shefte fur Schiffstechnik, Heft 2, pp. 81-88, 1953. Roshko, A. , "Some Measurements of Flow in a Rectangular Cutout. " NACA Tech. Note No. 3488, 1955. Tani, I. , Tuchi, M., and Komodo, H., "Experimental Investi- gation of Flow Separation Associated with a Step or a Groove, " Aeronautical Research Institute, University of Tokyo, Report No. 364, April, 1961. Seban, R. A., Emery, A., and Levy, A., "Heat Transfer to Separated and Reattached Subsonic Turbulent Flows Obtained Downstream of a Surface Step, " Journal of the Aero/Space Sciences, Vol. 26, No. 12, pp. 809-814, December,, 1959. 74 10. 11. 12. 13. 14. 15. 16. 17. 75 Seban, R. A. and Fox, J. , "Heat Transfer to the Air Flow in a Surface Cavity, " Proceedings of the 1961 International Heat Transfer Conference, ASME, p. 426. Chapman, D. R. , "A Theoretical Analysis of Heat Transfer in Regions of Separated Flow, -" NACA TN _3792, 1966. Burggraf, Odus R. , "A Model of Steady Separated Flow in Rec— tangular Cavities at High Reynolds Number, " Proceedings of the 1965 Heat Transfer and Fluid Mechanics Institute, " The Standord University Press, p. 190. Hinze, J. 0., 'Turbulence, McGraw-Hill Book Company, Inc., New York, p. 104, 1959. Abramovich, G. N. , The Theory of Tufbulent Jets, The M.I.T. Press, Cambridge (Mass.), 1963, pp. 61-63. Batchelor, G, K. , "On Steady Laminar Flow with Closed Stream- lines at Large Reynolds Numbers, " Journal of Fluid Mechanics, Vol. 1, p. 177, 1956. Flow Corporation "Model HWBZ Hot Wire Anemometer Theory and Instruction, " Flow Corp. Bulletin No. 37B, 1958. Boussinesq, J. , "Essai sur la theorie des eaux courantes, " Memoires presentes par divers savants a l'Academie des Sciences 23, 1877. APPENDIX 77 Table l.--Velocity data, h/b = 1.0. H, x-component of time mean velocity (ft./sec.) Z13 x/b = 0.15 x/b = 0.50 x/b i 0.85 5/b 49 4.10 50 Ag .30 .so .10 .30 .50 0.60 99.5 100.1 105 0 0.55 99.5 100.3 10u.8 0.50 99.5 100.0 10u.8 0.u5 97.0 100.0 . 10u.8 0.u0 99.2 95.0 100 0 96.9 105.0 102.0 0.35 99.3 100 0 10u.8 97.6 0.30 99.3 92.0 100.0 9u.0 105.0 97.5 0.25 95.8 102.5 9u.0 0.20 100.0 93.0 87.5 100.3 95.8 89.u 105.0 98.2 92.2 0.15 100.0 100.0 90.5 109.8 9u.8 86.3 0.10 100.0 87.0 81.5 100.3 86.0 77.3 105.1 89.u 83.0 0.05 88.5 78.0 70.0 91.5 75.7 9u.5 80.2 73.5 0 6u.0 63.5 55.5 56.1 60.5 56.1 71.2 6u.6 63.0 -0.05 25.5 29.5 18.0 33.6 27.2 28.u ul.5 H1.6 H2.3 —0.10 25.0 22.5 13.5 29.u 21.2 20.7 31.3 31.5 31.u -0.15 22.5 20.0 26.5 20.3 2u.u 23.0 23.3 —0.20 21.0 16.0 12.5 22.8 18.7 1u.2 2u.6 19.6 21.1 -0.25 10.0 15.u 13.3 21.3 17.5 2u.5 -0.30 19.5 10.0 11.5 12.u 11.2 8.1 20.u 16.7 22.u -0.90 17.5 10.0 11.0 11.u u.8 u.6 15.7 1u.7 19.9 —0.50 16.0 5.0 10.0 8.1 u.1 u 13.3 lu.7 19.1 78 Table 2.--Velocity data, h/b = 2.0. E, x-component of time mean velocity (ft./sec.) {:3 X/b = 0.15 X/b = 0.50 X/b = 0.85 5A; .[g .30 .So .\_0 .30 .so .10 go .50 0.60 100.0 100.5 0.55 99.5 100.3 100.5 0.50 99.7 100.3 100.5 0.U5 101.9 0.40 100.0 95.5 100.3 97.7 104.8 100.0 0.35 99.7 100.3 105.0 96.9 0.30 100.0 92.0 100.3 91.9 105.0 95.3 0.25 97.9 102.5 93.7 0.20 100.0 92.5 88.0 99.7 9“.“ 86.9 98.7 89.6 0.15 100.0 100.1 92.6 105.0 9u.7 86.4 0.10 100.0 83.5 78.5 100.3 85.0 76.5 104.7 88.0 81.5 0.05 87.0 76.5 69.5 90.4 77.8 96.3 80.” 69.6 0 50.0 56.5 50.0 60.1 57.6 51.9 74.2 61.6 59.6 -0.05 20.1 19.5 22.0 30.0 22.8 23.7 “2.0 36.2 36.7 -0.10 19.5 16.5 15.0 25.5 22.2 19.1 38.2 25.9 26.4 -0.15 15.1 1U.2 20.“ 19.2 31.8 20.“ 25.2 -0.20 13.5 12.5 8.0 15.3 18.“ 17.9 27.7 17.0 22.“ -0.25 12.5 11.6 22.7 15.1 20.“ -0.30 12.5 12.5 10.5 10.6 10.1 22.7 13.9 20.4 -O.UO 11.5 .5 7.5 9.7 7. 17.3 12.9 18.9 -0.50 10.5 .9 5 5.5 8.7 5. 10.1 12.3 16.5 79 Table 3.——Turbulence data. (h/b = 1.0) (h/b = 2.0) 1 g. —u'v' -u'v' E1_. —u'v' -u'v' 2L3 b b 1112 -—2 u2 I_L_u2 {I2 u2 2 a) a) 2 a) a) 0.10 0.10 .060 .061 0.075 .005 .0109 .089 .009 .010 .080 0.050 .010 .0299 .198 .010 .029 .150 0.025 .019 .0579 .205 .018 .059 .215 0 .021 .108 .223 .023 .115 .225 -.025 .015 .179 .200 .017 .185 .215 -.050 .006 .169 .178 .007 .173 .180 -.-75 .003 .090 .119 .009 .090 .129 -.100 .003 .099 .077 .003 .110 .115 -.125 .0025 .090 .057 .0025 .090 .075 —.150 .003 .107 .002 .110 .070 -.175 .0025 .102 .0025 .100 0.10 .060 .060 0.075 .005 .0117 .098 .005 .012 .110 0.050 .008 .0275 .138 .009 .029 .193 0.025 .0125 .0912 .182 .013 .091 .182 0 .016 .0885 .198 .017 .091 .203 -.025 .010 .129 .190 .011 .125 .193 —.050 .006 .173 .161 .007 .189 .172 —.075 .003 .096 .119 .005 .097 .115 -.100 .0025 .089 .077 .003 .092 .082 -.125 .0025 .092 .057 .002 .092 .057 -.150 .0025 .099 .002 0.075 .005 .0129 .089 .005 .0125 .090 0.05 .006 .0192 .127 .005 .019 .126 0.025 .010 .0355 .198 .010 .090 .152 0 .011 .066 .152 .012 .069 .155 -.025 .008 .109 .191 .009 .110 .191 -.050 .005 .170 .100 .005 .170 .110 -.075 .003 .106 .077 .003 .106 .079 -.100 .0025 .096 .057 .003 .090 .050 -.l25 .0025 .105 .057 .002 .110 .050 -.150 .0020 .0925 ' 80 Table 9.-—Temperature data (x/b = 0.5). h/b = 1.0 h/b = 1.0 h/b = 2 0 h/b = 2 0 (G/b = 0.28) (es/b = 0.58) (G/b = 0 38) (d/b = 0 76) 1 T T-TOO T T-TOO 1 T T-TOO T T-T b OF Tw—TD TW-TOC> b Tw-Tn Tw-T)o .25 80.0 .0099 76.0 0.0 .3 76.0 0.0 75.0 0.0 .20 80.0 .0099 76.0 0.0 .2 76.0 0.0 75.0 0.0 .15 80.1 .0055 78.0 .020 .1 78.2 .0022 77.9 0.29 .10 80.1 .0055 79.2 .032 0 89.6 .086 88.6 .136 .05 82.1 .0277 80.6 .096 —.1 98.9 .229 108.7 .337 0 83.1 .0388 83.2 .072 —.2 107.6 .316 110.0 .350 —.05 102.3 .251 109.6 .336 -.3 108.6 .326 113.7 .387 -.10 110.9 .391 119.3 .383 -.9 109.6 .376 115.6 .906 —.15 113.9 .379 117.0 .910 —.5 115.0 .390 116.0 .910 —.20 115.7 .900 117.7 .917 -.6 115.1 .391 117.0 .920 -.25 116.7 .911 118.5 .925 -.8 116.5 .905 117.7 .927 —.30 117.9 .918 118.0 .920 -1.0 118.5 .925 118.2 .932 -.90 117.0 .919 118.7 .927 -1.2 122.6 .966 123.2 .982 -.50 117.1 .915 119.1 .931 -1.9 126.8 .508 130.9 .559 -.60 117.5 .919 119.7 .937 -1.6 130.6 .596 133.0 .580 -.70 118.9 .929 119.6 .936 -1.8 133.1 .571 138.9 .639 —.80 117.9 .923 119.7 .937 -.90 118.0 .925 122.6 .966 Table 5.—-Loca1 cavity heat transfer (h/b 81 1.0). .2 o h = g/A (BTU/hr.ft F) Tw_Too Upstream Wall y/b 0/b=0.l6 6/b=0.28 6/b=0.36 —.0992 23.8 19.9 18.0 -.1311 23.8 19.7 17.7 -.2131 22.8 18.1 16.8 -.2951 22.9 16.9 16.0 -.3770 20.7 17.9 16.1 -.9590 20.6 18.1 16.5 -.5910 21.3 18.7 16.9 -.6230 22.7 20.3 17.2 -.7099 29.7 22.1 18.9 -.7869 25.8 22.1 19.8 -.8689 25.6 21.5 20.0 -.9508 29.0 20.3 19.6 Bottom x/b .0992 22.9 19.0 18.9 .1311 21.6 18.1 17.6 .2131 21.6 18.1 17.8 .2951 21.8 18.9 18.9 .3770 22.2 19.0 18.8 .9590 23.6 19.9 20.0 .5910 29.9 21.5 22.0 .6230 25.2 22.6 22.9 .7099 26.5 29.1 29.0 .7869 28.0 25.0 29.2 .8689 26.6 29.5 22.0 .9508 26.0 23.1 20.6 Downstream Wall y/b —.9508 29.0 53.5 38.2 -.8689 23.7 99.2 33.9 -.7869 23.8 92.6 29.0 —.7099 29.0 30.9 28.5 -.6230 29.5 26.9 26.0 —.5910 26.9 29.3 29.3 -.9590 31.6 22.9 23.9 -.3770 35.3 21.5 22.6 -.2951 98.9 20.8 20.0 -.2l31 59.3 20.3 19.8 -.1311 62.1 20.0 19.0 -.0992 63.9 20.8 19.0 82 Table 6.--Average cavity heat transfer. Stanton No. Based on (To'TaQ h uaa g Ta) Tw To Q —E = q/A b ft/sec.b oF oF oF BTU/HR 0CD uoo(To-Toa) 0.50 70.6 .296 77.0 251.0 137.0 3960 .0983 0.50 76.0 .297 77.7 299.9 139.5 3210 .0997 0.50106.0 .278 78.5 206.8 120.5 3175 .0916 0.50 71.5 .999 77.0 272.8 199.6 3320 .093 0.50 78.3 .996 77.0 270.0 193.2 3117 .0395 0.50107 0 .917 77.0 232.5 129.9 3090 .0365 1.0 80.3 .559 76.5 272.0 156.1 3125 .0932 1.0 86.5 .572 76.5 273.7 158.3 3070 .039 1.0 92.1 .586 76.5 273.5 155.3 2985 .038 1'0 103'” .373 77.5 208.2 139.5 3005 .0398 1.0 86.9 .386 77.5 229.8 193.0 2950 .0907 1.0 70.5 .900 77.5 299.2 152.0 2730 .0953 1.0 112.0 .161 77.0 157.6 108.5 2535 .0916 1.0 77.2 .173 77.0 199.7 120.0 2765 .0989 1.0 87.2 .162 77.0 182.5 116.0 2800 .0513 1.5 71.8 .60 78.0 293.9 160.5 1965 .0902 1.5 89.8 .579 78.0 218.8 135.8 2250 .099 1.5 110.3 .550 78.0 195.3 126.5 1790 .0378 1.5 89.3 .80 76.5 235.6 193.3 2170 .038 1.5 89.7 .78 76.5 237.7 192.7 2015 .036 1.5 109.3 .75 76.5 210.1 132.7 1875 .0365