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Recovery of reattached turbulent shear layers
Experimental Thermal and Fluid Science 17 (1998) 57±62
Recovery of reattached turbulent shear layers Srba Jovic Sterling Software, NASA Ames Research Center, MS 247-2, Moett Field, CA 94035-1000, USA Received 28 October 1996; received in revised form 7 May 1997; accepted 19 May 1997
Abstract Turbulence structure of the recovering ¯ow downstream of a backward facing step is investigated for ®ve dierent ¯ow conditions. Reynolds number Reh U0 h/v and d/h (representing a measure of a perturbation strength) appear to be the chief governing parameters of turbulence characteristics. Present measurements reveal that the skin-friction coecient magnitude decreases everywhere in the ¯ow (including the recirculating region) as the Reynolds number increases. The pressure coecient distribution C^p collapses on a single curve for x < xr in the examined parameter range. All C^p distributions appear to reach approximately constant values by x 1.5xr (xr is a mean reattachment length). The turbulence structure near the wall recovers to that of an equilibrium boundary layer by x 15h, while the outer region takes much longer distance (order of 100h). The slope of the Prandtl mixing length near the wall does not approach the equilibrium value of 0.41 until x 40h. The normalized maximum eddy viscosity in the outer layer appears to take more than 100h to approach the equilibrium value of 0.0168. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Turbulence; Recovering ¯ow; Shear layers
1. Introduction The fully developed turbulent structure of a constantpressure boundary layer is perturbed by a discontinuity in the boundary condition. An impervious wall abruptly ends at the step lip allowing the internal mixing-layer like ¯ow to develop downstream, imbedded in the original turbulent boundary layer. Further downstream, in the reattachment region, there is a new change of the boundary condition representing a new perturbation of the ¯ow. The mixing-layer like structure encounters an impervious wall in the reattachment region, and gradually begins to change its character, undergoing recovery to a structure characteristic of an equilibrium turbulent boundary layer (hereinafter TBL). The response of the turbulent structure to the imposed perturbation is not instantaneous across the entire ¯ow but is achieved rather gradually. Three dierent basic ¯ow structures, namely the mixing-layer, wall- and wake-layer like structures of TBL, compete in the recovery region. Depending upon the boundary conditions in the dierent ¯ow regions, one of the three ¯ow structures prevails. It appears that the turbulent structure near the wall recovers much faster to that of an equilibrium boundary layer than the structure in the outer part of the ¯ow. The fundamental complexities of the turbulent structure of this family of 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 4 9 - 8
turbulent ¯ows presents a real challenge for the available turbulence models. Numerous studies have been conducted on separated/ reattached ¯ows during the past four decades. The research has been conducted for many dierent geometric con®gurations. However, most of these studies have addressed separation induced by a backward-facing step. Extensive studies on separated ¯ow for a blunt plate have been made by Cherry et al. [9], and Kiya and Sasaki [21,22]. Ruderich and Fernholz [30], Castro and Haque [8], and Cutler and Johnston [10] studied the structure of a separated ¯ow behind a normal plate (fence) with a splitter plate. Durst and Tropea [13], Chandrsuda and Bradshaw [7], Kim et al. [20], Armaly et al. [3], Westphal et al. [32], Eaton and Johnston [14], Pronchick and Kline [29], Driver and Seegmiller [12], Adams and Johnston [1,2], just to name a few, have conducted extensive measurements of a separated ¯ow behind a backward-facing step. The objective of the present experiment is to present results and discuss dependence of turbulence properties on dierent ¯ow parameters, i.e. Reynolds number Reh U0 h/m (U0 is the reference velocity upstream of the step, h is the step height, and m is the kinematic viscosity) and perturbation strength, d/h (d is the boundary layer thickness upstream of the step).
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S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
2. Apparatus, techniques and conditions All measurements were performed in the middle plane of a wind tunnel comprised of a symmetric three-dimensional 9:1 contraction, a 1690 mm long ¯ow development section with dimensions 197 mm ´ 420 mm, a backward-facing step of height h that was varied over the range 17±38 mm and width of 42 cm, and a 205 cm long recovery section. The ¯ow was tripped at the inlet of the development section using 1.6 mm diameter wire followed by a 110 mm width of 40 grit emery paper to produce a fully developed TBL upstream of the step. The side walls diverged slightly outward to assure approximate zero-pressure gradient in the development and the recovery sections of the tunnel. The ¯ow conditions are listed in Table 1. The incoming boundary layer, measured 40 mm upstream of the step, was fully turbulent having a Reynolds number based on the momentum thickness, Reh , in the range 1650±3600 and a shape factor, H, of about 1.4 (decreasing with increasing Re). The Reynolds number Reh was varied in the range between 6800 and 37000 by varying the step height and the reference velocity (Table 1). The minimum step height was 17 mm and the maximum was 38 mm, where as the reference velocity was varied between 6.0 m/s and 14.6 m/s. The d/h, ranged from 0.82 to 2.0, while the expansion ratio, Er (ratio of the tunnel height downstream and upstream of the step), remained in the narrow range between 1.09 and 1.2. The aspect ratio (tunnel width/step height) varied from 11 for the largest step to 24.7 for the smallest step. This aspect ratio is suciently large to ensure adequate two-dimensionality in the separated region as recommended by de Brederode and Bradshaw [6]. The pressure gradient in the recovery region, for x > 9h say, was negligibly small so that the ¯ow recovered virtually under zero-pressure gradient. Mean static pressure was measured on the upper and lower (step-side) walls using a standard pressure transducer. The skin-friction coecient distribution downstream of the step was measured using a laser-oil interferometer [26,27]. This technique allowed direct measurements of the shear stress, both in the recirculating and the reattached regions of the ¯ow. Mean velocity and turbulence measurements were made with normal and X-wire probes. The sensor ®laments were made of 10% Rhodium±Platinum wire 2.5 lm in diameter and 0.6 mm (or 22 wall units in the upstream boundary layer) in length for the X-wire probe, Table 1 Summary of ¯ow parameters
and 1.25 lm in diameter and 0.3 mm (or 11 wall units) in length for the normal-wire probe. The spacing between crossed wires was 0.4 mm or 15 wall units. The aspect ratio, l/d, of the sensor ®laments was 240 for both probes. The usual 90° included angle of the crossed wires was replaced by the 110° angle. This angle was chosen to improve accuracy of the measurements in the regions with higher levels of local turbulence intensity. The constant temperature anemometers were operated at overheat ratios of 1.3 and had a frequency response of 25 kHz as determined by the square wave test. The normal-wire signal was low-pass ®ltered at 10 kHz and was digitized at 20 ksamples/s for 30 s. The X-wire signals were low-pass ®ltered at 6 kHz and were sampled at 12 ksamples/s for 30 s. Analog signals were digitized using a Tustin A/D converter with 15 bit (plus sign) resolution. The probes were calibrated using a static calibration procedure and calibration data of each hot-wire channel were ®tted with a fourth-order polynomial. In the1=2 case of ¯ows with high turbulence intensities, say U 2 =U > 0:25 say, the linear theory of hotwire breaks down introducing large errors in the Reynolds stresses. Following the method described by Muller [28], an improved data reduction scheme was introduced where higher-order terms in the series expansion of the hotwire response equation were taken into account. The maximum Reynolds stresses corrections occurred in the reattachment region of each studied case, nominally for x < 10h, and was on the order of 15%, 25% and 20% for u2 ; v2 , and ÿuv, respectively. Further downstream, the turbulence intensity in the ¯ow decreased, and consequently the corrections reduced to 5%, 10% and 10%, respectively. Using the method of Yavuzkurt [33] and Moat [25], the total uncertainty for u2 ; v2 , and ÿuv in the separated and reattachment regions where local turbulence intensity exceeds 30%, are estimated at 10%, 15%, and 18%, respectively. In the recovery region, the uncertainties reduce to 8% for all three measured stresses due to the reduction in the local turbulence levels. The uncertainty of turbulent kinetic energy is 12%. Derived turbulence quantities, eddy viscosity p mt ÿuv= @U =@y, Prandtl mixing length L ÿuv= @U =@y, and dL/dy have somewhat higher estimated uncertainties of 15%, 18%, 20%, respectively. As reported in Jovic and Driver [18], uncertainties of skin friction Cf are 8% and 15% in the recovery and the separated regions, respectively. All ®ve test cases were performed with the same X-wire and the same probe orientation enabling us to see dierence between the test cases caused by ¯ow physics rather than instrumentation uncertainties.
Case
Reh
d/h
Reh
U0 (m/s)
h (mm)
Er
xr /h
3. Results
1 2 3 4 5
6800 10400 25500 25500 37200
2.00 1.27 1.20 0.82 0.82
1650 1650 3600 2400 3600
6.0 6.0 14.6 10.0 14.6
17.0 26.0 26.0 38.0 38.0
1.09 1.14 1.14 1.20 1.20
5.35 6.35 6.90 6.90 6.94
3.1. Pressure coecient
s } n h
The wall static-pressure coecient is de®ned as Cp 2 p ÿ p0 = qU02
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
Fig. 1. Pressure distribution for dierent Reh . Symbols are given in Table 1.
59
Fig. 2. Skin-friction coecient distribution for dierent Reh . Symbols are given in Table 1. Additional symbols: (+) Reh 5000 [17]; ( ± ) DNS prediction for Reh 5100 [24].
where p is the wall static pressure at any x location and p0 is the reference wall static pressure measured on the top wall at x0 )272 mm upstream of the step. The distributions of transformed pressure-coecient C^p Cp ÿ Cp;min = 1 ÿ Cp;min along the bottom wall for ®ve dierent Reynolds numbers vs. x/xr are shown in Fig. 1. It appears that the pressure asymptotes to a nearly constant value, (@p/ @x 0) for x > 1.5xr downstream of the step for all Reh . Pressure gradients for dierent Reynolds numbers and conditions appear not to dier appreciably for different expansion ratios in the recirculating region (x < xr ). This indicates that the turbulence structure of the separated shear layer for dierent cases is not signi®cantly dependent on the adverse pressure gradient induced by the expansion in the given range of Reynolds numbers Reh and expansion ratios Er . 3.2. Skin-friction coecient Experimental skin-friction measurements for dierent Reynolds number ¯ows over a backward-facing step are shown in Fig. 2. Two additional distributions of Cf are included in the comparison; one is the direct numerical simulation (DNS) prediction of Le et al. [24] for Reh 5100, d/h 1 and Er 1.2 of the double-sided symmetric sudden expansion, and the second one is the experiment of Jovic and Driver [17,18] for the same geometry and ¯ow conditions (this experiment was used for the DNS validation). It is seen that the skin-friction coecient magnitude, de®ned as Cf 2sw /(qU20 ), decreases everywhere in the ¯ow, both inside of the recirculating zone as well as in the downstream recovery region, as the Reynolds number increases. The local minimum skin-friction coecient (Fig. 3) measured inside of the recirculating zone, Cf;min , varies as Rehÿ1=2 for the range of Reynolds numbers tested, as found by Adams and Johnston [1], Devenport and Sutton [11], and Jovic and Driver [18]. This ®nding suggests
Fig. 3. Cf;min as a function of Reh . Symbols: (h), corresponds to experiments in Table 1; ( ) [12]; (n), Reh 5000 experiment; ( ± ) line with a )1/2 slope; (n) other measurements of Jovic and Driver [18].
that the ¯ow near the wall in the recirculating region is a viscous-dominated, laminar-like ¯ow. 3.3. Reynolds stresses Fig. 4 shows the recovery of turbulence intensity 1=2 urms =us urms U 2 downstream of the reattachment point for Reh 37200. Near the wall at y y yus = v 20; urms =us falls from a value of about 4.0 to 3.0 as x increases to about x 15h and remains constant for the rest of the recovery region regardless of the turbulence levels in the outer layer. The urms =us pro®les are compared with the zero-pressure gradient boundary layers of Klebano [23] (Reh 7500) and Spalart [31] (Reh 1410). Figs. 5 and 6 examine the rate of recovery of urms =us in the near wall and outer wall regions. They show the variation with x of the nominal maximum value of turbulence intensity urms =us near the wall (at y y 20) and in the outer layer (maximum value occurs at y=d 0:5; in outer coordinates), respectively, for dierent ¯ow parameters.
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S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
Fig. 4. Pro®les of turbulence intensity in the recovery region for Reh 37200.
Fig. 5. Decay of turbulence intensity near the wall at y 20. Symbols are given in Table 1. Straight solid lines represent values of 2.7 and 3.0.
Fig. 6. Decay of maximum turbulence intensity in the outer layer. Symbols are given in Table 1. The solid lines are for visual aid.
It is seen that the turbulence intensity near the wall recovers faster to that of Klebano [23] under the action of the ``wall'' dynamics, than in the outer layer. In the far ®eld for x > 15h; urms =us falls to a value between 2.7 and 3.0 at y 20 (depending upon the local Reh and depicted with solid lines in Fig. 5), and remains nominally constant for the rest of the recovery region. This suggests that the turbulence structure soon recovers to that of an equilibrium TBL in a thin layer close to the wall, irrespective of high turbulence levels in the outer layer. This observation is consistent with the turbulence kinetic energy balance near the wall in the recovery region, where production and dissipation of turbulence kinetic energy nearly balance each other (not shown here, see [16,19]). In the near ®eld, x < 15h, the turbulence intensity deviates appreciably from the equilibrium value for high Reynolds numbers Reh (25500 and 37000) and strong perturbation d/h (0.82 and 1.2), emphasizing the non-equilibrium nature of the ¯ow in this region. For the case of weaker perturbation (d/h 1.27,2.0) and low Reynolds number (Reh 6800, 10400), urms =us is quickly dominated by the wall dynamics, approaching the equilibrium value in the early stages of recovery. The outer-layer peak turbulence intensity, urms =us , increases with increasing Reh and perturbation strength d/ h. Fig. 6 shows that, for increasing x; urms =us decays for all ®ve examined ¯ow conditions but with diminishing dierences between the individual experiments, so that, for x > 30h, all ®ve cases appear to collapse on a single curve approaching an equilibrium state. The decay of the maximum turbulent kinetic energy (TKE), k=Ue2 (Ue is the shear layer edge velocity), in the outer layer is plotted on a log±log scale in Fig. 7. The peak values of TKE approach a straight line, with a slope of )1.2, for Reh > 25500 and x > 30h. It appears that the rate of energy decay of the present experiment approaches that of the homogeneous turbulence. This value of )1.2 falls between the usually quoted value of
Fig. 7. Decay of maximum k in the outer layer. Symbols are given in Table 1. The solid lines are for visual aid.
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
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Fig. 8. Slope of the mixing length near the wall. Symbols are given in Table 1. The solid lines are for visual aid.
Fig. 9. Maximum eddy viscosity variation for dierent ¯ow conditions. Symbols are given in Table 1. The solid lines are for visual aid.
)1.0 [4], and )1.27 [15]. Possible explanation: high turbulent content in the outer region for x > 30h is being convected from the region with high turbulence energy production (x < xr ), while at the same time, the velocity gradients are small contributing very little to the production of new turbulent energy. The low production of TKE is observed from the analysis of individual terms of the TKE balance equation for high Reh and strong perturbation [19]. However, in the case of low Reynolds number (Reh 6800, 10400), TKE fails to approach the decay of homogeneous turbulence. The recovery of two more quantities, the slope of the Prandtl mixing length in the inner layer and the maximum eddy viscosity, was also examined. Fig. 8 shows the variation of the slope of the Prandtl mixing length in the thin layer near the wall (nominally y=d < 0:1). High values of the slope downstream of the reattachment for all Reh and d/h indicate unusually large eddy structures across the entire attached shear layer [5,7,10]. This is consistent with the production of large eddy structures in the mixing-layer like ¯ow in the separated region downstream of the step, and with the long life-time of these structures, especially in the outer layer where convection prevails. In the vicinity of the wall, the large eddies break into smaller eddies under the action of high shear. It appears that the slope approaches a characteristic zero-pressure TBL value of 0.41 (solid line in Fig. 8) for x > 40h. Maximum eddy viscosity in the outer layer, shown in Fig. 9, exhibits overall larger values for stronger perturbed ¯ows (d/h 0.82,1.2,1.27) and higher Reh . Interestingly, maximum eddy viscosity increases initially for all ®ve cases attaining its maximum and then decays, ultimately attaining an equilibrium value of 0.0168 (solid line in Fig. 9). This initial increase suggests that the mean velocity gradient decays faster than the shear stress, while downstream of the location of maximum eddy viscosity, the roles are exchanged. It appears that the location of the maximum eddy viscosity gradually shifts upstream for increasing Reh and perturbation strength d/h.
4. Conclusions Results from the present study show that the skinfriction coecient decreases everywhere as the Reh increases. The negative minimum Cf;min , measured inside ÿ1=2 of the recirculating region, decreases as Reh in the range of Reynolds number between roughly 5000 and 37000. It appears that the pressure coecient distribution C^p collapses on a single curve for x < xr . Furthermore, all C^p distribution appear to reach approximately constant values by x 1.5xr . The near-wall internal boundary layer recovers to the same state as in an equilibrium TBL by x 15h for Reh > 25500, and it recovers sooner for lower Reh cases. The outer region takes a much longer to recover (order of 100h). Decay of the maximum turbulent kinetic energy in the outer layer approaches that of a homogeneous turbulent ¯ow for x > 30h for Reh > 25500. The slope of the Prandtl mixing length (@L/@y) near the wall does not approach the equilibrium value of 0.41 until x 40h, whereas turbulent quantities such as kinetic energy, and eddy viscosity in the outer layer appear to take more than 100h to recover to equilibrium values re¯ecting the long life-time of large eddies generated upstream of reattachment. Turbulence modelers should be aware of the fact that there are large Reh eects in back step ¯ows which makes it dicult to develop high Reynolds number turbulent models using low Reynolds number data bases (either DNS or experimental).
Acknowledgements This research was supported by NASA Grant NCC2465 while the author was with Eloret Institute, and is gratefully acknowledged. The author wishes to thank Professor P. Bradshaw and Dr. D.M. Driver for their timely review of this paper and valuable suggestions.
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S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
References [1] E.W. Adams, J.P. Johnston, Part 1, Exp. Fluids 6 (1988) 400±408. [2] E.W. Adams, J.P. Johnston, Part 2, Exp. Fluids 6 (1988) 493±499. [3] B.F. Armaly, F. Durst, J.C.F. Pereira, B. Schonung, J. Fluid Mech. 127 (1983) 473. [4] G.K. Batchelor, Homogeneous Turbulence, University Press, Cambridge, 1956. [5] P. Bradshaw, F.Y.F. Wong, J. Fluid Mech. 52 (1971) 113±135. [6] V. de Brederode, P. Bradshaw, I.C. Aero Rep. 72-19, Imperial College, London, UK, 1972. [7] C. Chandrsuda, P. Bradshaw, J. Fluid Mech. 110 (1981) 171±194. [8] I.P. Castro, A. Haque, J. Fluid Mech. 179 (1987) 439±468. [9] N.J. Cherry, R. Hillier, M.E.M.P. Latour, J. Fluid Mech. 144 (1984) 13. [10] A. Cutler, J.P. Johnston, J. Fluid Mech. 200 (1989) 367. [11] L.W. Devenport, E.P. Sutton, AIAA J. 29 (1991) 25. [12] D.M. Driver, H.L. Seegmiller, AIAA J. 23 (1985) 163. [13] F. Durst, C. Tropea, Proceedings of the Third Symposium on Turbulent Shear Flows, Davis, CA, USA, 1981. [14] J.K. Eaton, J.P. Johnston, in: L.J. Bradbury, F. Durst, B.E. Launder, F.W. Schmidt, J.H. Whitelaw (Eds.), Turbulent Shear Flows 3, Springer, Berlin, 1982, p. 162. [15] M.M. Gibson, T. Dakos, Exp. Fluids 16 (1993) 146±154. [16] S. Jovic, Engineering Turbulence Modelling and Experiments, in: W. Rodi, F. Martelli (Eds.), Proceedings of the International
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
Symposium on Engineering Turbulence Modelling and Measurements, 31 May±2 June, Florence, Italy, 1993. S. Jovic, D.M. Driver, NASA TM-108807, 1994. S. Jovic, D.M. Driver, Exp. Fluids 18 (1995) 164±167. S. Jovic, NASA TM-110384, 1996. J. Kim, S.J. Kline, J.P. Johnston, Trans. ASME J. Fluids Engrg. 102 (3) (1980) 302. M. Kiya, K. Sasaki, J. Fluid Mech. 137 (1983) 83±113. M. Kiya, K. Sasaki, J. Fluid Mech. 154 (1985) 463±491. P.S. Klebano, NACA, TN. 3178, 1954. H. Le, P. Moin, J. Kim, Report TF-58, Thermoscience Division, Department of Mechanical Engineering, Stanford University, 1993. R.J. Moat, Exp. Thermal Fluid Sci. 1 (1988) 3. D. Monson, D. Driver, J. Szodruch, ICIASF'81 Record, International Congress on Instrumentation in Aerospace Simulation Facilities, 1981, pp. 232±243. D. Monson, H. Higuchi, AIAA J. 19 (1981) 739. U. Muller, J. Fluid Mech. 119 (1982) 155. S.W. Pronchick, S.J. Kline, Report MD-42, Thermosciences Division, Mechanical Engineering Dept., Stanford University, 1983. R. Ruderich, H.H. Fernholz, J. Fluid Mech. 163 (1986) 283±322. Ph.R. Spalart, J. Fluid Mech. 205 (1989) 319±340. R.V. Westphal, J.P. Johnston, J.K. Eaton, NASA CR-3765, 1984. S. Yavuzkurt, Trans. ASME J. Fluid Engrg. 106 (1984) 181.
Recovery of reattached turbulent shear layers Srba Jovic Sterling Software, NASA Ames Research Center, MS 247-2, Moett Field, CA 94035-1000, USA Received 28 October 1996; received in revised form 7 May 1997; accepted 19 May 1997
Abstract Turbulence structure of the recovering ¯ow downstream of a backward facing step is investigated for ®ve dierent ¯ow conditions. Reynolds number Reh U0 h/v and d/h (representing a measure of a perturbation strength) appear to be the chief governing parameters of turbulence characteristics. Present measurements reveal that the skin-friction coecient magnitude decreases everywhere in the ¯ow (including the recirculating region) as the Reynolds number increases. The pressure coecient distribution C^p collapses on a single curve for x < xr in the examined parameter range. All C^p distributions appear to reach approximately constant values by x 1.5xr (xr is a mean reattachment length). The turbulence structure near the wall recovers to that of an equilibrium boundary layer by x 15h, while the outer region takes much longer distance (order of 100h). The slope of the Prandtl mixing length near the wall does not approach the equilibrium value of 0.41 until x 40h. The normalized maximum eddy viscosity in the outer layer appears to take more than 100h to approach the equilibrium value of 0.0168. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Turbulence; Recovering ¯ow; Shear layers
1. Introduction The fully developed turbulent structure of a constantpressure boundary layer is perturbed by a discontinuity in the boundary condition. An impervious wall abruptly ends at the step lip allowing the internal mixing-layer like ¯ow to develop downstream, imbedded in the original turbulent boundary layer. Further downstream, in the reattachment region, there is a new change of the boundary condition representing a new perturbation of the ¯ow. The mixing-layer like structure encounters an impervious wall in the reattachment region, and gradually begins to change its character, undergoing recovery to a structure characteristic of an equilibrium turbulent boundary layer (hereinafter TBL). The response of the turbulent structure to the imposed perturbation is not instantaneous across the entire ¯ow but is achieved rather gradually. Three dierent basic ¯ow structures, namely the mixing-layer, wall- and wake-layer like structures of TBL, compete in the recovery region. Depending upon the boundary conditions in the dierent ¯ow regions, one of the three ¯ow structures prevails. It appears that the turbulent structure near the wall recovers much faster to that of an equilibrium boundary layer than the structure in the outer part of the ¯ow. The fundamental complexities of the turbulent structure of this family of 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 4 9 - 8
turbulent ¯ows presents a real challenge for the available turbulence models. Numerous studies have been conducted on separated/ reattached ¯ows during the past four decades. The research has been conducted for many dierent geometric con®gurations. However, most of these studies have addressed separation induced by a backward-facing step. Extensive studies on separated ¯ow for a blunt plate have been made by Cherry et al. [9], and Kiya and Sasaki [21,22]. Ruderich and Fernholz [30], Castro and Haque [8], and Cutler and Johnston [10] studied the structure of a separated ¯ow behind a normal plate (fence) with a splitter plate. Durst and Tropea [13], Chandrsuda and Bradshaw [7], Kim et al. [20], Armaly et al. [3], Westphal et al. [32], Eaton and Johnston [14], Pronchick and Kline [29], Driver and Seegmiller [12], Adams and Johnston [1,2], just to name a few, have conducted extensive measurements of a separated ¯ow behind a backward-facing step. The objective of the present experiment is to present results and discuss dependence of turbulence properties on dierent ¯ow parameters, i.e. Reynolds number Reh U0 h/m (U0 is the reference velocity upstream of the step, h is the step height, and m is the kinematic viscosity) and perturbation strength, d/h (d is the boundary layer thickness upstream of the step).
58
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
2. Apparatus, techniques and conditions All measurements were performed in the middle plane of a wind tunnel comprised of a symmetric three-dimensional 9:1 contraction, a 1690 mm long ¯ow development section with dimensions 197 mm ´ 420 mm, a backward-facing step of height h that was varied over the range 17±38 mm and width of 42 cm, and a 205 cm long recovery section. The ¯ow was tripped at the inlet of the development section using 1.6 mm diameter wire followed by a 110 mm width of 40 grit emery paper to produce a fully developed TBL upstream of the step. The side walls diverged slightly outward to assure approximate zero-pressure gradient in the development and the recovery sections of the tunnel. The ¯ow conditions are listed in Table 1. The incoming boundary layer, measured 40 mm upstream of the step, was fully turbulent having a Reynolds number based on the momentum thickness, Reh , in the range 1650±3600 and a shape factor, H, of about 1.4 (decreasing with increasing Re). The Reynolds number Reh was varied in the range between 6800 and 37000 by varying the step height and the reference velocity (Table 1). The minimum step height was 17 mm and the maximum was 38 mm, where as the reference velocity was varied between 6.0 m/s and 14.6 m/s. The d/h, ranged from 0.82 to 2.0, while the expansion ratio, Er (ratio of the tunnel height downstream and upstream of the step), remained in the narrow range between 1.09 and 1.2. The aspect ratio (tunnel width/step height) varied from 11 for the largest step to 24.7 for the smallest step. This aspect ratio is suciently large to ensure adequate two-dimensionality in the separated region as recommended by de Brederode and Bradshaw [6]. The pressure gradient in the recovery region, for x > 9h say, was negligibly small so that the ¯ow recovered virtually under zero-pressure gradient. Mean static pressure was measured on the upper and lower (step-side) walls using a standard pressure transducer. The skin-friction coecient distribution downstream of the step was measured using a laser-oil interferometer [26,27]. This technique allowed direct measurements of the shear stress, both in the recirculating and the reattached regions of the ¯ow. Mean velocity and turbulence measurements were made with normal and X-wire probes. The sensor ®laments were made of 10% Rhodium±Platinum wire 2.5 lm in diameter and 0.6 mm (or 22 wall units in the upstream boundary layer) in length for the X-wire probe, Table 1 Summary of ¯ow parameters
and 1.25 lm in diameter and 0.3 mm (or 11 wall units) in length for the normal-wire probe. The spacing between crossed wires was 0.4 mm or 15 wall units. The aspect ratio, l/d, of the sensor ®laments was 240 for both probes. The usual 90° included angle of the crossed wires was replaced by the 110° angle. This angle was chosen to improve accuracy of the measurements in the regions with higher levels of local turbulence intensity. The constant temperature anemometers were operated at overheat ratios of 1.3 and had a frequency response of 25 kHz as determined by the square wave test. The normal-wire signal was low-pass ®ltered at 10 kHz and was digitized at 20 ksamples/s for 30 s. The X-wire signals were low-pass ®ltered at 6 kHz and were sampled at 12 ksamples/s for 30 s. Analog signals were digitized using a Tustin A/D converter with 15 bit (plus sign) resolution. The probes were calibrated using a static calibration procedure and calibration data of each hot-wire channel were ®tted with a fourth-order polynomial. In the1=2 case of ¯ows with high turbulence intensities, say U 2 =U > 0:25 say, the linear theory of hotwire breaks down introducing large errors in the Reynolds stresses. Following the method described by Muller [28], an improved data reduction scheme was introduced where higher-order terms in the series expansion of the hotwire response equation were taken into account. The maximum Reynolds stresses corrections occurred in the reattachment region of each studied case, nominally for x < 10h, and was on the order of 15%, 25% and 20% for u2 ; v2 , and ÿuv, respectively. Further downstream, the turbulence intensity in the ¯ow decreased, and consequently the corrections reduced to 5%, 10% and 10%, respectively. Using the method of Yavuzkurt [33] and Moat [25], the total uncertainty for u2 ; v2 , and ÿuv in the separated and reattachment regions where local turbulence intensity exceeds 30%, are estimated at 10%, 15%, and 18%, respectively. In the recovery region, the uncertainties reduce to 8% for all three measured stresses due to the reduction in the local turbulence levels. The uncertainty of turbulent kinetic energy is 12%. Derived turbulence quantities, eddy viscosity p mt ÿuv= @U =@y, Prandtl mixing length L ÿuv= @U =@y, and dL/dy have somewhat higher estimated uncertainties of 15%, 18%, 20%, respectively. As reported in Jovic and Driver [18], uncertainties of skin friction Cf are 8% and 15% in the recovery and the separated regions, respectively. All ®ve test cases were performed with the same X-wire and the same probe orientation enabling us to see dierence between the test cases caused by ¯ow physics rather than instrumentation uncertainties.
Case
Reh
d/h
Reh
U0 (m/s)
h (mm)
Er
xr /h
3. Results
1 2 3 4 5
6800 10400 25500 25500 37200
2.00 1.27 1.20 0.82 0.82
1650 1650 3600 2400 3600
6.0 6.0 14.6 10.0 14.6
17.0 26.0 26.0 38.0 38.0
1.09 1.14 1.14 1.20 1.20
5.35 6.35 6.90 6.90 6.94
3.1. Pressure coecient
s } n h
The wall static-pressure coecient is de®ned as Cp 2 p ÿ p0 = qU02
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
Fig. 1. Pressure distribution for dierent Reh . Symbols are given in Table 1.
59
Fig. 2. Skin-friction coecient distribution for dierent Reh . Symbols are given in Table 1. Additional symbols: (+) Reh 5000 [17]; ( ± ) DNS prediction for Reh 5100 [24].
where p is the wall static pressure at any x location and p0 is the reference wall static pressure measured on the top wall at x0 )272 mm upstream of the step. The distributions of transformed pressure-coecient C^p Cp ÿ Cp;min = 1 ÿ Cp;min along the bottom wall for ®ve dierent Reynolds numbers vs. x/xr are shown in Fig. 1. It appears that the pressure asymptotes to a nearly constant value, (@p/ @x 0) for x > 1.5xr downstream of the step for all Reh . Pressure gradients for dierent Reynolds numbers and conditions appear not to dier appreciably for different expansion ratios in the recirculating region (x < xr ). This indicates that the turbulence structure of the separated shear layer for dierent cases is not signi®cantly dependent on the adverse pressure gradient induced by the expansion in the given range of Reynolds numbers Reh and expansion ratios Er . 3.2. Skin-friction coecient Experimental skin-friction measurements for dierent Reynolds number ¯ows over a backward-facing step are shown in Fig. 2. Two additional distributions of Cf are included in the comparison; one is the direct numerical simulation (DNS) prediction of Le et al. [24] for Reh 5100, d/h 1 and Er 1.2 of the double-sided symmetric sudden expansion, and the second one is the experiment of Jovic and Driver [17,18] for the same geometry and ¯ow conditions (this experiment was used for the DNS validation). It is seen that the skin-friction coecient magnitude, de®ned as Cf 2sw /(qU20 ), decreases everywhere in the ¯ow, both inside of the recirculating zone as well as in the downstream recovery region, as the Reynolds number increases. The local minimum skin-friction coecient (Fig. 3) measured inside of the recirculating zone, Cf;min , varies as Rehÿ1=2 for the range of Reynolds numbers tested, as found by Adams and Johnston [1], Devenport and Sutton [11], and Jovic and Driver [18]. This ®nding suggests
Fig. 3. Cf;min as a function of Reh . Symbols: (h), corresponds to experiments in Table 1; ( ) [12]; (n), Reh 5000 experiment; ( ± ) line with a )1/2 slope; (n) other measurements of Jovic and Driver [18].
that the ¯ow near the wall in the recirculating region is a viscous-dominated, laminar-like ¯ow. 3.3. Reynolds stresses Fig. 4 shows the recovery of turbulence intensity 1=2 urms =us urms U 2 downstream of the reattachment point for Reh 37200. Near the wall at y y yus = v 20; urms =us falls from a value of about 4.0 to 3.0 as x increases to about x 15h and remains constant for the rest of the recovery region regardless of the turbulence levels in the outer layer. The urms =us pro®les are compared with the zero-pressure gradient boundary layers of Klebano [23] (Reh 7500) and Spalart [31] (Reh 1410). Figs. 5 and 6 examine the rate of recovery of urms =us in the near wall and outer wall regions. They show the variation with x of the nominal maximum value of turbulence intensity urms =us near the wall (at y y 20) and in the outer layer (maximum value occurs at y=d 0:5; in outer coordinates), respectively, for dierent ¯ow parameters.
60
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
Fig. 4. Pro®les of turbulence intensity in the recovery region for Reh 37200.
Fig. 5. Decay of turbulence intensity near the wall at y 20. Symbols are given in Table 1. Straight solid lines represent values of 2.7 and 3.0.
Fig. 6. Decay of maximum turbulence intensity in the outer layer. Symbols are given in Table 1. The solid lines are for visual aid.
It is seen that the turbulence intensity near the wall recovers faster to that of Klebano [23] under the action of the ``wall'' dynamics, than in the outer layer. In the far ®eld for x > 15h; urms =us falls to a value between 2.7 and 3.0 at y 20 (depending upon the local Reh and depicted with solid lines in Fig. 5), and remains nominally constant for the rest of the recovery region. This suggests that the turbulence structure soon recovers to that of an equilibrium TBL in a thin layer close to the wall, irrespective of high turbulence levels in the outer layer. This observation is consistent with the turbulence kinetic energy balance near the wall in the recovery region, where production and dissipation of turbulence kinetic energy nearly balance each other (not shown here, see [16,19]). In the near ®eld, x < 15h, the turbulence intensity deviates appreciably from the equilibrium value for high Reynolds numbers Reh (25500 and 37000) and strong perturbation d/h (0.82 and 1.2), emphasizing the non-equilibrium nature of the ¯ow in this region. For the case of weaker perturbation (d/h 1.27,2.0) and low Reynolds number (Reh 6800, 10400), urms =us is quickly dominated by the wall dynamics, approaching the equilibrium value in the early stages of recovery. The outer-layer peak turbulence intensity, urms =us , increases with increasing Reh and perturbation strength d/ h. Fig. 6 shows that, for increasing x; urms =us decays for all ®ve examined ¯ow conditions but with diminishing dierences between the individual experiments, so that, for x > 30h, all ®ve cases appear to collapse on a single curve approaching an equilibrium state. The decay of the maximum turbulent kinetic energy (TKE), k=Ue2 (Ue is the shear layer edge velocity), in the outer layer is plotted on a log±log scale in Fig. 7. The peak values of TKE approach a straight line, with a slope of )1.2, for Reh > 25500 and x > 30h. It appears that the rate of energy decay of the present experiment approaches that of the homogeneous turbulence. This value of )1.2 falls between the usually quoted value of
Fig. 7. Decay of maximum k in the outer layer. Symbols are given in Table 1. The solid lines are for visual aid.
S. Jovic / Experimental Thermal and Fluid Science 17 (1998) 57±62
61
Fig. 8. Slope of the mixing length near the wall. Symbols are given in Table 1. The solid lines are for visual aid.
Fig. 9. Maximum eddy viscosity variation for dierent ¯ow conditions. Symbols are given in Table 1. The solid lines are for visual aid.
)1.0 [4], and )1.27 [15]. Possible explanation: high turbulent content in the outer region for x > 30h is being convected from the region with high turbulence energy production (x < xr ), while at the same time, the velocity gradients are small contributing very little to the production of new turbulent energy. The low production of TKE is observed from the analysis of individual terms of the TKE balance equation for high Reh and strong perturbation [19]. However, in the case of low Reynolds number (Reh 6800, 10400), TKE fails to approach the decay of homogeneous turbulence. The recovery of two more quantities, the slope of the Prandtl mixing length in the inner layer and the maximum eddy viscosity, was also examined. Fig. 8 shows the variation of the slope of the Prandtl mixing length in the thin layer near the wall (nominally y=d < 0:1). High values of the slope downstream of the reattachment for all Reh and d/h indicate unusually large eddy structures across the entire attached shear layer [5,7,10]. This is consistent with the production of large eddy structures in the mixing-layer like ¯ow in the separated region downstream of the step, and with the long life-time of these structures, especially in the outer layer where convection prevails. In the vicinity of the wall, the large eddies break into smaller eddies under the action of high shear. It appears that the slope approaches a characteristic zero-pressure TBL value of 0.41 (solid line in Fig. 8) for x > 40h. Maximum eddy viscosity in the outer layer, shown in Fig. 9, exhibits overall larger values for stronger perturbed ¯ows (d/h 0.82,1.2,1.27) and higher Reh . Interestingly, maximum eddy viscosity increases initially for all ®ve cases attaining its maximum and then decays, ultimately attaining an equilibrium value of 0.0168 (solid line in Fig. 9). This initial increase suggests that the mean velocity gradient decays faster than the shear stress, while downstream of the location of maximum eddy viscosity, the roles are exchanged. It appears that the location of the maximum eddy viscosity gradually shifts upstream for increasing Reh and perturbation strength d/h.
4. Conclusions Results from the present study show that the skinfriction coecient decreases everywhere as the Reh increases. The negative minimum Cf;min , measured inside ÿ1=2 of the recirculating region, decreases as Reh in the range of Reynolds number between roughly 5000 and 37000. It appears that the pressure coecient distribution C^p collapses on a single curve for x < xr . Furthermore, all C^p distribution appear to reach approximately constant values by x 1.5xr . The near-wall internal boundary layer recovers to the same state as in an equilibrium TBL by x 15h for Reh > 25500, and it recovers sooner for lower Reh cases. The outer region takes a much longer to recover (order of 100h). Decay of the maximum turbulent kinetic energy in the outer layer approaches that of a homogeneous turbulent ¯ow for x > 30h for Reh > 25500. The slope of the Prandtl mixing length (@L/@y) near the wall does not approach the equilibrium value of 0.41 until x 40h, whereas turbulent quantities such as kinetic energy, and eddy viscosity in the outer layer appear to take more than 100h to recover to equilibrium values re¯ecting the long life-time of large eddies generated upstream of reattachment. Turbulence modelers should be aware of the fact that there are large Reh eects in back step ¯ows which makes it dicult to develop high Reynolds number turbulent models using low Reynolds number data bases (either DNS or experimental).
Acknowledgements This research was supported by NASA Grant NCC2465 while the author was with Eloret Institute, and is gratefully acknowledged. The author wishes to thank Professor P. Bradshaw and Dr. D.M. Driver for their timely review of this paper and valuable suggestions.
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