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Effect of a transverse square groove on a turbulent boundary layer
Experimental Thermal and Fluid Science 20 (1999) 1±10
www.elsevier.nl/locate/etfs
Eect of a transverse square groove on a turbulent boundary layer Sutardi, C.Y. Ching
*
Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. JohnÕs, NF, Canada A1B 3X5 Received 23 November 1998; accepted 13 July 1999
Abstract The eect of a transverse square groove on a turbulent boundary layer has been investigated at two values of momentum thickness Reynolds numbers Rh 1000 and 3000 using hot-wire anemometry. The ratio of the groove width to the boundary layer thickness (w/d1 ) is approximately 0.07. The wall shear stress (sw ) is estimated from the velocity pro®les using the Clauser chart method and by assuming a power law velocity distribution. The smooth wall results indicate the Clauser chart is better suited at higher Rh , whereas the power law is more suited at lower Rh . The eect of the groove on sw at the lower Rh is not signi®cant. At the higher Rh , there is a sudden increase in sw just downstream of the groove, followed by an undershoot and an oscillatory relaxation back to the smooth wall value. The mean velocity pro®les (U ) are not aected by the presence of the groove at both Rh . There is, however, a small reduction in the turbulence intensity in the near-wall region y K 10 at both Rh . There is a strong correlation between the sw and wake parameter (p) distributions, with an increase in p for a decrease in sw . The growth of the internal layer immediately downstream of the groove is very rapid at both Rh , followed by a much slower growth beyond x=w J 7. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Transverse square groove; Turbulence; Square groove; Boundary layer
1. Introduction Despite being studied for many years, there are many aspects of turbulent boundary layers that are still not fully understood. It has been suggested that transverse square grooves, optimally sized and located, could result in a reduction of skin friction drag. To determine the most important scaling factors, however, the eect of a single transverse square groove on the turbulent boundary layer needs to be clearly understood. Roshko [1] and Haugen and Dhanak [2] investigated the eect of a rectangular groove, with depth to width ratio (d/w) ranging up to 3.0, on a turbulent boundary layer. In both studies, the groove width was much larger than the boundary layer thickness 4:0 6 w=d 6 10:0, and the objective was to study the turbulent momentum transfer mechanism in the separated ¯ow region. Their results indicate that the contribution of the pressure drag to the total drag is very signi®cant, while the friction drag on the main surface and the groove wall is relatively small. The de¯ection of the boundary layer separation into the groove is believed responsible for *
Corresponding author. Fax: +1-709-737-4042. E-mail address: [email protected] (C.Y. Ching)
the formation of the high-pressure coecient on the down stream edge of the groove wall. Haugen and Dhanak [2] found a single stable vortex inside the groove if the ratio d/w equals unity. When d/w is increased, the number of vortices inside the groove was found to be of the order of the d/w ratio. In these studies, the entire boundary layer is aected because of the very large size of the groove relative to the boundary layer thickness. More recently, there have been several studies on the perturbation of much smaller grooves on a turbulent boundary layer to determine the eect of the groove on the near-wall turbulence structure. Choi and Fujisawa [3] studied the eect of a transverse square groove with w=d 0:4 and reported a skin-friction drag reduction of about 1%, while ignoring the pressure drag on the groove walls. They found a small sudden decrease in the wall shear stress (sw ) just downstream of the groove. The relaxation of sw back to the smooth wall value occurred at about 100w downstream from the groove trailing edge. The turbulence intensity above the groove was found to be higher than that for a corresponding smooth wall, while the wake parameter (p) was higher than that on a smooth wall for x=w 6 100. Pearson et al. [4] investigated the shear stress distribution downstream of a single square groove. They deduced that there was an
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
overshoot in sw just down stream of the groove, followed by an undershoot and an oscillatory relaxation back to the smooth wall value. Through ¯ow visualization, they determined that there were out ¯ows (ejections) from and in¯ows to the groove. The ¯ow visualization results were qualitatively similar to those of Ching et al. [5] and Elavarasan et al. [6]. The ejections of ¯uid from the groove are very likely due to the passage of the quasi-streamwise vortices over the groove. The pressure minimum at the center of these vortices is likely to ``pump'' the ¯uid out of the groove. Elavarasan et al., [6] found an increase in friction drag of about 3.4% for a surface with grooves spaced s=w 20 apart, where s is the groove spacing. Although there was an increase in total friction drag, a local decrease in sw was observed immediately downstream of the groove, similar to the observations of Pearson et al. [4]. Tani et al. [7] found that the skin friction on a smooth surface with grooves spaced 10 to 40w apart (hereafter referred to as a sparse d-type surface) was lower than that on a corresponding smooth-wall over a certain range of Reynolds numbers 8:4 105 K Rx K 15 105 . The grooves were found to reduce the total drag by up to 3%, at least in the low Reynolds number range [7]. Coustols and Savill [8] reported that a 2±3% net drag reduction could be achieved if the grooves are spaced 20w apart. The bene®cial modi®cation of the near-wall turbulence structure, such as a weakening of the quasi-streamwise vortices as they pass over the groove, is believed to be responsible for the skin friction drag reduction. Matsumoto [9] investigated turbulent boundary layers over sparse d-type surfaces, and suggested a possibility of skin friction drag reduction for the cases s=w 10 and 20. To obtain a skin friction drag reduction, the spacing s/w was found to be a function of Rh . For example, for s=w 10, a drag reduction was obtained for Rh < 3700, whereas for s=w 20, a reduction in Cf was determined for Rh < 5200. The drag calculations of Elavarasan et al. [6] and Choi and Fujisawa [3] are contradictory, highlighting the need for a more detailed parametric study of sparse d-type surfaces. Also, no studies have been performed to determine an optimum groove pro®le. Clearly, the relative size and shape of the transverse groove will have a signi®cant eect on the near-wall turbulence structure of the overlying shear layer. While sharp upper corners may be necessary, rounding of the base may reduce the cavity ¯ow losses. Additional data, especially on the signi®cance of w/d to the perturbation of the boundary layer is desirable. The objective of the present study is to determine the eect of a single transverse square groove on the characteristics of a turbulent boundary layer over a ¯at plate. Experiments were performed in a low-speed wind tunnel using hot-wire anemometry at two dierent Reynolds numbers. The eects of the groove on the skin friction, mean velocity, and turbulence intensity are determined. The turbulent wake parameter and development of the internal layer downstream of the groove are also studied.
2. Experimental details and methodology The experiments were performed in an open-circuit low-speed wind tunnel at Memorial University of Newfoundland. The test section is 1 m ´ 1 m and is over 20 m long. The roof of the tunnel is adjusted to maintain a zero pressure gradient along the test section. Only the initial 5.5 m length of the test section was used for this investigation. A centrifugal blower driven by a 19 kW motor is used in the wind tunnel. The air passes through a screened diuser and a large settling chamber with three single-piece precision screens. The air is accelerated into the test section through a 5:1 contraction. The maximum free stream velocity in the test section is about 15 m/s. The free stream turbulence intensity is less than 0.5% at all velocities. The velocity in the test section is changed using motorized variable angle inlet vanes on the blower. Experiments were performed at two free stream velocities (U1 ) of 2 and 5.5 m/s, corresponding to Reynolds numbers, based on the momentum thickness just upstream of the groove, of 1000 and 3000, respectively. The boundary layer thickness (d) at x=w 1 is 76 and 69 mm at U1 2:0 and 5.5 m/s, corresponding to a Rd of 9800 and 24 500, respectively. At each free stream velocity, measurements were made on a smooth wall ¯at plate and a ¯at plate with a single transverse square groove (hereafter referred to as the grooved wall). A roughness strip, consisting of a 100 mm wide sand paper (series 0811) and a 1.5 mm diameter cylindrical rod was used to trip the boundary layer at the leading edge of the plate (Fig. 1). The ¯at plate is made of 25 mm thick acrylic and is mounted horizontally on the ¯oor of the wind tunnel. After the initial measurements on the smooth wall, a transverse square groove (w/d 1, w 5 mm) was cut on the plate at a distance of 2635 mm from the leading edge. The present experimental conditions and ¯ow parameters together with those previous experiments related to the present study are given in Table 1. Hot-wire anemometry is used for the velocity measurements. The hot-wire probe is traversed in the wallnormal direction using a specially designed traversing mechanism using a Mitutoyo height gauge. The traverse is installed on rails mounted on the roof of the tunnel. This traverse has a maximum span of approximately 46 cm and a minimum linear division of 0.01 mm.
Fig. 1. Schematic of test-plate showing a single transverse square groove.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
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Table 1 Experimental conditions and ¯ow parametersa
U1 (m/s) Rh w/d1 d/w a
Choi and Fujisawa [3]
Elavarasan et al. [6]
Pearson et al. [4]
7.0 ~1200 ~0.4 1.0
0.4 1300 ~0.125 1.0
0.4 1320 ~0.17 1.0
Present study Low velocity
High velocity
2.0 1000 0.066 1.0
5.5 3000 0.072
Note: d1 is the boundary layer thickness at x/w 1.
A 5-lm DANTEC 55P05 boundary layer type probe was used for the boundary layer measurements. The velocity measurements were made using a DANTEC 55M01 standard bridge. The hot-wire signals were digitized using a 12 bit Keithley 570 System Analog to Digital (A/D) converter, interfaced to a 486 DX-AT personal computer. For the hot-wire calibrations, the velocity was measured using a pitot-static tube connected to a dierential pressure transducer (Furness type FCO34) with a range 0.00±0.254 cm water. The wall shear stress (sw ) was estimated from the velocity pro®les using the Clauser-chart (log-law) method and by assuming a power-law velocity distribution. An accurate estimation of the wall shear stress is necessary to predict the skin friction drag associated with boundary layers. The relation between the wall shear stress (sw ), and the friction velocity (us ) can be expressed as 2
sw q us :
1
Following BarenblattÕs [10] argument of incomplete similarity, Ching et al. [11] showed that us can be expressed as: ( 1= 1a ) 1 exp 3=2a us U1 ; 2 exp 3=2a C where a and C are calculated as follows: a
3 ; 2 ln Rd
where j and B are the Karman and the smooth-wall constants, respectively, and are assumed independent of the Reynolds number. White [13] proposed 0.41 and 5.0 for j and B, respectively. Sill [14] found that the experimental results give B 5:0 to 5.5, while Reynolds [15] proposed j to be 0.385 10%. Zagarola and Smits [16], however, found slightly higher values of j 0.436 and B 6.13, for a turbulent pipe ¯ow. The wall shear stress can also be calculated from the mean velocity gradient at the wall. oU sw l 6 : oy y0
This method, however, requires accurate velocity data very close to the wall (y K 3:0. Due to the eect of heat conduction from the hot-wire to the wall, this results in a spurious increase in U above the sub-layer relation U y , and cannot be used in the present study. A complete uncertainty analysis of the results was performed using the method outlined by Yavuzkurt [18], Moat [19], and Coleman and Steele [20]. The uncertainties are calculated at 95% con®dence and are given in Table 2. 3. Results and discusions
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1 5 4 C p lnRd : 2 3 Djenidi et al. [12] showed that us determined from Eq. (2) was within 0.57% to that obtained from Preston tube measurements for Rh 940. It was argued that at low Reynolds numbers Rh K 1500, say), the eect of viscosity is not insigni®cant, and a power-law velocity distribution is more appropriate than a log-law velocity distribution. The Clauser-chart method assumes a universal mean velocity pro®le in the overlap region. The Clauser-chart method is only appropriate if the Reynolds number is suciently high Rh J 1500) for a log-law region to exist. The general form of the log-law is u y U 1 s ln B; 5 us j m
The smooth-wall U pro®les at the lower Rh are compared with the power-law distributions (only two pro®les are shown for clarity) in Fig. 2. There is a good collapse of the pro®les, and the power-law is in good agreement with the experimental data in the entire outer region. In the inner region y K 30, the power law distribution is not valid, and cannot be used to describe the mean velocity pro®les. In this ®gure, U and y are normalized using us deduced from the power-law. The Table 2 Experimental uncertainties Flow variable Mean velocity (U) Fluctuating velocity (u) Boundary layer thickness (d) Skin friction coecient (Cf )
Uncertainty (%) Rh 1000
Rh 3000
3.09 3.09 0.62 9.25
1.40 1.40 0.28 4.73
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 2. Power-law and log-law ®ts to mean velocity pro®les in the overlap region for a smooth-wall boundary layer. Experiment: D, Rh 830; h, Rh 1250; Power-law ®t: A, Rh 830; ± ± ±, Rh 1250; Log-law ®t: A, Rh 830; 1250; DNS [17]: ± ± ±, Rh 670 ; ± ± , Rh 1410.
experimental data are also in good agreement with the smooth-wall DNS data of Spalart [17], and the small dierences in the outer region could be attributed to the dierence in Reynolds numbers. It is more dicult to use the Clauser chart method at the lower Rh since the log region in this case is very narrow [11,17]. The log-law ®ts the data well only in the range 30 K y K 100, and leads to a higher uncertainty in the estimation of sw using the log-law in this case. The present results support the incomplete similarity hypothesis of Barenblatt [10], which states that at low Reynolds numbers, the friction velocity can be deduced from the power-law distribution in the overlap region [12]. Unlike for the lower Rh , the power-law ®t, especially in the wake region, is poor at the higher Rh (Fig. 3). In this Reynolds number range Rh > 2000, it is dicult to obtain a good ®t to the experimental data using a power-law approximation. The estimation of sw using the power law method is based on the extrapolation of the power law ®t into the boundary layer thickness (d). The power-law ®ts the data well only in the range 30 K y K 300. This is in agreement with the pipe ¯ow data of Zagarola et al. [21], in which the power-law ®ts the data well only in the range 50 K y K 500. The log
Fig. 3. Power-law and log-law ®ts to mean velocity pro®les in the overlap region for a smooth-wall boundary layer. Experiment: D, Rh 2450; s, Rh 4230; Power-law ®t: A, Rh 2450; ± ± ±, Rh 4230; Log-law: A, Rh 2450; 4230.
region is much broader at the higher Rh , making it easier to implement the Clauser chart method. In the current study, the most appropriate values of j and B at the higher Reynolds number Rh 3000, are 0.44 and 6.13, respectively. While these are higher than the traditionally used values in turbulent boundary layers [13±15], they are similar to those obtained by Zagarola and Smits [16] for turbulent pipe ¯ow. The constants j and B are obtained from the best ®t to the experimental data, and there is no formal basis to obtain unique values for j and B. The value of us calculated from the power-law is approximately 3% and 8% higher than that calculated from the log-law at the lower and higher Rh , respectively. At the lower Rh , the power-law ®ts the data over a much broader region than the log law, and hence the power-law is employed to calculate us . On the other hand, at the higher Rh , the power-law deviates in the wake region, and the log region is much broader. This makes it easier to ®t the log-law, and therefore, at the higher Rh , the log-law is used to estimate us . The normalized skin friction coecient (Cf /Cf;0 ) distributions downstream of the groove are presented in Figs. 5 and 6 where Cf;0 is the local skin-friction coecient on the corresponding smooth-wall at the same x/w location (Fig. 4). At the lower Rh , the change in Cf /Cf;0 in the vicinity of the groove is very small (1.3%), and within the experimental uncertainty (Fig. 5). It is very likely that the relative groove size w=d1 0:066 at Rh 1000 is too small to perturb the boundary layer signi®cantly. At the higher Rh , a small sharp rise (5%) in Cf /Cf;0 immediately downstream of the groove is discernible, which is greater than the experimental uncertainty. The sharp rise in Cf /Cf;0 can be attributed to the local intense favorable pressure gradient that emanates from the downstream edge of the groove [4]. The sharp rise is followed by a decrease in Cf /Cf;0 below the smooth-wall value in the range 30 K x=w K 110. The decrease in Cf /Cf;0 is then followed by an oscillatory relaxation back to Cf;0 . At x=w J 180, Cf is essentially the same as Cf;0 . The results at the higher Rh are qualitatively similar to the results of Pearson et al. [4] and Elavarasan et al. [6]. The sharp rise in Cf /Cf;0 of the
Fig. 4. Skin friction coecient on the smooth-wall. Symbols: s, Rh 1000; h, Rh 3000. The lines are plotted only for convenience.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 5. Development of Cf /Cf;0 in the streamwise direction at Rh 1000. The line is plotted only for convenience.
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grooves. Nevertheless, the results provide some insight into the behavior of Cf due to a transverse square groove. If drag reduction is to be achieved, the sharp rise in Cf immediately downstream of the groove needs to be reduced. This could be achieved by optimizing the groove size and shape to reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove. The groove should, however, still provide the bene®cial undershoot in Cf . Also, the area under the undershoot should be greater than the area under the increase in Cf immediately downstream of the groove. The optimum groove spacing (s/w) is obtained when the next groove is located at the end of the undershoot, so that the oscillatory overshoot beyond that is eliminated. It is obvious that w/d1 plays an important role in the relaxation of Cf downstream of the groove and the parameter needs to be optimized to obtain the most bene®cial Cf distribution. Ching and Parsons [22] showed that the total surface drag, in addition to w/d1 , is also a function of Reynolds number and groove spacing to width ratio (s/w). The mean velocity, U U =uT , pro®les on the grooved-wall at Rh 1000 and 3000 are presented in Figs. 7 and 8, respectively. In both cases, the data compares well with the DNS pro®le, and are similar to the smooth wall pro®les (not shown for clarity). The eect of the groove on the velocity pro®les is negligible, and is probably due to the weak perturbation of the groove on the boundary layer. The above results are dierent from the results of Pearson et al. [4], and
Fig. 6. Development of Cf /Cf;0 in the streamwise direction at Rh 3000. The line is plotted only for convenience.
current study, however, is not as intense as in the study of Pearson et al. This dierence may be attributed to the dierence in w/d1 of the two studies. In the current study, w=d1 0:072 at Rh 3000, while in the study of Pearson et al. [4], w=d1 0:17 at Rh 1320. An approximate estimate of the drag can be obtained by integrating Cf along the streamwise direction. At the higher Rh , the drag on the grooved-wall is almost the same as that on the smooth-wall. This is because the increase in Cf immediately downstream of the groove is oset by the decrease in the range 40 K x=w K 100. In the present study, the friction and pressure drag on the groove walls was not measured. It may be assumed, however, to be negligibly small because of the very small size of the groove (w=d 0:07). Choi and Fujisawa [3], using w=d 0:4, found that the contribution of the pressure drag inside the groove to the total drag to be insigni®cant. They did not measure the skin friction distribution on the groove walls. Roshko [1] and Haugen and Dhanak [2] found that the skin friction drag due to the groove walls, on the other hand, is very small compared to the pressure drag. With the present data, it is dicult to make any conclusions on drag reduction by transverse square
Fig. 7. Mean velocity pro®les over a grooved-wall at Rh 1000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 8. Mean velocity pro®les over a grooved-wall at Rh 3000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
Fig. 9. Streamwise turbulence intensity pro®les over a grooved-wall at Rh 1000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
Elavarasan et al. [6]. In those two studies, there was a signi®cant shift in the U pro®les due to the presence of the groove. The reason for this is the large dierence in the magnitude of the overshoot and undershoot in Cf . In the current study, (Cf /Cf;0 )max is about 1.05, while (Cf / Cf;0 )min is about 0.975. On the other hand, (Cf /Cf;0 )max obtained by Pearson et al. [4] and Elavarasan et al. [6] is about 3.0, and 1.5, respectively, while (Cf /Cf;0 )min is about 0.5. This illustrates the signi®cant eect of w/d1 on Cf /Cf;0 . At the higher Rh of the present study, w/d1 is about 0.072, while Pearson et al. [4] and Elavarasan et al. [6] had w=d1 0:17 and 0.125, respectively. The eect of w/d on Cf /Cf;0 is illustrated in Table 3, and it may be conjectured, that w/d1 must be larger than 0.1 to signi®cantly aect the boundary layer. The streamwise turbulence intensity, u ( u/us ), pro®les along the streamwise direction for the groovedwall at Rh 1000 are shown in Fig. 9. As the streamwise distance increases, the experimental pro®les more closely match the DNS pro®le. This is because of the increase in Rh resulting in a better match between the experimental and DNS Rh , and also the relaxation of the boundary
layer after the perturbation due to the groove. For example, the pro®les at the last two measurement locations (x=w 181 and 401) are in good agreement with the DNS pro®le. For these pro®les, Rh is 1250 and 1650 compared to 1410 for the DNS data. While the eect of the groove on u is not discernible at the lower Rh , at the higher Rh , there is a slight decrease in u in the inner region in the vicinity of the groove (Fig. 10). At locations x=w ÿ5 and ÿ1, u is lower than the DNS data. A direct comparison between the smooth-and grooved-wall u pro®les at a location just downstream of the groove x=w 1 at Rh 1000 and 3000 reveal slight dierences between the two cases (Figs. 11 and 12). At both Rh , in the near-wall region y K 10; u on the grooved-wall is lower than the smooth-wall value at that location. The presence of the groove is likely to weaken the streamwise vorticity in the near-wall region, and result in a decrease in u at that x/w location. The groove attenuates the turbulence intensity in this region, and this eect seems to be con®ned to the near-wall region. In the region y J 10; u pro®les on the smoothand grooved-wall are indistinguishable. The overall
Table 3 Eect of w/d on (Cf /Cf;0 )max and (Cf /Cf;0 )min Pearsons et al. [4] Elavarasan et al. [6] Present study
Rh
w/d
(Cf /Cf;0 )max
(Cf /Cf;0 )min
1320 1300 3000
0.170 0.125 0.072
3.00 1.50 1.05
0.500 0.500 0.975
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
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Fig. 12. Streamwise turbulence intensity pro®les on the smooth- and grooved-wall at Rh 3000 and x/w 1. Symbols: s, smooth-wall; e, grooved-wall; AA, DNS [17], Rh 1410.
Fig. 10. Streamwise turbulence intensity pro®les over a grooved-wall at Rh 3000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
eect of the cavity on u , however, is very weak, because the ratio w/d1 is very small in this case. The streamwise distributions of the wake parameter (p) on the smooth-and grooved-wall at Rh 1000 and 3000 are shown in Figs. 13 and 14, respectively. The value of p is calculated using the following equation: j U1 1 us d ÿ ln ÿB : 7 p 2 us j m
Fig. 13. Wake parameter (p) distribution on the smooth- and groovedwall at Rh 1000. Symbols: s, smooth-wall; e, grooved-wall.
The constants j and B are 0.41 and 5.0, respectively, at Rh 1000, and 0.44 and 6.13, respectively, at Rh 3000. At the lower Rh , there is an increase in p with x for both the smooth- and grooved-wall, though
Fig. 14. Wake parameter (p) distribution on the smooth- and groovedwall at Rh 3000. Symbols: s, smooth-wall; e, grooved-wall.
Fig. 11. Streamwise turbulence intensity pro®les on the smooth- and grooved-wall at Rh 1000 and x/w 1. Symbols: s, smooth-wall; e, grooved-wall; ± ± ±, DNS [17], Rh 670; AA, Rh 1410.
the increase for the grooved-wall is less pronounced. For the smooth-wall, p increases from 0.3 to 0.5 as x/w increases from ÿ10 to 400, while the increase in the case of the grooved-wall is only very slight. Matsumoto [9] found that p was constant at 0.53 and 0.59 for a turbulent boundary layer over a sparse d-type surface with s/w 10 and 20, respectively. Choi and Fujisawa [3] found that p varied with x in a range 0:35 K p K 0:40 for a turbulent boundary layer downstream of a transverse square groove with w=d 0:4.
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
At Rh 3000, the value of p on the smooth-wall increases with x/w from 0.63 to 0.65. This is higher than the value at the lower Rh because Cf is lower at Rh 3000. Over the grooved-wall, there is a decrease in p in the vicinity of the groove. The decrease is followed by an increase and subsequent relaxation back to the smooth-wall value. There is a strong correlation between the p and Cf distributions (Fig. 6) over the grooved wall at the higher Rh . The value of p on the grooved wall is lower than that on the smooth-wall for x=w K 30, and higher in the range 30 K x=w K 100, which is completely opposite to the Cf /Cf;0 distribution. In the region x=w J 100, the p-distributions on the smooth- and grooved-wall are almost the same with a slight monotonic increase with x. The p-distribution at Rh 3000 corroborates the results of Matsumoto [9] and Tani et al. [7], that a reduction in Cf results in an increase in p and vice versa. Representative energy spectra on the smooth-and grooved-wall at the two Rh are compared in Fig. 15. The spectra are at the streamwise location immediately downstream of the cavity (x=w 1) and at the y-location where the turbulence intensity is maximum (y 13). There is no distinguishable dierence between the spectra for the two con®gurations at both Rh . As Rh increases, the high wave number component of energy increases, while the low wave number component decreases. It may be concluded that the groove has no signi®cant eect on the spectral distribution of the turbulent kinetic energy. The growth of the internal layer (di ) as a response of the turbulent boundary layer to the presence of the groove is shown in Fig. 16. In the present study, the height of the internal layer is de®ned as the location of the slope change of the mean velocity pro®le when U/U1 is plotted against y1=2 [6]. The mean velocity pro®le inside the internal layer near the step change is linear when plotted in the form U/U1 versus y1=2 . In the region outside of the internal layer, the velocity pro®le when plotted on the same axes will have a linear pro®le with much lower slope than that inside the internal layer. The
Fig. 15. Energy spectra of the streamwise velocity ¯uctuation at x/ w 1, and y 13. Symbols: At Rh 1000, ÿ ÿ ÿ ÿ, smooth-wall; ÿ ÿ A ÿ ÿ, grooved-wall; at Rh 3000, AAA, smooth-wall; A A A, grooved-wall.
Fig. 16. The internal layer growth on the grooved-wall at Rh 1000 and 3000: s, Rh 1000; e, Rh 3000. Lines are drawn only for convenience.
height of the internal layer in this instance was obtained using the ``knee'' method of Antonia and Luxton [23]. The internal layer at Rh 1000 is approximately twice as high as that at Rh 3000 (Fig. 16). In the range x=w K 7, there is a rapid growth of the internal layers, with (ddi /dx) about 0.093 and 0.041 mm/mm at the lower and higher Rh , respectively. The initial rate of growth of di at the lower Rh is almost twice that at the higher Rh . The subsequent rate of growth of di beyond x=w J 7, however, is much slower, and approximately the same at both Rh . The growth of the internal layers in the present study is compared to the data of Elavarasan et al. [6] in Fig. 17. The data are presented in a semilogarithmic scale to enhance the region immediately downstream of the cavity. While there is a dierence in the height of the internal layers, the rate of growth of di is approximately the same. 4. Practical signi®cance There has been a substantial research eort devoted to studying methods of turbulent skin friction drag
Fig. 17. The internal layer growth on the grooved-wall at Rh 1000 and 3000: s, Rh 1000; h, Rh 3000; D, data of Elavarasan et al. (1996, Rh 1300). Lines are drawn only for convenience.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
reduction. Both active and passive methods are currently being studied. The response of a turbulent boundary layer to a transverse square groove suggests that it may be possible to optimize the groove geometry and size to obtain a small drag reduction. A greater understanding of the interaction between the grooves and the boundary layer structure, however, needs to be obtained to determine the important scaling parameters. In addition, there have been several studies where momentum has been added to the boundary layer through 2-dimensional slots or grooves for active control of the boundary layer. If a static groove can be tuned to reduce drag, even slightly, it would reduce the overall energy requirements necessary for eective active control schemes. 5. Conclusions and recommendations The eect of a transverse square groove on a turbulent boundary layer under a zero pressure gradient has been studied at two dierent Rh (1000 and 3000). The main conclusions can be summarized as follows. (1) The smooth wall results indicate that it is more appropriate to use the power law to estimate us from the mean velocity pro®le at low Rh ( 6 1500). The log law, however, can be used more eectively at higher Rh . The values for the log law constants, j and B, at the higher Rh are 0.44 and 6.13. These are higher than those previously used in turbulent boundary layers, but are similar to those obtained by Zagarola and Smits [16] for turbulent pipe ¯ows. (2) The eect of the groove on Cf is insigni®cant at the lower Rh . At the higher Rh , the eect of the groove is more pronounced. There is an increase in Cf over the smooth-wall value immediately downstream of the groove, followed by a decrease and a subsequent oscillatory relaxation back to the smooth-wall value. While these results are qualitatively similar to those of Pearson et al. [4], the magnitude of the increase is much less in the present study. This can be attributed to the smaller w/d1 of the present study. (3) While it is dicult to make any conclusions on the overall drag, the Cf distribution suggests that if drag reduction is to be achieved, the sharp rise in Cf immediately downstream of the groove needs to be reduced. This could be achieved by optimizing the groove size and shape to reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove. The groove should, however, still provide the bene®cial undershoot in Cf . Also, the area under the undershoot should be greater than the area under the increase in Cf immediately downstream of the groove. The optimum groove spacing (s/w) is obtained when the next groove is located at the end of the undershoot, so that the oscillatory overshoot beyond that is eliminated. (4) The mean velocity pro®les are not aected by the presence of the groove at both Rh . The ratio w/d1 is probably too small to alter the mean velocity pro®les. It
9
may be conjectured that w/d1 must be greater than 0.1 for it to have a signi®cant eect. (5) The streamwise turbulence intensity is reduced over the grooved-wall in the inner region (y K 10) at both Rh . The groove attenuates the turbulence intensity in this region, and this eect seems to be con®ned to the near-wall region. The location of u max remains unchanged at y 13 for both the smooth-and groovedwall at both Rh . (6) There is a strong correlation between the wake parameter (p) and Cf distributions at Rh 3000. This correlation is similar to that obtained by Matsumoto [9], where p increases as Cf decrease. (7) There is no signi®cant eect of the groove on the energy spectra, at least at the location where the turbulence intensity is maximum. The eect of Rh on the spectra is to increase the high wave number component of energy, and to reduce the low wave number component as Rh increases. (8) The height of the internal layer at the lower Rh is approximately twice as that at the higher Rh . The internal layer grows rapidly immediately downstream of the groove (x=w K 7), with the initial rate of growth at the lower Rh almost twice that at the higher Rh . Beyond x/w 7, the growth of the internal layer is much slower, with the rates being almost equal at both Rh . In the present study, the eect of the groove on the skin friction, mean velocity and turbulence intensity is less pronounced compared with the studies of Pearson et al. [4], Elavarasan et al. [6], and Choi and Fujisawa [3]. This dierence may be attributed to the dierence in w/d1 of the present study and the previous three studies. In the present study, w=d1 0:072, while in the previous three studies, w=d1 0:17, 0.125, and 0.4, respectively. It is important to determine whether there is an optimum w/d1 for the most bene®cial Cf distribution. It is also desirable to determine the eect of dierent groove shapes and Reynolds numbers, especially in the context of reducing the sharp increase in Cf immediately downstream of the groove. While keeping the basic dimensions of the groove (d and w) constant, dierent groove shapes as suggested in Fig. 18 can be investigated. This may potentially reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove and reduce the sharp rise in Cf immediately downstream of the groove.
Fig. 18. Alternative groove shapes.
10
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Nomenclature B y-intercept on the Clauser-chart, dimensionless 2 Cf skin friction coecient, (Cf 2sw = qU1 )), dimensionless Cf;0 skin friction coecient on the smooth-wall, dimensionless d groove depth, m E(k1 ) turbulence energy spectra, m3 /s2 k1 one-dimensional wave length, m Reynolds number based on Rd d Rd U1 d=m), dimensionless Rh Reynolds number based on h Rh U1 h=m, dimensionless s distance between grooves, m U mean velocity in the x-direction, m/s U normalized streamwise mean velocity (U/ us ), dimensionless U1 free stream velocity, m/s u ¯uctuation of velocity in the x-direction, m/s u normalized streamwise ¯uctuating velocity (u/us ), dimensionless us friction velocity ( sw =q0:5 , m/s w groove width, m x streamwise coordinate measured from the groove trailing edge, m y wall-normal coordinate, m y normalized wall-normal coordinate (yus /m), dimensionless Greek Symbols d boundary layer thickness, m d1 boundary layer thickness at x=w 1, m di internal layer thickness, m j Karman constant, dimensionless m kinematic viscosity, m2 /s p wake parameter, dimensionless h momentum thickness, m sw wall shear stress, N/m2
References [1] A. Roshko, Some measurements of ¯ow in a rectangular cut-out, NACA Technical Note 3488, 1955. [2] R.L. Haugen, A.M. Dhanak, Momentum transfer in turbulent separated ¯ow past a rectangular cavity, J. App. Mech. 33 (1966) 641±646. [3] K.S. Choi, N. Fujisawa, Possibility of drag reduction using d-type roughness, Appl. Sci. Res. 50 (1993) 315±324.
[4] B.R. Pearson, R. Elavarasan, R.A. Antonia, The response of a turbulent boundary layer to a square groove, J. Fluids Eng. 119 (1997) 466±469. [5] C.Y. Ching, R. Elavarasan, R.A. Antonia, Visualization of near wall region in a turbulent boundary layer over a d-type roughness, in: J.P. Crowder (Ed.), Proceedings of the Seventh International Symposium on Flow Visualization, Seatle, 1995, pp. 398±403. [6] R. Elavarasan, C.Y. Ching, R.A. Antonia, Turbulent boundary layer over a smooth wall with widely separated transverse square cavities, Appl. Sci. Res. 55 (1996) 227±243. [7] I. Tani, H. Munakata, A. Matsumoto, K. Abe, Turbulence management by groove roughness, in: H.W. Liepmann, R. Narasimha (Eds.), Turbulence Management and Relaminarisation, IUTAM Symposium, Bangalore, India, 1987, pp. 161±172. [8] E. Coustols, A.M. Savill, Turbulent skin friction drag reduction by active and passive means special course on skin friction drag reduction, AGARD Report 786 (1991) 8.1±8.80. [9] A. Matsumoto, Some features of turbulent boundary layers over grooved rough walls, Trans. Jpn Soc. Aeronautical Space Sci. 37 (115) (1994) 27±41. [10] G.I. Barenblatt, Scaling laws for fully developed shear ¯ows: Part 1. Basic hypotheses and analysis, J. Fluid Mech. 248 (1993) 521± 529. [11] C.Y. Ching, L. Djenidi, R.A. Antonia, Low-Reynolds-number eects in a turbulent boundary layer, Experiments in Fluids 19 (1995) 61±68. [12] L. Djenidi, Y. Dubief, R.A. Antonia, Advantages of using a power law in a low Rh turbulent boundary layer, Experiments in Fluids 22 (1997) 348±350. [13] F.M. White, Fluid mechanics, 2nd ed., McGraw-Hill, New York, 1986. [14] B.L. Sill, Velocity pro®les in the turbulent boundary layers, in: R.E.A. Arndt, H.G. Stefan, C. Farell, S.M. Peterson (Eds.), Advancements in Aerodynamics Fluid Mechanics and Hydraulics, ASCE, New York, 1986, pp. 9±18. [15] A.J. Reynolds, Turbulent ¯ows in engineering, 1st ed., Wiley, London, 1974. [16] M.V. Zagarola, A.J. Smits, Scaling of the mean velocity pro®le for turbulent pipe ¯ow, Phys. Rev. Lett. 78 (2) (1997) 239±242. [17] P.R. Spalart, Direct simulation of a turbulent boundary layer up to Rh 1410, J. Fluid Mech. 187 (1988) 61±98. [18] S. Yavuzkurt, A guide to uncertainty analysis of hot-wire data, J. Fluids Eng. 106 (1984) 181±186. [19] R.J. Moat, Contributions to the theory of single sample uncertainty analysis, J. Fluids Eng. 104 (1982) 250±260. [20] H.W. Coleman, G.W. Steele Jr, Experimentation and Uncertainty Analysis for Engineers, Wiley, New York, 1989. [21] M.V. Zagarola, A.E. Perry, A.J. Smits, Log laws or power laws: the scaling in the overlap region, Phys. Fluids 9 (7) (1997) 2094± 2100. [22] C.Y. Ching, B.L. Parsons, Drag characteristics of a turbulent boundary layer over a ¯at plate with transverse square grooves, Experiments in Fluids 26 (3) (1999) 273±275. [23] R.A. Antonia, R.E. Luxton, The response of a turbulent boundary layer to a step change in surface roughness: Part 1, Smooth to rough, J. Fluid Mech. 48 (4) (1971) 721±761.
www.elsevier.nl/locate/etfs
Eect of a transverse square groove on a turbulent boundary layer Sutardi, C.Y. Ching
*
Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. JohnÕs, NF, Canada A1B 3X5 Received 23 November 1998; accepted 13 July 1999
Abstract The eect of a transverse square groove on a turbulent boundary layer has been investigated at two values of momentum thickness Reynolds numbers Rh 1000 and 3000 using hot-wire anemometry. The ratio of the groove width to the boundary layer thickness (w/d1 ) is approximately 0.07. The wall shear stress (sw ) is estimated from the velocity pro®les using the Clauser chart method and by assuming a power law velocity distribution. The smooth wall results indicate the Clauser chart is better suited at higher Rh , whereas the power law is more suited at lower Rh . The eect of the groove on sw at the lower Rh is not signi®cant. At the higher Rh , there is a sudden increase in sw just downstream of the groove, followed by an undershoot and an oscillatory relaxation back to the smooth wall value. The mean velocity pro®les (U ) are not aected by the presence of the groove at both Rh . There is, however, a small reduction in the turbulence intensity in the near-wall region y K 10 at both Rh . There is a strong correlation between the sw and wake parameter (p) distributions, with an increase in p for a decrease in sw . The growth of the internal layer immediately downstream of the groove is very rapid at both Rh , followed by a much slower growth beyond x=w J 7. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Transverse square groove; Turbulence; Square groove; Boundary layer
1. Introduction Despite being studied for many years, there are many aspects of turbulent boundary layers that are still not fully understood. It has been suggested that transverse square grooves, optimally sized and located, could result in a reduction of skin friction drag. To determine the most important scaling factors, however, the eect of a single transverse square groove on the turbulent boundary layer needs to be clearly understood. Roshko [1] and Haugen and Dhanak [2] investigated the eect of a rectangular groove, with depth to width ratio (d/w) ranging up to 3.0, on a turbulent boundary layer. In both studies, the groove width was much larger than the boundary layer thickness 4:0 6 w=d 6 10:0, and the objective was to study the turbulent momentum transfer mechanism in the separated ¯ow region. Their results indicate that the contribution of the pressure drag to the total drag is very signi®cant, while the friction drag on the main surface and the groove wall is relatively small. The de¯ection of the boundary layer separation into the groove is believed responsible for *
Corresponding author. Fax: +1-709-737-4042. E-mail address: [email protected] (C.Y. Ching)
the formation of the high-pressure coecient on the down stream edge of the groove wall. Haugen and Dhanak [2] found a single stable vortex inside the groove if the ratio d/w equals unity. When d/w is increased, the number of vortices inside the groove was found to be of the order of the d/w ratio. In these studies, the entire boundary layer is aected because of the very large size of the groove relative to the boundary layer thickness. More recently, there have been several studies on the perturbation of much smaller grooves on a turbulent boundary layer to determine the eect of the groove on the near-wall turbulence structure. Choi and Fujisawa [3] studied the eect of a transverse square groove with w=d 0:4 and reported a skin-friction drag reduction of about 1%, while ignoring the pressure drag on the groove walls. They found a small sudden decrease in the wall shear stress (sw ) just downstream of the groove. The relaxation of sw back to the smooth wall value occurred at about 100w downstream from the groove trailing edge. The turbulence intensity above the groove was found to be higher than that for a corresponding smooth wall, while the wake parameter (p) was higher than that on a smooth wall for x=w 6 100. Pearson et al. [4] investigated the shear stress distribution downstream of a single square groove. They deduced that there was an
0894-1777/99/$ - see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 9 ) 0 0 0 3 1 - X
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
overshoot in sw just down stream of the groove, followed by an undershoot and an oscillatory relaxation back to the smooth wall value. Through ¯ow visualization, they determined that there were out ¯ows (ejections) from and in¯ows to the groove. The ¯ow visualization results were qualitatively similar to those of Ching et al. [5] and Elavarasan et al. [6]. The ejections of ¯uid from the groove are very likely due to the passage of the quasi-streamwise vortices over the groove. The pressure minimum at the center of these vortices is likely to ``pump'' the ¯uid out of the groove. Elavarasan et al., [6] found an increase in friction drag of about 3.4% for a surface with grooves spaced s=w 20 apart, where s is the groove spacing. Although there was an increase in total friction drag, a local decrease in sw was observed immediately downstream of the groove, similar to the observations of Pearson et al. [4]. Tani et al. [7] found that the skin friction on a smooth surface with grooves spaced 10 to 40w apart (hereafter referred to as a sparse d-type surface) was lower than that on a corresponding smooth-wall over a certain range of Reynolds numbers 8:4 105 K Rx K 15 105 . The grooves were found to reduce the total drag by up to 3%, at least in the low Reynolds number range [7]. Coustols and Savill [8] reported that a 2±3% net drag reduction could be achieved if the grooves are spaced 20w apart. The bene®cial modi®cation of the near-wall turbulence structure, such as a weakening of the quasi-streamwise vortices as they pass over the groove, is believed to be responsible for the skin friction drag reduction. Matsumoto [9] investigated turbulent boundary layers over sparse d-type surfaces, and suggested a possibility of skin friction drag reduction for the cases s=w 10 and 20. To obtain a skin friction drag reduction, the spacing s/w was found to be a function of Rh . For example, for s=w 10, a drag reduction was obtained for Rh < 3700, whereas for s=w 20, a reduction in Cf was determined for Rh < 5200. The drag calculations of Elavarasan et al. [6] and Choi and Fujisawa [3] are contradictory, highlighting the need for a more detailed parametric study of sparse d-type surfaces. Also, no studies have been performed to determine an optimum groove pro®le. Clearly, the relative size and shape of the transverse groove will have a signi®cant eect on the near-wall turbulence structure of the overlying shear layer. While sharp upper corners may be necessary, rounding of the base may reduce the cavity ¯ow losses. Additional data, especially on the signi®cance of w/d to the perturbation of the boundary layer is desirable. The objective of the present study is to determine the eect of a single transverse square groove on the characteristics of a turbulent boundary layer over a ¯at plate. Experiments were performed in a low-speed wind tunnel using hot-wire anemometry at two dierent Reynolds numbers. The eects of the groove on the skin friction, mean velocity, and turbulence intensity are determined. The turbulent wake parameter and development of the internal layer downstream of the groove are also studied.
2. Experimental details and methodology The experiments were performed in an open-circuit low-speed wind tunnel at Memorial University of Newfoundland. The test section is 1 m ´ 1 m and is over 20 m long. The roof of the tunnel is adjusted to maintain a zero pressure gradient along the test section. Only the initial 5.5 m length of the test section was used for this investigation. A centrifugal blower driven by a 19 kW motor is used in the wind tunnel. The air passes through a screened diuser and a large settling chamber with three single-piece precision screens. The air is accelerated into the test section through a 5:1 contraction. The maximum free stream velocity in the test section is about 15 m/s. The free stream turbulence intensity is less than 0.5% at all velocities. The velocity in the test section is changed using motorized variable angle inlet vanes on the blower. Experiments were performed at two free stream velocities (U1 ) of 2 and 5.5 m/s, corresponding to Reynolds numbers, based on the momentum thickness just upstream of the groove, of 1000 and 3000, respectively. The boundary layer thickness (d) at x=w 1 is 76 and 69 mm at U1 2:0 and 5.5 m/s, corresponding to a Rd of 9800 and 24 500, respectively. At each free stream velocity, measurements were made on a smooth wall ¯at plate and a ¯at plate with a single transverse square groove (hereafter referred to as the grooved wall). A roughness strip, consisting of a 100 mm wide sand paper (series 0811) and a 1.5 mm diameter cylindrical rod was used to trip the boundary layer at the leading edge of the plate (Fig. 1). The ¯at plate is made of 25 mm thick acrylic and is mounted horizontally on the ¯oor of the wind tunnel. After the initial measurements on the smooth wall, a transverse square groove (w/d 1, w 5 mm) was cut on the plate at a distance of 2635 mm from the leading edge. The present experimental conditions and ¯ow parameters together with those previous experiments related to the present study are given in Table 1. Hot-wire anemometry is used for the velocity measurements. The hot-wire probe is traversed in the wallnormal direction using a specially designed traversing mechanism using a Mitutoyo height gauge. The traverse is installed on rails mounted on the roof of the tunnel. This traverse has a maximum span of approximately 46 cm and a minimum linear division of 0.01 mm.
Fig. 1. Schematic of test-plate showing a single transverse square groove.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
3
Table 1 Experimental conditions and ¯ow parametersa
U1 (m/s) Rh w/d1 d/w a
Choi and Fujisawa [3]
Elavarasan et al. [6]
Pearson et al. [4]
7.0 ~1200 ~0.4 1.0
0.4 1300 ~0.125 1.0
0.4 1320 ~0.17 1.0
Present study Low velocity
High velocity
2.0 1000 0.066 1.0
5.5 3000 0.072
Note: d1 is the boundary layer thickness at x/w 1.
A 5-lm DANTEC 55P05 boundary layer type probe was used for the boundary layer measurements. The velocity measurements were made using a DANTEC 55M01 standard bridge. The hot-wire signals were digitized using a 12 bit Keithley 570 System Analog to Digital (A/D) converter, interfaced to a 486 DX-AT personal computer. For the hot-wire calibrations, the velocity was measured using a pitot-static tube connected to a dierential pressure transducer (Furness type FCO34) with a range 0.00±0.254 cm water. The wall shear stress (sw ) was estimated from the velocity pro®les using the Clauser-chart (log-law) method and by assuming a power-law velocity distribution. An accurate estimation of the wall shear stress is necessary to predict the skin friction drag associated with boundary layers. The relation between the wall shear stress (sw ), and the friction velocity (us ) can be expressed as 2
sw q us :
1
Following BarenblattÕs [10] argument of incomplete similarity, Ching et al. [11] showed that us can be expressed as: ( 1= 1a ) 1 exp 3=2a us U1 ; 2 exp 3=2a C where a and C are calculated as follows: a
3 ; 2 ln Rd
where j and B are the Karman and the smooth-wall constants, respectively, and are assumed independent of the Reynolds number. White [13] proposed 0.41 and 5.0 for j and B, respectively. Sill [14] found that the experimental results give B 5:0 to 5.5, while Reynolds [15] proposed j to be 0.385 10%. Zagarola and Smits [16], however, found slightly higher values of j 0.436 and B 6.13, for a turbulent pipe ¯ow. The wall shear stress can also be calculated from the mean velocity gradient at the wall. oU sw l 6 : oy y0
This method, however, requires accurate velocity data very close to the wall (y K 3:0. Due to the eect of heat conduction from the hot-wire to the wall, this results in a spurious increase in U above the sub-layer relation U y , and cannot be used in the present study. A complete uncertainty analysis of the results was performed using the method outlined by Yavuzkurt [18], Moat [19], and Coleman and Steele [20]. The uncertainties are calculated at 95% con®dence and are given in Table 2. 3. Results and discusions
3
1 5 4 C p lnRd : 2 3 Djenidi et al. [12] showed that us determined from Eq. (2) was within 0.57% to that obtained from Preston tube measurements for Rh 940. It was argued that at low Reynolds numbers Rh K 1500, say), the eect of viscosity is not insigni®cant, and a power-law velocity distribution is more appropriate than a log-law velocity distribution. The Clauser-chart method assumes a universal mean velocity pro®le in the overlap region. The Clauser-chart method is only appropriate if the Reynolds number is suciently high Rh J 1500) for a log-law region to exist. The general form of the log-law is u y U 1 s ln B; 5 us j m
The smooth-wall U pro®les at the lower Rh are compared with the power-law distributions (only two pro®les are shown for clarity) in Fig. 2. There is a good collapse of the pro®les, and the power-law is in good agreement with the experimental data in the entire outer region. In the inner region y K 30, the power law distribution is not valid, and cannot be used to describe the mean velocity pro®les. In this ®gure, U and y are normalized using us deduced from the power-law. The Table 2 Experimental uncertainties Flow variable Mean velocity (U) Fluctuating velocity (u) Boundary layer thickness (d) Skin friction coecient (Cf )
Uncertainty (%) Rh 1000
Rh 3000
3.09 3.09 0.62 9.25
1.40 1.40 0.28 4.73
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 2. Power-law and log-law ®ts to mean velocity pro®les in the overlap region for a smooth-wall boundary layer. Experiment: D, Rh 830; h, Rh 1250; Power-law ®t: A, Rh 830; ± ± ±, Rh 1250; Log-law ®t: A, Rh 830; 1250; DNS [17]: ± ± ±, Rh 670 ; ± ± , Rh 1410.
experimental data are also in good agreement with the smooth-wall DNS data of Spalart [17], and the small dierences in the outer region could be attributed to the dierence in Reynolds numbers. It is more dicult to use the Clauser chart method at the lower Rh since the log region in this case is very narrow [11,17]. The log-law ®ts the data well only in the range 30 K y K 100, and leads to a higher uncertainty in the estimation of sw using the log-law in this case. The present results support the incomplete similarity hypothesis of Barenblatt [10], which states that at low Reynolds numbers, the friction velocity can be deduced from the power-law distribution in the overlap region [12]. Unlike for the lower Rh , the power-law ®t, especially in the wake region, is poor at the higher Rh (Fig. 3). In this Reynolds number range Rh > 2000, it is dicult to obtain a good ®t to the experimental data using a power-law approximation. The estimation of sw using the power law method is based on the extrapolation of the power law ®t into the boundary layer thickness (d). The power-law ®ts the data well only in the range 30 K y K 300. This is in agreement with the pipe ¯ow data of Zagarola et al. [21], in which the power-law ®ts the data well only in the range 50 K y K 500. The log
Fig. 3. Power-law and log-law ®ts to mean velocity pro®les in the overlap region for a smooth-wall boundary layer. Experiment: D, Rh 2450; s, Rh 4230; Power-law ®t: A, Rh 2450; ± ± ±, Rh 4230; Log-law: A, Rh 2450; 4230.
region is much broader at the higher Rh , making it easier to implement the Clauser chart method. In the current study, the most appropriate values of j and B at the higher Reynolds number Rh 3000, are 0.44 and 6.13, respectively. While these are higher than the traditionally used values in turbulent boundary layers [13±15], they are similar to those obtained by Zagarola and Smits [16] for turbulent pipe ¯ow. The constants j and B are obtained from the best ®t to the experimental data, and there is no formal basis to obtain unique values for j and B. The value of us calculated from the power-law is approximately 3% and 8% higher than that calculated from the log-law at the lower and higher Rh , respectively. At the lower Rh , the power-law ®ts the data over a much broader region than the log law, and hence the power-law is employed to calculate us . On the other hand, at the higher Rh , the power-law deviates in the wake region, and the log region is much broader. This makes it easier to ®t the log-law, and therefore, at the higher Rh , the log-law is used to estimate us . The normalized skin friction coecient (Cf /Cf;0 ) distributions downstream of the groove are presented in Figs. 5 and 6 where Cf;0 is the local skin-friction coecient on the corresponding smooth-wall at the same x/w location (Fig. 4). At the lower Rh , the change in Cf /Cf;0 in the vicinity of the groove is very small (1.3%), and within the experimental uncertainty (Fig. 5). It is very likely that the relative groove size w=d1 0:066 at Rh 1000 is too small to perturb the boundary layer signi®cantly. At the higher Rh , a small sharp rise (5%) in Cf /Cf;0 immediately downstream of the groove is discernible, which is greater than the experimental uncertainty. The sharp rise in Cf /Cf;0 can be attributed to the local intense favorable pressure gradient that emanates from the downstream edge of the groove [4]. The sharp rise is followed by a decrease in Cf /Cf;0 below the smooth-wall value in the range 30 K x=w K 110. The decrease in Cf /Cf;0 is then followed by an oscillatory relaxation back to Cf;0 . At x=w J 180, Cf is essentially the same as Cf;0 . The results at the higher Rh are qualitatively similar to the results of Pearson et al. [4] and Elavarasan et al. [6]. The sharp rise in Cf /Cf;0 of the
Fig. 4. Skin friction coecient on the smooth-wall. Symbols: s, Rh 1000; h, Rh 3000. The lines are plotted only for convenience.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 5. Development of Cf /Cf;0 in the streamwise direction at Rh 1000. The line is plotted only for convenience.
5
grooves. Nevertheless, the results provide some insight into the behavior of Cf due to a transverse square groove. If drag reduction is to be achieved, the sharp rise in Cf immediately downstream of the groove needs to be reduced. This could be achieved by optimizing the groove size and shape to reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove. The groove should, however, still provide the bene®cial undershoot in Cf . Also, the area under the undershoot should be greater than the area under the increase in Cf immediately downstream of the groove. The optimum groove spacing (s/w) is obtained when the next groove is located at the end of the undershoot, so that the oscillatory overshoot beyond that is eliminated. It is obvious that w/d1 plays an important role in the relaxation of Cf downstream of the groove and the parameter needs to be optimized to obtain the most bene®cial Cf distribution. Ching and Parsons [22] showed that the total surface drag, in addition to w/d1 , is also a function of Reynolds number and groove spacing to width ratio (s/w). The mean velocity, U U =uT , pro®les on the grooved-wall at Rh 1000 and 3000 are presented in Figs. 7 and 8, respectively. In both cases, the data compares well with the DNS pro®le, and are similar to the smooth wall pro®les (not shown for clarity). The eect of the groove on the velocity pro®les is negligible, and is probably due to the weak perturbation of the groove on the boundary layer. The above results are dierent from the results of Pearson et al. [4], and
Fig. 6. Development of Cf /Cf;0 in the streamwise direction at Rh 3000. The line is plotted only for convenience.
current study, however, is not as intense as in the study of Pearson et al. This dierence may be attributed to the dierence in w/d1 of the two studies. In the current study, w=d1 0:072 at Rh 3000, while in the study of Pearson et al. [4], w=d1 0:17 at Rh 1320. An approximate estimate of the drag can be obtained by integrating Cf along the streamwise direction. At the higher Rh , the drag on the grooved-wall is almost the same as that on the smooth-wall. This is because the increase in Cf immediately downstream of the groove is oset by the decrease in the range 40 K x=w K 100. In the present study, the friction and pressure drag on the groove walls was not measured. It may be assumed, however, to be negligibly small because of the very small size of the groove (w=d 0:07). Choi and Fujisawa [3], using w=d 0:4, found that the contribution of the pressure drag inside the groove to the total drag to be insigni®cant. They did not measure the skin friction distribution on the groove walls. Roshko [1] and Haugen and Dhanak [2] found that the skin friction drag due to the groove walls, on the other hand, is very small compared to the pressure drag. With the present data, it is dicult to make any conclusions on drag reduction by transverse square
Fig. 7. Mean velocity pro®les over a grooved-wall at Rh 1000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Fig. 8. Mean velocity pro®les over a grooved-wall at Rh 3000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
Fig. 9. Streamwise turbulence intensity pro®les over a grooved-wall at Rh 1000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
Elavarasan et al. [6]. In those two studies, there was a signi®cant shift in the U pro®les due to the presence of the groove. The reason for this is the large dierence in the magnitude of the overshoot and undershoot in Cf . In the current study, (Cf /Cf;0 )max is about 1.05, while (Cf / Cf;0 )min is about 0.975. On the other hand, (Cf /Cf;0 )max obtained by Pearson et al. [4] and Elavarasan et al. [6] is about 3.0, and 1.5, respectively, while (Cf /Cf;0 )min is about 0.5. This illustrates the signi®cant eect of w/d1 on Cf /Cf;0 . At the higher Rh of the present study, w/d1 is about 0.072, while Pearson et al. [4] and Elavarasan et al. [6] had w=d1 0:17 and 0.125, respectively. The eect of w/d on Cf /Cf;0 is illustrated in Table 3, and it may be conjectured, that w/d1 must be larger than 0.1 to signi®cantly aect the boundary layer. The streamwise turbulence intensity, u ( u/us ), pro®les along the streamwise direction for the groovedwall at Rh 1000 are shown in Fig. 9. As the streamwise distance increases, the experimental pro®les more closely match the DNS pro®le. This is because of the increase in Rh resulting in a better match between the experimental and DNS Rh , and also the relaxation of the boundary
layer after the perturbation due to the groove. For example, the pro®les at the last two measurement locations (x=w 181 and 401) are in good agreement with the DNS pro®le. For these pro®les, Rh is 1250 and 1650 compared to 1410 for the DNS data. While the eect of the groove on u is not discernible at the lower Rh , at the higher Rh , there is a slight decrease in u in the inner region in the vicinity of the groove (Fig. 10). At locations x=w ÿ5 and ÿ1, u is lower than the DNS data. A direct comparison between the smooth-and grooved-wall u pro®les at a location just downstream of the groove x=w 1 at Rh 1000 and 3000 reveal slight dierences between the two cases (Figs. 11 and 12). At both Rh , in the near-wall region y K 10; u on the grooved-wall is lower than the smooth-wall value at that location. The presence of the groove is likely to weaken the streamwise vorticity in the near-wall region, and result in a decrease in u at that x/w location. The groove attenuates the turbulence intensity in this region, and this eect seems to be con®ned to the near-wall region. In the region y J 10; u pro®les on the smoothand grooved-wall are indistinguishable. The overall
Table 3 Eect of w/d on (Cf /Cf;0 )max and (Cf /Cf;0 )min Pearsons et al. [4] Elavarasan et al. [6] Present study
Rh
w/d
(Cf /Cf;0 )max
(Cf /Cf;0 )min
1320 1300 3000
0.170 0.125 0.072
3.00 1.50 1.05
0.500 0.500 0.975
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
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Fig. 12. Streamwise turbulence intensity pro®les on the smooth- and grooved-wall at Rh 3000 and x/w 1. Symbols: s, smooth-wall; e, grooved-wall; AA, DNS [17], Rh 1410.
Fig. 10. Streamwise turbulence intensity pro®les over a grooved-wall at Rh 3000. Symbols: ´, x=w ÿ11; , ÿ5; n, ÿ1; e, 0.4; d, 1; D, 1.8; +, 13; h, 33; r, 81; N, 181; s, 401; AA, DNS [17], Rh 1410.
eect of the cavity on u , however, is very weak, because the ratio w/d1 is very small in this case. The streamwise distributions of the wake parameter (p) on the smooth-and grooved-wall at Rh 1000 and 3000 are shown in Figs. 13 and 14, respectively. The value of p is calculated using the following equation: j U1 1 us d ÿ ln ÿB : 7 p 2 us j m
Fig. 13. Wake parameter (p) distribution on the smooth- and groovedwall at Rh 1000. Symbols: s, smooth-wall; e, grooved-wall.
The constants j and B are 0.41 and 5.0, respectively, at Rh 1000, and 0.44 and 6.13, respectively, at Rh 3000. At the lower Rh , there is an increase in p with x for both the smooth- and grooved-wall, though
Fig. 14. Wake parameter (p) distribution on the smooth- and groovedwall at Rh 3000. Symbols: s, smooth-wall; e, grooved-wall.
Fig. 11. Streamwise turbulence intensity pro®les on the smooth- and grooved-wall at Rh 1000 and x/w 1. Symbols: s, smooth-wall; e, grooved-wall; ± ± ±, DNS [17], Rh 670; AA, Rh 1410.
the increase for the grooved-wall is less pronounced. For the smooth-wall, p increases from 0.3 to 0.5 as x/w increases from ÿ10 to 400, while the increase in the case of the grooved-wall is only very slight. Matsumoto [9] found that p was constant at 0.53 and 0.59 for a turbulent boundary layer over a sparse d-type surface with s/w 10 and 20, respectively. Choi and Fujisawa [3] found that p varied with x in a range 0:35 K p K 0:40 for a turbulent boundary layer downstream of a transverse square groove with w=d 0:4.
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
At Rh 3000, the value of p on the smooth-wall increases with x/w from 0.63 to 0.65. This is higher than the value at the lower Rh because Cf is lower at Rh 3000. Over the grooved-wall, there is a decrease in p in the vicinity of the groove. The decrease is followed by an increase and subsequent relaxation back to the smooth-wall value. There is a strong correlation between the p and Cf distributions (Fig. 6) over the grooved wall at the higher Rh . The value of p on the grooved wall is lower than that on the smooth-wall for x=w K 30, and higher in the range 30 K x=w K 100, which is completely opposite to the Cf /Cf;0 distribution. In the region x=w J 100, the p-distributions on the smooth- and grooved-wall are almost the same with a slight monotonic increase with x. The p-distribution at Rh 3000 corroborates the results of Matsumoto [9] and Tani et al. [7], that a reduction in Cf results in an increase in p and vice versa. Representative energy spectra on the smooth-and grooved-wall at the two Rh are compared in Fig. 15. The spectra are at the streamwise location immediately downstream of the cavity (x=w 1) and at the y-location where the turbulence intensity is maximum (y 13). There is no distinguishable dierence between the spectra for the two con®gurations at both Rh . As Rh increases, the high wave number component of energy increases, while the low wave number component decreases. It may be concluded that the groove has no signi®cant eect on the spectral distribution of the turbulent kinetic energy. The growth of the internal layer (di ) as a response of the turbulent boundary layer to the presence of the groove is shown in Fig. 16. In the present study, the height of the internal layer is de®ned as the location of the slope change of the mean velocity pro®le when U/U1 is plotted against y1=2 [6]. The mean velocity pro®le inside the internal layer near the step change is linear when plotted in the form U/U1 versus y1=2 . In the region outside of the internal layer, the velocity pro®le when plotted on the same axes will have a linear pro®le with much lower slope than that inside the internal layer. The
Fig. 15. Energy spectra of the streamwise velocity ¯uctuation at x/ w 1, and y 13. Symbols: At Rh 1000, ÿ ÿ ÿ ÿ, smooth-wall; ÿ ÿ A ÿ ÿ, grooved-wall; at Rh 3000, AAA, smooth-wall; A A A, grooved-wall.
Fig. 16. The internal layer growth on the grooved-wall at Rh 1000 and 3000: s, Rh 1000; e, Rh 3000. Lines are drawn only for convenience.
height of the internal layer in this instance was obtained using the ``knee'' method of Antonia and Luxton [23]. The internal layer at Rh 1000 is approximately twice as high as that at Rh 3000 (Fig. 16). In the range x=w K 7, there is a rapid growth of the internal layers, with (ddi /dx) about 0.093 and 0.041 mm/mm at the lower and higher Rh , respectively. The initial rate of growth of di at the lower Rh is almost twice that at the higher Rh . The subsequent rate of growth of di beyond x=w J 7, however, is much slower, and approximately the same at both Rh . The growth of the internal layers in the present study is compared to the data of Elavarasan et al. [6] in Fig. 17. The data are presented in a semilogarithmic scale to enhance the region immediately downstream of the cavity. While there is a dierence in the height of the internal layers, the rate of growth of di is approximately the same. 4. Practical signi®cance There has been a substantial research eort devoted to studying methods of turbulent skin friction drag
Fig. 17. The internal layer growth on the grooved-wall at Rh 1000 and 3000: s, Rh 1000; h, Rh 3000; D, data of Elavarasan et al. (1996, Rh 1300). Lines are drawn only for convenience.
Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
reduction. Both active and passive methods are currently being studied. The response of a turbulent boundary layer to a transverse square groove suggests that it may be possible to optimize the groove geometry and size to obtain a small drag reduction. A greater understanding of the interaction between the grooves and the boundary layer structure, however, needs to be obtained to determine the important scaling parameters. In addition, there have been several studies where momentum has been added to the boundary layer through 2-dimensional slots or grooves for active control of the boundary layer. If a static groove can be tuned to reduce drag, even slightly, it would reduce the overall energy requirements necessary for eective active control schemes. 5. Conclusions and recommendations The eect of a transverse square groove on a turbulent boundary layer under a zero pressure gradient has been studied at two dierent Rh (1000 and 3000). The main conclusions can be summarized as follows. (1) The smooth wall results indicate that it is more appropriate to use the power law to estimate us from the mean velocity pro®le at low Rh ( 6 1500). The log law, however, can be used more eectively at higher Rh . The values for the log law constants, j and B, at the higher Rh are 0.44 and 6.13. These are higher than those previously used in turbulent boundary layers, but are similar to those obtained by Zagarola and Smits [16] for turbulent pipe ¯ows. (2) The eect of the groove on Cf is insigni®cant at the lower Rh . At the higher Rh , the eect of the groove is more pronounced. There is an increase in Cf over the smooth-wall value immediately downstream of the groove, followed by a decrease and a subsequent oscillatory relaxation back to the smooth-wall value. While these results are qualitatively similar to those of Pearson et al. [4], the magnitude of the increase is much less in the present study. This can be attributed to the smaller w/d1 of the present study. (3) While it is dicult to make any conclusions on the overall drag, the Cf distribution suggests that if drag reduction is to be achieved, the sharp rise in Cf immediately downstream of the groove needs to be reduced. This could be achieved by optimizing the groove size and shape to reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove. The groove should, however, still provide the bene®cial undershoot in Cf . Also, the area under the undershoot should be greater than the area under the increase in Cf immediately downstream of the groove. The optimum groove spacing (s/w) is obtained when the next groove is located at the end of the undershoot, so that the oscillatory overshoot beyond that is eliminated. (4) The mean velocity pro®les are not aected by the presence of the groove at both Rh . The ratio w/d1 is probably too small to alter the mean velocity pro®les. It
9
may be conjectured that w/d1 must be greater than 0.1 for it to have a signi®cant eect. (5) The streamwise turbulence intensity is reduced over the grooved-wall in the inner region (y K 10) at both Rh . The groove attenuates the turbulence intensity in this region, and this eect seems to be con®ned to the near-wall region. The location of u max remains unchanged at y 13 for both the smooth-and groovedwall at both Rh . (6) There is a strong correlation between the wake parameter (p) and Cf distributions at Rh 3000. This correlation is similar to that obtained by Matsumoto [9], where p increases as Cf decrease. (7) There is no signi®cant eect of the groove on the energy spectra, at least at the location where the turbulence intensity is maximum. The eect of Rh on the spectra is to increase the high wave number component of energy, and to reduce the low wave number component as Rh increases. (8) The height of the internal layer at the lower Rh is approximately twice as that at the higher Rh . The internal layer grows rapidly immediately downstream of the groove (x=w K 7), with the initial rate of growth at the lower Rh almost twice that at the higher Rh . Beyond x/w 7, the growth of the internal layer is much slower, with the rates being almost equal at both Rh . In the present study, the eect of the groove on the skin friction, mean velocity and turbulence intensity is less pronounced compared with the studies of Pearson et al. [4], Elavarasan et al. [6], and Choi and Fujisawa [3]. This dierence may be attributed to the dierence in w/d1 of the present study and the previous three studies. In the present study, w=d1 0:072, while in the previous three studies, w=d1 0:17, 0.125, and 0.4, respectively. It is important to determine whether there is an optimum w/d1 for the most bene®cial Cf distribution. It is also desirable to determine the eect of dierent groove shapes and Reynolds numbers, especially in the context of reducing the sharp increase in Cf immediately downstream of the groove. While keeping the basic dimensions of the groove (d and w) constant, dierent groove shapes as suggested in Fig. 18 can be investigated. This may potentially reduce the intense favorable pressure gradient that emanates from the downstream edge of the groove and reduce the sharp rise in Cf immediately downstream of the groove.
Fig. 18. Alternative groove shapes.
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Sutardi, C.Y. Ching / Experimental Thermal and Fluid Science 20 (1999) 1±10
Nomenclature B y-intercept on the Clauser-chart, dimensionless 2 Cf skin friction coecient, (Cf 2sw = qU1 )), dimensionless Cf;0 skin friction coecient on the smooth-wall, dimensionless d groove depth, m E(k1 ) turbulence energy spectra, m3 /s2 k1 one-dimensional wave length, m Reynolds number based on Rd d Rd U1 d=m), dimensionless Rh Reynolds number based on h Rh U1 h=m, dimensionless s distance between grooves, m U mean velocity in the x-direction, m/s U normalized streamwise mean velocity (U/ us ), dimensionless U1 free stream velocity, m/s u ¯uctuation of velocity in the x-direction, m/s u normalized streamwise ¯uctuating velocity (u/us ), dimensionless us friction velocity ( sw =q0:5 , m/s w groove width, m x streamwise coordinate measured from the groove trailing edge, m y wall-normal coordinate, m y normalized wall-normal coordinate (yus /m), dimensionless Greek Symbols d boundary layer thickness, m d1 boundary layer thickness at x=w 1, m di internal layer thickness, m j Karman constant, dimensionless m kinematic viscosity, m2 /s p wake parameter, dimensionless h momentum thickness, m sw wall shear stress, N/m2
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