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An examination of a 3D corner-step experiment
Experimental Thermal and Fluid Science 17 (1998) 132±138
An examination of a 3D corner-step experiment Sheldon D. Stokes a, Mark N. Glauser
b,*
, Thomas B. Gatski
c
a
Ph.D. Candidate, Clarkson University, Potsdam, NY 13699, USA Associate Professor, Clarkson University Potsdam, NY 13699, USA Senior Research Scientist, NASA Langley Research Center, Hampton, VA 23666, USA b
c
Received 6 November 1996; received in revised form 29 April 1997; accepted 8 May 1997
Abstract This paper presents results of LDV measurements in a complex three-dimensional (3D) turbulent ¯ow. The experimental con®guration studied, shown in Fig. 1, provides a complex 1 parameter (step height) three-dimensional ¯ow with two-dimensional (2D) relaxation limits in the spanwise direction (that is, 2D channel ¯ow and 2D backstep). Note that this ¯ow is more complex than the simple 2D back-step described by Eaton and Johnston, 1981, AIAA Journal, vol. 19, pp. 1093±1100. However, when spanwise limits are taken out from either side of the centerline the ¯ow relaxes to two well studied 2D ¯ows. On the block side, the spanwise limit results in a 2D turbulent channel ¯ow. On the step side, the spanwise limit results in the 2D back-step ¯ow. In the center region of the facility, however the ¯ow is strongly 3D. The data presented here, was taken with a three component laser Velocimetry system, thus all three velocity components can be measured simultaneously. The results presented here clearly indicate that this ¯ow provides a complex 3D ¯ow for turbulence model calibration with the added attraction of the 2D spanwise limits. In the region within two stepheights either side in span of the streamwise step edge, secondary ¯ows are observed. In addition, a spanwise evolving streamwise reattachment length is seen. The prediction of the secondary motions along with the spanwise evolving streamwise reattachment length found here will provide a challenging test for any turbulence model. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Laser Doppler velocimetry; 3D corner step; 2D channel ¯ow; 2D backstep; Step height; Turbulence model calibration
1. Introduction The development of fundamentally sound turbulence models for prediction in complex non-equilibrium threedimensional 3D turbulent ¯ows requires comparison with simple 3D experiments (Bradshaw, 1987; Gatski, 1996). Many experimental and numerical studies have been performed in 3D turbulent ¯ow ®elds. For example, Graham (1969) studied the eect of end plates on the two dimensionality of a vortex wake. Seale (1982) examined fully developed ¯ow in a simulated rod bundle. Speziale (1982) studied the secondary turbulent ¯ow in pipes of non-circular cross sections. Several other ¯ows have been studied by various authors (Madabhushi and Vanka, 1991; Shiono and Knight, 1991; Anthony and Willmarth, 1992; Gessner et al., 1993; Huser and Biringen, 1993; Speziale et al., 1993). For a comprehensive review of 3D turbulent secondary ¯ows, see Bradshaw (1987) and references therein. The 3D corner-step ¯ow studied here has several distinct advantages over the previously mentioned *
Corresponding author.
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 5 8 - 9
studies. First, it is a simple geometry which is easily described by units of step height. All the angles are normal, and in this case, all the dimensions are whole step multiples. These properties make model grid generation easier, and geometry duplication in other tunnels as simple as possible. The second and main advantage of this ¯ow is while it is strongly 3D in the center, it has 2D limits in span. In the negative spanwise limit (Fig. 1), the ¯ow collapses to a turbulent channel ¯ow. In the positive spanwise limit, the ¯ow collapses to a 2D backstep. We can look at the turbulent kinetic energy equation to see the eect of taking the 2D spanwise limits. The turbulent kinetic energy equation (Eq. (1)), written in index notation, is shown below, where the overbar denotes average quantities, capital letters denote mean quantities and small letters denote ¯uctuating quantities (Tennekes and Lumley, 1972). @ui ui =2 @ 1 ui ui uj @ui ui =2 Uj ÿv ÿ puj 2 @xj @xj q @xj ÿ ui uj
@Ui @ui @ui ÿv : @xj @xj @xj
1
S.D. Stokes et al. / Experimental Thermal and Fluid Science 17 (1998) 132±138
Fig. 1. 3D corner-step diagram.
The expanded kinetic energy equation (Eq. (2)) is shown below. Note all terms are retained due to the 3D nature of the ¯ow in the center region. U1
@K @K @K U2 U3 @x1 @x @x3 2 1 @pu1 @pu2 @pu3 ÿ @x2 @x3 q @x1 @ u1 u1 u1 u2 u2 u1 u3 u3 u1 2 @x1 @ u1 u1 u2 u2 u2 u2 u3 u3 u2 2 @x2 @ u1 u1 u3 u2 u2 u3 u3 u3 u3 2 @x3 @K @K @K @U1 ÿ u1 u1 ÿv @x1 @x1 @x2 @x3 @U1 @U2 @U1 @U3 ÿ u1 u3 ÿ u1 u2 @x2 @x1 @x3 @x1 @U2 @U2 @U3 @U3 ÿ u3 u3 ÿ u2 u3 ÿ u2 u2 @x2 @x3 @x2 @x3 @u1 @u1 @u1 @u1 @u1 @u1 @u2 @u2 ÿv @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1
that kinetic energy equation relaxes to the equation shown below (Eq. (4)): @K @K 1 @pu1 @pu2 U1 U2 ÿ @x2 @x1 @x2 q @x1 @ u1 u1 u1 u2 u2 u1 u3 u3 u1 2 @x1 @ u1 u1 u2 u2 u2 u2 u3 u3 u2 @K @K ÿv 2 @x2 @x1 @x2 @U1 @U1 @U2 ÿ u1 u2 ÿ u1 u1 @x1 @x2 @x1 @U2 @u1 @u1 @u1 @u1 @u1 @u1 ÿv ÿ u2 u2 @x2 @x1 @x1 @x2 @x2 @x3 @x3 @u2 @u2 @u2 @u2 @u2 @u2 @u3 @u3 @u3 @u3 @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1 @x2 @x2 @u3 @u3 : 4 @x3 @x3 Hence, it can be seen, that the equations also reduce in complexity as the ¯ow relaxes from a fully 3D ¯ow in the center, to two more fundamental ¯ows at the edges of the tunnel in span. First order statistics in several planes will be shown in this paper. Measurements have been taken well into the various spanwise positions to verify that the 2D spanwise limits have been obtained. Second order statistics at the spanwise limits will be compared to the second order statistics taken in the same tunnel in the backstep con®guration. The results show that the second order statistics are relaxing to those seen in the respective 2D ¯ows.
@u2 @u2 @u2 @u2 @u3 @u3 @u3 @u3 @u3 @u3 ; @x2 @x2 @x3 @x3 @x1 @x1 @x2 @x2 @x3 @x3 2
K
133
u1 u1 u2 u2 u3 u3 : 2
As the ¯ow relaxes to a fully developed 2D channel ¯ow, it becomes homogeneous in the streamwise and spanwise direction and U3 0, so the above equation relaxes to the equation shown below (Eq. (3)): @K 1 @pu2 @ u 1 u1 u2 u2 u 2 u2 u3 u3 u2 ÿ U1 2 @x1 q @x2 @x2 @K @U1 @u1 @u1 @u1 @u1 ÿ u1 u2 ÿv ÿv @x2 @x1 @x1 @x2 @x2 @x2 @u1 @u1 @u2 @u2 @u2 @u2 @u2 @u2 @u3 @u3 @x3 @x3 @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1 @u3 @u3 @u3 @u3 : 3 @x2 @x2 @x3 @x3
As the ¯ow relaxes to a 2D backstep ¯ow, it becomes homogeneous in the spanwise direction and U3 0, so
2. Experimental setup The experiment was conducted in NASA Langley's low Reynolds Number Backstep/3D corner-step facility. This facility was constructed initially to be used to validata LES (large eddy simulation) models. It is capable of running at a Reynolds numbers (based on step height) of 10,000±60,000. The tunnel is 36 in. wide, and has a step height of 3 in. The expansion ratio is 1:2. The ¯ow was determined to be fully developed upstream of the step region, from previous experiments (Stokes, 1995). The backstep facility was easily converted to the 3D corner-step facility by adding a streamwise step to the backstep. This streamwise step divides the tunnel in span, such that one side is a turbulent channel ¯ow and the other side is a backstep ¯ow. The step height was maintained at 3 in. to ease the comparison to backstep data obtained in this same facility (Stokes, 1995). The main data acquisition system consists of a three component laser velocimeter system. The system uses a 10 W Coherent Innova-90 Argon-Ion laser. The optics are the TSI modular system. The 476.5 and 496.5 nm wavelengths are used to measure the U and V velocities respectively. The 514.5 line is used for the W compo-
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nent. Each component uses a Bragg cell to frequency shift the signals to resolve direction. The signal from each component is sent to three Macrodyne particle counters. The Macrodyne LVABI interface box acts like a buer to store up to 65,535 samples of data, and more importantly, is used here to force coincidence of the three signals. This insures that the signals received were all taken within a very short time window, and thus measured from the same particle. There is a fourth LV component that monitors changes in mean velocity and turbulence intensity, upstream of the 3D corner-step region. The LVABI is interfaced to a Power Macintosh 8500 computer, the control center for the whole facility. The optics system is mounted on a Klinger rail system, and traversed by Klinger 1 meter translation stages. The data acquisition system is set up as a cube that surrounds the test section of the tunnel that can move in the streamwise and spanwise directions using the Klinger translation stages. There is an inner cube attached to the outer cube with four of the translation stages. This inner cube moves in a wall-normal (vertical) direction. The laser and the three main components or the LV system are attached to the top of the inner cube. The entire cube and optics assembly is moved (via IEEE computer interface) to investigate any portion of the ¯ow. The fourth component is ®xed to the tunnel upstream. The database that was developed contains all three velocity components, and thus also contains the complete Reynolds stress tensor, and higher order moments. The statistics were calculated from 1000 point sample data sets at each spatial position. The collection times to gather 1000 points at each location ranged from 20 s to 3 min. A typical time integral scale in this ¯ow is 0.0075 s, so that on average, we have more than 10 integral scales between points. Therefore each sample can be considered statistically independent (Tennekes and Lumley, 1972). This results in a maximum statisticalp un certainty for the ®rst moments of 3.2% u0 =u 1= n 0 assuming u =u 1 where n number of samples]. Little would be gained by taking more points, especially given the 3D grid in this experiment. Table 1 summarizes the uncertainties in the experiment. For more discussion of the uncertainties in the experiment, refer to Stokes, 1995.
into three regions, ®rst the center of the tunnel (in span) has been investigated, concentrating on the region around the streamwise step. This region should contain the largest secondary, and most 3D ¯ow. Data was also taken at the center of the backstep and channel ¯ow regions to determine whether the ¯ow is asymptoting to a 2D channel ¯ow and a 2D backstep ¯ow. Data has been taken at ten streamwise stations, and at two stations in span in the center of the backstep region and another two stations in span in the channel ¯ow region. In the center region, above the spanwise step, only ®ve spanwise stations were investigated, but that number was increased as the investigation proceeded downstream. This was done to insure that the expanding secondary ¯ow was captured. By the tenth station downstream, eleven spanwise stations were investigated in this center region. Five wall-normal positions were investigated at each streamwise and spanwise station in the channel region. Ten wall-normal positions were investigated at each streamwise and spanwise station in the backstep region. Fig. 2 shows the spanwise and wall-normal mean velocity components at four streamwise locations. The horizontal axis is spanwise position (normalized by stepheight). The vertical axis is wall-normal position (normalized by stepheight). There are four streamwise positions shown in this ®gure. The top plot shows the mean velocity components one stepheight upstream of the spanwise step. It can be seen that there is very little secondary motion upstream of the spanwise step, showing the 2D character of the incoming ¯ow.
3. Results A database of information on the 3D corner-step ¯ow has been gathered. This database can be broken Table 1 Uncertainty contributions Variable
Uncertainty
Position Instantaneous velocity First moments Second moments
2 mm 0.00237 m/s 3.5% (max) 6%
Fig. 2. Secondary mean velocity ®eld.
S.D. Stokes et al. / Experimental Thermal and Fluid Science 17 (1998) 132±138
The second plot from the top, two stepheights downstream of the step, shows the developing secondary velocity ®eld. The secondary mean ¯ow ®eld can be seen in the backstep region, at the same height as the streamwise step. This ®eld is impinging on the corner of the streamwise step, causing a ¯ow in the negative direction in Z, in the channel region. The third plot from the top, ®ve stepheights downstream of the step, shows the secondary ¯ow ®eld developing farther. The secondary mean ¯ow ®eld has grown, both in magnitude and in space. This secondary ¯ow is now impinging more on the corner of the streamwise step, and causing an increased secondary ¯ow in the negative spanwise direction in the channel ¯ow portion. The bottom plot, six stepheights downstream of the step, shows the progression of this secondary mean ¯ow ®eld. At this point the secondary ¯ow has grown quite large, and is interacting with the top and bottom tunnel walls. From this point downstream, secondary velocities look similar as the ¯ow progresses downstream, due to the interaction with the tunnel walls. The evolution of this secondary vortical ¯ow can be seen from the results presented (Fig. 2). This secondary ¯ow is also being convected downstream by the primary ¯ow. In fact, the eect of this secondary ¯ow is largely swamped by the convective eects of the mean streamwise ¯ow. Examining the outside most spanwise station in the channel region (Fig. 2), it can be seen that there is very little secondary ¯ow at three stepheights or more into the channel region. This is an indication that the mean velocity has relaxed to a 2D channel ¯ow. Examining the outermost stations in the backstep region, it can be seen that there is a small amount of secondary ¯ow past three stepheights downstream of the step. This indicates that the ¯ow is not quite 2D yet. But the magnitudes of the secondary motion are fairly small, indicating that the ¯ow is approaching the 2D limit in the backstep region as well. Fig. 3 shows the streamwise and wall-normal mean velocity vectors. Four spanwise stations are shown, starting in the channel region and progressing into the backstep region. The horizontal axis is streamwise position (normalized by stepheight). The vertical axis is wallnormal position (normalized by stepheight). The top plot, three stepheights in span into the channel region, shows the characteristic pro®le of a turbulent channel ¯ow. As can be expected the pro®le is largely invariant of streamwise position. This indicates that the ¯ow is unaected by the actions of the streamwise step, and thus is 2D. The second plot from the top, directly above the streamwise step edge (zero stepheights in span), shows the evolution of the primary ¯ow evolution. As the ¯ow proceeds downstream, the increasing eect of the secondary mean ¯ow ®eld can be seen. The lowest wall-normal velocity point (Z 0.125 stepheights) is largely parallel to the bottom of the tunnel upstream of the spanwise step, and slowly is forced downward by the action of secondary ¯ow. The eect of the secondary ¯ow seems to be the strongest at ®ve stepheights down-
135
Fig. 3. Primary mean velocity ®eld at four spanwise locations.
stream. The secondary velocity component in the wall normal direction is the most negative at this point. The third plot from the top, one stepheight in span into the backstep region, shows the ¯ow separation, and it's evolution. The typical backstep type of ¯ow pattern can be seen in this ®gure. There is a separated shear layer, after the spanwise step, that evolves downstream and eventually impinges on the lower wall. This causes a recirculation region behind the streamwise step. In this case there is also the action of the streamwise step and the secondary mean ¯ow ®eld acting on the primary ¯ow. The action of these two factors tends to push the shear layer toward the lower wall of the tunnel, and shrink the recirculation region. In this ®gure, it can be seen that the reattachment point is at about ®ve stepheights downstream of the spanwise step. The bottom plot, three stepheights in span into the backstep region, shows the typically seen backstep ¯ow. At this far region in span, the eect of the streamwise step and the secondary mean ¯ow ®eld is much smaller, so the ¯ow looks much more like the traditional 2D backstep. It can be seen that the reattachment point is at about seven stepheights down-stream. This can be compared to backstep data taken in the same facility that shows the reattachment point to be about 7.5 stepheights downstream (Fig. 4). So the ¯ow is approaching two dimensionality in the ®rst order statistics. Fig. 5 shows the streamwise and spanwise mean velocities at two ®xed wall-normal planes. The planes
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S.D. Stokes et al. / Experimental Thermal and Fluid Science 17 (1998) 132±138
Fig. 4. Backstep mean velocities.
shown here are 0.75 stepheights (top plot), and 0.083 stepheights (bottom plot) above the bottom of the channel portion. The horizontal axis is streamwise position
Fig. 5. 3D corner-step top view at two wall-normal positions.
(normalized by stepheight). The vertical axis is spanwise position (normalized by stepheight). This is a view of the velocity ®eld as seen from above the tunnel looking down. The grey area is the tunnel ¯oor in the channel region and the input region. The white area is where the tunnel ¯oor falls away for the backstep region. In the top plot, it can be seen that the ¯ow is largely unaected by the presence of the streamwise step until about ®ve stepheights downstream. This is expected because this wall normal slice is very near the top of the tunnel. However, the eect of the secondary ¯ow can be seen from ®ve stepheights downstream. This is due to the growth of the area of in¯uence in the secondary ¯ow. The secondary ¯ow begins to interact with the top wall past ®ve stepheights downstream (Fig. 2). This can be seen by the turning of the vectors near the corner of the streamwise step. The outermost vectors in span on the channel-¯ow side are all parallel to the input mean ¯ow direction, indicating that the ¯ow has reached the desired 2D limit. The vectors outermost in span on the backstep region are slightly aected by the secondary ¯ow ®eld, indicating that the desired 2D limit in this direction has not yet been achieved at these stations in span. In the bottom plot, where this plotted plane is just above the bottom of the channel portion, the magnitude of the vectors on the channel side of the ¯ow are much smaller than those on the backstep side of the ¯ow. This is due to the boundary layer eects on the channel side. This can also be seen in Fig. 3 (top plot), which shows the channel ¯ow from the side. Note the strong shear ¯ow in span at y/h 0.083 (Fig. 5), indicating the 3D character of the ¯ow in the region near the streamwise step. If the outermost spanwise vectors in the channel side of the ¯ow are examined, it can be seen that they are parallel to the input mean ¯ow direction, indicating that the ¯ow is 2D at this wall-normal position (as it was seen at y/h 0.75). Examining the outermost spanwise vectors on the backstep side, it can be seen that they are slightly eected by the secondary ¯ow, but only to a small degree. The mean velocities have relaxed to two well-known 2D ¯ows as we move out in span in the tunnel. Ideally, in the spanwise limit, all the higher order statistical moments will also relax to the same values found in the 2D ¯ows. To test this assumption, the normal Reynolds stresses in the 3D corner-step ¯ow have been compared to the normal Reynolds stresses in the 2D backstep ¯ow measured in the same tunnel. Since the inlet ¯ow is a fully developed homogeneous channel ¯ow just upstream of the step region, it is invariant in the streamwise direction, as well as the spanwise direction (excluding side wall eects). Hence, the pro®les measured in the center of the tunnel just upstream of the step region can be used as a benchmark to compare the channel ¯ow spanwise limit. Fig. 6 shows the Reynolds stresses one stepheight upstream of the step, along the tunnel centerline with the corner step removed. It also shows the Reynolds stresses in the channel ¯ow region three stepheights downstream from the spanwise step, and
S.D. Stokes et al. / Experimental Thermal and Fluid Science 17 (1998) 132±138
Fig. 6. 3D corner-step/backstep comparison.
three stepheights in span into the channel ¯ow region. It can be seen that the second order statistics agree very well, indicating that the ¯ow has relaxed to a channel ¯ow in this limit. This graph also shows that the channel ¯ow is homogeneous in the streamwise direction, indicating that it is fully developed prior to the spanwise step. The good agreement of the two data sets also indicates that the ¯ow is homogeneous in span, and that side wall eects do not appear to be eecting the second moments. Fig. 7 shows a comparison between the 3D cornerstep and backstep ¯ow second order statistics for the backstep region. In the 3D corner-step ¯ow, data is taken three stepheights downstream of the spanwise step,
Fig. 7. 3D corner-step/backstep comparison.
137
and three stepheights in span into the backstep region (the center of the backstep region). The backstep data was also taken three stepheights downstream of the step, and on the centerline. As was seen in the channel ¯ow comparison, the backstep ¯ow second order statistics are also very similar between the 3D corner-step and backstep ¯ow, indicating that the second order statistics are very close to relaxing to those in the 2D backstep ¯ow. Note however that the ww and vv have not relaxed as closely as uu, in the recirculation region (region where x=h < 7). Fig. 8 shows the turbulent kinetic energy at two spanwise stations in the backstep region. The horizontal axis is kinetic energy. The vertical axis is wall-normal position (normalized by stepheight). It can be seen that the wall-normal pro®les show the same general shape as the pro®les seen in the 2D backstep (Stokes). However the magnitude of the kinetic energy now shows a spanwise dependence indicating the 3D character of this ¯ow. Note how the turbulent kinetic energy at three stepheights in span into the backstep region has a larger value near the wall than at one stepheight. This is due to the larger back¯ow velocity at three stepheights in span, hence greater mean shear and thus larger turbulence production. This can be seen by comparing the lower two plots in Fig. 3. Fig. 9 shows the Reynolds shear stress at two spanwise stations into the backstep region. The horizontal axis is shear stress magnitude. The vertical axis is wallnormal position (normalized by stepheight). The shear stress pro®les also have the same shape as the 2D backstep (Stokes), but in this case they show a spanwise dependence again indicating the 3D character of this ¯ow. The interesting features of this ¯ow also relate to the equations in several ways. The bottom graph in Fig. 5, shows that the ¯ow is statistically inhomogeneous in the spanwise direction (Z direction) around the center region of the tunnel. This implies that the kinetic energy (KE) equation for the center region (Eq. (2)) must contain all the terms relating to the gradients in Z. These
Fig. 8. Turbulent kinetic energy.
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streamwise step edge, secondary ¯ows are observed. In addition, a spanwise evolving streamwise reattachment length is seen in the backstep region. This database, with its complete Reynolds stress tensor, and higher order moments, should provide a valuable tool for the testing and calibration of a variety of turbulence models in 3D ¯ows. Acknowledgements We would like to thank NASA Langley, and William Sellers for the materials and facility with which this research was completed, as well as the needed ®nancial backing. References Fig. 9. Reynolds shear stress.
terms reduce to zero as the ¯ow becomes statistically homogeneous at the spanwise limits. Thus the KE equations for the channel ¯ow and the backstep ¯ow do not have any terms relating to this Z gradient as depicted in Eq. (3) and Eq. (4), respectively. The top graph in Fig. 3 shows the velocity pro®le in the channel region. From this graph, it can be seen that the ¯ow is statistically homogeneous in the X direction. Hence, Eq. (3) does not contain any terms that relate to the velocity gradient in X. However, the middle two graphs in Fig. 4 (center region of this ¯ow), clearly show the statistical inhomogeneity in the X direction, so all the terms relating to the X direction velocity gradient are retained in Eq. (2). Likewise, examining the bottom graph in Fig. 3, it can be seen that the backstep region also contains strong velocity gradients in X. Hence, Eq. (4) must contain all terms relating to the velocity gradients in X. 4. Conclusion This 3D corner-step ¯ow provides a complex 3D ¯ow that relaxes to two well-known 2D ¯ows. It provides a complex set of ¯ow characteristics that will challenge the abilities of any turbulence model. The ®rst and second order statistics in this ¯ow have been shown here. The data shows that the ¯ow has relaxed to the 2D channel ¯ow, by three stepheights in span into the channel region. The data also shows that the ¯ow is close to relaxing to the 2D backstep by three stepheights (in span) into the backstep region. In the region within two stepheights either side in span of the
Anthony, D.G., Willmarth, W.W., 1992. Turbulence measurements in a round jet beneath a free surface. J. Fluid Mech. 243, 699±720. Bradshaw, P., 1987. Turbulent secondary ¯ows. Ann. Rev. Fluid Mech. 19, 53±74. Eaton, J.K., Johnson, J.P., 1981. A review of research on subsonic turbulent ¯ow reattachment. AIAA J. 19 (9), 1093±1100. Gatski, T.B., 1996. Turbulent Flows-Model Equations and Their Solution, in: Handbook of Computational Fluid Mechanics, Academic Press, New York, pp. 340±415. Gessner, F.B., Eppich, H.M., Lund, E.G., 1993. Reynolds number eects on the near-wall structure of turbulent ¯ow along a streamwise corner. Near-Wall Turbulent Flows. Elseiver, Amsterdam, pp. 965±978. Graham, J.M.R., 1969. The eect of end-plates on the two-dimentionality of a vortexwake. Aero Quarterly 20, 237±246. Huser, A., Biringen, S., 1993. Direct numerical simulation of turbulent ¯ow in a square duct. J. Fluid Mech. 257, 65±95. Madabhushi, R.K., Vanka, S.P., 1991. Large eddy simulation of turbulence-driven secondary ¯ow in a square duct. Phys. Fluids A 3, 2734±2753. Seale, W.J., 1982. Measurements and predictions of fully developed turbulent ¯ow in a simulated rod bundle. J. Fluid Mech. 123, 399± 423. Shiono, K., Knight, D.W., 1991. Turbulent open-channel ¯ows with variable depth across the channel. J. Fluid Mech. 222, 617±646. Speziale, C.G., 1982. On turbulent secondary ¯ows in pipes of noncircular cross section. Int. J. Eng. Sci. 20, 863±872. Speziale, C.G., 1986. The dissapation rate correlation and turbulent secondary ¯ow in ducts. J. Fluids Eng. 108, 118±120. Speziale, C.G., So, R.M.C., Younis, B.A., 1993. On the prediction of turbulent secondary ¯ows. Near-Wall Turbulent Flows. Elsevier, Amsterdam, pp. 105±132. Stokes, S.D., 1995. An Experimental investigation Of The 2d Backstep And Two Step Flow Geometry Using Laser Doppler Velocimetry. M.S. Thesis Clarkson University. Tennekes, H., Lumley, J.L., A First Course in Turbulence. MIT Press, Cambridge, MA, pp. 27±58.
An examination of a 3D corner-step experiment Sheldon D. Stokes a, Mark N. Glauser
b,*
, Thomas B. Gatski
c
a
Ph.D. Candidate, Clarkson University, Potsdam, NY 13699, USA Associate Professor, Clarkson University Potsdam, NY 13699, USA Senior Research Scientist, NASA Langley Research Center, Hampton, VA 23666, USA b
c
Received 6 November 1996; received in revised form 29 April 1997; accepted 8 May 1997
Abstract This paper presents results of LDV measurements in a complex three-dimensional (3D) turbulent ¯ow. The experimental con®guration studied, shown in Fig. 1, provides a complex 1 parameter (step height) three-dimensional ¯ow with two-dimensional (2D) relaxation limits in the spanwise direction (that is, 2D channel ¯ow and 2D backstep). Note that this ¯ow is more complex than the simple 2D back-step described by Eaton and Johnston, 1981, AIAA Journal, vol. 19, pp. 1093±1100. However, when spanwise limits are taken out from either side of the centerline the ¯ow relaxes to two well studied 2D ¯ows. On the block side, the spanwise limit results in a 2D turbulent channel ¯ow. On the step side, the spanwise limit results in the 2D back-step ¯ow. In the center region of the facility, however the ¯ow is strongly 3D. The data presented here, was taken with a three component laser Velocimetry system, thus all three velocity components can be measured simultaneously. The results presented here clearly indicate that this ¯ow provides a complex 3D ¯ow for turbulence model calibration with the added attraction of the 2D spanwise limits. In the region within two stepheights either side in span of the streamwise step edge, secondary ¯ows are observed. In addition, a spanwise evolving streamwise reattachment length is seen. The prediction of the secondary motions along with the spanwise evolving streamwise reattachment length found here will provide a challenging test for any turbulence model. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Laser Doppler velocimetry; 3D corner step; 2D channel ¯ow; 2D backstep; Step height; Turbulence model calibration
1. Introduction The development of fundamentally sound turbulence models for prediction in complex non-equilibrium threedimensional 3D turbulent ¯ows requires comparison with simple 3D experiments (Bradshaw, 1987; Gatski, 1996). Many experimental and numerical studies have been performed in 3D turbulent ¯ow ®elds. For example, Graham (1969) studied the eect of end plates on the two dimensionality of a vortex wake. Seale (1982) examined fully developed ¯ow in a simulated rod bundle. Speziale (1982) studied the secondary turbulent ¯ow in pipes of non-circular cross sections. Several other ¯ows have been studied by various authors (Madabhushi and Vanka, 1991; Shiono and Knight, 1991; Anthony and Willmarth, 1992; Gessner et al., 1993; Huser and Biringen, 1993; Speziale et al., 1993). For a comprehensive review of 3D turbulent secondary ¯ows, see Bradshaw (1987) and references therein. The 3D corner-step ¯ow studied here has several distinct advantages over the previously mentioned *
Corresponding author.
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 5 8 - 9
studies. First, it is a simple geometry which is easily described by units of step height. All the angles are normal, and in this case, all the dimensions are whole step multiples. These properties make model grid generation easier, and geometry duplication in other tunnels as simple as possible. The second and main advantage of this ¯ow is while it is strongly 3D in the center, it has 2D limits in span. In the negative spanwise limit (Fig. 1), the ¯ow collapses to a turbulent channel ¯ow. In the positive spanwise limit, the ¯ow collapses to a 2D backstep. We can look at the turbulent kinetic energy equation to see the eect of taking the 2D spanwise limits. The turbulent kinetic energy equation (Eq. (1)), written in index notation, is shown below, where the overbar denotes average quantities, capital letters denote mean quantities and small letters denote ¯uctuating quantities (Tennekes and Lumley, 1972). @ui ui =2 @ 1 ui ui uj @ui ui =2 Uj ÿv ÿ puj 2 @xj @xj q @xj ÿ ui uj
@Ui @ui @ui ÿv : @xj @xj @xj
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Fig. 1. 3D corner-step diagram.
The expanded kinetic energy equation (Eq. (2)) is shown below. Note all terms are retained due to the 3D nature of the ¯ow in the center region. U1
@K @K @K U2 U3 @x1 @x @x3 2 1 @pu1 @pu2 @pu3 ÿ @x2 @x3 q @x1 @ u1 u1 u1 u2 u2 u1 u3 u3 u1 2 @x1 @ u1 u1 u2 u2 u2 u2 u3 u3 u2 2 @x2 @ u1 u1 u3 u2 u2 u3 u3 u3 u3 2 @x3 @K @K @K @U1 ÿ u1 u1 ÿv @x1 @x1 @x2 @x3 @U1 @U2 @U1 @U3 ÿ u1 u3 ÿ u1 u2 @x2 @x1 @x3 @x1 @U2 @U2 @U3 @U3 ÿ u3 u3 ÿ u2 u3 ÿ u2 u2 @x2 @x3 @x2 @x3 @u1 @u1 @u1 @u1 @u1 @u1 @u2 @u2 ÿv @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1
that kinetic energy equation relaxes to the equation shown below (Eq. (4)): @K @K 1 @pu1 @pu2 U1 U2 ÿ @x2 @x1 @x2 q @x1 @ u1 u1 u1 u2 u2 u1 u3 u3 u1 2 @x1 @ u1 u1 u2 u2 u2 u2 u3 u3 u2 @K @K ÿv 2 @x2 @x1 @x2 @U1 @U1 @U2 ÿ u1 u2 ÿ u1 u1 @x1 @x2 @x1 @U2 @u1 @u1 @u1 @u1 @u1 @u1 ÿv ÿ u2 u2 @x2 @x1 @x1 @x2 @x2 @x3 @x3 @u2 @u2 @u2 @u2 @u2 @u2 @u3 @u3 @u3 @u3 @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1 @x2 @x2 @u3 @u3 : 4 @x3 @x3 Hence, it can be seen, that the equations also reduce in complexity as the ¯ow relaxes from a fully 3D ¯ow in the center, to two more fundamental ¯ows at the edges of the tunnel in span. First order statistics in several planes will be shown in this paper. Measurements have been taken well into the various spanwise positions to verify that the 2D spanwise limits have been obtained. Second order statistics at the spanwise limits will be compared to the second order statistics taken in the same tunnel in the backstep con®guration. The results show that the second order statistics are relaxing to those seen in the respective 2D ¯ows.
@u2 @u2 @u2 @u2 @u3 @u3 @u3 @u3 @u3 @u3 ; @x2 @x2 @x3 @x3 @x1 @x1 @x2 @x2 @x3 @x3 2
K
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u1 u1 u2 u2 u3 u3 : 2
As the ¯ow relaxes to a fully developed 2D channel ¯ow, it becomes homogeneous in the streamwise and spanwise direction and U3 0, so the above equation relaxes to the equation shown below (Eq. (3)): @K 1 @pu2 @ u 1 u1 u2 u2 u 2 u2 u3 u3 u2 ÿ U1 2 @x1 q @x2 @x2 @K @U1 @u1 @u1 @u1 @u1 ÿ u1 u2 ÿv ÿv @x2 @x1 @x1 @x2 @x2 @x2 @u1 @u1 @u2 @u2 @u2 @u2 @u2 @u2 @u3 @u3 @x3 @x3 @x1 @x1 @x2 @x2 @x3 @x3 @x1 @x1 @u3 @u3 @u3 @u3 : 3 @x2 @x2 @x3 @x3
As the ¯ow relaxes to a 2D backstep ¯ow, it becomes homogeneous in the spanwise direction and U3 0, so
2. Experimental setup The experiment was conducted in NASA Langley's low Reynolds Number Backstep/3D corner-step facility. This facility was constructed initially to be used to validata LES (large eddy simulation) models. It is capable of running at a Reynolds numbers (based on step height) of 10,000±60,000. The tunnel is 36 in. wide, and has a step height of 3 in. The expansion ratio is 1:2. The ¯ow was determined to be fully developed upstream of the step region, from previous experiments (Stokes, 1995). The backstep facility was easily converted to the 3D corner-step facility by adding a streamwise step to the backstep. This streamwise step divides the tunnel in span, such that one side is a turbulent channel ¯ow and the other side is a backstep ¯ow. The step height was maintained at 3 in. to ease the comparison to backstep data obtained in this same facility (Stokes, 1995). The main data acquisition system consists of a three component laser velocimeter system. The system uses a 10 W Coherent Innova-90 Argon-Ion laser. The optics are the TSI modular system. The 476.5 and 496.5 nm wavelengths are used to measure the U and V velocities respectively. The 514.5 line is used for the W compo-
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nent. Each component uses a Bragg cell to frequency shift the signals to resolve direction. The signal from each component is sent to three Macrodyne particle counters. The Macrodyne LVABI interface box acts like a buer to store up to 65,535 samples of data, and more importantly, is used here to force coincidence of the three signals. This insures that the signals received were all taken within a very short time window, and thus measured from the same particle. There is a fourth LV component that monitors changes in mean velocity and turbulence intensity, upstream of the 3D corner-step region. The LVABI is interfaced to a Power Macintosh 8500 computer, the control center for the whole facility. The optics system is mounted on a Klinger rail system, and traversed by Klinger 1 meter translation stages. The data acquisition system is set up as a cube that surrounds the test section of the tunnel that can move in the streamwise and spanwise directions using the Klinger translation stages. There is an inner cube attached to the outer cube with four of the translation stages. This inner cube moves in a wall-normal (vertical) direction. The laser and the three main components or the LV system are attached to the top of the inner cube. The entire cube and optics assembly is moved (via IEEE computer interface) to investigate any portion of the ¯ow. The fourth component is ®xed to the tunnel upstream. The database that was developed contains all three velocity components, and thus also contains the complete Reynolds stress tensor, and higher order moments. The statistics were calculated from 1000 point sample data sets at each spatial position. The collection times to gather 1000 points at each location ranged from 20 s to 3 min. A typical time integral scale in this ¯ow is 0.0075 s, so that on average, we have more than 10 integral scales between points. Therefore each sample can be considered statistically independent (Tennekes and Lumley, 1972). This results in a maximum statisticalp un certainty for the ®rst moments of 3.2% u0 =u 1= n 0 assuming u =u 1 where n number of samples]. Little would be gained by taking more points, especially given the 3D grid in this experiment. Table 1 summarizes the uncertainties in the experiment. For more discussion of the uncertainties in the experiment, refer to Stokes, 1995.
into three regions, ®rst the center of the tunnel (in span) has been investigated, concentrating on the region around the streamwise step. This region should contain the largest secondary, and most 3D ¯ow. Data was also taken at the center of the backstep and channel ¯ow regions to determine whether the ¯ow is asymptoting to a 2D channel ¯ow and a 2D backstep ¯ow. Data has been taken at ten streamwise stations, and at two stations in span in the center of the backstep region and another two stations in span in the channel ¯ow region. In the center region, above the spanwise step, only ®ve spanwise stations were investigated, but that number was increased as the investigation proceeded downstream. This was done to insure that the expanding secondary ¯ow was captured. By the tenth station downstream, eleven spanwise stations were investigated in this center region. Five wall-normal positions were investigated at each streamwise and spanwise station in the channel region. Ten wall-normal positions were investigated at each streamwise and spanwise station in the backstep region. Fig. 2 shows the spanwise and wall-normal mean velocity components at four streamwise locations. The horizontal axis is spanwise position (normalized by stepheight). The vertical axis is wall-normal position (normalized by stepheight). There are four streamwise positions shown in this ®gure. The top plot shows the mean velocity components one stepheight upstream of the spanwise step. It can be seen that there is very little secondary motion upstream of the spanwise step, showing the 2D character of the incoming ¯ow.
3. Results A database of information on the 3D corner-step ¯ow has been gathered. This database can be broken Table 1 Uncertainty contributions Variable
Uncertainty
Position Instantaneous velocity First moments Second moments
2 mm 0.00237 m/s 3.5% (max) 6%
Fig. 2. Secondary mean velocity ®eld.
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The second plot from the top, two stepheights downstream of the step, shows the developing secondary velocity ®eld. The secondary mean ¯ow ®eld can be seen in the backstep region, at the same height as the streamwise step. This ®eld is impinging on the corner of the streamwise step, causing a ¯ow in the negative direction in Z, in the channel region. The third plot from the top, ®ve stepheights downstream of the step, shows the secondary ¯ow ®eld developing farther. The secondary mean ¯ow ®eld has grown, both in magnitude and in space. This secondary ¯ow is now impinging more on the corner of the streamwise step, and causing an increased secondary ¯ow in the negative spanwise direction in the channel ¯ow portion. The bottom plot, six stepheights downstream of the step, shows the progression of this secondary mean ¯ow ®eld. At this point the secondary ¯ow has grown quite large, and is interacting with the top and bottom tunnel walls. From this point downstream, secondary velocities look similar as the ¯ow progresses downstream, due to the interaction with the tunnel walls. The evolution of this secondary vortical ¯ow can be seen from the results presented (Fig. 2). This secondary ¯ow is also being convected downstream by the primary ¯ow. In fact, the eect of this secondary ¯ow is largely swamped by the convective eects of the mean streamwise ¯ow. Examining the outside most spanwise station in the channel region (Fig. 2), it can be seen that there is very little secondary ¯ow at three stepheights or more into the channel region. This is an indication that the mean velocity has relaxed to a 2D channel ¯ow. Examining the outermost stations in the backstep region, it can be seen that there is a small amount of secondary ¯ow past three stepheights downstream of the step. This indicates that the ¯ow is not quite 2D yet. But the magnitudes of the secondary motion are fairly small, indicating that the ¯ow is approaching the 2D limit in the backstep region as well. Fig. 3 shows the streamwise and wall-normal mean velocity vectors. Four spanwise stations are shown, starting in the channel region and progressing into the backstep region. The horizontal axis is streamwise position (normalized by stepheight). The vertical axis is wallnormal position (normalized by stepheight). The top plot, three stepheights in span into the channel region, shows the characteristic pro®le of a turbulent channel ¯ow. As can be expected the pro®le is largely invariant of streamwise position. This indicates that the ¯ow is unaected by the actions of the streamwise step, and thus is 2D. The second plot from the top, directly above the streamwise step edge (zero stepheights in span), shows the evolution of the primary ¯ow evolution. As the ¯ow proceeds downstream, the increasing eect of the secondary mean ¯ow ®eld can be seen. The lowest wall-normal velocity point (Z 0.125 stepheights) is largely parallel to the bottom of the tunnel upstream of the spanwise step, and slowly is forced downward by the action of secondary ¯ow. The eect of the secondary ¯ow seems to be the strongest at ®ve stepheights down-
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Fig. 3. Primary mean velocity ®eld at four spanwise locations.
stream. The secondary velocity component in the wall normal direction is the most negative at this point. The third plot from the top, one stepheight in span into the backstep region, shows the ¯ow separation, and it's evolution. The typical backstep type of ¯ow pattern can be seen in this ®gure. There is a separated shear layer, after the spanwise step, that evolves downstream and eventually impinges on the lower wall. This causes a recirculation region behind the streamwise step. In this case there is also the action of the streamwise step and the secondary mean ¯ow ®eld acting on the primary ¯ow. The action of these two factors tends to push the shear layer toward the lower wall of the tunnel, and shrink the recirculation region. In this ®gure, it can be seen that the reattachment point is at about ®ve stepheights downstream of the spanwise step. The bottom plot, three stepheights in span into the backstep region, shows the typically seen backstep ¯ow. At this far region in span, the eect of the streamwise step and the secondary mean ¯ow ®eld is much smaller, so the ¯ow looks much more like the traditional 2D backstep. It can be seen that the reattachment point is at about seven stepheights down-stream. This can be compared to backstep data taken in the same facility that shows the reattachment point to be about 7.5 stepheights downstream (Fig. 4). So the ¯ow is approaching two dimensionality in the ®rst order statistics. Fig. 5 shows the streamwise and spanwise mean velocities at two ®xed wall-normal planes. The planes
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Fig. 4. Backstep mean velocities.
shown here are 0.75 stepheights (top plot), and 0.083 stepheights (bottom plot) above the bottom of the channel portion. The horizontal axis is streamwise position
Fig. 5. 3D corner-step top view at two wall-normal positions.
(normalized by stepheight). The vertical axis is spanwise position (normalized by stepheight). This is a view of the velocity ®eld as seen from above the tunnel looking down. The grey area is the tunnel ¯oor in the channel region and the input region. The white area is where the tunnel ¯oor falls away for the backstep region. In the top plot, it can be seen that the ¯ow is largely unaected by the presence of the streamwise step until about ®ve stepheights downstream. This is expected because this wall normal slice is very near the top of the tunnel. However, the eect of the secondary ¯ow can be seen from ®ve stepheights downstream. This is due to the growth of the area of in¯uence in the secondary ¯ow. The secondary ¯ow begins to interact with the top wall past ®ve stepheights downstream (Fig. 2). This can be seen by the turning of the vectors near the corner of the streamwise step. The outermost vectors in span on the channel-¯ow side are all parallel to the input mean ¯ow direction, indicating that the ¯ow has reached the desired 2D limit. The vectors outermost in span on the backstep region are slightly aected by the secondary ¯ow ®eld, indicating that the desired 2D limit in this direction has not yet been achieved at these stations in span. In the bottom plot, where this plotted plane is just above the bottom of the channel portion, the magnitude of the vectors on the channel side of the ¯ow are much smaller than those on the backstep side of the ¯ow. This is due to the boundary layer eects on the channel side. This can also be seen in Fig. 3 (top plot), which shows the channel ¯ow from the side. Note the strong shear ¯ow in span at y/h 0.083 (Fig. 5), indicating the 3D character of the ¯ow in the region near the streamwise step. If the outermost spanwise vectors in the channel side of the ¯ow are examined, it can be seen that they are parallel to the input mean ¯ow direction, indicating that the ¯ow is 2D at this wall-normal position (as it was seen at y/h 0.75). Examining the outermost spanwise vectors on the backstep side, it can be seen that they are slightly eected by the secondary ¯ow, but only to a small degree. The mean velocities have relaxed to two well-known 2D ¯ows as we move out in span in the tunnel. Ideally, in the spanwise limit, all the higher order statistical moments will also relax to the same values found in the 2D ¯ows. To test this assumption, the normal Reynolds stresses in the 3D corner-step ¯ow have been compared to the normal Reynolds stresses in the 2D backstep ¯ow measured in the same tunnel. Since the inlet ¯ow is a fully developed homogeneous channel ¯ow just upstream of the step region, it is invariant in the streamwise direction, as well as the spanwise direction (excluding side wall eects). Hence, the pro®les measured in the center of the tunnel just upstream of the step region can be used as a benchmark to compare the channel ¯ow spanwise limit. Fig. 6 shows the Reynolds stresses one stepheight upstream of the step, along the tunnel centerline with the corner step removed. It also shows the Reynolds stresses in the channel ¯ow region three stepheights downstream from the spanwise step, and
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Fig. 6. 3D corner-step/backstep comparison.
three stepheights in span into the channel ¯ow region. It can be seen that the second order statistics agree very well, indicating that the ¯ow has relaxed to a channel ¯ow in this limit. This graph also shows that the channel ¯ow is homogeneous in the streamwise direction, indicating that it is fully developed prior to the spanwise step. The good agreement of the two data sets also indicates that the ¯ow is homogeneous in span, and that side wall eects do not appear to be eecting the second moments. Fig. 7 shows a comparison between the 3D cornerstep and backstep ¯ow second order statistics for the backstep region. In the 3D corner-step ¯ow, data is taken three stepheights downstream of the spanwise step,
Fig. 7. 3D corner-step/backstep comparison.
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and three stepheights in span into the backstep region (the center of the backstep region). The backstep data was also taken three stepheights downstream of the step, and on the centerline. As was seen in the channel ¯ow comparison, the backstep ¯ow second order statistics are also very similar between the 3D corner-step and backstep ¯ow, indicating that the second order statistics are very close to relaxing to those in the 2D backstep ¯ow. Note however that the ww and vv have not relaxed as closely as uu, in the recirculation region (region where x=h < 7). Fig. 8 shows the turbulent kinetic energy at two spanwise stations in the backstep region. The horizontal axis is kinetic energy. The vertical axis is wall-normal position (normalized by stepheight). It can be seen that the wall-normal pro®les show the same general shape as the pro®les seen in the 2D backstep (Stokes). However the magnitude of the kinetic energy now shows a spanwise dependence indicating the 3D character of this ¯ow. Note how the turbulent kinetic energy at three stepheights in span into the backstep region has a larger value near the wall than at one stepheight. This is due to the larger back¯ow velocity at three stepheights in span, hence greater mean shear and thus larger turbulence production. This can be seen by comparing the lower two plots in Fig. 3. Fig. 9 shows the Reynolds shear stress at two spanwise stations into the backstep region. The horizontal axis is shear stress magnitude. The vertical axis is wallnormal position (normalized by stepheight). The shear stress pro®les also have the same shape as the 2D backstep (Stokes), but in this case they show a spanwise dependence again indicating the 3D character of this ¯ow. The interesting features of this ¯ow also relate to the equations in several ways. The bottom graph in Fig. 5, shows that the ¯ow is statistically inhomogeneous in the spanwise direction (Z direction) around the center region of the tunnel. This implies that the kinetic energy (KE) equation for the center region (Eq. (2)) must contain all the terms relating to the gradients in Z. These
Fig. 8. Turbulent kinetic energy.
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streamwise step edge, secondary ¯ows are observed. In addition, a spanwise evolving streamwise reattachment length is seen in the backstep region. This database, with its complete Reynolds stress tensor, and higher order moments, should provide a valuable tool for the testing and calibration of a variety of turbulence models in 3D ¯ows. Acknowledgements We would like to thank NASA Langley, and William Sellers for the materials and facility with which this research was completed, as well as the needed ®nancial backing. References Fig. 9. Reynolds shear stress.
terms reduce to zero as the ¯ow becomes statistically homogeneous at the spanwise limits. Thus the KE equations for the channel ¯ow and the backstep ¯ow do not have any terms relating to this Z gradient as depicted in Eq. (3) and Eq. (4), respectively. The top graph in Fig. 3 shows the velocity pro®le in the channel region. From this graph, it can be seen that the ¯ow is statistically homogeneous in the X direction. Hence, Eq. (3) does not contain any terms that relate to the velocity gradient in X. However, the middle two graphs in Fig. 4 (center region of this ¯ow), clearly show the statistical inhomogeneity in the X direction, so all the terms relating to the X direction velocity gradient are retained in Eq. (2). Likewise, examining the bottom graph in Fig. 3, it can be seen that the backstep region also contains strong velocity gradients in X. Hence, Eq. (4) must contain all terms relating to the velocity gradients in X. 4. Conclusion This 3D corner-step ¯ow provides a complex 3D ¯ow that relaxes to two well-known 2D ¯ows. It provides a complex set of ¯ow characteristics that will challenge the abilities of any turbulence model. The ®rst and second order statistics in this ¯ow have been shown here. The data shows that the ¯ow has relaxed to the 2D channel ¯ow, by three stepheights in span into the channel region. The data also shows that the ¯ow is close to relaxing to the 2D backstep by three stepheights (in span) into the backstep region. In the region within two stepheights either side in span of the
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