Direct numerical simulation of turbulent channel flow with wall velocity ...

Direct numerical simulation of turbulent channel flow with wall velocity disturbances P. Orlandi, S. Leonardi, R. Tuzi, and R. A. Antonia Citation: Physics of Fluids 15, 3587 (2003); doi: 10.1063/1.1619137 View online: http://dx.doi.org/10.1063/1.1619137 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/15/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Direct numerical simulation of drag reduction in a turbulent channel flow using spanwise traveling wave-like wall deformation Phys. Fluids 25, 105115 (2013); 10.1063/1.4826887 Direct numerical simulation of turbulent channel flows with boundary roughened with virtual sandpaper Phys. Fluids 18, 031701 (2006); 10.1063/1.2183806 Numerical simulation of particle-laden turbulent channel flow Phys. Fluids 13, 2957 (2001); 10.1063/1.1396846 Direct numerical simulation of particle-laden rotating turbulent channel flow Phys. Fluids 13, 2320 (2001); 10.1063/1.1383790 Direct numerical simulation of turbulent channel flow up to Re τ =590 Phys. Fluids 11, 943 (1999); 10.1063/1.869966

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PHYSICS OF FLUIDS

VOLUME 15, NUMBER 12

DECEMBER 2003

ARTICLES

Direct numerical simulation of turbulent channel flow with wall velocity disturbances P. Orlandi,a) S. Leonardi, and R. Tuzi Dipartimento di Meccanica e Aeronautica, Universita` La Sapienza, Via Eudossiana 16, I-00184, Rome, Italy

R. A. Antonia Discipline of Mechanical Engineering, University of Newcastle, Newcastle, New South Wales 2308, Australia

共Received 30 January 2003; accepted 18 August 2003; published 13 October 2003兲 This paper considers the effect of applying nonzero velocity fluctuations along a flat smooth wall, mainly with the aim of understanding how conditions at the wall interact with the outer turbulent flow. Such a study is expected to be of use in formulating effective strategies for wall turbulence control. Three direct numerical simulations of a turbulent plane channel flow are carried out, starting with a flow field with no-slip conditions. Each simulation evolves by imposing 共on one wall兲 only one nonzero velocity component. When a nonzero longitudinal velocity fluctuation u 1 is applied, drag reduction occurs. With a nonzero spanwise velocity fluctuation u 3 , the flow is very similar to that in an unperturbed channel. However, the use of a nonzero wall-normal velocity fluctuation u 2 results in structural changes similar to those observed in a direct numerical simulation of a turbulent flow over a rough surface. From the present simplified simulations, the inference is that the salient characteristics of rough wall flows reflect mainly the presence of a nonzero wall-normal normal velocity distribution at the interface between the roughness cavities and the external flow. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1619137兴

I. INTRODUCTION

In fully turbulent rough-wall flows, the size of the roughness is usually large enough to preclude any simplifications to the Navier–Stokes equations. Leonardi et al.3 performed direct numerical simulations of a channel flow with square bars 共on one of the walls兲 aligned along the spanwise direction, transversely to the flow. From the pressure distribution around the bars, they were able to quantify the importance of the form drag, relative to the skin-frictional drag; such an exercise would be difficult to carry out in a laboratory experiment. Leonardi et al.4 were also able to assess how low and high-speed streaks are disrupted by the roughness elements. Contour plots in the near-wall region of the normal component of vorticity indicated that structures become shorter and wider relative to a smooth wall, i.e., the wall layer tends towards isotropy, which is consistent with the laboratory observations of Antonia and Krogstad.5 Leonardi et al.3,4 performed a number of simulations by varying the ratio w/k of the cavity 共w is the longitudinal distance between the trailing edge of one bar and the leading edge of the consecutive bar and k is the height of the cavity兲. The magnitude of k ⫹ (⫽ku ␶ / v , u ␶ is the friction velocity and v is the kinematic viscosity of the fluid兲 was greater than 50, so that the flow may be considered as fully rough. The intensity of both spanwise and normal velocity fluctuations increased in the wall region. The increase is very large and approximately linear from w/k⫽1 up to w/k⫽7; there is no

Most wall flows encountered in engineering applications develop over rough surfaces. Consequently, a good understanding of the physics of the flow in the vicinity of the roughness is an essential prerequisite when designing, for example, surfaces that optimize the convective heat transfer. Direct numerical simulations 共DNSs兲 have already proven to be most useful for probing the physics of turbulent smoothwall flows. There is now increasing optimism that direct numerical simulations should be able to provide new insight into the structure of turbulence over rough walls. The effect of roughness can be studied analytically when the interest is in the stability of the flow in the presence of only a small deformation of the surface. For example, Cabal et al.1 considered the instability of wavy surfaces and found that a two-dimensional disturbance leads to the formation of three-dimensional vortical structures near the wall. The latter structures are the precursor of the well-documented streaky structures in turbulent smooth-wall flows. In an earlier paper, Floryan2 noted that the results obtained using geometrical disturbances were similar to those caused by applying transpiration at the wall. a兲

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significant change between w/k⫽7 and w/k⫽10. If the main interest is in the flow above the roughness, one could suppose, as a first approximation, that the effect of the cavity on the outer flow could be reproduced by simply imposing an appropriate velocity distribution along the imaginary plane which contains the crests of the elements, i.e., the velocity distribution at the interface between the cavity and the overlying flow would be the main factor which dictates the changes that occur above the cavity. This velocity distribution will of course change as w/k is varied. From the simulations of Leonardi et al.,3 it is not possible to ascertain which of the three velocity components at the cavity interface was mainly responsible for the structural changes observed above the cavity. The present paper aims to clarify this issue. More generally, we expect the information generated by this study should contribute to formulating effective wall turbulence control strategies and strategies to design more efficient roughness surfaces. Recently, this topic has received much interest and several strategies have been suggested. For instance, Choi et al.6 succeeded in obtaining a significant drag reduction, albeit with the use of a type of control that would be difficult to implement in an experiment. On the other hand, Jimenez et al.,7 aiming to reproduce a channel flow over a porous surface, imposed pressure gradient distributions along the wall. With this strategy, which is equivalent to imposing a normal velocity at the wall, the outcome was an increase in drag. Here we performed direct numerical simulations of a channel flow with plane smooth boundaries, starting with an initial no-slip conditions flow field. Each simulation evolves by imposing 共on one wall兲 only one nonzero velocity component. The spatial distribution of this nonzero component is the same as for the DNS of Leonardi et al.3,4 Whereas in the latter 共hereafter referred to as the ‘‘real’’兲 case, there is a non-negligible interaction between the interior of the cavity and the external field, in the present situation, only the boundary conditions can affect the flow. The simulations were done at h ⫹ ⫽u ␶ h/ v ⫽180 where h is the channel halfwidth. This Reynolds number is sufficient for turbulence to be sustained. With a reasonable number of grid points, useful insight into turbulence modifications can be inferred from low cost simulations. II. NUMERICAL PROCEDURE

The incompressible nondimensional Navier–Stokes and continuity equations are 1 ⳵ 2U i ⳵ U i ⳵ U iU j ⳵P ⫽⫺ ⫹⌸ ␦ i1 ⫹ ; ⫹ ⳵t ⳵x j ⳵xi Re ⳵ x 2j

ⵜ•U⫽0, 共1兲

where Re⫽(Uch/v) is the Reynolds number, U c is the centerline velocity, ⌸ is the pressure gradient required to maintain a constant flow rate, U i is the component of the velocity vector in the i direction, P is the pressure, x 1 , x 2 , and x 3 are the streamwise, wall-normal and spanwise directions, respectively. The Navier–Stokes equations have been discretized in an orthogonal coordinate system using a staggered central

second-order finite-difference approximation. In the inviscid case with free-slip conditions, energy is conserved. More details of the numerical method can be found in Orlandi.8 The discretized system is advanced in time using a fractional-step method with viscous terms treated implicitly and convective terms explicitly. The large sparse matrix resulting from the implicit terms is inverted by performing fast Fourier transform 共FFTs兲 in the homogeneous directions and applying tridiagonal solvers in the nonhomogeneous directions. The reduced wavenumber is used instead of the real wavenumber in order to maintain the second-order accuracy of staggered finite differences. At each time step, the momentum equations are advanced using the pressure at the previous step, yielding an intermediate nonsolenoidal velocity field. A scalar quantity ⌽ projects the nonsolenoidal field onto a solenoidal one. A hybrid low-storage third-order Runge–Kutta scheme is used to advance the equations in time. The code has been parallelized in MPI and the parallelization has been done by dividing the channel in horizontal layers each with at least 20 points in the normal direction. Only low Reynolds number flows are considered so that the simulations do not require a heavy parallelization. We have used only four processors in a machine, assembled in-house, with fast ethernet access. The large data transfer, required to transpose matrices, slows down the calculation and reduces the efficiency. Nonetheless, outputs can be obtained within times comparable to those in large supercomputer centers where the codes may be queued for relatively long times. A simulation with a 192⫻192⫻400 grid 共including temperature兲 requires approximately 144 s for one time step, which is subdivided into three substeps. To simulate 100 time units, about 8000 steps are needed with ⌬t evaluated by imposing the maximum value 共⫽1兲 of the Courant–Friedrich–Lewis number 共CFL兲. III. FLOW CONFIGURATION

Leonardi et al.3 carried out rough wall simulations for w/k⫽0.33,0.6,1,2.07,3,4,5.5,7,8,9,10,19. Here, we consider w/k⫽3, a geometry for which the total drag was found to be entirely contributed by the form drag, the frictional resistance being negligible. This value of w/k happens to coincide with that used in a number of experiments, either in a boundary layer or a channel flow, at higher Re 共Moore,9 Liu et al.,10 Furuya et al.,11 Antonia and Luxton,12 Perry et al.13兲. The results of Leonardi et al.4 are consistent, both in terms of the flow patterns around the elements and relative distributions of the Reynolds stresses above the elements, with those obtained in the cited experiments. The fact that the effect of Re does not seem important is not surprising since the drag is supplied by the shape of the elements. Note that Furuya et al.11 used circular rods and were able to determine the pressure distribution around the rod 共this can be done by using only one pressure tap and rotating the rod through 360 deg兲; the resulting dependence of their form drag on w/k is quite similar to that in the Leonardi simulations. Figure 1共b兲 provides a qualitative picture of the effect of the cavity on the normal vorticity at y/h⫽0.05 共y is the distance from the



DNS of turbulent channel flow with wall velocity

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FIG. 1. Wall normal vorticity contours; 共a兲 unperturbed wall; 共b兲 rough wall at a distance y⫽0.05.

crests of the square bars兲. There is no evidence of the elongated coherent structures which characterize a flat wall. A comparison between the ␻ 2 contours and those for the other two vorticity components corroborates one of the main conclusions that Smalley et al.14 inferred from Reynolds stress anisotropy invariant maps, i.e., the introduction of the roughness improves the near-wall isotropy. The distribution of velocity at the interface between the cavities and the outer flow, obtained at a particular time, has been used to impose the boundary condition on one wall. In the real simulation, this distribution varies with time; here, it is kept constant for the complete time evolution. It may be argued that the results could depend on the particular realization that has been chosen. Later, it will be shown that, for the most interesting case (u 2 distribution兲, two different wall boundary conditions yield the same statistically steady state. If the velocity distributions for a different w/k is used as input, the present simplified simulations should also reproduce the trend observed in the real case. This has been verified by carrying out another simulation starting from the same condition and with a u 2 disturbance taken from a velocity field by Leonardi et al.3,4 at w/k⫽7. The disturbance

was calculated in the same manner as for w/k⫽3. The choice w/k⫽7 was made because experiments 共Furuya et al.11兲 have indicated that, for this value the maximum roughness function ⌬U ⫹ is achieved. We would like to repeat that the purpose of this paper is not to reproduce the roughness effects in an exact manner, but to understand, through these simplifications, how the near-wall turbulence reacts to wall disturbances. However, since the velocity disturbances are taken from real simulations, it is important to clarify the type of roughness used in the latter. For w/k⫽3 and for w/k ⫽7, k ⫹ is 70 and 90, respectively. These are values typical of fully rough flows, for which the wall friction is independent on Re. The contours in Fig. 2共c兲 show that the streamwise velocity is positive over a large fraction of the cavity and that there are only a few narrow regions where it is negative. These regions are so narrow that they are not clearly discernible; their locations correspond to the white regions. The other two velocity components have equally distributed positive and negative values. The u 2 contours 关Fig. 2共b兲兴 suggest that the cavity interacts with the external flow to produce local inflows and ejections, the latter being more intense


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FIG. 2. Velocity contours at the interface, 共positive solid, negative dashed兲 from Leonardi et al. 共Ref. 4兲: 共a兲 u 3 , 共b兲 u 2 , 共c兲 u 1 . Contour interval for 共a兲 and 共b兲 ⌬⫽0.05; for 共c兲 ⌬⫽0.1.

close to the leading edge of the cavity. Velocity contours for w/k⫽7 are not shown but to give an idea of the differences relative to w/k⫽3, we report the dimensionless velocity minima and maxima. 共For i⫽1,2,3 and for w/k⫽3 ⫺0.205, 0.832; ⫺0.360, 0.354; ⫺0.476, 0.486; for w/k⫽7 ⫺0.393, 0.944; ⫺0.505, 0.595; ⫺0.700, 0.710.兲 From these contours, some preliminary considerations can be made. For example, the u 1 distribution has a mean value different from zero. The global effect is equivalent to that obtained by translating the wall in the same direction as

the external flow; this motion reduces the velocity gradient at the wall and in the near-wall region. The simulations of Perot and Moin15 and Orlandi and Leonardi,16 as well as the experiment by Uzkan and Reynolds,17 indicated that this reduces the production of turbulent energy and leads to drag reduction. It is difficult to speculate on possible effects which may arise from the other two velocity components. To overcome this, three different simulations were carried out, each with one of the three components imposed and the other two set to zero. The results are compared with Leonardi et al.4 in




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FIG. 3. Pressure gradient time history, unperturbed, u3 , • • • • u 1 . The thicker chain–dot and chain–dash lines correspond to different u 2 disturbances, •• •• •• •• u 2 from the real simulation with w/k⫽7.

order to understand which of the three boundary conditions are most relevant to rough wall research. The simulation strategy consists in initiating the calculation using a coarse mesh 共e.g., 48⫻96⫻64兲 with no-slip boundaries; after a further mesh refinement 共e.g., 64⫻128 ⫻128兲, a simulation with a 96⫻128⫻200 grid evolved between t⫽200 and t⫽300, also with no-slip conditions. At t⫽300, the field was saved and from this latter starting condition, four simulations were launched over a further 100 time units. One had no-slip conditions, while each of the other three had one of the velocity distributions in Fig. 2 imposed at the lower wall while the other two components were set equal to zero. IV. RESULTS

In Fig. 3, the time history of the pressure gradient shows the following interesting features. As noted previously, imposing u 1 results in drag reduction. The effect of u 3 is marginal, the pressure gradient remaining almost similar to that with no-slip. In these two cases, the frequency with which ⌸ varies is small, resulting in decreased turbulence activity near the wall and more ordered vortical structures. On the other hand, when u 2 is applied, the frequency of ⌸ increases, so that smaller vortical structure are generated near the wall. The use of a different distribution of u 2 共Fig. 3兲, taken at a different time in the Leonardi et al.4 simulation, leads to essentially the same statistically steady state. The contours of the u 2 disturbance for the other case are not shown but to give an idea of the differences between the two cases the minimum and maximum dimensionless velocity values are 共⫺0.360,0.354兲 and 共⫺0.394,0.446兲 for the first and second cases, respectively. To investigate whether these simplified calculations reproduce the trend of the real simulation, Fig. 3 shows that the u 2 distribution for w/k⫽7, as in the real simulation, produces a greater pressure gradient together

with higher fluctuations. The amplitude of the pressure gradient fluctuations are a first indication that there are large modifications to the flow structures. A. Mean velocity

Calculated mean velocity profiles have been scaled in two different ways. In Fig. 4共a兲, the scaling is with u ␶ , obtained by averaging the pressure gradient in Fig. 3; this means that the friction on both walls is taken into account. Although this scaling is not appropriate, we are showing it because, in experiments, the total drag is easily inferred from the streamwise pressure gradient 共the drag on each wall is difficult to measure, especially if the wall is rough兲. The used grid is more than sufficient to reproduce the no-slip results of Kim et al.18 When u 3 is disturbed, profiles on either side of the centreline are close to each other and to the no-slip Kim et al.18 profile; this is expected from Fig. 3 and implies that the wall friction is essentially unaffected when the perturbation is applied to u 3 . In the other two cases, the profiles are shifted upwards in the drag reducing case (u 1 disturbance兲 and downwards for the drag increasing case (u 2 disturbance兲. This shift of the profiles on the side with the disturbance 共open symbols兲, reflects changes to the friction on the wall where the velocities have been assigned. The consequence is that on the other side 共closed symbols兲, the log law has a different slope. Figure 4共a兲 shows that for the u 1 disturbance, the linear, buffer and log regions remain discernible, but the slope of the log region changes. For the u 2 disturbance, the profile no longer has the characteristics of wall-bounded flows. This is a first rough indication of a certain kind of isotropisation. Confirmation and quantification of this will be provided later through the modifications to the turbulent stresses and flow structures. A more appropriate way to normalize the velocity profiles is to use the friction velocity on each wall. Figure 4共b兲


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FIG. 4. Velocity profiles in wall coordinates: 共a兲 scaled with the total friction, 共b兲 scaled with the local friction. Kim et al. 共Ref. 18兲, 具u⫹ 1 典 ⫹ ⫽(1/0.41)log y ⫹5.5, — – — real DNS Leonardi et al. 共Ref. 3兲, w/k⫽3, •• •• •• •• w/k⫽7. Velocity disturbance: 䊐 u 3 , 䊊 u 2 , 䉭 u 1 (w/k⫽3), inverted triangles u 2 for w/k⫽7. Solid symbols unperturbed wall, open symbols perturbed wall.

shows that near the no-slip wall, all the profiles collapse well, whereas large differences occur on the other side. With u 3 , the profile exhibits a slight downward shift, relative to the ‘‘normal’’ log law. This shift is due to the slight increase in drag. When u 1 is perturbed, the mean velocity at the wall is nonzero. Although the mean velocity gradient is decreased, it is still sufficiently large to produce turbulent en-

ergy near the wall and generate the characteristic streaky structures. After subtraction of the mean velocity at y⫽0, the normalized mean velocity coincides with that over the unperturbed wall. When u 2 is assigned, 具 u 1 典 is zero at the wall; the slope is the same and the downward shift, characteristic of rough wall flows, is the same as in Fig. 4共a兲. This indicates that the friction on the perturbed wall is much greater




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FIG. 5. rms profiles; left in wall coordinates 共a兲–共c兲, right not normalized 共d兲–共f兲; Kim et al. 共Ref. 18兲, • • • • DNS with rough elements 关Leonardi et al. 共Ref. 3兲兴, 䊐 u 3 , 䊊 u 2 , 䉭 u 1 , solid symbols unperturbed wall, open symbols perturbed wall.

than that on the unperturbed wall. The change of the shape of the profile reflects the modifications to the turbulent structures. Comparison with the profile obtained from the real DNS of Leonardi et al.4 shows that the u 2 disturbance shifts the profiles in the same direction. There is poor quantitative agreement due to the absence of the form drag in the present simulation. We would like to point out that the profile of the real DNS has been obtained by subtracting the value of 具 u 1 典 at the interface ( 具 u 1 典 0 ) and by taking the distance y from the location of the interface at the position x 20 . This operation produces profiles that are different from those usually obtained in experiments, where x 20 can be evaluated but 具 u 1 典 0 cannot. Leonardi et al.4 discussed different ways by which the virtual origin x 20 can be estimated. To confirm that the u 2

disturbance is the one that produces the main effect of the roughness, Fig. 4共b兲 shows that the shift ⌬ 具 u ⫹ 1 典 in the two simplified simulations based on u 2 is the same as that for the real simulations. From this, we can speculate that even for w/k⫽7, the effect of the other two velocity components is minor and it did not seem worthwhile to implement simulations with u 3 and u 1 disturbances. From the previous observations, we conclude that the downward shift characteristic of rough wall flows, as found in several experiments, e.g., Perry et al.13 and in numerical simulations 共Leonardi et al.3兲, can be attributed mainly to the normal transpiration generated by the rough wall interacting with the external turbulence field. Before describing the effect on turbulent stresses, we would like to briefly address the question of whether the


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FIG. 6. 共a兲 Absolute value of turbulent stress profiles, legend as Fig. 5.

FIG. 7. Vorticity rms, 共a兲 upper wall, 共b兲 lower wall. Lines real simulation 共w/k⫽3兲, symbols present simulation 共u 2 disturbance兲: , 䊐 ␻ 3; , 䊊 ␻ 2; • • • , 䉭 ␻ 1.




FIG. 8. Two point correlations: left streamwise, right spanwise,

results are affected by the choice of the grid. The grid refinement check is given in the Appendix. From the results given there, we believe that the 96⫻128⫻200 simulations are sufficient to reproduce the modifications to the turbulent structures which contain the major portion of the energy. Clearly, if the focus is on the smallest scales, a 192⫻192⫻400 grid is probably required. However, if the objective is to understand the effect the roughness has on low-order statistics, scales close to the Kolmogorov scale should not play a substantial role; they can then be discarded or poorly reproduced by the numerics 共see the spectra in the Appendix兲. B. Turbulent Reynolds stresses

In Figs. 5共c兲–5共e兲 the normal stresses scaled with the appropriate u ␶ are plotted and compared with the Kim

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unperturbed wall, 䊐 u 3 , 䊊 u 2 , 䉭 u 1 .

et al.18 data and the Leonardi et al.4 profiles, the latter normalized by the appropriate friction. For a quantitative comparison, we have to consider that in the real simulation, at w/k⫽3, the form drag represents the total drag. In the present simulations, there is no form drag, so that only a qualitative comparison is appropriate. In Fig. 5共a兲, the longitudinal stress shows that only with a u 2 disturbance is the rms reduced near the wall 共it increases far from the wall兲. The reduction of the normalized rms is larger in the real case due to the greater u ␶ . Far from the wall, in the real case, the bigger rms is due to the geometry used by Leonardi et al.3 In fact, in the real simulation, the distance between the upper wall and the interface was equal to 1.8; here, it is 2. With the u 1 disturbance, as discussed previously, the mean velocity is different from zero at the wall, but its value is such that it


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FIG. 9. Streamwise velocity contours at y ⫹ ⬇15 scaled by the local rms: 共a兲 Unperturbed, 共b兲 u 3 , 共c兲 u 1 , 共d兲 u 2 . Contour interval is 0.4.

FIG. 10. Pressure contours at y ⫹ ⬇15 scaled by the local rms: 共a兲 Unperturbed, 共b兲 u 3 , 共c兲 (u 1 , 共d兲 u 2 . Contour interval is 0.4.

does not affect the turbulence production and the maximum occurs at the same dimensionless distance from the wall. We can speculate that, near the wall, the imposed steady u 1 disturbances, which interact with the external flow, generate very small scales which are immediately dissipated and their effect on the energy containing scales is negligible. The rms spanwise velocity, with the u 3 disturbance, increases near the wall due to the input conditions, in similar fashion to what occurs in Fig. 5共a兲 for u 3⬘ ⫹ 共in this section, u i⬘ represents 具 u 2i 典 1/2). The u 1 disturbance does not affect u 1⬘ ⫹ ; on the other hand, u 2 produces a large effect on u 1⬘ ⫹ . To appreciate the magnitude of this effect, it must be recalled that the profiles in Fig. 5共b兲 have been normalized by u ␶ , which is quite large for the u 2 disturbance. The large modi-

fications to the Reynolds stresses can be better appreciated by considering non-normalized quantities. In the very near-wall region, u 2⬘ is only slightly affected by the u 3 and u 1 disturbances. This moderate growth disappears within a short distance from the wall, and for y ⫹ ⬎10, the profile for the unperturbed case is recovered. As expected from the discussion of Fig. 4共b兲 and Figs. 5共a兲 and 5共b兲, the u 2 disturbance affects even this stress significantly; its peak value is close to that of the real simulation. But, because of the absence of the form drag in the present simulations, we fully expect the non-normalized profiles to have smaller values than in the real case 关Fig. 5共e兲兴. From Figs. 5共a兲–5共c兲, the interesting conclusion is that the turbulence modifications close to the rough wall are due to the normal



FIG. 11. u ⬘1 and usx3 contours in x 1 ⫺x 2 planes at the center of the ⬘ channel; 共a兲 and 共b兲共u 1⬘ and u 2⬘ for w/k⫽3, 共c兲 and 共d兲共u 1⬘ and u 2⬘ for w/k⫽7兲, contour interval is 0.05.

velocities generated by the roughness elements. From a design viewpoint, we can speculate that an effective roughness shape is one which enhances the normal velocities in the vicinity of the roughness elements. As discussed earlier, the rms profiles reflect the effect of the roughness best when they are not normalized by wall variables. Figures 5共d兲–5共f兲 underline that very large variations in all three normal stresses occur only with the u 2 perturbation. Some of these large variations are in good quantitative agreement with the results from the real case. To quantify some of the discrepancies, it is important to remember that the real case was performed in a channel of size 2h,


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with elements of size 0.2h. The interface between the cavities and the channel was located at x 2 ⫽⫺0.8h, so that the distance between the fictitious and the upper walls was 1.8h. This smaller distance may explain why the effect of the roughness in the real case extends to a greater distance inside the channel. The solid lines in Figs. 5共d兲–5共f兲 show that, at the opposite wall, the rms of u ⬘1 is more affected than the other two, but the effect is small. Figures 5共d兲–5共f兲 show that the u 1 disturbance reduces all stresses and this is related to the reduced turbulence production due to a decrease in the mean shear. u 3 has a negligible effect. This is very interesting because Choi et al.6 using a DNS with different types of control concluded that the effects from u 3 or u 2 control were approximately equal. The different results in the two simulations are related to the different ways of imposing the boundary conditions. For Choi et al.,6 the wall boundary conditions were driven by sensors at an optimal distance from the wall (y ⫹ ⫽15). Here, on the contrary, the boundary conditions are passive, i.e., they are not correlated in any way with what occurs in the outer field and in particular where the turbulence is generated. The u 2 disturbance case is somewhat similar to the simulations of Jimenez et al.7 who aimed to reproduce flows past porous plates. In both cases, the drag increases and this can be of interest in several applications which rely on an increase in the mixing or in the heat transfer. In aeronautical applications, the interest is often in energizing the turbulent layer to delay the separation on the wing. In their simulation, Jimenez et al.7 introduced a parameter accounting for the porosity and imposed a periodic pressure gradient distribution at the wall. Although the latter is a relatively basic investigation with general applications, the present study aims at explaining features of rough wall flows which could be exploited for designing more efficient rough surfaces. The trend observed for the normal stresses holds for the turbulent shear stress. Figure 6 shows that, near the upper wall, the stress remains unchanged while, near the perturbed wall, the u 2 perturbation produces a large increase. The comparison with the real case shows that the shape and trend are reproduced but the quantitative difference should be related to the different geometrical conditions. The rms vorticities are the quantities which best characterize the anisotropy of wall bounded flows and should therefore be sensitive to changes that occur near the wall. Figure 7共a兲 shows that the profiles of the present simulation compare well with those of the real simulation and this is an indication that disturbances from one wall do not affect the other wall. On the other hand, Fig. 7共b兲 shows that the vorticity anisotropy in both simulations is reduced by imposing disturbances. The large values of 具 ␻ ⬘3 2 典 1/2 in the present simulation are due to the no slip assumption on u 1 at the wall 共this is different from zero in the real simulation兲. Near a solid wall, 具 ␻ 1⬘ 2 典 1/2 has a minimum at a distance from the wall where the center of the streamwise vortices is located. These are the vortices generating the low and high speed streaks that are representative of the wall anisotropy. The minimum disappears, suggesting that the long streaks become shorter and more inclined to the wall.


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FIG. 12. Normal stress profiles; lines coarse 96⫻128⫻200, symbols fine 192⫻192⫻400 共䊊 u 1 , 䊐 u 3 , 䉭 u 2 兲.

C. Flow structures

The changes to the wall turbulence statistics are related to modifications to the near-wall structures. Very close to the wall, these modifications are directly affected by the wall boundary conditions, and since these are steady, there is not enough space for the structure to form. This explains why some of the normal stresses are very large near the wall. In all cases, the peaks of the streamwise rms value 关Fig. 5共a兲兴 are located at a distance of about 15 wall units, where the turbulent energy production is maximum. It is at this position that visualizations and evaluations of the longitudinal and spanwise correlations have been made. In Figs. 8共a兲– 8共c兲, two-point velocity correlations are shown for the unperturbed and perturbed simulations R ii (r)⫽ 具 u i⬘ (r)u i⬘ (0) 典 / 具 u ⬘i (0)u ⬘i (0) 典共here u i⬘ indicates fluctuations兲. The streamwise and spanwise correlations are shown on the left and right sides, respectively. For the perturbed simulations, the length of the streaks is reduced for u 3 and more especially u 2 disturbances. From the spanwise correlations, a well defined minimum is no longer discernible and this implies that the high-speed and low-speed streaks are less defined. These observations are confirmed by the q ⬘1 ⫽u 1⬘ / 具 u ⬘1 2 典 1/2 visualizations in Figs. 9共a兲–9共d兲. These figures yield only a qualitative estimate about the character of the structures, in particular their spanwise and streamwise coherence. When the wall is partially in motion 关Fig. 9共c兲兴, the structures are more ordered with a better alignment in the streamwise direction. Comparison between Figs. 9共a兲 and 9共b兲 shows that these do not differ appreciably, as could be inferred from the two-point correlations in Fig. 8. The u 2 disturbance, on the other hand, shows that the coherence of the streaks is lost and visualizations of the other two velocity components reveal that all three velocity components have more similar distributions. This sort of isotropization of the wall region was quantified via Reynolds stress

anisotropic maps by Smalley et al.;14 this represented an important corroboration of the earlier laboratory observations by Antonia and Krogstad.5 The correlations confirm this trend; for the u 2 disturbance, the spanwise correlations no longer show a distinct minimum corresponding to the alternate occurrence of high and low-speed streaks. Even the streamwise correlations exhibit this behavior, their intersection with the zero axis being nearly the same as for the unperturbed flow. Contours of pressure fluctuations are representative of the large scales. These can thus give a global idea of the modifications to the vorticity field at y ⫹ ⬇15 where the energy production is maximum. These 关Figs. 10共a兲–10共c兲兴 show that the disturbances u 2 and u 1 affect the structures in a different manner. The u 1 disturbance has a large influence, with positive pressure peaks recurring periodically in the streamwise direction. The location of these peaks correspond to that of the leading edge of the roughness element, where high negative gradients of the u 1 disturbance occur 关Fig. 2共c兲兴. The pressure contours show that, even with the u 1 disturbance, there is a greater coherence in the spanwise direction relative to the u 2 disturbance 关Fig. 10共c兲兴. This is due to the reduced variations along x 3 due to the u 1 disturbance. The outcome is that more ordered structures form, as shown in Fig. 9共c兲. Flow visualizations, not shown here, of q 2⬘ indicate a periodicity in the streamwise direction when the u 1 disturbance is applied. As reported earlier, these are related to the highest negative gradients. From the continuity equation, positive ⳵ u 2 / ⳵ x 2 is produced to offset the negative ⳵ u 1 / ⳵ x 1 . The oscillatory behavior of the R 22 correlation in Fig. 8共c兲 reflects the behavior of the instantaneous distribution of q 2⬘ . The modifications to the wall structures can also be inferred from contours of fluctuating velocities in x 1 ⫺x 2




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FIG. 13. One-dimensional energy spectra; 共a兲 streamwise, 共b兲 spanwise directions: lines fine, symbols coarse 共䊊 E 11 , 䊐 E 33 , 䉭 E 22 兲.

planes. These planes show the differences between the two walls and here we focus on the fluctuating normal and streamwise velocity components. Contours of u 1⬘ are shown in Figs. 11共a兲 and 11共c兲 for w/k⫽3 and w/k⫽7. Near the unperturbed wall, the low and high speed streaks have al-

most the same inclination and intensity. Near the perturbed wall, the u 2 disturbance increases this inclination, and for w/k⫽7, almost the entire channel is affected. Contours of u 2⬘ in Figs. 11共b兲 and 11共d兲 show very clearly the trend towards isotropy due to the roughness.


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V. CONCLUSIONS

Although experiments have provided significant insight into the behavior of turbulent flows over different types of rough surfaces, progress has been relatively slow, partly due to the limitations of experiments in providing reliable information on the details of the flow in the vicinity of the roughness elements. In direct numerical simulations, this latter difficulty can be overcome, at least for relatively simple roughness geometries. The strategy outlined in this paper is an attempt to understand the effect of the roughness on the flow by applying appropriate boundary conditions on a plane surface, albeit those provided by a simulation of the flow over a particular type of rough surface. The purpose here is not necessarily to ‘‘mimic’’ the flow over rough surfaces but, rather, to gain some insight into the underlying physics of such flows without the complication of dealing with the rough wall geometry directly. The observation that the normal velocity is the driving parameter which leads to a drag increase is of particular interest in the context of optimizing the shape of a rough surface. The high velocity gradients at the leading edge of the elements were the main factor behind the modifications to the wall structures. The direct use of concentrated regions of positive and negative u 2 can be exploited for the purpose of energizing the turbulence. On the other hand, strong negative gradients of u 1 across narrow localized regions, will lead to a reduction in drag. Although the realizability of these conditions, e.g., in the laboratory, may not be straightforward, the present observations and conclusions should nevertheless be useful in guiding the formulation of turbulence wall-control strategies. A more interesting issue is to investigate the Reynolds number dependence; the computer limitations do not allow DNSs to be carried out at high Re. Work is in progress to introduce into the code subgrid models which will allow results to be obtained at larger values of Re. ACKNOWLEDGMENTS

The support of a MURST 60% grant is acknowledged. We would also like to thank the Super Computer Center of San Diego where some of the simulations were performed. R.A.A. acknowledges the support of the Australian Research Council. APPENDIX: ACCURACY CHECKS

In view of the skepticism which is sometimes leveled at the use of finite difference schemes, it is important to establish that the present results are grid independent. It is equally important to keep in perspective the objective of the simulations. If the interest is in studying the smallest scales, the grid check should show that one-dimensional spectra in the homogeneous directions collapse also in the dissipation range; a rule usually used is that the smallest grid should be 1.5 times the Kolmogorov scale. Since the Kolmogorov scale depends strongly on the resolution, the mean dissipation rate should also be grid independent. We would like to recall that, in energy conservative schemes, the simulations are stable and diverge only when the CFL restriction is violated. At any

given Reynolds number, there may be a contribution from the insufficient resolved dissipation rate. This has a negligible effect if any of the k 2i E ii (k i ) has a small plateau at high wavenumbers. This is a stricter requirement than that derived from analyzing E ii (k i ). On the other hand, if the aim is to study the modifications of the energy containing scales, the check can be limited to the second-order statistics and it may not be necessary to have correct spectra at high wavenumbers. One could interpret this approach as equivalent to carrying out a LES where the inadequately resolved small scales act as the subgrid model. To show that the 96⫻128⫻200 grid is satisfactory for capturing the modifications due to different velocity distributions on one wall, a grid refinement was implemented for the u 2 disturbance. In this case, we have observed that small 共energy containing兲 scales are generated and large variations occur. Through different stages similar to those described for the coarse simulations, a simulation with a 192⫻192⫻400 grid evolved for 100 time units and the statistics were evaluated from fields that were saved at an interval of two time units. The normal stresses 共Fig. 12兲, as produced by the two grids, are in satisfactory agreement. The largest differences are in 具 u 1⬘ 典 ; on the other hand, the other two stresses coincide perfectly in the wall region. Since the largest variations among the three turbulent stresses with different disturbances occur for u 2⬘ and u 3⬘ , we expect that the coarse grid simulations are satisfactory. To assess which scales are affected by the insufficient resolution, one-dimensional spectra at a distance y ⫹ ⬇15 were calculated and are shown in Fig. 13. The large scales are well represented by the coarse simulations. Figure 13共a兲 shows a small energy pile-up around k⫽90 due to the insufficient resolution, and, moreover, it appears that even the dissipative range is, on the whole, reproduced reasonably well. The spanwise spectra in Fig. 13共b兲 show that, for 96 points, the dissipative range is barely reproduced but the large scales can be considered to be sufficiently well resolved for the present purpose. 1

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