1 Introduction

Due to the close relationship between near-wall streamwise vortices (NWSV) and the high frictional drag in wall shear turbulent flow,many active control schemes have been developed to manipulate the organized structures for drag reduction^{[1, 2, 3]},e.g.,the micro-electro-mechanical system (MEMS) technology^{[4, 5]}. Before the design of active control systems with the MEMS technology,the detection of the coherent structures by wall measurable information,especially the detection of the NWSV which plays a leading role in turbulence self-sustaining,needs to be deeply understood. The wall pressure fluctuation p' _w is usually used as a detection signal to inhibit the NWSV^{[6, 7]}. To reveal the spatial relation between p' _w and the NWSV,Kim et al.^[8] used the two-point correlation function R in a turbulent boundary layer. The correlation coefficient R between the wall variable θ_w(x,z) and the streamwise vorticity ω_x(x,y,z) is defined by

where rms is an abbreviation of root-mean-square,and x,y,and z represent the streamwise, wall normal,and spanwise coordinates,respectively. The contours of the correlation function R_p'wωx are shown in Fig. 1. A significant correlation can be found between p'_w and the NWSV. The largest value of Rp'_wω_x occurs at the location (Δx+,y+,Δz+) around (−20,15,±20).

As shown in Fig. 1,the wall region with high pressure (p'_w > 0) has a good correspondence with the sweep events attributed to the upstream NWSV,while the wall region with low pressure (p'_w < 0) has a good correspondence with the upwash of the fluids due to the vortical structures. However,the correlation between p'_w at the detecting point (Δx⁺=0,Δz⁺=0) and the overhead streamwise vorticity is very weak. The value is only about 0.05. Different from the wall shear stress studied by Kravchenko et al.^[9],the pressure is a scalar. Therefore,in the aspect of R_p'wωx, the correlation between p'_w and the overhead NWSV which can rotate in different directions will be missed.

In this paper,the spatial relation between p'_w and the NWSV is further investigated in a turbulent channel via the direct numerical simulation (DNS) method. The pseudo-spectral method is used for the spatial discretization of the channel flow,and a 3rd-order time splitting method is adopted for the advance of time. The DNS method is well validated by Xu et al.^[10], which has also been used to produce a high quality turbulence database by Deng and Xu^[11], Deng et al.^[12],and Fang et al.^{[13, 14]}. The Reynolds number is

where U_m is the mean bulk velocity,and H is the half channel width. It is in accord with

where u_τ is the friction velocity,and ν is the kinetic viscosity. The computational domain spans 4π × 2 × 2π in the x-,y-,and z-directions with 128×129×128 grids,respectively. All the statistical results are counted in a simulation time period of 1 000H/U_m.

2 Conditional correlation

To get the exact relationship between p'_w and the NWSV,the conditional correlations between p'_w and the strong vortical structures at y⁺ = 15 rotating in different directions are performed,respectively. In the conditional averaging,only the streamwise vortices ω_x(x,y,z) satisfying

are taken into the calculation,where ω_x(x,y⁺ = 15,z) is the streamwise vorticity on the plane at y⁺=15 with the same x and z as ω_x(x,y,z). The vertical distance y⁺=15 is generally considered to be the strongest position of the NWSV. The conditional correlation between p'_w and the strong positive NWSV can be written as Similarly,the conditional correlation R_p'wωx^N between p'_w and the strong negative NWSV can also be written in the same way.

The conditional correlations R_p'wωx^Nand R_p'wωx^N are antisymmetrical about the coordinate z. Therefore,only R_p'wωx^N is analyzed in detail. Figure 2 exhibits the contours of R_p'wωx^N in several

cross sections with different Δx⁺. Different from the non-conditional correlation function,the contours of R_p'wωx^N are not symmetric about the coordinate z any more. As shown in Fig. 2,the correlation coefficient is the largest in the slice through the detecting point,is smaller in the upstream slices,and is the smallest in the downstream slices. The large correlation coefficient at the wall is induced by the shear effect between the NWSV and the wall. Therefore,the correlation contours above the wall are our main concern. Figure 2 is only for the positive streamwise vorticity. In this figure,the positive R_p'wωx^N contour represents the relevancy between the clockwise NWSV and the positive wall pressure at the location of (Δx=0,Δz=0), while the negative R_p'wωx^N represents the relevancy between the clockwise NWSV and the region when p'_w <0 at (Δx=0,Δz=0). In the upstream,the positive/negative correlation appears on the left/right side of Δz=0,which means that the high/low wall pressure region has a good correspondence with the sweep/ejection events due to the clockwise NWSV in the upstream. In the region of

the largest value of negative R_p'wωx^N appears just above the detecting point,indicating that the low wall pressure is mostly associated with the overhead clockwise NWSV. In the downstream of Δx⁺ > 54,the condition is just opposite to that in the upstream.

The schematic of the NWSV and p'_w is shown in Fig. 3. The high wall pressure region is correlated closely with the downwash of the fluids due to the upstream NWSV,while the wall region with low pressure is closely related with the upwash of the fluids attributed to the upstream NWSV. In the region around the measured location,the low wall pressure is correlated mostly with the overhead (clockwise/counterclockwise) NWSV. For simplification, the spatial relation between p'_w and the downstream NWSV,which is opposite to Figs. 3(a) and 3(b),is not given in the schematic.

3 Statistics of sweep and ejection events

To examine the correspondence between p'_w and the NWSV in another aspect,the contributions of the sweep/ejection motion to the Reynolds shear stress are conditionally analyzed. Above these locations in the upstream/downstream,i.e.,

and the measured point

the Reynolds shear stresses from the ejection motion and the sweep motion satisfying

are counted,respectively. Statistics without differentiating

at the measured point is also given for comparison. The product of u_rmsv_rms at the corresponding y⁺ is used to normalize the Reynolds shear stress.

Figure 4 shows the contributions of the ejection motion and the sweep motion to the Reynolds shear stresses above

As shown in Fig. 4,when p'_w <0 at the measured point,the upstream ejection motion increases and the sweep motion decreases; when p'_w >0,the upstream sweep motion increases,and the ejection motion decreases. Therefore,it can be concluded that the wall region with high pressure has a better correspondence with the upstream sweep motion,while the wall region with low pressure corresponds to the upstream ejection event better.

Figure 5 shows the contributions of the ejection motion and the sweep motion to the Reynolds shear stresses just above the measured point. From the figure,we can see that when p'_w <0, both the sweep event and the ejection event increase obviously,while when p'_w >0,both of them reduce. This indicates that the turbulence above the low pressure region is more active.

Figure 6 shows the contributions of the ejection motion and the sweep motion to the Reynolds stresses above the locations of

in the downstream of the measured point. In the downstream,the condition is just opposite to that in the upstream,but the quantity of the variation is smaller. At the location of (Δx⁺ = 108,Δz⁺ = 0),almost no correspondence exists between the ejection/sweep motion and the wall pressure fluctuations.

4 View of spatial relation between p'_w and NWSV

To show the relationship between the NWSV and p'_w more clearly,a conditional average of the flow variable φ(x,y,z) is performed. The coordinate values without the superscript + are normalized by the half channel width H. In the conditional average,only the flow samples satisfying

are taken into calculation,where ω_x(x⁽ⁱ⁾,y⁺ = 15,z⁽ⁱ⁾) is the streamwise vorticity at the central point of the plane at y⁺=15 of the flow sample. The conditional average related with the strong positive NWSV is defined by where

The conditional average is a three-dimensional result with a domain of l_x × 1 × l_z. in the x-, y-,and z-directions,respectively. The conditional average ‹φ(Δx,y,Δz)›^N,related with the strong negative NWSV,can also be written in the same way as Eq. (3) with the flow samples satisfying

The iso-surface of ‹ω_x›^P = 1.0 and the contours of the conditional average ‹P'›^P at different Δx⁺ are shown in Fig. 7. A local minimum pressure can be found in the core of the NWSV, and the centrifugal force is balanced by the pressure force. Therefore,the local minimum is normally used as the identification of a vortex^[15]. From Fig. 7,we can also observe that the spanwise motion of the fluid is slowed down by the shear effect of the wall on the near-wall boundary of the vortices. Therefore,a smaller pressure gradient is needed here to balance the centrifugal force. Thus,although the wall-pressure below the NWSV is substantially bigger than the pressure in the core of the NWSV,it is fairly lower than the other part of the wall. To some extent,the low wall-pressure region seems to be a wall-trace of the overhead NWSV, which brings a local minimum pressure in its core.

A plane view of the iso-surface ‹ω_x›^P = 1.0 and the contours of the conditional average ‹P'_w›^P are shown in Fig. 8. As exhibited in Fig. 8,the iso-surface of ‹ω_x›^P = 1.0 tilts to the negative z-direction,which agrees well with the NWSV identified by Jeong et al.^[16] using λ₂. Here,λ₂ is the second largest eigenvalue of S²+Ω²,where Ω and S represent the rotation tensor and the deformation rate tensor,respectively. Since the streamwise structures with positive ωx are the symmetric counterparts of the structures with negative ω_x,only the conditional average associated with positive ω_x is shown here. Below the strong streamwise vortex,a low wall pressure region forms with a bigger tilting angle. The wall region with high pressure mainly arises on the downwash side of the upstream NWSV.

Inferred from all the above analysis,a full schematic of the spatial relation between the NWSV and P'_w is shown in Fig. 9. The high pressure region and the low pressure region on the wall appear in a staggered pattern,following the alternate positive/negative NWSV in the streamwise direction. Based on Fig. 9,a more detailed explanation of the conditional correlation in 2 is given here. In Fig. 9,three streamwise vortices,including the NWSV A in the upstream,the NWSV B in the middle,and the NWSV C in the downstream,are sketched. It can be observed that the low pressure regions formed on the wall are located just below the NWSV. Therefore,a strong conditional correlation between the low wall-pressure and the overhead NWSV can be obtained in Fig. 2. For the convenience of illustration,let us only focus on the high and low wall pressure regions near the NWSV B. Due to the tilt of the NWSV,the high pressure region on the wall corresponds well with the upstream sweep motion. This is because that the adjacent vortices and the wall region with low pressure have a good correspondence with the upstream ejection motion due to the overhead NWSV. Hence, a close association between the high/low pressure region on the wall and the sweep/ejection motion in the upstream can be obtained. In the same way,the low wall pressure region also corresponds with the downstream sweep motions induced by the overhead NWSV,while the high wall pressure region has a weaker correspondence with the up-wash of fluids due to the downstream NWSV C. Therefore,an inverse correspondence of the wall pressure and the sweep/ejection motion is induced downstream of the detecting point. In summary,attributed to the relative positions of the NWSV and the high/low wall-pressure regions,the asymmetry correlations between the wall region with high/low pressure and the ejection/sweep motion are induced.

5 Conclusions

The spatial relation between the wall pressure and the NWSV is further investigated,using the DNS database of a fully developed turbulent channel flow at Re_τ=180. The results show that the region with low wall pressure is mostly associated with the overhead NWSV. In addition,the wall region with high pressure has a good correspondence with the sweep/ejection motion attributed to the upstream/downstream NWSV. The region with low pressure has a good correspondence with the upwash/downwash of fluids in the upstream/downstream vortical structures. According to the new findings,a more complete view of the spatial relation between the wall pressure and the NWSV is proposed,which provides a significant basis for the detection of the NWSV in turbulent control.

Acknowledgements

The authors would like to express their gratitude to Professor Chunxiao XU and Professor Guixiang CUI in Tsinghua University for their help in this work.