The Response of Turbulent Channel Flow to Standing Wave-Like Wall Motion

Abstract

The effect of wall deformations on wall turbulence is a topic of fundamental importance for the advancement of flow control strategies aimed at reducing the frictional drag. Dynamic wall deformations in the form of wall-normal of in-plane oscillations represents an area that is yet to be fully explored, and that can open a new realm of drag control strategies, as well as provide fresh insights into the structure of wall turbulence. In this study, we present several results from direct numerical simulations aimed at understanding the response of wall turbulence to standing wave-like wall motion, arranged in a checkerboard pattern. More specifically, here we target a low Reynolds number turbulent channel flow, featuring standing wave-like dynamic wall deformations on both bottom and top walls, with parameterizations in terms of the frequency of oscillations, roughness height, and spacing. We quantify the effect that this type of wall motion can have on the skin friction drag, various turbulent statistics, and turbulent flow structures. In addition, taking advantage of the periodicity and the symmetry of the flow, an improved phase-averaging procedure is introduced, which enhances the smoothness of the data. The results show that this type of dynamic wall deflection can have a significant effect on the turbulent flow in proximity to the wall, and that the variation of the spatial wavenumbers of the wall deflections can make a big difference. A slight total drag reduction, in the order of 2%, was observed for some combinations of wavenumbers and frequencies, especially for the highest streamwise wavenumber case.

1. Introduction

Flow control targeting the reduction of turbulent frictional drag is a challenging problem and is being actively pursued by the fluid dynamics community, with techniques involving experimental, numerical or/and theoretical approaches. Most previous control strategies for turbulent boundary layers targeted the alteration of energy carried by the streamwise-oriented, viscosity-dominated structures in the form of high- and low-speed streaks in the near wall region, in an attempt to reduce the frictional drag. The list of most popular techniques includes, among others, wall blowing and suction, wall deformations or oscillations, and wall actuators.

Wall motion in the form of wall-normal or spanwise oscillations can have a significant effect on wall turbulence dynamics, altering the wall shear stress distribution and consequently modifying the frictional drag. Numerous previous studies showed that methodologies capable of manipulating the parameters associated with wall deformations, if performed properly, can lead to substantial drag reduction. These techniques require and rely on sophisticated mathematical, computational and experimental tools that are used to investigate either the onset of transition to turbulence or the fully-turbulent boundary layer. Both passive and active flow control strategies have been considered over the years in an attempt to reduce the turbulent friction drag. The latter strategy relies on using a certain form of external energy aimed at manipulating the wall geometry or the flow itself. Although the focus of this study is on active flow control, worth mentioning are several passive flow control techniques based on wall deformations, which can be in the form of riblets (e.g., Garcia-Mayoral and Jimenez [1] or Sasamori et al. [2]), dimples (e.g., Ligrani et al. [3], Lienhart et al. [4], or Tay et al. [5]), surface waviness (e.g., Karniadakis & Choi [6], Hoepffner & Fukagata [7], or Nakanishi et al. [8]), or two-dimensional roughness elements (e.g., Fong et al. [9] or Duan et al. [10]) that have been applied in a number of studies to delay transition in boundary layers or to reduce the skin friction drag in wall turbulence. These are just a few examples from the vast literature that exists in this area of research.

Active flow control techniques involving boundary layers can target either the delay of transition from laminar to turbulent flow or the reduction of the skin friction drag in wall turbulence. Here, we focus our attention on the latter. Control based on active wall deformations aimed at counteracting streaks in turbulent boundary layers or channel flows has been successfully applied to reduce the frictional drag, although the reduction was not as high as in the case of controlled wall blowing and suction (Choi et al. [11]). In one of the earliest studies, performed by Carlson & Lumley [12], the effect of wall deformation on turbulent structures at the wall was considered; an actuator was used to control one pair of coherent structures near the wall. Another example is the work of Endo et al. [13], who conducted a DNS study that targeted a feedback control scheme involving wall deformations, aimed at reducing the skin friction drag in a turbulent channel flow. Kang & Choi [14] investigated the potential of reducing the skin-friction drag in a turbulent channel flow via a control scheme based on active wall deformations. A reduction of the frictional drag on the order of 13–17% was achieved by their approach. Mani et al. [15] used a deformable skin actuated by active materials for turbulent boundary layers control, claiming large reduction of the skin friction drag. It was based on a generalized actuation principle that was capable of generating a traveling sine wave along the surface of an active skin. Amitay et al. [16] experimentally investigated a boundary layer control mechanism based on a localized dynamic roughness, using a pair of piezoelectrically driven oscillating surfaces. In a series of studies, McKeon et al. [17,18] investigated experimentally the response of a turbulent boundary layer flow to dynamic roughness perturbations. Among other things, they showed that both the frequency response as a result of wall forcing and the change to the mean flow profile are dependent on the input frequency and amplitude. More recently, Gibeau and Ghaemi [19] conducted experiments to investigate whether wall-normal surface deformations are able to control very large scale motions in a turbulent boundary layer.

Other approaches involving wall motion have been considered, such as for example spanwise or streamwise wall oscillations, which experienced a high level of success. A sizable body of research performed in this area was reviewed by Karniadakis and Choi [6] or by Quadrio [20]. Hack and Zaki [21], for example, performed direct numerical simulations to study the effect of a spanwise oscillating flat plate on the bypass breakdown to turbulence; they found that the transition onset can be delayed and the transition region can be significantly extended. Spanwise wall oscillations have also been shown to attenuate effectively the turbulence intensity in wall-bounded flows, thereby producing a sustained reduction of turbulent wall friction. Worth mentioning is also the work of Choi et al. [22] who performed an experimental investigation of the effect of spanwise-wall oscillation on the skin friction drag to confirm previous results from numerical simulations. They obtained as high as 45% drag reduction as a result of some optimizations, and attributed this to the mean velocity gradient reduction due to the spanwise vorticity generated by the Stokes layer. In Meysonnat et al. [23], an experimental-numerical investigation of spanwise transversal wall surface waves was conducted in an attempt to reduce the frictional drag, but only a very small reduction was achieved. Bird et al. [24] conducted an experimental study targeting the control of turbulent boundary layers using streamwise traveling waves of spanwise wall velocity, which was achieved by utilizing a novel active surface based on kagome lattice geometry. The highest drag reduction was within 21.5% for a $R{e}_{\tau }=1125$ turbulent boundary layer. Other worth mentioning studies in this area are Quadrio et al. [25], Skote [26], Moarref and Jovanovic [27], Touber and Leschziner [28], Deshpande et al. [29], or Rouhi et al. [30].

In this study, we conduct direct numerical simulations of low Reynolds number turbulent channel flows to study the response of wall turbulence and frictional drag to dynamic wall deformations in the form of standing wave-like motion featuring a checkerboard pattern, with deflections oscillating at certain frequencies. The parameter space of interest include wall deflection amplitude, packing density (quantified in terms of streamwise and spanwise wavenumbers), and frequencies of wall oscillations. The objective is to analyze the effect of these deformations on turbulent structures and a number of statistical quantities, and to identify a combination of parameters that can eventually reduce the friction drag. For simplicity we consider a turbulent channel flow, where both bottom and top walls feature this type of dynamic wall motion. By targeting a range of frequencies, spanwise/streamwise wavenumbers, and wall deflection amplitudes, we quantify the effect that this type of dynamic wall motion has on the skin friction, various turbulent statistics, and turbulent flow structures.

2. Numerical Approach and Simulation Settings

We solve for the non-dimensional incompressible Navier-Stokes equations using the entropically damped artificial compressibility (EDAC) formulations of Clausen [31]. with the momentum equation given as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{1}{Re}}{\displaystyle \frac{{\partial }^2{u}_i}{\partial x_i\partial x_j}}+F_1{\delta }_{i1}\end{array}$$

(1)

where $u_i$ is the i-th component of the velocity vector, $x_i$ is the i-th spatial coordinate, with $x_1$ in the streamwise direction, $x_2$ in the woll-normal direction and $x_3$ in the spanwise direction, p is the pressure, $Re$ is the Reynolds number based on the channel height and mean centerline velocity, and $F_1{\delta }_{i1}$ is a pressure gradient term (quantified as the total drag divided by the volume of the channel) to maintained a constant bulk velocity (${\delta }_{ij}$ is the Kronecker delta symbol). In the EDAC approach, an entropy balance is utilized to close the system of equations, assuming an isentropic behavior and minimizing density fluctuations. Thus, the EDAC equation for pressure follows as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial p}{\partial t}}+u_j{\displaystyle \frac{\partial p}{\partial x_j}}=-{\displaystyle \frac{1}{M}}{\displaystyle \frac{\partial u_j}{\partial x_j}}+{\displaystyle \frac{1}{Re}}{\displaystyle \frac{{\partial }^{2}p}{\partial x_j\partial x_j}}\end{array}$$

(2)

where M is the Mach number (more details and discussions about this method can be found in Clausen [31]). This approach has been tested and validated for turbulent channel flows at $R{e}_{\tau }=180$ and 395 by Kajzer and Pozorski [32]. The numerical algorithm involves high-order finite difference approximations for the spatial derivatives and explicit time marching. The time integration is performed using a second order Adams-Bashforth method, while spatial derivatives are discretized using dispersion-relation-preserving schemes of Tam and Webb [33]. No-slip boundary condition for velocity is imposed at the bottom and top walls, and periodic boundary conditions are imposed in both the streamwise and spanwise directions.

The wall deformations are imposed using the displacement equation

$$\begin{array}{c}\hfill y_w(x,z,t)=A\sin \left(\omega t\right)\sin \left({\displaystyle \frac{2\pi k_{x}x}{L_x}}\right)\sin \left({\displaystyle \frac{2\pi k_{z}z}{L_z}}\right),\end{array}$$

(3)

where A is the amplitude (maximum wall displacement), $\omega$ is the angular frequency of the oscillation, $k_x$ and $k_z$ are the wavenumbers corresponding to the streamwise and spanwise directions, respectively, and $L_x$ and $L_z$ are the lengths of the computational domain in the streamwise and spanwise directions, respectively. Since wall deflections will be described by a combination of smooth functions, a body-fitted mesh will be employed to discretize the flow domain, with the governing equations written in generalized curvilinear coordinates, in Figure 1, an isometric view of the flow domain and the mesh is displayed, where deformations on the top and bottom walls can be noticed.

Direct numerical simulations for a turbulent channel flow with both bottom and top wall deformations have been conducted in different conditions, at the Reynolds number based on the centerline velocity and channel height of $Re=4300$, which corresponds to the Reynolds number based on friction velocity of $R{e}_{\tau }=300$. The mesh resolution consists of 300 points in the streamwise x direction, which corresponds to a domain length of $L_x=2\pi$, 200 points in the spanwise direction, corresponding to a domain width of $L_z=\pi$, and 100 points in the wall-normal direction, corresponding to a domain height of $\delta =1$. The first grid point from the wall corresponds to $\mathsf{\Delta }{y}^{+}=0.3$, while the mesh resolutions in the streamwise and spanwise directions correspond to approximately $\mathsf{\Delta }{x}^{+}=10$ and $\mathsf{\Delta }{z}^{+}=6$, respectively ($\mathsf{\Delta }{y}^{+}$ at the centerline is in the same order as $\mathsf{\Delta }{z}^{+}$).

We consider different combinations of streamwise and spanwise wavenumbers, $k_x$ and $k_z$, scaled by the channel height $\delta$, as given in

Table 1. Parameter space.

Case	${\mathit{k}}_{\mathit{x}}$	${\mathit{k}}_{\mathit{x}}^{+}$	${\mathit{k}}_{\mathit{z}}$	${\mathit{k}}_{\mathit{z}}^{+}$	$\mathit{St}$	A	${\mathit{A}}^{+}$
1	1	0.009	8	0.073	0.08	0.015	3.28
2	1	0.009	8	0.073	0.08	0.030	6.57
3	2	0.018	8	0.073	0.08	0.015	3.28
4	2	0.018	8	0.073	0.08	0.030	6.57
5	4	0.036	8	0.073	0.08	0.015	3.28
6	4	0.036	8	0.073	0.08	0.030	6.57
7	8	0.073	8	0.073	0.08	0.015	3.28
8	8	0.073	8	0.073	0.08	0.030	6.57
9	1	0.009	8	0.073	0.16	0.015	3.28
10	1	0.009	8	0.073	0.16	0.030	6.57
11	2	0.018	8	0.073	0.16	0.015	3.28
12	2	0.018	8	0.073	0.16	0.030	6.57
13	4	0.036	8	0.073	0.16	0.015	3.28
14	4	0.036	8	0.073	0.16	0.030	6.57
15	8	0.073	8	0.073	0.16	0.015	3.28
16	8	0.073	8	0.073	0.16	0.030	6.57
17	1	0.009	8	0.073	0.32	0.015	3.28
18	1	0.009	8	0.073	0.32	0.030	6.57
19	2	0.018	8	0.073	0.32	0.015	3.28
20	2	0.018	8	0.073	0.32	0.030	6.57
21	4	0.036	8	0.073	0.32	0.015	3.28
22	4	0.036	8	0.073	0.32	0.030	6.57
23	8	0.073	8	0.073	0.32	0.015	3.28
24	8	0.073	8	0.073	0.32	0.030	6.57

(in this table, the corresponding wavenumbers in viscous units, $k_x^{+}$ and $k_z^{+}$, are also included), and three Strouhal numbers of $0.08$, $0.16$ and $0.32$ (Strouhal number is defined as $St=f\delta /U_0$, where $f=\omega /2\pi$ is the frequency, $\delta$ is the channel height, and $U_0$ is the centerline velocity). In addition, we also consider two wall deflection amplitudes A of $0.015$ and $0.03$ which corresponds to $A^{+}$ of $3.28$ and $6.57$ in viscous units, respectively; the first is less than the viscous sublayer upper limit ($y^{+}=5$), while the second is greater. All parameters used in the suite of numerical simulations are listed in Table 1. Figure 2 displays the wall layouts for the four wavenumber combinations, where black color represents hills and white color represents dimples. The spanwise wavenumber was kept constant at the level that approximately resembles the spanwise separation of the high- and low-speed streaks that are typical in proximity to the wall at the particular Reynolds number that is considered here, while the streamwise wavenumber was varied from a level that is similar to the spanwise separation to a level that is eight times larger.

3. Results and Discussion

In Figure 3, iso-surfaces of Q-criterion ($Q=1/2\left[\right|\mathsf{\Omega }{|}^2-\left|\mathbf{S}{|}^2\right]$, where $\mathbf{S}=1/2[\nabla \mathbf{v}+{(\nabla \mathbf{v})}^T]$ is the rate-of-strain tensor, and $\mathsf{\Omega }=1/2[\nabla \mathbf{v}-{(\nabla \mathbf{v})}^T]$ is the vorticity tensor) colored by the streamwise velocity component are plotted for two configurations, corresponding to the flat surface and the largest streamwise wavenumber $k_x=8$, with wall deflection amplitude of $A=0.03$. These iso-surfaces show that more intense turbulence is generated in proximity to the wall by the dynamic wall deformations, for this particular combination of wavenumbers. Similar iso-surfaces corresponding to the lowest streamwise wavenumber ($k_x=2$), not included here, do not show a significant difference compared to the flat surface case.

We consider a triple decomposition of the instantaneous velocity components in the form:

$$\begin{array}{c}\hfill u_i(x_i,t)=\langle u_i\left(x_i\right)\rangle +{\tilde{u}}_i(x_i,t)+u_i^{″}(x_i,t),\end{array}$$

(4)

where

$$\begin{array}{c}\hfill \langle u_i\left(x_i\right)\rangle ={\displaystyle \frac{1}{T}}{\int }_0^T{u}_i(x_i,t)dt\end{array}$$

(5)

is the time averaged velocity (or mean velocity) over a time interval of length T, ${\tilde{u}}_i(x_i,t)$ is the deterministic velocity fluctuation, which is essentially the periodic component of the fluctuation, and $u_i^{″}(x_i,t)$ is the turbulent velocity fluctuation. We also denote $u_i^{\prime }(x_i,t)={\tilde{u}}_i(x_i,t)+u_i^{″}(x_i,t)$ as the total turbulent velocity fluctuation, which will be used to calculate the Reynolds stress components, and $u_i^p(x_i,t)=\langle u_i\left(x_i\right)\rangle +{\tilde{u}}_i(x_i,t)$ as the phase averaged velocity calculated via

$$\begin{array}{c}\hfill u_i^p(x_i,t)=\underset{N\to \infty }{\lim }{\displaystyle \frac{1}{N}}\sum _{n=0}^N{u}_i(x_i,t+n\widehat{T}),\end{array}$$

(6)

where $\widehat{T}$ is the period associated with wall oscillations and N is the number of realizations. Since it is often challenging to save sufficient time history data for phase averaging, we took advantage of the periodicity in the streamwise and spanwise directions to improve the smoothness of the data, by taking phase averaging in these two directions. The idea is illustrated in Figure 4, where we defined a rectangular cell with the sides equal to the streamwise and spanwise wavelengths of wall deflections. In addition, taking advantage of the symmetry with respect to the top and bottom walls, we took an average of the two half subdomains (caution needs to be taken here because the wall-normal velocity component is anti-symmetric, requiring a sign change). So, the updated phase-averaging equation can be written as

$$\begin{array}{c}\hfill u_i^p(x_i,t)=\underset{N\to \infty }{\lim }{\displaystyle \frac{1}{2N\left(2k_x\right)k_z}}\sum _{n=0}^N\sum _{n_x=1}^{2k_x}\sum _{n_y=1}^2\sum _{n_z=1}^{k_z}{(-1)}^{(i+1)(n_y-1)}{u}_i(x+n_x{\lambda }_x,y+(n_y-1)(\delta -y),z+n_z{\lambda }_z,t+n\widehat{T}),\end{array}$$

where $y\in [0,\delta /2]$ (half of the channel height), ${\lambda }_x=2\pi /k_x$ and ${\lambda }_z=2\pi /k_z$ are the streamwise and spanwise wavelengths, respectively, and the factor ${(-1)}^{(i+1)(n_y-1)}$ in front of $u_i$ ensures that the $u_2$ velocity component takes the negative sign when adding the top half of the domain in the averaging procedure. Since this phase averaging procedure depends on $k_x$, which varies from 1 to 8, smoother data is obtained for cases that correspond to higher values of $k_x$ (for example, for $k_x=1$, the averaging in the streamwise and spanwise directions is taken over $2k_x{k}_z=16$ cells, while for $k_x=8$ this averaging is taken over 128 cells).

In Figure 5, we plot profiles of Reynolds stress components $\langle u^{\prime }{u}^{\prime }\rangle$ and $\langle u^{\prime }{v}^{\prime }\rangle$ scaled by the friction velocity squared for all combinations of wavenumbers, and superposed on the profile corresponding to the flat surface ($(u_1,u_2,u_3)$ and $(u,v,w)$ represent the same velocity vector and are used inter-changeably here). This figure compares cases with different combinations of wavenumbers. We observe the same effect that was observed in the mean velocity profiles plots, in terms of the amplitude of the wall deflections. Also, it can be observed that more elongated surface deformations have weaker effects on the Reynolds stress components, while less elongated wall deformations have a more significant effect on turbulence intensity, quantified by $\langle u^{\prime }{u}^{\prime }\rangle$ profiles. The effect of wall deflections on the vertical turbulent momentum flux $\langle u^{\prime }{v}^{\prime }\rangle$ is fairly similar. However, for this component we observe a switch in sign in close proximity to the wall, especially for higher streamwise wavenumbers and larger wall deflection, which is an indication that there is downward vertical momentum flux in the downstream of wall protuberances. This downward momentum flux enhances mixing between the near wall and outer flow regions which can lead to increases in drag. Similar Reynolds stress profiles are displayed in Figure 6, except that the curves that are superposed represent different frequencies, while keeping the same wavenumbers and wall deflection ($k_x=8$ and $k_z=8$ and $A=0.03$). This figure shows that the frequency of the wall deflection does not make a big difference on the turbulence intensity or vertical turbulent momentum flux, although there is a significant departure from the flat surface level.

Typical longitudinal and transverse autocorrelations in a point located in proximity to the wall ($y^{+}=30$) have been calculated and plotted in Figure 7 for all wavenumber combinations. This location was chosen to be roughly where the Reynolds stresses become predominant in comparison to viscous stresses. At the same time, we wanted this point to be as close as possible to the wall deformations, and at a distance that is much greater than the wall deflection amplitude. One piece of information that these autocorrelations provides is the turbulent integral length scale obtained by taking the integral of the autocorrelation (i.e., the area under the curve). With wall deflections activated, the top Figure 7a indicates that for lower streamwise wavenumbers of the wall deflection turbulent structures become larger in the longitudinal direction, while for higher streamwise wavenumbers turbulent structures decrease in size. The transverse autocorrelation curves in Figure 7b shows that the transverse extent of flow structures become smaller for all combinations of streamwise wavenumbers.

Probability density function (PDF) of longitudinal velocity increment ${\mathsf{\Delta }}_{r}u(x_i,t)=u(x_i+e_{1}r,t)-u(x_i,t)$, where $e_1$ is the unit vector component along the streamwise x-direction and r the spatial shift, has been calculated at the same distance from the wall, $y^{+}=30$. PDF results are included in Figure 8 for different wavenumbers (top) and different frequencies (bottom). Figure 8a reveals that as the streamwise wavenumber is increased the PDF curves become wider, while we found that the velocity derivative skewness increases slightly with increasing the streamwise wavenumber, which is consistent with previous studies that showed that wall actuation may increase the skewness (see, for example, Duvvuri and McKeon [34] or Deshpande et al. [29]). Also, the PDF corresponding to the lowest streamwise wavenumber ($k_x=1$) is very close to the PDF corresponding to the flat wall. These later observations suggest that there is increased mixing in the cases with large streamwise wavenumbers (or less elongated wall deformations). Figure 8b is consistent with Figure 6 in the sense that the frequency does not seem to have a significant effect on the PDF.

In Figure 9, we quantify total drag force calculated as

$$\begin{array}{c}\hfill D={\displaystyle \frac{1}{T}}{\int }_0^T{\int }_0^{L_x}{\int }_0^{L_z}[{\tau }_w(x,y_w,z,t)\cos \left(\theta \right)+p(x,y_w,z,t)\sin \left(\theta \right)]dzdxdt,\end{array}$$

(7)

which include both the frictional drag and the pressure drag (note that the latter is zero for the flat plate), for both wall deflection amplitudes, all combinations of streamwise and spanwise wavenumbers, and all three frequencies. $y_w$ represents either the bottom or top wall location, ${\tau }_w$ is the wall shear stress, p is pressure, and $\theta$ is the angle between the grid line at the wall and the x axis direction, calculated as $\theta =ta{n}^{-1}[(y_{wi+1,k}-y_{wi,k})/(x_{wi+1,k}-x_{wi,,k})]$ ($x_{wi,k}$ and $y_{wi,k}$ are grid coordinates at the wall). All three plots in Figure 9 show a slight decrease in the total drag force for the highest streamwise wavenumber, $k_x=8$, in the order of $0.8$% for $A=0.015$ and $1.8$% for $A=0.03$. All the other cases show either a slight increase, such as $k_x=2$ and $k_x=4$ in Figure 9c, or no deviation from the flat wall total drag, such as $k_x=1$, $k_x=2$ and $k_x=4$ in Figure 9a,b. According to Equation (7) the total drag includes both frictional and pressure drag forces, although almost all of the contribution comes from the frictional drag, especially for cases corresponding to small streamwise wavenumbers. The slight reductions in drag force seen in some of the cases in Figure 9 is the result of the wall-normal velocity component induced by wall oscillations, rather than the wall deformation. Also, the more significant drag reduction for the highest streamwise wavenumber may be due to the ability of the wall deformations to breakdown the streaks that are evolving in proximity to the wall and are often responsible for the increase of frictional drag in wall turbulence. For all considered frequencies, these deformations correspond to a time scale (i.e., period of oscillations) smaller than or in the same order of magnitude as the time scale of the viscosity dominated structures. This means that the wall oscillations can convert low-speed into high-speed streaks at a higher rate, especially at smaller streamwise wavenumbers, for which there is more space for the flow to adapt to the wall deformations, as shown in the bottom panel of Figure 10. On the other hand, the top panel of the same figure show that for the highest $k_x$ the flow is less likely to adapt to the wall oscillations, so the streaky structures, which are primordially responsible for the increase in the wall shear stress, do not affect the near wall region as much.

Finally, we show several results in terms of the phase averaged velocity calculated using Equation (7). In Figure 10, contour plots of the phase-averaged streamwise velocity component is illustrated for three values of the streamwise wavenumber of $k_x=2$, $k_x=4$ and $k_x=8$ and at one instant in time (note that the phase averaged data is periodically transient with period $\widehat{T}=2\pi /\omega$). It appears that for the highest wavenumber the flow does not have sufficient time to ‘adapt’ to the dynamics of the wall deflections (low-speed regions of the flow seem to be trapped in between protuberances), while for the lowest wavenumber it does have time to adapt. Small regions of flow reversal have been observed in the downstream of wall protuberances, especially for the highest wall deflection amplitude. Consecutive time instances (over a period) of the phase-averaged wall-normal velocity component, that are included in Figure 11 along a plane cutting the peaks of protuberances, show that there exist periodic fluctuations that convect in the opposite direction, to the left, while the mean flow is moving to the right.

In Figure 12, the effect of wall motion on the flow in proximity to the wall is quantified by plotting consecutive time instances (over a period) of the phase-averaged streamwise velocity (regular scale is used in the horizontal direction to be able to plot negative values of $y^{+}$, which would not be shown in a logarithmic scale). For the highest streamwise wavenumber deflection, the impact on the phase-averaged flow extends up to $y^{+}=10$ for $A=0.015$, and up to approximately $y^{+}=15$ for $A=0.03$. This impact seems to increase as the streamwise wavenumber decreases: it goes up to $y^{+}=20$ for $k_x=4$ and as high as $y^{+}=40$ for $k_x=2$. This conclusion drawn from the last result can be correlated with findings associated with Figure 10, where it was shown that the mean flow ’adapts’ to the wall shape oscillation when the streamwise wavenumber is small. This ’adaptation’ of the mean flow to wall oscillations provides a more effective engine that promotes vertical oscillations of the flow, extending to the upper layers.

4. Concluding Remarks

To summarize, we conducted a series of direct numerical simulations to study the response of a low Reynolds number turbulent channel flow and the associated frictional drag to standing wave-like wall deflections featuring a chekerboard pattern. In the parametric study, the wall deflection amplitude, packing density (quantified in terms of streamwise and spanwise wavenumbers), and frequencies of the wall oscillation were varied. The instantaneous flow was decomposed into three components representing the time-averaged flow, the deterministic fluctuating flow and the turbulent fluctuating flow, where the summation of the first and the second components makes the phase-averaged flow. Various statistical quantities indicated that this type of dynamic wall deflection can have a significant effect, and that the variation of the spatial wavenumbers of the deflections makes a big difference for example, the Reynolds stress profiles showed an increase of as high as $30\%$ compared to the flat surface case. A slight total drag reduction, in the order of $2\%$, was observed for some combinations of wavenumbers and frequencies, especially for the highest streamwise wavenumber case. A procedure to improve the smoothness of the phase-averaging was utilized, where phase averaging in time was combined with averaging in the periodic directions and in the wall-normal directions. Various phase-averaged quantities have been presented and discussed, including contour plots of streamwise or wall-normal velocity components or profiles of phase-averaged streamwise velocity component.

This is a limited numerical study represented by a canonical turbulent channel flow case and at low Reynolds number. Although the reduction of total drag for some cases may seem insignificant, this investigation may open some avenues towards future flow control research, and towards the understanding of non-equilibrium in wall turbulence. Future work will target higher Reynolds numbers and an analysis of non-equilibrium effects in turbulent boundary layers, which represents a more appealing problem.