On Drag Reduction by Spanwise Wall Oscillation in Compressible Turbulent Channel Flow

Abstract

In this chapter, we summarize our progress in the investigation of drag reduction (DR) by spanwise wall oscillation (SWO). Reliable direct numerical simulation (DNS) models have been established for compressible turbulent channel flow. A large amount of 39.3% drag reduction has been achieved for supersonic compressible flows. It is found that the compressible effect is modest with respect to the turbulent effect for the skin-fiction in compressible turbulent channel flows. The reduced DR is mainly because of the decreasing turbulent contribution, and the related small compressible term also slightly decreases with the increase ${W_m}^{+}$. More DNS cases with different maximum wall velocities, oscillation periods, and flow Reynold numbers for compressible cases should be analyzed. The optimal combination with the highest drag reduction efficiency has significant importance on real applications, which deserves to be studied in detail to characterize the underlying mechanisms.

1. Introduction

Flow control aiming at reducing turbulent skin friction is a challenging theoretical and technological problem, which is of great importance for energy saving, aircraft designation and pipeline transportation. Knowledge of the dynamics of near-wall turbulent structures can provide a thorough understanding of the underlying physics and has been developed into various drag reduction control methods in the past decades [1]. However, typical passive controls without energy input, such as riblets [2], random wall roughness [3], polymers [4], superhydrophobic surfaces [5], and dimples [6], are hard to continuously sustain in practical applications [7]. Spanwise wall oscillation (SWO), as an efficient active control technique with significant drag reduction efficiency, has received increasing attention in recent years.

Among the intriguing research work, Jung et al. [8] and Akhavan et al. [9] were the first to adopt direct numerical simulation (DNS) to investigate the suppression of turbulence when a spanwise oscillating wall was imposed. In their simulations, as demonstrated by Kim et al. [10], the periodic boundary conditions were adopted in homogeneous directions. At the channel walls, the usual no-slip and no-penetration conditions were applied. The wall boundary condition for the spanwise component of velocity was:

$$W=W_{m}sin\left(\mathit{\omega t}\right)=W_{m}sin\left(\frac{2\pi }{T}t\right)$$E1

i.e. the two walls move in phase with a spanwise velocity $W$ which is a sinusoidal function of time t with prescribed amplitude ${W_m}^{+}$ and oscillation frequency $\omega$ (or period $T$). They observed that 40% skin-friction drag could be achieved in a numerical channel at ${\mathit{Re}}_{\tau }=180$, and the optimal wall oscillation period was found to be ${T_{\mathit{opt}}}^{+}\approx 100$. Following this work, Baron and Quadrio [11] further considered both energetic costs and benefits of this active technique, and showed that around 10% net energy saving was possible by spanwise wall oscillation, which was a similar level to the passive drag reduction controls, such as riblets. This drag reduction performance by the spanwise wall oscillation was confirmed experimentally by Laadhari et al. [12], Di Cicca et al. [13], and Ricco [14] for the flat plate wall. In the present project, we also impose these no-slip time-dependent boundary conditions at the walls to study the drag reduction by SWO in turbulent channel flow.

Most studies on drag reduction by SWO have focused on the two control parameters, i.e., the maximum wall velocity ${W_m}^{+}$ and the oscillation frequency ${\omega }^{+}$, and explored a scaling law of the drag reduction. It was concluded that the drag reduction value was not simply scaled by the maximum wall displacement ${D_m}^{+}$ nor the oscillation frequency ${\omega }^{+}$. Choi and Graham [15] experimentally studied the drag reduction in a circular pipe at two Reynolds numbers, ${\mathit{Re}}_{\tau }=650$ and 1000, and found that the maximum wall velocity ${W_m}^{+}$ gave a better scaling than the oscillation frequency ${\omega }^{+}$. Choi et al. [16] proposed a combined number $V_c$ with a thickness $l^{+}$, an acceleration rate $a^{+}$, the maximum wall velocity ${W_m}^{+}$, and the Reynolds number ${\mathit{Re}}_{\tau }$: ${V_c}^{+}=a^{+}{l}^{+}/{W_m}^{+}{\mathit{Re}}_{\tau }^{-0.2}$, and the authors found that the drag reduction was scaled in a quadratic form of ${V_c}^{+}$, i.e., $\mathrm{DR}=1000{V_c}^{+}+50{V_c}^{+}$. Similarly, Quadrio and Ricco [17] proposed a combined parameter $S^{+}=a^{+}{l}^{+}/{W_m}^{+}$ without the consideration of the Reynolds number effect. The authors found that the drag reduction was scaled linearly well with $S^{+}$, when the oscillation period was small, i.e., $T^{+}$ < 150, which was also the accurate prediction regime of the model-based approach by Moarref and Jovanović [18]. Ricco and Quadrio [19] further explored the parameter space $T^{+}$ - ${W_m}^{+}$ and $T^{+}$ - ${D_m}^{+}$ for the drag reduction region and also for the net energy saving condition at $T^{+}$ < 150, where the linear scaling correlation is held.

In spite of their various captivating features, the drag reduction mechanism of spanwise oscillations has been partially understood [1]. Table 1 presents different models coexisted in the literature. Akhavan et al. [9] and Baron and Quadrio [11] pointed out that the spanwise wall oscillation generated the Stokes layer, and it shifted the position of the low-speed streaks relative to the quasi-streamwise vortices. Dhanak and Si [27] further considered a single coherent quasi-streamwise vortex dipole in a spanwise oscillation flow and argued that the spanwise wall oscillation deformed the quasi-streamwise vortices and increased the mixing of the high-speed and low-speed streaks, resulting in the drag reduction. Nevertheless, Galionis and Hall [28] theoretically studied the unstable Görtler vortex on a concave surface subjected to the spanwise wall oscillation. The growth rate of the most amplified Görtler vortex was found to be significantly reduced. Negi et al. [29] used a localized volume forcing to generate the low-speed streaks in a laminar boundary layer and studied the interaction between the low-speed streaks and the spanwise wall oscillation. The authors found that the drag reduction values had a better correlation with the wall-normal velocity fluctuations.

Source	Mechanism
Choi and Graham [15]	Negative spanwise vorticity is created in both positive and negative movement of spanwise wall oscillation, reducing velocity gradient in the near wall region.
Xu and Huang [20]	The global turbulence suppression is caused by the sustained attenuation of the pressure strain term in the turbulent budget.
Duggleby et al. [21]	The coherent vorticity structures are pushed away from the wall into higher speed flow, causing a shorter time interaction between the propagating wave modes and the roll modes, thus less Reynolds shear stress production.
Ricco et al. [22]	Turbulent dissipation rate is enhanced during the transient process, which leads to drag reduction in the new quasi equilibrium state.
Touber and Leschziner [23]	The unsteady cross-flow straining causes major spanwise distortions in the streaks, and the reduction in wall normal and shear stresses.
Agostini et al. [24]	The drag reduction process is linked to the change in the Stokes strain in the upper region of the viscous sublayer where streaks are strong.
Yakeno et al. [25]	Spanwise wall oscillation suppresses the near-wall streamwise vortices rotating in the opposite direction, and also tilt the structures into spanwise direction.
Gatti and Quadrio [26]	The large structures, which contribute to turbulent friction via the Reynolds shear stress produced in the outer region, do not directly jeopardize the drag reducing action-based spanwise forcing.

Table 1.

Drag reduction mechanisms by spanwise wall oscillation in literature.

DNS provides an opportunity to critically examine the coherent structures in the turbulent flow due to its ability to simulate highly accurate 3D flow fields. With the visualization technique, Blesbois et al. [30] found that in a spanwise oscillating turbulent boundary layer, there were infinitely long structures with certain angles to the mean flow, and the angle and amplitude of the structures jumped suddenly at a certain instant during the oscillation period, which was consistent with the conditioned streaks angle in the DNS by Touber and Leschziner [23]. Yakeno et al. [25] performed the ensemble average of the quasi-streamwise vortices and the associated quadrant events, and argued that the drag reduction for the cases with small oscillation periods was due to the suspension of the Q2 event; while the drag increase for the cases with large oscillation periods was due to the enhancement of the Q4 event. Touber and Leschziner [23] compared the turbulent statistics between the spanwise wall oscillation case and the no control case in much detail and pointed out that the origin of the statistics changes was the spanwise distortion of the near wall streaks. This mechanism was further supported by Agostini et al. [24] at a higher Reynolds number, and the drag reduction by the spanwise wall oscillation was due to the increase of turbulent enstrophy and dissipation in the transient process [22]. Recently, Gatti and Quadrio [26] and Agostini and Leschziner [31] focused on the impact of footprints of large-scale outer structures on the near-wall layer in the presence of drag-reducing spanwise wall motion. It was noted that the large structures, which contribute to turbulent friction via the Reynolds shear stress produced in the outer region [32], do not directly jeopardize the drag-reducing action of the wall-based spanwise forcing. However, the physical mechanisms underlying the SWO are still not clear, with unanswered questions such as how the oscillation affects the near-wall dynamics and how the resultant self-sustaining cycle impacts the suppression of streamwise vortex generation and hence reduction of drag.

It has been pointed out that streamwise vortices play an important role in all self-sustaining mechanisms of near-wall turbulence [33]. There are different explanations for how these eddies are created and maintained. For example, Hamilton et al. [34] reported that the nonlinear advection term is mainly responsible for flow direction vortices due to flow direction disturbances, while Schoppa and Hussain [35] claimed that the vortex stretching term is mainly responsible for flow direction vortices. However, streamwise vortices are formed and maintained autonomously by a self-sustaining process, which involves wall-layer streaks and instabilities associated with them [36]. In addition, the current literature on SWO largely investigated incompressible flow in turbulent boundary layer, channel, and pipe flows. It was illustrated that the local self-sustaining process also existed for the turbulence in the inner region of compressible turbulent boundary layers for different Mach numbers [33]. Relatively few investigations have been carried out on high-speed compressible flows. Fang et al. [37, 38] and Ni et al. [39] employed large eddy simulation (LES) and DNS to study SWO with compressible turbulent channel flow at Ma = 0.5 and a supersonic turbulent boundary layer at Ma = 2.9, respectively. In these studies, they found that SWO simultaneously suppresses momentum transportation and heat transportation in a consistent manner. Previous studies of compressible wall-bounded flows have led to the widely accepted notion that the main effect of flow compressibility resides in the incurred mean density variations [33].

Recently, Yao and Hussain [40, 41] presented a new dynamical vortex definition for compressible flows, which is derived directly from the compressible momentum (Navier–Stokes) equation and identifies the coherent local pressure minimum in a plane. Based on this work, they further performed DNS for isothermal channel flow with SWO to investigate subsonic and supersonic flows at bulk Mach numbers of 0.8 and 1.5, respectively [42]. Their results suggested that significant drag reduction can still be achieved via SWO even when the compressibility effect is considered. Therefore, an attempt to extend the research work to account for compressibility effects at transitional Mach numbers and reveal the detailed drag reduction mechanism by SWO in these flows urgently needs to be addressed.

2. Research progress and results

2.1 Establishment of a reliable DNS model for turbulent channel flow

As shown in Figure 1, the turbulent channel flow consists of two infinitely large parallel plates which are at a finite distance of 2$h$ apart. Periodic boundary conditions are applied in the streamwise ($x$) and spanwise ($z$) directions, respectively. The conservation equations for solving compressible flow problems include continuity, momentum, and energy equations. For a Newtonian fluid behaving as an ideal gas, these equations can be expressed as,

Figure 1.

The schematic of turbulent channel flow with spanwise wall oscillation.

$$\frac{\partial \rho }{\partial t}+\frac{\partial \rho u_i}{\partial x_i}=0,$$E2

$$\frac{\partial \rho u_i}{\partial t}+\frac{\partial \rho u_i{u}_j}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial {\sigma }_{\mathit{ij}}}{\partial x_j}+f{\sigma }_{i1},$$E3

$$\frac{\partial \mathit{\rho E}}{\partial t}+\frac{\partial \rho u_{j}H}{\partial x_j}=-\frac{\partial q_j}{\partial x_j}+\frac{\partial {\sigma }_{\mathit{ij}}{u}_i}{\partial x_j}+{\mathit{fu}}_1,$$E4

where $u_i$ is the velocity component in the ith direction, $\rho$ the density, $p$ the pressure, $E=c_{v}T+u_i{u}_i/2$ the total energy per unit mass, and $H=E+p/\rho$ is the total enthalpy. $\gamma =c_p/c_v=1.4$ is the specific heat ratio. $q_j$ and ${\sigma }_{\mathit{ij}}$ are the components of the heat flux vector and the viscous stress tensor, respectively,

$${\sigma }_{\mathit{ij}}=\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\frac{\partial u_k}{x_k}{\delta }_{\mathit{ij}}\right),$$E5

$$q_j=-\kappa \frac{\partial T}{\partial x_j},$$E6

where the dependence of the viscosity coefficient on temperature is accounted for through Sutherland’s law and $k=c_p\mu /P_r$ is the thermal conductivity, with $P_r=0.72$. The forcing term $f$ in Eq. (3) is evaluated at each time step in order to discretely enforce constant mass-flow-rate in time, and the corresponding power spent is added to the right-hand-side of the total energy equation.

As shown in Table 2, the physical parameters used in our DNS are presented. The computational domain is $12h\times 2h\times 6h$ with $320\times 160\times 210$grid points in the streamwise (x), wall-normal (y), and spanwise (z) directions, respectively. The grid spacing is constant in the wall-parallel directions, and an error-function mapping is used to cluster grid points towards the walls. This domain size is compared to previous studies for compressible turbulent channel flow [44], and is also dictated by the need to accommodate the large eddies which become energetically relevant at sufficiently high Reynolds numbers [48, 49]. At a fixed bulk Reynolds number ${\mathit{Re}}_b=h{\rho }_b{u}_b/{\mu }_w=3000$, we have carried out DNS of subsonic and supersonic flows at bulk Mach numbers ${\mathit{Ma}}_b=u_b/c_w=0.8$ and 1.5, respectively. Here, $u_b$ and ${\rho }_b$ are the bulk channel velocity and density, and $c_w$ is the speed of sound evaluated at the wall.

	$R_b$	${{Re}}_{\tau }$	Domain size $L_x\times L_x\times L_z$	Grid points $n_x\times n_x\times n_z$	$\mathrm{\Delta }{x}^{+}$	$\mathrm{\Delta }{y}^{+}$	$\mathrm{\Delta }{z}^{+}$
Coleman et al. [43]	3000	222	$4\mathit{\pi h}\times 2h\times 4/3\mathit{\pi h}$	$144\times 119\times 80$	19	$0.1\sim 8.0$	12
Morinishi et al. [44]	3000	218	$4\mathit{\pi h}\times 2h\times 4/3\mathit{\pi h}$	$120\times 180\times 120$	23	$0.36\sim 5.1$	7.6
Foysi et al. [45]	3000	221	$4\mathit{\pi h}\times 2h\times 4/3\mathit{\pi h}$	$192\times 150\times 128$	14.46	$0.84\sim 5.02$	7.23
Sun et al. [46]	3000	220	$4\mathit{\pi h}\times 2h\times 2\mathit{\pi h}$	$160\times 160\times 160$	17.3	$0.05\sim 5.5$	8.6
Modesti and Pirozzoli [47]	3000	215	$6\mathit{\pi h}\times 2h\times 2\mathit{\pi h}$	$512\times 128\times 256$	8.0	-	5.9
Present DNS	3000	200	$12h\times 2h\times 6h$	$320\times 160\times 210$	7.46	$0.19\sim 4.1$	5.7

Table 2.

Setup of compressible channel DNS for supersonic flows at bulk Mach number $M_b=1.5$, with the friction Mach number $M_{\tau }=u_{\tau }/c_{\omega }=0.079$ and the heat flux coefficient $B_q=q_{\omega }/\left({\rho }_{\omega }{C}_p{u}_{\omega }\right)=0.048$.

In addition, it is noted that the presence of the walls in y direction may lead to failure in the conservation of mass in co-located flow solvers [43, 50], which is typically fixed by adding a source term to the continuity equations at the wall nodes in such a way that the integrated density remains discretely constant in time. In this work, we prefer to stagger the first node off the wall in such a way that the latter coincides with an intermediate node, where the convective fluxes are identically zero. Hence, correct telescoping of the numerical fluxes is guaranteed, and no net mass variation can occur. A further benefit of this approach is that, for a given distance of the first grid point from the wall, the maximum allowable time step associated with the vertical grid spacing is doubled. In order to maximize the spectral resolution in the streamwise direction, all simulations are performed in a convective frame of reference [51], in which the bulk velocity is zero. All computations are initiated with a parabolic velocity profile, superposed random perturbations, and uniform values of density and temperature.

Foysi et al. [45] studied the compressible turbulent channel by using DNS. They also observed a change in the Reynolds normal stresses at high Mach numbers, similar to that of the drag-reducing flow. In this flow case, the results are compared with DNS data obtained by Coleman et al. [43] (see Table 2), which uses a Fourier-Legendre spectral discretization along with a hybrid implicit-explicit third-order four-substep time-advance algorithm. Figure 2 compares the mean velocity and the Reynolds stresses distributions in wall units with the results in database by Coleman et al. [43]. It is seen that good agreements have been obtained for the mean velocity, in spite of the fact that the Reynolds stresses distributions are not accurately modeled. However, both the trends in wall-normal direction approximately coincide with the standard results, which provide convincing evidence for the effectiveness of the current DNS solver for compressible turbulent flows. Therefore, the spanwise oscillation velocity can be imposed on the walls to investigate the drag reduction performance for compressible turbulent channel flows.

Figure 2.

Comparison of (a) mean streamwise velocity profile and (b) root mean square (rms) values of the velocity fluctuations in compressible turbulent channel flow for ${\mathit{Re}}_b=3000$ at ${\mathit{Ma}}_b=1.5$.

2.2 Assessment of turbulent drag reduction via SWO

For compressible turbulent channel flows, we have established different oscillatory cases to investigate the effects of maximal wall velocity ${W_m}^{+}$ and oscillation period $T^{+}$, as shown in Table 3. The friction coefficient $C_f$ is the ratio between the boundary mean shear stress and the mean kinetic energy per unit volume. With nondimensionalized quantities, this can be written as,

Cases	${W_m}^{+}$	$T^{+}$	${10}^3{C}_f$	$\%P_{\mathit{sav}}$	${10}^3{C}_L$	${10}^3{C}_T$	${10}^3{C}_C$	${10}^3{C}_{CT}$
cCase0	0	0	7.81 (7.7)	0	1.87	5.61	0.33	0.00103
cW6T100	6	100	5.99	23.3	1.870	3.94	0.24	0.00966
cW12T50	12	50	5.59	28.4	1.78	3.55	0.25	0.0112
cW12T100	12	100	5.42	30.6	1.71	3.57	0.12	0.0187
cW12T200	12	200	5.92	24.1	1.76	3.98	0.16	0.0205
cW18T100	18	100	1.72	39.3	1.62	3.02	0.071	0.0261

Table 3.

Power budget data for compressible oscillatory turbulent channel flows for ${\mathit{Re}}_b=3000$. The data inside the braces are the results from Modesti and Pirozzoli [52].

$${\left.C_f=\frac{2}{Re}\frac{\widehat{c}\left(u\right)}{\widehat{c}y}\right|}_{y=1,1},$$E7

where $\mathit{Re}=2h{\rho }_0{U}_b/\mu \left(T_0\right)$ is the bulk Reynolds number, ${\rho }_0$ is the initial uniform density, and $\mu$($T_0$) is the molecular viscosity at the wall temperature $T_0$.

Based on the work by Gomez et al. [53], for a compressible turbulent flow, we need to define two averaging operators. The Favre average permits us to transform the mean of a product into the product of a mean as noted by Coleman et al. [43]. In the case of the channel flow, the Reynolds operator consists in averaging on $x$-$z$ space variables and $t$ time variable. The single prime and the double prime denote the turbulent fluctuations with respect to Reynolds and Favre averages, respectively. The difference between the Reynolds-averaged and the Favre-averaged quantities can be written as,

$$⟨f⟩-\left\{f\right\}=⟨f^{"}⟩=-\frac{⟨{\rho }^{\prime }{f}^{\prime }⟩}{⟨\rho ⟩}=-\frac{⟨{\rho }^{\prime }{f^{\prime }}^{\prime }⟩}{⟨\rho ⟩}.$$E8

The difference between Reynolds-averaged and Favre-averaged quantities is mainly relevant in the near-wall region by using DNS [50]. Therefore, the operator definitions give the following relations:

$$\frac{\partial ⟨\cdot ⟩}{\partial x}=\frac{\partial ⟨\cdot ⟩}{\partial z}=\frac{\partial ⟨\cdot ⟩}{\partial t}=0,$$E9

$$\frac{\partial \left\{\cdot \right\}}{\partial x}=\frac{\partial \left\{\cdot \right\}}{\partial y}=\frac{\partial \left\{\cdot \right\}}{\partial t}=0.$$E10

Hereafter, by assuming (i) constant flow rate, (ii) homogeneity in the streamwise $\left(x\right)$ and the spanwise $\left(z\right)$ directions, (iii) no-slip conditions at the wall surfaces $y$= −1 and + 1, and (iv) symmetry with respect to the center plane. Then, the momentum equation averaged along homogeneous directions $x$-$z$ and time $t$ variables yields,

$$\frac{\partial <\rho >\left\{\mathit{uw}\right\}}{\partial y}=\frac{1}{Re}\frac{\partial <{\tau }_{\mathit{xy}}>}{\partial y}-f_1,$$E11

where $f_1$ is the force gradient term, defined by the following relation:

$$f_1=-\frac{1}{2}\left({\left.\frac{2}{Re}\frac{\partial ⟨u⟩}{\partial y}\right|}_{y=-1,1}\right)=-\frac{1}{2}{C}_f$$E12

We can finally obtain the following relationship:

$$\begin{array}{l}{C}_f=\frac{6}{\underset{C_L}{\underbrace{Re}}}+\underset{C_T}{\underbrace{6{\int }_{-1}^{0}y⟨\rho ⟩\left\{u"{v^{\prime }}^{\prime }\right\}\mathit{dy}}}+\underset{C_C}{\underbrace{\frac{6}{Re}{\int }_{-1}^0-y⟨\tilde{\mu }⟩\frac{\partial ⟨u⟩}{\partial y}\mathit{dy}}}\\ +\underset{C_{\mathit{CT}}}{\underbrace{\frac{6}{Re}{\int }_{-1}^0-y⟨{\mu }^{\prime }\left(\frac{\partial u^{\prime }}{\partial y}+\frac{\partial w^{\prime }}{\partial y}\right)⟩\mathit{dy}}}.\end{array}$$E13

This relation shows that the skin friction coefficient can be split into four contributing terms: the laminar contribution $C_L$, the turbulent contribution $C_T$, the compressible contribution $C_{\mathit{CT}}$, and the compressible-turbulent interaction term $C_{\mathit{CT}}$. The turbulent term $C_T$ is proportional to the weighted average of the Reynolds stress, where the weight is linearly decreasing with the distance from the wall. As remarked by Fukagata et al. [54], this explains why the frictional drag observed in wall turbulence is mainly due to the turbulence wall structures which occur closer to the wall than the position of the maximum Reynolds stress. The compressible term $C_C$ is proportional to the mean viscosity fluctuation due to thermal variations and the mean wall normal velocity gradient. This term is obtained by a weighted average with the linearly decreasing weight as for $C_T$. The last contributing term $C_{\mathit{CT}}$ is proportional to the weighted average of the mean product between viscosity and velocity fluctuation gradients. The skin friction coefficient with the four components for different cases are given in Table 3.

It is clear that the measured friction coefficient for nonoscillatory case (cCase0) compared very well with the value reported by Modesti and Pirozzoli [52] (within a 0.2% difference). This result also agrees well with the simulation data by Gomez et al. [53] (within a 1% difference). In addition, Figure 3 presents $P_{\mathit{sav}}\%$ as a function of maximum wall velocity ${W_m}^{+}$ and oscillation period $T^{+}$. The total drag-reduction performance as a function of ${W_m}^{+}$ and $T^{+}$, is similar to that in incompressible flows, namely for a given value of ${W_m}^{+}$, the highest drag reductions are attained at $T^{+}=100$, and that drag reduction appears to increase monotonically with ${W_m}^{+}$ for a fixed period of oscillation. These findings are in good agreement with the observations by Fang et al. [37, 38] and Ni et al. [39].

Figure 3.

%$P_{\mathit{sav}}$ as a function of (a) maximum wall velocity ${W_m}^{+}$ at $T^{+}=100$ and of (b) oscillation period $T^{+}$ at ${W_m}^{+}=12$.

Figure 4 further shows the four components of the skin friction coefficient for a function of maximum wall velocity ${W_m}^{+}$ and oscillation period $T^{+}$. It is clear that there is a remarkable decrease of the turbulent contribution $C_T$ under the action of oscillatory walls, and this tendency is the most obvious for the cases with larger ${W_m}^{+}$ at $T^{+}=100$. The turbulent contribution dominates the overall drag reduction performance, in spite of the fact that the compressible term $C_C$ due to thermal variations and the mean wall normal velocity gradient slightly decreases with the increasing ${W_m}^{+}$. This implies that the compressible effect is modest with respect to the turbulent effect for the skin-fiction in compressible turbulent channel flows, which is identical to the findings by Gomez et al. [53]. In addition, this phenomenon also repeats for the laminar contribution $C_L$. Interestingly, the compressible-turbulent interaction term $C_{\mathit{CT}}$ slightly increases for larger ${W_m}^{+}$. As remarked by Fukagata et al. [54] and Li et al. [55], the frictional drag observed in wall turbulence is mainly due to the turbulence wall structures which occur closer to the wall than the position of the maximum Reynolds stress. The compressible term is proportional to the mean viscosity fluctuation due to thermal variations and the mean wall normal velocity gradient [56]. This term is obtained by weighted average with the linearly decreasing weight as for the turbulent term. In addition, the compressible contribution to the skin friction is concentrated in a region closer to the wall than that of the Reynolds stress contribution, which is more largely extended through the channel. This can be physically explained by the fact that the thermal boundary layer is thinner than the kinetic boundary layer.

Figure 4.

The four components of skin friction coefficient $C_L$, $C_T$, $C_C$, $C_{\mathit{CT}}$ as a function of (a) maximum wall velocity ${W_m}^{+}$ at $T^{+}=100$ and of (b) oscillation period $T^{+}$ at ${W_m}^{+}=12$.

3. Challenges encountered

DNS is a simulation in computational fluid dynamics in which the Navier–Stokes equations are numerically solved without any turbulence model. This means that the whole range of spatial and temporal scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational grid, from the smallest dissipative scales (Kolmogorov microscales), up to the integral scale, associated with the motions containing most of the kinetic energy. Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the cases in our simulations, we have more than 10 million grid points. The simulations for compressible cases were run for more than $5.0\times {10}^5$ CPU core hours to get statistically stable results.

The methodology for solving incompressible flow problems using pressure-based algorithms has been well developed. For our studies, these algorithms have to be extended to allow for the simulation of compressible flows in various Mach number regimes. Incompressible flow solutions do not generally require the solution of the energy equation. However, compressibility affects both hydrodynamics and thermodynamics, necessitating the simultaneous solution of the continuity, momentum, and energy equations.

The dependence of density on pressure and temperature, which is expressed via an equation of state, further complicates the velocity-pressure coupling present in incompressible flows. The derivation of the pressure correction equation now involves a density correction that introduces a convection-like term to the equation, in addition to the diffusion-like term introduced by the velocity correction. Another difficulty is the complex boundary conditions introduced that arise in compressible flow problems.

4. Conclusions and recommendations

We have established a reliable DNS model for compressible turbulent channel flow, and the skin-friction drag reduction by the spanwise motion generated by the wall movement has been studied. The aim is to bring skin-friction drag reduction control by spanwise wall oscillation into real engineering applications. It is found that the reduced drag reduction (DR) is mainly due to the decrease turbulent contribution under the action of oscillatory walls. Additionally, the small compressible term, related to thermal variations and the mean wall normal velocity gradient, also slightly decreases with increasing ${W_m}^{+}$.

However, more DNS cases with different maximum wall velocity, oscillation period, and flow Reynold number for compressible cases should be analyzed. The optimal combination with highest drag reduction efficient has a significant importance on real applications, which deserves to be studied in detail to characterize the underlying mechanisms. In addition, the near-wall streaks are conditioned from the turbulent fields. Understanding the modulation of the exact coherent structure by the spanwise wall oscillation can provide a direct view of how the regeneration cycle is weakened by the spanwise motion. At the flight Reynolds number, the contribution to skin-friction from the very large scale motions is very important, and it is an interesting topic to understand how to control the very large scale motions effectively from the outer region for compressible flows.

Acknowledgments

The authors gratefully acknowledge the financial support from the QCY Innovative and Entrepreneurial Talent Programme of Shaanxi Province and the Young Talent Support Plan of Xi’an Jiaotong University - XJTU. Computational resources were granted by the High-Performance-Computer Center of XJTU.

Conflict of interest

The authors declared that there is no conflict of interest.