1 Introduction

Porous materials (or porous media) are widely used in many fields of applied science and engineering due to their functions such as filtration,separation,sound absorption ^[1] ,and heat transfer ^[2] .

Sine porous surfaces can modulate turbulent,the fluid flow through a porous medium is one of the most common topics and has become an interesting separate subject of study ^[3] . At different porosities and with different Darcy numbers,the effects of porous surface on the modulation of turbulent flow are different. Bruneau and Mortazavi ^[4] simulated the flow past around the two-dimensional (2D) bluff body which was wrapped by a porous material,and found that the drag coefficient of the bluff body dropped. Naito and Fukagata ^[5] reported the same trend in conjunction with drag reduction by a simulation on the flow around the threedimensional (3D) cylinder wrapped by a porous medium. The results indicated that the velocity and pressure fluctuations around the cylinder were weakened distinctly as a result of the porous medium. However,Jim´enez et al. ^[6] found that the turbulent level and the skin friction were to increase rather than to decrease when the turbulent shear flow was over a porous wall. By now,both points of view are complementary,and the detailed mechanism still remains unclear.

To simulate the flow efficiently in porous and fluid media,it is necessary to solve the Darcy equations and the Navier-Stokes equations in these two media simultaneously. This is quite hard to handle since it needs to couple two simulations and to put forward the right condition at the interface of the two media. Jim´enez et al. ^[6] treated the wall-normal velocity as a transpiration velocity which was proportional to the local wall pressure fluctuations. However,this method is a simplified mathematical method,and cannot recover exactly the impact from the porous wall. Hahn et al. ^[7] presented a boundary condition at the interface between a permeable block and the turbulent channel flow,and investigated the characteristics of the wall-bounded turbulence with permeable walls. The results showed that the viscous sublayer thickness decreased and the near-wall vortex structures were significantly weakened by the porous wall. In addition, the porous wall reduced the turbulence intensities,the Reynolds shear stress,the pressure and vorticity fluctuations,and the skin-friction. However,their method dealt with the porous medium and the fluid,respectively,and the condition at the interface of the two media is really a troublesome problem.

In fact,each medium can be considered as a porous medium. It is understandable to treat fluid as a porous medium of infinite permeability and solid as a porous medium of zero permeability. In this work,the lattice Boltzmann method (LBM) is used to solve the porous wall-bounded turbulent flow.

The LBM appears to be an attractive approach to compute complex flows ^[8] ,such as turbulence flow ^{[9, 10]} and microscale flow ^[11] . It has been successfully used in many areas of computational physics,e.g.,aeroacoustics ^[12] . All in all,the LBM is a rather efficient,robust method which is very easy to implement. Traditional computational fluid dynamics (CFD) methods use to solve the conservation equations of macroscopic properties (mass,momentum,and energy) numerically. The LBM is based on the discrete-velocity Boltzmann equation. In general, propagation and collision of dummy particles are the main interpretation processes in the above procedure.

In this paper,the turbulence modulation in a porous wall-bounded turbulent channel flow is further explored. The physical model and the mathematical formulation are described in Section 2. For the flow inside the porous medium,the volume-averaged method ^[13] is adopted. We also present how the lattice Boltzmann equation containing the Darcy-Brinkman-Forchheimer acting force term ^{[14, 15]} is deduced in this section. Some statistical quantities of the impermeable wallbounded turbulent flow and validations are given in Section 3. The turbulence statistics are presented in Section 4. In Section 5,the turbulent structures of the velocity and vorticity instantaneous field are shown to explain the mechanism for the drag reduction of the porous medium. Lastly,a summary is followed in Section 6. 2 Problem and method 2.1 Governing equation

We consider the turbulent flow between two parallel planes (see Fig. 1). The widths of the computational domain are 2πH ,πH ,and 2H in the streamwise,spanwise,and wall-normal directions,respectively. Here,we set the porous layer close to the two walls. The porous layer’s thickness is δ,which is far less than the distance between the two plates. The porosity of the porous layer is ε,and the permeability is K.

For the isothermal incompressible flow in the porous medium,according to the BrinkmanForchheimer equation ^[16] ,the governing equations are given as follows:

where F is the total force which contains the medium resistance and the driving force terms ^{[16, 17]}, i.e., in which v is the kinematic viscosity,and

For the flow inside the porous medium,this model is averaged over a small spatial volume with dimensions large enough to smooth in the homogeneities at pore scales or small enough to remain the flow dynamics. According to the Ergun empirical formulas ^[18] ,the structure function F_ε and the permeability K are expressed as follows: where d _p is the particle diameter of the porous medium.

It is worth pointing out that the porosity tends to be 1.0 and the permeability is approximately infinite except in the porous medium domain. Meanwhile,the force F can be treated as G,and E_q. (1) is equivalent to the standard Navier-Stokes equations in the fluid domain. In other words,the governing equations are the Navier-Stokes equations in general. 2.2 Lattice Boltzmann force model

The lattice Bhatnagar-Gross-Krook (LBGK) ^[19] is a single relaxation model,which is one of the most widely used lattice Boltzmann equation (LBE) models. In fact,it has the secondorder accuracy in time and space if the flow is incompressible and the numerical viscosity can be absorbed into the physical viscosity ^[20] .

According to the LBGK model including the forcing term ^[21] ,this paper deduces the D3Q19 LBE model which solves the 3D porous wall-bounded turbulent flow as follows:

in which the equilibrium distribution function can be expressed as follows: and the forcing term is

The macroscopic density and the velocity are defined by

where c ₀ and c ₁ are parameters,and v is a temporary velocity expressed by

2.3 Chapman-Enskog expansion derivation

It needs to verify the equivalence between the LBE model and the corresponding NavierStokes equations. With the definition of the equilibrium distribution function,one can easily obtain the following moments:

From the LBE model (see E_q. (4)),the arising macro-dynamical behavior can be found from a multi-scale analysis by use of an expansion parameter λ,which is proportional to the ratio of the lattice spacing to a characteristic macroscopic length. To do so,the following expansions are introduced:

Expanding f _i (x + c _i δ _t,t + δ _t ) and applying the above multi-scale expansions to the resulting continuous equation,one can obtain the following equations in the consecutive order of the parameter λ:

where

According to the definitions

and the moment equations (8) and (9),we have

Taking the moments of Eq. (15),we can obtain the following macroscopic equations on the time scale t₁ = εt and the space scale x ₁ = εx:

where Π ₍₀₎ is the zeroth-order momentum flux tensor given by

The first-order momentum flux can be simplified by E_q. (18). After some standard algebra,we obtain

where the terms of order O(M ³ ) are neglected. The macroscopic equations on the time scale t² = λ ² t are derived by taking the moments of E_q. (16). With the aid of E_q. (18),the final equations can be written as follows: where

Combining the macroscopic equations on t₁ and t₂ ,one can finally obtain the macroscopic equation as E_q. (1). So far,we have finished the verification by the series expansion. 3 Model validation

The Boltzmann method is easy to be programmed,and has high performance on parallel computing. It is really an efficient way since it is convenient to deal with complex boundaries. Therefore,we simulate the porous wall-bounded turbulent flow by the LBM.

Firstly,to validate the reliability of the current computational procedure,we calculate an impermeable wall-bounded turbulent flow and compare the obtained results with some typical direct numerical simulation (DNS) results. The periodic boundary condition is adopted in the streamwise and spanwise directions,and the correctional bounce-back scheme boundary condition (the 2nd-order accuracy) is used in the wall-normal direction. To save the computational resource and evolve the turbulent flow field as soon as possible,we choose the time development mode and set the initial velocity field ^[22] as follows:

where the symbol “+” stands for the velocity normalized by the wall shear velocity. Obviously, the velocity field appears as linear-logarithmic distribution. Therefore,the von constant k is 0.4,and the other two parameters B and y _w are 5.2,and 11.6,respectively. To arouse the turbulence,we also add perturbation to the initial velocity field. At the first time-step, several vortices are superposed along the streamwise or the spanwise. The random perturbation is also added to the initial velocity field satisfying the divergence-free condition ^[23] . The perturbation immediately propagates into all the three velocity components,but approximate solenoidality is still maintained ^[22] .

Table 1 shows the statistical parameters of the wall-bounded turbulence. All the parameters are obtained after the flow field full being developed. InTable 1 ,y_min ⁺ is the mesh size in the near wall region normalized by the wall unit,u _t ,u _c ,and u_m are the wall shear velocity,the mean centerline velocity,and the mean bulk velocity in the streamwise,respectively,Re_τ = u _τ H/v is the wall Reynolds number,Re_c = H_c H/v is the centerline Reynolds number,_Re_m = U_mH/v is the mean bulk Reynolds number,c f = (2T _w )/(ρU_m ² ) is the drag coefficient.

The validation case is carried out in the uniform meshes and the number of the total meshes is about 10% less than that of Kim et al. ^[24] . The statistics’ errors between the two cases are still within 1%. Therefore,the current method can do simulate the wall-bounded turbulence under lower simulation cost. Through analyzing the velocity statistics (see Fig. 2),in the linear and logarithmic regions,the velocity curve almost coincides with that of Kim et al. ^[24] .

For the second-order statistics,the root-mean-square velocity fluctuations in the spanwise and wall-normal directions all display good consistency with the results of Kim et al. ^[24] and Moser et al. ^[25] ,and the streamwise velocity fluctuation shows a little offset probably on account of two kinds of boundary conditions (see Fig. 3). In addition,the Reynolds stress agrees well with that of Kim et al. ^[24] (see Fig. 4). In Fig. 4,the total stress is the sum of and .

The common high-order statistics data are the skewness and the flatness factors ^[24] . They all reflect the degrees of the turbulence deviating from the normal distribution. If the Say skewness factor S(u_i ^′ ) is positive,the positive probability of u_i ^′ is more likely. The flatness factor is used to the measure frequency of the velocity fluctuations,and it usually characterizes the intensity of the turbulent intermittency. Figure 5 shows the statistical results of the velocity fluctuations. From the figures,we can find that the skewness and flatness factors of the wall-normal velocity coincide well with that of Kim et al. ^[24] .

In view of the above analysis,it can be confirmed that our calculation based on the LBM is high efficient and reliable for the prediction of statistical quantities of the turbulent flow with a porous surface.

4 Parametric study

The second part of this paper will give a description of the porous wall-bounded turbulence. There are mainly two parameters for the porous walls,i.e.,the permeability K and the porosity ε. Here,we replace the permeability with the Darcy number expressed by D_a = K/H ² . In addition,the thickness δ of the porous walls is rather small compared with the distance between both walls,and we assume that

4.1 Effects of Darcy number

Firstly,we discuss the turbulent transport at different Darcy numbers (the porosity of wall is 0.2). We set several cases listed inTable 2 ,in which Case 0 stands for the conventional wall-bounded turbulence and Cases 1,2,and 3 stand for the porous wall-bounded turbulence at different Darcy numbers. The statistical results obtained from Cases 1 and 2 show that the mean velocities and the central velocities in the streamwise direction are all lower than those obtained form Case 0 while higher than those obtained from Case 3.

Figure 6 shows the streamwise mean velocities along the normal direction. We can obviously find that the mean velocity increases in the central region and decreases near the wall when the Darcy number decreases. We can find that there exists a cross point on the curves of the mean profile. In fact,it is significant in the flow of the turbulent boundary layer. Usually,its location stands for the interface area between the viscous sublayer and the logarithmic law layer.

The porous walls can modulate the velocity fluctuations. Figures 7(a),7(b),and 7(c) show the distributions of the streamwise,wall-normal,and spanwise root-mean-square velocity fluctuations normalized by the wall shear velocity at different Darcy numbers. When the Darcy number increases,the streamwise velocity fluctuation decreases in the near wall region and increases in the far wall area; and the spanwise and wall-normal velocity fluctuations in all region increase obviously. In addition,the Reynolds stress shows the same variation trend (see Fig. 7(d)).

Figure 8 shows the skewness and flatness factors of the velocity fluctuations at different Darcy numbers. From the figure,we can find that the flatness factors of the streamwise and wallnormal velocity fluctuations decreases when the Darcy numbers increases (see Figs. 8(a),8(b), and 5),and the skewness factor doesn’t display a change obviously. Under the same situations, the skewness factor of the streamwise velocity fluctuation decreases,while the skewness factor of the wall-normal velocity fluctuation increases. Therefore,we can draw a conclusion that the porous walls can actually weaken the sweeping movement in the near wall region.

Some high-order statistics of the Reynolds stress u ^′ v ^′ are also analyzed to investigate the influence of the porous wall on the Reynolds stress. From Figs. 8(c) and 8(d),when the Darcy number becomes larger,the amplitude of the skewness factor in the near wall region gets smaller. Moreover,the differences among the flatness factors of the Reynolds stress from different porous walls are obvious. This indicates that the Darcy number has a remarkable impact on the Reynolds stress in the whole flow field. 4.2 Effects of porosity

According to Section 2.1,we know that the force term F in the porous region can be expressed by the porosity,the Darcy number,etc. To study how the porosity affects the wallbounded turbulence,we firstly discuss the functional relation between the fluid drag and the porosity of the porous walls. Substituting E_q. (3) into E_q. (2),we can get

Obviously,there exists a nonlinear functional relation between F and ε. According to the cases of this work,we can assign the parameters as follows (all the parameters are set in the lattice unit):

Now,plot three graphs (Fx ~ ε) at different Darcy numbers (see Fig. 9). As shown in Fig. 9, when the Darcy number is 5×10 ⁻² ,the force in the streamwise direction increases when the porosity increases; when the Darcy number is 5×10 ⁻⁴ (hereinafter it is referred to as the criticalDarcy number),the force increases at first,and decreases later; and when the Darcy number is 5×10 ⁻⁶ ,the force is monotonically decreasing.

Now,we design some new cases according to the above functional relation between the drag and the porosity (see Table 3 ). Cases 4 and 5 are under different porosities,and their Darcy numbers are both 5×10 ⁻⁴ . It is easy to find that Case 4 displays a better drag reduction effect than Case 5. By comparing Case 3 with Case 6,we can find that the porous walls at larger porosity get better drag reduction effects. According to these simulations,the maximum drag reduction coefficient is 4.5% under Case 6.

Figure 10 shows the streamwise mean velocities for four Cases 3,4,5,and 6. Case 4 reveals the largest mean velocity in the near wall region,and Case 6 displays the largest velocity in the far wall area. It is worth mentioning that with the same Darcy number,increasing porosity doesn’t mean obtaining larger mean velocity.

We examine the root-mean-square velocity fluctuations and Reynolds stresses for different porous wall-bounded turbulent flows. Figure 11(a) shows that Case 4 has smaller velocity fluctuations than Case 5 in all directions,and has smaller Reynolds stress than Case 5 (see Fig. 11(b)).

From Case 2,Case 4,and Case 5,we can find that the drag coefficient increases firstly,and then decreases when the porosity increases with the same Darcy number. This phenomenon can be explained by the root-mean-square velocity fluctuations and Reynolds stresses. When the Darcy number is small enough,e.g.,Da 5 × 10 ⁻⁶ ,the streamwise velocity fluctuation decreases and the spanwise and wall-normal velocity fluctuations increase when the porosity increases (see Fig. 11(c)). Even so,the Reynolds stress in the fluid region is reduced (see Fig. 11(d)). 5 Analysis of velocity and vorticity instantaneous fields

Figure 12 shows the contours of the streamwise instantaneous velocity in the xz-plane for Cases 0,1,and 6,respectively. By comparing Fig. 12(b) with Fig. 12(a),the contour of Fig. 12(b) displays more smaller structures,which appears as distortions of the large-eddy contours. However,the streamwise coherence of Case 6 in Fig. 12(c) displays stronger velocity correlation in the streamwise direction. The low-speed streaks of the near-wall flow structures in Fig. 12(c) are considerably increased,and they are longer and more regular in the geometric profile than those in Fig. 12(a). Consequently,this is a typical feature of drag-reduced flows,which is independent of the actual cause of drag reduction. On the contrary,Fig. 12(b) does not reveal these features. The streamwise instantaneous velocity contours in the xz-plane in the far wall area are shown in Fig. 13. Figure 13(c) shows more high-speed streaks than Fig. 13(b). That is to say,Case 6 has a larger mean velocity in the core area than Case 1,which is an intuitive behavior to drag reduction.

We have already known that shear is the major characteristic in the turbulence boundary layer. Since the conventional vorticity criterion (ω = rot V ) is disabled in this kind flow,we distinguish the vortex structures in the near wall region by the Q-criterion ^[26] . The criterion is defined as

where W_ij and S_ij are,respectively,the symmetric and skew-symmetric velocity gradient tensors expressed by

Figure 14 shows the vortex structures in the near wall region of conventional and the porous wall-bounded turbulence. From Case 6,we can find that the coherent structures,the scales, and the attack angles obtained from Case 6 are all reduced with respect to those obtained from Case 0. On the contrary,Case 1 exactly gets enhancive coherent structures. According to Kolmogorov’s K41 theory ^[27] ,the dissipative eddies (small scale magnitude) always dissipate the energy from large scale vortexes. Obviously,since the turbulence of Case 6 has more micro vortexes than those of Case 1,the former will dissipative less energy,and then the drag reduction can be achieved.

6 Conclusions

We investigate the characteristics of turbulent flow and the turbulence modulations of wallbounded turbulence in the presence of porous walls by means of a direct numerical simulation by the LBM. At different porosities and with different Darcy numbers,pronounced turbulence modulations are observed. From the analysis,the following conclusions can be made:

(i) The porous surface can modulate the turbulence intensity and the Reynolds shear stresses in near wall region,which can affect the streamwise coherence of the near wall flow structures. In the view of physics,the porous medium is essentially to generate an intermediate flow,which controls the vortex flows and weakens the boundary layer.

(ii) The mean velocity increases when the Darcy number decreases. In addition,when the Darcy number decreases,the streamwise velocity fluctuation increases in the near wall region, and the growth trend slows down. As a result,the turbulent anisotropy weakens the wall-normal and spanwise velocity fluctuations and the Reynolds stresses.

(iii) When the Darcy number is in the order of O(10 ⁻⁴ ),the streamwise mean velocities and velocity fluctuations are indeterminate. The present results show that very small porosity may also result in weak velocity fluctuations and Reynolds stresses in the near wall region.

(iv) The drag coefficient is damped in the porous wall-bounded turbulent flow when the Darcy number is in the order of O(10 ⁻⁶ ). The streamwise low-speed streaks become longer and more regular than those of the impermeable wall-bounded flow,and appear stronger velocity correlation in the streamwise direction. At the same time,the coherent structures of the vorticity field and the scale and attack angles of vortex in the near wall region are reduced.