Higher-Order Homogenized Boundary Conditions for Flows Over Rough and Porous Surfaces

Abstract

We derive a homogenized macroscopic model for fluid flows over ordered homogeneous porous surfaces. The unconfined free flow is described by the Navier–Stokes equation, and the Darcy equation governs the seepage flow within the porous domain. Boundary conditions that accurately capture mass and momentum transport at the contact surface between these two domains are derived using the multiscale homogenization technique. In addition to obtaining the generalized version of the widely used Beavers–Joseph slip condition for tangential velocities, the present work provides an accurate formulation for the transpiration velocity and pressure jump at fluid–porous interfaces; these two conditions are essential for handling two- and three-dimensional flows over porous media. All the constitutive parameters appearing in the interface conditions are computed by solving a set of Stokes problems on a much smaller computational domain, making the formulations free of empirical parameters. The tensorial form of the proposed interface conditions makes it possible to handle flows over isotropic, orthotropic, and anisotropic media. A subset of interface conditions, derived for porous media, can be used to model flows over rough walls. The accuracy of the proposed macroscopic model is numerically quantified for flows over porous and rough walls by comparing the results from our homogenized model with those obtained from geometry-resolved microscopic simulations.

Appendices

Appendix 1: Relation Between Gradients of Fast Flow Velocity and Full Velocity

One of the essential steps in arriving at the final interface conditions is to relate the gradients of fast flow velocities ($U_{i,j}$) to that of full velocities ($\left\langle u_i\right\rangle _{,j}$). Corresponding leading order ${\mathcal {O}}(\epsilon )$ and higher-order ${\mathcal {O}}(\epsilon ^1)$ relations are described in equations (47) and (58), respectively. In this appendix, we provide validation of these expressions. We compute $\left\langle u_i\right\rangle _{,j}$ by ensemble average of DNS data, and other terms in these expressions are computed from effective simulations with no-slip conditions at the interface (Eq. 22).

We carry out this validation on the example of isotropic circular inclusions, and the results are presented in Fig. 18. From the figure, it is evident that there is a hierarchy of amplitudes presented in all the derivatives at the interface. The velocity derivative $\partial u_1/\partial x_2$ corresponding to shear has the largest magnitude, followed by the derivatives $\partial u_1/\partial x_1$ and $\partial u_2/\partial x_2$ , which are order of magnitude smaller, and finally, the derivative $\partial u_2/\partial x_1$ is two orders of magnitude smaller compared to the $\partial u_1/\partial x_2$ . We see from figure 18 that ${\mathcal {O}}(\epsilon )$ approximation provides only the largest derivative $\partial u_1/\partial x_2$, while the higher-order ${\mathcal {O}}(\epsilon ^1)$ approximation provides a reasonable approximation to $\partial u_1/\partial x_1$ and $\partial u_2/\partial x_2$ also. However, $\partial u_2/\partial x_1$ is still out of reach. This serves to demonstrate the better accuracy of the higher-order approximation to gradients of velocity.

Fig. 18

Approximation of velocity gradients with leading order ${\mathcal {O}}(\epsilon )$ and higher-order ${\mathcal {O}}(\epsilon ^2)$ approximation. Averaged DNS implies $\left\langle u_i\right\rangle _{,j}$, ${\mathcal {O}}(\epsilon )$ and ${\mathcal {O}}(\epsilon ^2)$ approximation imply right-hand side of equations (47) and (58), respectively

Appendix 2: Relation Between Porous and Rough Wall Interface Conditions

The velocity boundary condition for the porous wall, given in Eq. (65), reads

$$\begin{aligned} \left\langle u_i^+\right\rangle ={\mathcal {L}}_{ij}s_j+\left( \overline{{\mathcal {M}}}_{ijk} +{\mathcal {K}}_{ik}\left\langle B_j^-\right\rangle \right) d_{jk}-{\mathcal {K}}_{ij}p^-_{,j} +{\mathcal {O}}(\epsilon ^2), \end{aligned}$$

(78)

where

$$\begin{aligned} \overline{{\mathcal {M}}}_{ijk}={\mathcal {M}}_{ijk} -{\mathcal {L}}_{ik}{\mathcal {L}}_{mj}n_m-{\mathcal {L}}_{im}{\mathcal {L}}_{mj}n_k -{\mathcal {L}}_{il}\left\langle M^+_{ljk}\right\rangle _{,m}n_m -{\mathcal {L}}_{il}\left\langle M^+_{mjk}\right\rangle _{,l}n_m \end{aligned}$$

(79)

In order to use this condition for a rough wall, we need write quantities defined in porous domain ($\left\langle p^-\right\rangle _{,j}$ and $\left\langle B_j^-\right\rangle$) in terms of their equivalent free fluid quantities. This is because in the effective simulations of rough wall flows, we cut off the domain below the interface, and hence these quantities are undefined.

In order to do so, we consider the pressure jump condition given in Eq. (65),

$$\begin{aligned} \llbracket p \rrbracket ={\mathcal {B}}_ks_k+{\mathcal {O}}(\epsilon ). \end{aligned}$$

(80)

Taking derivative along the interface tangential direction of the above expression, we obtain

$$\begin{aligned} \left( P+\left\langle p^+\right\rangle \right) _{,j}-\left\langle p^-\right\rangle _{,j}={\mathcal {B}}_kd_{kj} +{\mathcal {O}}(\epsilon ^2). \end{aligned}$$

(81)

Substituting the above expression in Eq. (78), we get the desired result,

$$\begin{aligned} \left\langle u_i^+\right\rangle ={\mathcal {L}}_{ij}s_j+\left( \overline{{\mathcal {M}}}_{ijk} +{\mathcal {K}}_{ik}\left\langle B_j^+\right\rangle \right) d_{jk} -{\mathcal {K}}_{ij}\left\langle p\right\rangle _{,j}+{\mathcal {O}}(\epsilon ^2), \end{aligned}$$

(82)

where $\left\langle p\right\rangle =P+\left\langle p^+\right\rangle$, which is the quantity defined in the free fluid region. The above expression is used to arrive at the boundary conditions for rough walls given in Eq. (8).

We stress that the derivative of pressure, defined in Eq. (81), is valid only along the direction tangent to the interface. Derivative of ${\mathcal {B}}$ along the interface normal direction is undefined, which means it is not possible to relate wall-normal pressure gradient across either sides of the interface using such an expression.

Appendix 3: Performance of TR Model for Porous Media

In the order of convergence study for isotropic porous medium presented in Sect. 5.1.1, we replace BJ model by TR model, and the results are presented in Fig. 19. The tangential velocity formulations of TR model is same as that of ${\mathcal {O}}(\epsilon ^0)-$model. Hence, the convergence rate as well as errors produced by TR model is approximately same as that of ${\mathcal {O}}(\epsilon ^0)-$model. However, for transpiration velocity TR model exhibits a surprising behavior; it produces more accurate results than the higher-order model while maintaining the same order of convergence as shown in Fig. 19b. This might be due to a more accurate pressure jump representation by the TR model than ${\mathcal {O}}(\epsilon ^1)-$model (results not shown here). The reason for this superior behavior is unclear and will be probed in future.

Fig. 19

Convergence plots for porous medium with isotropic solid inclusions