Wake Prediction in 3D Porous–Fluid Flows: A Numerical Study Using a Brinkman Penalization LBM Approach

Abstract

The simulation of the wake past porous obstacles is numerically challenging because since it requires both an accurate model of the porous medium and a good grid resolution in the fluid domain. In this study a single-domain Brinkman penalization technique in a LBM framework is employed to investigate wake prediction in 3D porous–fluid flows. First we assess the ability of the Brinkman model to predict porous flow features with different permeability values (Darcy numbers). In particular the flow over a porous bed is studied to characterize the ability of this approach to predict the thickness of the porous boundary layer (called Brinkman boundary layer) and the slip velocity at the porous/fluid interface. The porous boundary layer is well described with this model and a finer grid resolution at the wall is needed for low permeability values (low Darcy numbers). Then the impact of these quantities on the wake prediction of a 3D porous sphere is studied for various Darcy and Reynolds numbers. It is shown that wake transitions are well recovered and some further studies are made in terms of grid resolution. Some preliminary results are presented for higher Reynolds numbers.

Appendix 1: Validation of the LBM Results for the Sphere Case at \(Re=300\) with Respect to Another Numerical Method: The Semi-Lagrangian Vortex Method

1.1 Implementation of the Brinkman Penalization Technique in the Semi-Lagrangian Vortex Method Framework

In the semi-Lagrangian Vortex Method (denoted as “VM” in the sequel), one solves the incompressible Navier–Stokes equations in their adimensionalized velocity(${\textbf{u}}$)—vorticity(${\varvec{\omega }}$) formulation. Adding the Brinkman penalization forcing term in these equations gives the so-called penalized vorticity-transport-equation (Kevlahan and Ghidaglia 2001):

$$\begin{aligned} \dfrac{\partial {\varvec{\omega }}}{\partial t} +({\textbf{u}}\cdot \nabla ) {\varvec{\omega }}-({\varvec{\omega }}\cdot \nabla ) {\textbf{u}}=\frac{1}{Re} \Delta {\varvec{\omega }}-\nabla \times {\textbf{F}}_{\textbf{p}}&\qquad {\text{in}} \ \Omega \times (0,t) \end{aligned}$$

(29)

$$\begin{aligned} \Delta {\textbf{u}}= -\nabla \times {\varvec{\omega }}&\qquad {\text{in}} \ \Omega \times (0,t), \end{aligned}$$

(30)

In the present VM framework, the system of equations (29)–(30) is solved with a fractional step technique (where the diffusive, convective and stretching effects are handled successively within one time step) and by using a semi-Lagrangian approach where the convection of the vorticity field is performed in a Lagrangian way and all the other substeps are resolved on a grid using classical Eulerian schemes (finite differences, spectral) thanks to the regular remeshing of the Lagrangian particles on the grid. The reader is referred to Mimeau et al. (2016) for more details.

The resolution of the penalization equation $\partial _{t} {\varvec{\omega }}= - \nabla \times {{\textbf{F}}_{\textbf{p}}} = - \nabla \times (\lambda \chi {\textbf{u}})$ is one of the fractional step of the global algorithm, and such equation is solved on the grid with an implicit Euler scheme for time integration (with time step $\Delta t$) and a finite differences scheme for the discretization of the curl operator. The conservative formulation of the scheme allowing to update the discrete penalized vorticity therefore writes:

$$\begin{aligned} {\varvec{\omega }}^{n+1} = {\varvec{\omega }}^{n} + \nabla \times \left( \dfrac{-\lambda \chi \Delta t \ {\textbf{u}}^{n}}{1 + \lambda \chi \Delta t} \right) \end{aligned}$$

(31)

1.2 Comparison Between LBM and VM Results

This section aims at comparing the Brinkman-penalization-LBM results presented in this paper to the one obtained with the VM framework described above. One recalls that the LBM and VM implementations are completely independent one from the other and come from two distinct solvers. In absence of validation results in literature to confront the present LBM results with (e.g. the unsteady flow past a porous sphere at $Re=300)$, the VM outcomes are employed here to reinforce their robustness. The computational setup in the VM framework is exactly the same as in the LBM one (Figs. 18 and 19).

Fig. 18

Flow past a porous sphere at $Re=300$. Comparison of the Da number influence on the kinetic energy between the VM and LBM algorithms

Fig. 19

Flow past a porous sphere at $Re=300$. Influence of the Da number on the zero-level contours of the mean streamwise velocity ${\overline{u}}_{x}$ (computed on the $t^{*} \in [50, 125]$ time-range). Comparison between LBM (top row) and VM (bottom row) approaches. (Note: the slight discrepancy of the VM contour with respect to the LBM one at $Da=10^{-3}$ in the XY plane may be explained by the different behaviors of the VM and mean solutions on the chosen time window (cf Fig. 18): on the $t^{*} \in [50, 125]$ time-range, the LBM mean solution covers exactly two periods starting from the lowest energy state, which is not the case for the VM solution on the same time-range, explaining the slight shift of the mean VM contour under the plane-symmetry axis)