A pseudo-compressible variational multiscale solver for turbulent incompressible flows

Abstract

In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with $512^3$ cells on a 12 cores 64 GB ram machine.

Appendix

Here we illustrate the matrix-vector implementation in 2D for linear FEs for simplicity. The nodal shape function associated to node a is denoted by $N^a(x)$ and its value at gauss point b is represent as ${ N_{b}^a }$. Every node a is represented by the lexicographical index $(i_a,j_a)$, with $i_a,j_a = 0,1$. Further, given the $\alpha$-th axis direction, with $\alpha = 1,2$, we define the function $tw(a,\alpha ) = tw(i_a,j_a,\alpha )$ as $(1-i_a,j_a)$ for $\alpha = 1$ and $(i_a,1-j_a)$ for $\alpha = 2$. We use a nodal quadrature for the numerical integration, which leads to a lumped mass matrix. As a result, using the expression of the nodal shape functions and their derivatives on nodes, we get:

$$\begin{aligned} N^a(x_b)= & {} \delta _{a b}, \quad N^a(x_b) \delta _{a b}\quad \partial _\alpha N^a(x_b) \\= & {} - \partial _\alpha N^{tw(a,\alpha )}(x_b), \partial _\alpha N^a(x_{tw(a,\beta )}) \\= & {} \delta _{\alpha \beta } \partial _\alpha N^a(x_{a}), \end{aligned}$$

where $x_b$ denotes the coordinates of the node b and $\delta _{\alpha \beta }$ the Kronecker delta. Using these expressions, we can implement efficiently all the terms in our matrix-free formulation using the following expressions. The viscous term is implemented as follows:

$$\begin{aligned}&\int _K \partial _\beta u^\alpha (x) \partial _\beta N^a (x) \mathrm{d}{\varOmega }= \sum _{b=1}^4 \partial _\beta u^\alpha (x_b) \partial _\beta N^a_b \left| J \right| w_b \\&\quad = (\partial _\beta u^\alpha (x_a) + \partial _\beta u^\alpha (x_{tw(a,\beta )}) ) \partial _\beta N^a_a \left| J \right| w_a. \end{aligned}$$

The nonlinear convective term is implemented as:

$$\begin{aligned}&\int _K a_\beta \partial _\beta u^\alpha (x) a_\gamma \partial _\gamma N^a (x) \mathrm{d}{\varOmega }\\&\quad = \sum _{b=1}^4 (a_\beta \partial _\beta u^\alpha ) (x_b) a_\gamma (x_b) \partial _\gamma N^a_b \left| J \right| w_b \\&\quad = ((a_\beta \partial _\beta u^\alpha a_\gamma )(x_a) \\&\qquad + \; (a_\beta \partial _\beta u^\alpha a_\gamma ) (x_{tw(a,\gamma )}) )\partial _\gamma N^a_a \left| J \right| w_a. \end{aligned}$$

We proceed analogously for the rest of terms.