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**Chapter 4**

*Lunji Song*

**Abstract**

A Fully Discrete SIPG Method for

To simulate incompressible Navier–Stokes equation, a temporal splitting scheme in time and high-order symmetric interior penalty Galerkin (SIPG) method in space discretization are employed, while the local Lax-Friedrichs flux is applied in the discretization of the nonlinear term. Under a constraint of the Courant–Friedrichs– Lewy (CFL) condition, two benchmark problems in 2D are simulated by the fully discrete SIPG method. One is a lid-driven cavity flow and the other is a circular cylinder flow. For the former, we compute velocity field, pressure contour and vorticity contour. In the latter, while the von Kármán vortex street appears with Reynolds number 50≤ *Re* ≤400, we simulate different dynamical behavior of circular cylinder flows, and numerically estimate the Strouhal numbers comparable to the existing experimental results. The calculations on vortex dominated flows are

Solving Two Classes of Vortex

carried out to investigate the potential application of the SIPG method.

**Keywords:** Navier–stokes equations, von Kármán vortex street, discontinuous

ber (i.e. non-rarefied) fluid dynamics. Phenomena described with the Navier– Stokes equations include boundary layers, shocks, flow separation, turbulence, and vortices, as well as integrated effects such as lift and drag. Analytical solutions of real flow problems including complex geometries are not available, therefore numerical solutions are necessary. The Navier–Stokes equation has been investigated by many scientists conducting research on numerical schemes for approxi-

The Navier–Stokes equations are a concise physics model of low Knudsen num-

There exist two ways to provide reference data for such problems: One consists in the measurement of quantities of interest in physical experiments and the other is to perform careful numerical studies with highly accurate discretizations. With the prevalence and development of high-performance computers, advanced numerical algorithms are able to be tested for the validation of approaches and codes and for

Among discontinuous Galerkin (DG) methods, primal schemes and mixed methods are distinguished. The former depend on appropriate penalty terms of the discontinuous shape functions, while the latter rely on the mixed methods as the

high-order convergence behavior of delicate discretizations.

Dominated Flows

Galerkin method, interior penalty

mation solutions (see [1–8]).

**1. Introduction**

**81**
