High-order kinetic flow solver based on the flux reconstruction framework

The goal of this paper is to develop a high-order numerical method based on the kinetic inviscid flux (KIF) method and flux reconstruction (FR) framework. The KIF aims to find a balance between the excellent merits of the gas-kinetic scheme (GKS) and the lower computational costs. The idea of KIF can be viewed as an inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and Ohwada have made the fundamental contribution. The combination of totally thermalized transport (TTT) scheme and kinetic flux vector splitting (KFVS) method are achieved in KIF. Using a coefficient which is related to time step and averaged collision time, KIF can adjust the weights of TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous splitting, KIF is very suitable and easy to integrate into the existing framework. The well-understood FR framework is used widely for the advantages of robustness, economical costs, and compactness. The combination of KIF and FR is originated by three motivations. The first purpose is to develop a high-order method based on the gas-kinetic theory. The second reason is to keep the advantages of GKS. The last aim is that the designed method should be more efficient. In present work, we use the KIF method to replace the Riemann flux solver applied in the interfaces of elements. The common solution at the interface is computed according to the gas-kinetic theory, which makes the combination of KIF and FR scheme more reasonable and available. The accuracy and performance of present method are validated by several numerical cases. The Taylor-Green vortex problem has been used to verify its potential to simulate turbulent flows.

Figure 1

The discontinuous solution polynomials and interface common solutions within elements ${\mathrm{\Omega }}_{i-1}, {\mathrm{\Omega }}_i$, and ${\mathrm{\Omega }}_{i+1}$.

Figure 2

The demonstration of solution correction: (a) Corrected solution beside the interface and (b) the solution correction within element ${\mathrm{\Omega }}_i$.

Figure 3

The demonstration of flux correction: (a) Corrected flux beside the interface and (b) the flux correction within element ${\mathrm{\Omega }}_i$.

Figure 4

The $h$ criterion grid refinement tests for Sod shock tube problem: (a) density, (b) $u$-velocity, (c) pressure, and (d) temperature distributions at $t=0.2$.

Figure 5

The results of Sod shock tube problem with different values of $s_0+\kappa$: (a) density, (b) $u$-velocity, (c) pressure, and (d) temperature distributions at $t=0.2$. The length scale $h$ is set as $1/64$.

Figure 6

The comparison between FRKIF and second-order gas-kinetic scheme: (a) density, (b) $u$-velocity, (c) pressure, and (d) temperature distributions at $t=0.2$.

Figure 7

The zoom in view of density distribution shown in Fig. 6.

Figure 8

The $h$ criterion grid refinement tests for Lax problem: (a) density, (b) $u$-velocity, (c) pressure, and (d) temperature distributions at $t=0.14$.

Figure 9

Shu-Osher problem: The distribution of density-wave at $t=1.8$.

Figure 10

Shock vortex interaction problem: The computational domain and the boundary conditions. In the simulation, Cartesian grid is used, and the element length scale of mesh $h$ equals $1/100$.

Figure 11

Shock vortex interaction problem: (a) The initial status and (b) the contour of pressure at $t=0.3$.

Figure 12

Shock vortex interaction problem: (a) The contours of pressure at $t=0.6$ and (b) the contours of pressure at $t=0.8$.

Figure 13

The computational domain and boundary conditions used in the simulation of supersonic flow past a circular cylinder.

Figure 14

Numerical results of the supersonic flow over a circular cylinder: (a) density and (b) pressure.

Figure 15

The sensor values in the simulation of supersonic flow over a circular cylinder: (a) full domain and (b) zoom in view.

Figure 16

The pressure coefficient on the surface of circular cylinder.

Figure 17

The triangular grid used in the simulation of lid-driven cavity flow (816 triangular elements).

Figure 18

The velocity profile of the lid-driven cavity flow at $\mathrm{Re}=400$: (a) $u$-velocity profiles at $x=0.5$ and (b) $v$-velocity profiles at $y=0.5$.

Figure 19

The velocity profile of the lid-driven cavity flow at $\mathrm{Re}=1000$: (a) $u$-velocity profiles at $x=0.5$ and (b) $v$-velocity profiles at $y=0.5$.

Figure 20

The velocity profile of the lid-driven cavity flow at $\mathrm{Re}=3200$: (a) $u$-velocity profiles at $x=0.5$ and (b) $v$-velocity profiles at $y=0.5$.

Figure 21

The “Grid 2” used in the simulation of laminar flow over a flat plate.

Figure 22

Blasius incompressible laminar flat plate: (a) $u$-velocity and (b) $v$-velocity profiles at $x/L=0.1$.

Figure 23

Blasius incompressible laminar flat plate: (a) $u$-velocity and (b) $v$-velocity profiles at $x/L=0.3$.

Figure 24

Blasius incompressible laminar flat plate: (a) $u$-velocity and (b) $v$-velocity profiles at $x/L=0.5$.

Figure 25

Blasius incompressible laminar flat plate: The skin friction coefficient distribution along the flat plate.

Figure 26

The kinetic energy and the dissipation rate: (a) $E_k$ and (b) $\epsilon \left(E_k\right)$. nXpY denotes that the grid size is $X^3$ and the order of polynomial is Y.

Figure 27

Isosurfaces of $Q$ ($Q=0.5$) criterion colored by velocity magnitude: (a) $t=3$, (b) $t=5$, (c) $t=7$, and (d) $t=9$.