Implementation of dual time-stepping strategy of the gas-kinetic scheme for unsteady flow simulations

In our study, the dual time-stepping strategy of the gas-kinetic scheme is constructed and used for the simulation of unsteady flows. In comparison to the previous implicit gas-kinetic scheme, both the inviscid and viscous flux Jacobian are considered in our work, and the linear system of the pseudo-steady-state is solved by applying generalized minimal residual algorithm. The accuracy is validated by several numerical cases, the incompressible flow around blunt bodies (stationary circular cylinder and square cylinder), and the transonic buffet on the NACA0012 airfoil under hybrid mesh. The numerical cases also demonstrate that the present method is applicable to approach the fluid flows from laminar to turbulent and from incompressible to compressible. Finally, the case of acoustic pressure pulse is carried out to evaluate the effects of enlarged time step, and the side effect of enlarged time step is explained. Compared with the explicit gas-kinetic scheme, the proposed scheme can greatly accelerate the computation and reduce the computational costs for unsteady flow simulations.

Figure 1

The finite volumes at both sides of the $J\text{th}$ interface.

Figure 2

The computational domain of flow around a circular cylinder. The diameter of circular cylinder is $d=1$, and the outer diameter of the computational domain is $50d$.

Figure 3

The definition of length of recirculation bubble ($L_w$).

Figure 4

Mean velocity at the central line.

Figure 5

Length of recirculation bubble vs Reynolds numbers.

Figure 6

Pressure coefficient on the cylinder surface at Reynolds numbers ${\text{Re}}_{\infty }=60,80,100$.

Figure 7

Time history of the streamlines past a circular cylinder. $t$ represents the period.

Figure 8

Computational domain and boundary conditions for the simulation of incompressible flow around a square cylinder.

Figure 9

Grids of full domain and near view in the simulation of incompressible flow around a square cylinder.

Figure 10

The definition of the length of recirculation region ($x_R$).

Figure 11

Streamwise velocity profiles in the wake centerline of a square cylinder.

Figure 12

Streamwise velocity profiles at four positions in the wake of a square cylinder.

Figure 13

Time history of the streamlines in the wake of a square cylinder. $t$ represents the period.

Figure 14

Computational grids for NACA0012 airfoil.

Figure 15

The evolution of pressure coefficient in a period. $t$ represents the period.

Figure 16

The $\lambda$-shock structure over a NACA0012 airfoil.

Figure 17

Unsteady response to the physical time step. $a_{\infty }$ represents the sound speed of the free stream flow.

Figure 18

Horizontal pressure ($p^{\prime }=(p-p_{\infty })/p_{\mathrm{ref}}$) profiles at $y=0.5$ and the nondimensional time $\overline{t}$ is 0.32. The exact solution is referred to in the literature [54].