THESIS
2022
1 online resource (xvii, 111 pages) : illustrations (some color)
Abstract
With the development of high-speed computers and the demand from challenging
problems in engineering and industries, high-order numerical methods are
prevailing in computational fluid dynamics. Most current high-order schemes
for the Euler and Navier-Stokes equations are based on the Riemann solver to
evaluate the flux and the multi-stage Runge-Kutta time stepping techniques for
temporal discretization. Different from them, the gas-kinetic scheme (GKS) is
based on the Bhatnagar-Gross-Krook model which gives a time-dependent evolution
solution. The high-order GKS (HGKS) has been systematically developed
in the past decades, with good performance in continuum flow simulations. The
current HGKS are efficient, accurate, and robust. Although the HGKS within
the finite volume framework has ac...[
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With the development of high-speed computers and the demand from challenging
problems in engineering and industries, high-order numerical methods are
prevailing in computational fluid dynamics. Most current high-order schemes
for the Euler and Navier-Stokes equations are based on the Riemann solver to
evaluate the flux and the multi-stage Runge-Kutta time stepping techniques for
temporal discretization. Different from them, the gas-kinetic scheme (GKS) is
based on the Bhatnagar-Gross-Krook model which gives a time-dependent evolution
solution. The high-order GKS (HGKS) has been systematically developed
in the past decades, with good performance in continuum flow simulations. The
current HGKS are efficient, accurate, and robust. Although the HGKS within
the finite volume framework has achieved great success, there are few schemes
based on the difference formulation. In this thesis, we will aim to construct the
HGKS under difference frameworks for the Euler and Navier-Stokes solutions.
Firstly, the conservative finite difference HGKS is constructed for inviscid and
viscous flows. Rather than the use of the discontinuous gas distribution function
in the finite volume framework, the continuous fluxes at the node points will be
kinetically split and then utilized to achieve high-order spatial accuracy through the reconstruction. The fifth-order WENO-Z reconstruction will be implemented
on those kinetic-splitting fluxes to improve the resolution of the scheme. The two-stage
fourth-order (S2O4) time stepping method, together with the second-order GKS flux, is used to have high-order temporal discretization. Numerical tests for
flows with vortexes and shocks have validated the robustness and accuracy of the
new method. For the finite volume one using the same WENO-Z reconstruction,
S2O4 time-stepping method, and the GKS flux, the CPU time is 4 times as
much as that of the finite difference scheme. And the efficiency tests regarding
the CPU time vs. the errors show that the current scheme is more efficient than
the same order finite volume version of HGKS as well. However, the above finite
difference GKS cannot be directly extended to the unstructured mesh because
the conservative property cannot be satisifed automatically. Thus, we turn to a
new class of methods called spectral difference methods.
Then spectral difference gas-kinetic scheme (SDGKS) has been successfully developed
on the unstructured quadrilateral meshes. It's high-order, conservative, and efficient, with the advantages of both finite volume and finite difference methods,
such as flexibility in geometry and efficiency in computation. The S2O4 time
marching technique is also adopted since the derivatives required in S2O4 can be
easily obtained by the time-accurate GKS flux function. In this way, compared
with the traditional RK methods, 60% of iteration stages are saved. The efficiency is higher than the traditional schemes for viscous
flows. Stability analysis for the linear advection equation shows that the new scheme is linearly stable.
The numerical simulations agree well with the analytic and reference solutions.
The new difference gas-kinetic schemes in this thesis, are generally high-order
accurate, efficient, robust, and conservative, with possible further improvement
in the future.
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