虚拟单元有限体积WENO5格式及其应用

THE APPLICATION OF FVWENO5 SCHEME AND GHOST CELL METHOD

本文在笛卡尔网格上给出一种五阶有限体积加权基本无振荡格式首先在二十五个单元构成的空间大模板上构造五次不完全多项式;将此大模板划分为九个子模板,并在其上构造三次不完全多项式;计算线性权,光滑指示器和非线性权;利用三阶TVD Runge-Kutta时间离散方法得到时空一致高精度格式.虽然该格式具有较高数值精度但不能直接应用于具有复杂拓扑结构物体的可压缩绕流问题.为降低该格式对网格的要求,本文采用ST和GBCM两种浸入边界虚拟单元方法处理物面边界条件,将有限体积高精度格式同虚拟单元方法相结合,能有效降低格式构造和网格生成的复杂性.文中给出的多个经典复杂物体绕流问题的数值计算充分表明了本方法的可靠性和有效性.

In this paper we investigate a new fifth order finite volume weighted essentially non- oscillatory （FVWENO） scheme on Cartesian meshes. The main procedure is as follows. Firstly, an incomplete fifth degree polynomial which has the same cell average of variableson all cells is reconstructed on the big spatial stencil including twenty-five cells. Then the big spatial stencil is divided into nine smaller ones and each of them has nine cells, and associated incomplete third degree polynomial is constructed on every smaller spatial sten- cil. After doing so, the linear weights, the smoothness indicators and the nonlinear weights are computed. Together with the application of the third order TVD Runge-Kutta time discretization method, the fully high order accurate finite volume scheme both in space and time is obtained. Although the scheme is high order accurate, it is not easily applied to simulate the compressible flow around a complex body. For the sake of dealing with this difficulty, two kinds of immersed boundary ghost cell methods, such as ST and GBCM meth- ods, are applied to confine the boundary condition nearby a complex body region. Together with the FVWENO5 scheme and the immersed boundary ghost cell methods, the simulation of compressible flow problem around a complex body surface is fulfilled, and simultaneous- ly reducing the complexities of constructing the high order accurate finite volume scheme and generating the computational meshes. Some classical numerical tests containing the flow around a complex body are provided to verify the dependability and validity of these methods.