Extension of domain‐free discretization method to simulate compressible flows over fixed and moving bodies

Abstract

SUMMARYThis paper is the first endeavour to present the local domain-free discretization (DFD) method for the solution of compressible Navier-Stokes/Euler equations in conservative form. The discretization strategy of DFD is that for any complex geometry, there is no need to introduce coordinate transformation and the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior dependent points are updated at each time step to impose the wall boundary condition by the approximate form of solution near the boundary. Some points inside the solution domain are constructed for the approximate form of solution, and the flow variables at constructed points are evaluated by the linear interpolation on triangles. The numerical schemes used in DFD are the finite element Galerkin method for spatial discretization and the dual-time scheme for temporal discretization. It is well known that the analytical method takes two separate steps to get the closed-form solution of a PDE. In the first step, a general solution is pursued which is only based on the given PDE. Then in the second step, the expression of the general solution is substituted into the boundary conditions to determine the unknown coefficients in the general solution. Clearly, the first step does not involve the solution domain. The solution domain (geometry of the problem) is * Correspondence to: C. Shu, Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore. † E-mail: mpeshuc@nus.edu.sg only involved in the second step when the boundary condition is implemented. So the analytical method can be well applied to both regular and irregular domain problems.In contrast, the conventional numerical method solves the PDE by directly coupling it with the boundary condition. In other words, the numerical solution is obtained in just one step. In this step, the PDE is discretized on the solution domain with proper implementation of the boundary condition. We can see clearly that the discretization of the PDE in a numerical method is problem dependent. Due to this feature, some numerical methods such as finite difference schemes can only be applied to regular domain problems. When they are applied to solve irregular domain problems, the coordinate transformation is a must. In general, the process of coordinate transformation is very complicated, and problem dependent. To overcome the drawbacks of conventional numerical methods that strongly couple the PDE with the solution domain, the DFD method was developed from the process of analytical method.In the DFD, the implementation of the boundary condition and discretization of the governing equation are treated separately. The selected point for numerical discretization of PDE is inside the solution domain, but the discrete form of the PDE at the selected point may involve some points outside the solution domain, which serve as the role to implement the boundary condition. The key process in the...

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

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Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

“…Step 2: Use equation (6) to get the density distribution function at time level n tt = (initially setting 0 F α = ) and compute the macro variables using equations (11) and (23); Step 3: Solve equation system (21) to get the velocity corrections at all boundary points and use equation (17) to get velocity corrections at Eulerian points;…”

“…However, due to irregular 4 structures of the cut cells, the calculation of fluxes at the interface of cut cells requires complicated treatment, which may bring inconvenience and affect the computational efficiency. Recently, Zhou et al [17] proposed an efficient Cartesian grid method, namely, the local domain-free discretization (DFD) method, for simulation of compressible flows around moving boundaries. In the local DFD method, the boundary information is transferred to an adjacent point to the boundary through low order interpolation.…”

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

“…In this paper, we extend the local DFD method developed in [5] to the simulation of threedimensional compressible flows governed by Euler equations in conservative form. According to [12], the concept of the so-called 'modified osculating plane' is adopted, with which the local DFD can be easily implemented in three dimensions.…”

“…These points are associated with the thin body, the width of which is smaller than two or one grid interval. The technique for the treatment of these multi-valued points has been first suggested by Dadone and Grossman [12] and can also be considered an extension of that used in two-dimensional flow conditions [5]. The difference is that the body may be thin in several directions in three-dimensional conditions.…”