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

Abstract

SUMMARYIn this paper, the domain-free discretization method (DFD) is extended to simulate the three-dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that 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 by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of 'osculating plane' is adopted, with which the local DFD can be easily implemented for the three-dimensional case. Geometry-adaptive tetrahedral mesh is employed for three-dimensional calculations. Finally, we validate the DFD method for threedimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary-conforming grid was displayed and the results show that the present DFD results compare very well with the reference data.

An immersed boundary method for simulation of inviscid compressible flows

“…In Figure 18, the computed profiles of surface pressure coefficient at six spanwise sections are presented and compared with experimental data [32] and numerical results obtained by Navier-Stokes solver on adaptive Octree Cartesian grid with 342,289 hexahedral cells and 1,105,732 faces [33], Euler solver on adaptive unstructured mesh with 173,412 nodes [34], and DFD Euler solver with 157,304 computational nodes [10]. Although all calculations overpredicted the expansion region on the upper surface, global agreement can still be observed obviously.…”

“…The density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively, which were proposed by Dadone and Grossman in the ghost‐cell method . We extend this methodology to the three‐dimensional case with ease by using a local coordinate system introduced in and . Just as the ghost‐cell method and the DFD method, there is no special treatment for cut cells in the present method.…”

“…Also, the irregular cells with very small size have an adverse impact on the stability and conservation of the solver. To address this 777 the three-dimensional case with ease by using a local coordinate system introduced in [6] and [10]. Just as the ghost-cell method and the DFD method, there is no special treatment for cut cells in the present method.…”

“…In the method proposed by Dadone and Grossman for inviscid compressible flows [5,6], simplified momentum equation and constant distribution of entropy and total enthalpy in the direction normal to wall are applied to compute the pressure, density, and tangential velocity at the centers of ghost cells. Tseng and Ferziger [7] and Ghias et al [8] have applied the ghost-cell approach to simulate viscous flows, and the overall second-order accuracy of the base solver can be preserved.The group of Shu and Zhou recently proposed a non-boundary-conforming method, named domain-free discretization (DFD) method, to simulate compressible and incompressible flows containing moving or deforming bodies [9][10][11]. In this method, a partial differential equation is discretized at all mesh points inside the solution domain, but the discrete form may involve some exterior points.…”

“…These points are usually located in the vicinity of thin surfaces or sharp corners, such as the trailing edge of an airfoil. The discussion and treatment of these multi-valued points can be found in [6,9,10], from which we can see that handling the multi-valued points is a tedious task in the implementation of the ghost-cell method and the DFD method, especially for three-dimensional complex geometries.The classical immersed boundary method (IBM) was first introduced to study blood flow in the human heart by Peskin in the early 1970s [13], in which the external force field was explicitly introduced in the momentum equation to satisfy the no-slip condition at the vascular boundary. A different approach that does not require the explicit addition of discrete forces to the governing equations was proposed by and Fadlun et al [15].…”