Research On Spatial Discretization Schemes For Finite Volume Method On Unstructured Grids

In order to improve the computational accuracy,robustness and convergence of un-structured finite volume method which has been commonly used in computational fluid dynamics analyses,the presented thesis is investigating the spatial discretization schemes for unstructured finite volume method.Therefore,two major issues,the upwind schemes for the computation of convective flux and the spatial reconstruction methods,are dis-cussed in this thesis.The two major issues are divided into four specific topics,on which the investigations are made.Firstly,the rotated upwind schemes are investigated,which had been deemed as shock stable and dissipation tunable method.The dissipative terms of rotated schemes are anal-ysed,and then the conjecture that the dissipation is intensified during the introducing of rotation procedure is made.The designed numerical test cases show that the diffusive effects of rotated schemes are increasing while using the given increasing rotation angles.At the same time,rotated schemes show shock stable results on unstructured triangular grid while the fixed maximum rotated angle is used.A novel rotation strategy is hence designed.In this new strategy,rotation angle of rotated scheme is considered as the parameter tuning dissipation effects.The rotation angle magnitude is defined based on pressure discontinuities in flow fields,and thus the numerical dissipation is regulated.With Roe scheme applied,the strategy has effectively removed instabilities in strong shock wave simulations,and maintained good accuracy in viscous flow simulations.In conclusion,the intensified dissipation deteriorates the accuracy of viscous flow simulations and improves the shock stability.Furthermore,the dissipation of rotated schemes is monotonously in-tensifying during the increasing of the rotation angle,and thus the presented new rotation strategy could be used for regulating dissipation.Secondly,a simple,physical meaningful novel hybrid scheme is designed based on the convective upwind and split pressure framework.The scheme is designed based on SLAU scheme and van Leer-Hanel scheme.Thereinto,the fluxes of mass equation and energy equation,and the pressure splitting in momentum flux,are the same as those of SLAU scheme.Only the dissipation term in momentum flux is modified by introducing the char-acter of van Leer-Hanel scheme.Compared with SLAU scheme,only a scalar function needs to be calculated for the hybrid scheme as extra cost,and the hybrid scheme is show-ing good performances in the shock stability,viscous accuracy and low Mach number flow simulation by adaptively tuning dissipation.Numerical test cases and the construction of hybrid scheme are both supporting the viewpoints that transversal momentum perturba-tion on shock waves is the cause of shock instabilities,and the accurate approximation on contact discontinuity reduces the ability of upwind schemes to suppress the perturbation.Therefore,the upwind schemes that produce accurate results for contact discontinuity are potentially producing shock instabilities.In order to improve the spatial accuracy of unstructured finite volume method,gradi-ent reconstruction methods are investigated.Several circumstances,under which common used methods may be suffering accuracy deterioration,are discussed.In general,the lack of the reconstruction stencils is an important reason that causes accuracy loss.There-fore,a,gradient reconstruction method based on vertex weighted-least-squares method is designed.The method is named as VWLSQ(n)method which introduces sufficient computation stencils,and the stencil invalidation caused by distance weight is alleviated.The method has unified the gradient reconstruction for inviscid and viscous flow simula-tions by the vertex-based computation,and thus the weight-least-squares calculations are reduced,especially on unstructured triangle/tetrahedron grids.Numerical results show that the VWLSQ(1)method that uses one-degree inverse distance weight shows lower computation error and better convergence,especially on triangular grids.For the monotonicity of flow variables reconstruction are slope limiters used,by which the computation stability could be guaranteed.Slope limiters and gradient reconstruction methods are two parts of the spatial reconstruction of unstructured finite volume method.Inspired by the unstructured MLP limiter and MLP condition,strick-and weak-MLP condition are presented,for which the maximum/minimum principle are proved.Espe-cially,the strict-MLP condition is strictly maintaining monotonicity in one-dimensional reconstruction.Therefore,a MLP-pw limiter is constructed,in which the strict-and weak-MLP conditions are tuned by a pressure weight function.Thereinto,the strict-MLP condition is applied in the vicinity of shock waves to improve the stability and conver-gence of the computations of strong shock waves.The weak-MLP condition is applied in regions that show small pressure gradient to reduce the numerical dissipation in simula-tions of contact discontinuity and smooth flows.In the numerical results,the MLP-pw limiter shows significant improvement in computational convergence in hypersonic flow simulations.Meanwhile,because the limitation effects is restricted in regions of high pres-sure discontinuities,the MLP-pw limiter also shows low dissipation in many simulation conditions.Finally,three-dimensional transonic,supersonic and hypersonic test cases are used to integrally verify all the methods introduced in this thesis.The results prove that the presented spatial discretization schemes could be used for the applications of real-life engineering problems.