Supersonic flow has complex flow characteristics, such as shock waves, shock waves/complex flow interactions, viscous/inviscid interactions, flow transitions. In order to capture the shock wave without oscillation, the numerical dissipation is usually introduced into the numerical methods. Meanwhile, in order to resolve other smallscale structures in the flow field, the method is required to have high accuracy and low dissipation. The high accuracy and high resolution shock-capturing schemes in computational fluid dynamics are developing all the time for dealing with this contradiction.
The weighted essentially non-oscillatory (WENO) scheme is a prominent representative of the high accuracy shock-capturing schemes in recent years, which has been continuously developed and widely used in CFD since it was proposed. In this paper, we further develop the WENO scheme with novel idea and advanced properties, such as perturbational WENO scheme, multistep WENO scheme. In addition, these methods are also applied to simulate the transition of supersonic flat-plate boundary layer in this work.
1) A perturbational weighted essentially non-oscillatory scheme is proposed and developed. In order to improve the accuracy and dissipative dispersion characteristics of the WENO scheme, the traditional methods are basically realized by improving the smoothness indicator of the sub-stencil or developing new method for calculating weights.
However, the perturbational WENO scheme improves the accuracy of the sub-flux function by using the numerical perturbation sub-stencils numerical flux function, and then the perturbation weighted scheme is constructed by weighting the perturbed numerical fluxes. Firstly, the accuracy of the numerical flux in the conservative form of the hyperbolic conservation equation is analyzed, and a lemma about the accuracy of numerical flux is given. The high order sub-stencil numerical fluxes are obtained by using the idea of numerical perturbation algorithm. And then, the final numerical flux of a fifth order perturbational WENO (P-WENO) scheme is constructed by a convex combination of the new perturbed fluxes. Theoretical analysis shows that the P-WENO scheme relaxes
the requirement of necessary and sufficient conditions for fifth-order convergence on the weights. That is, the P-WENO scheme is easier to obtain the fifth-order convergence, which effectively solves the problem that the fifth-order WENO scheme is difficult to obtain low dissipation near discontinuity while achieving fifth-order accuracy at critical point. Finally, the new method is generalized and a seventh-order P-WENO scheme isdeveloped. The perturbation weighting method provides a novel idea for developing theWENO schemes.
2) A high performance fifth-order multistep WENO (HM-WENO) scheme is developed to improve the accuracy of traditional fifth-order WENO scheme at transition point (connecting a smooth region and a discontinuity point). First, the accuracy of the fifthorder WENO scheme at transition point is analyzed, and then a more effective multistep WENO scheme is developed based on the analysis of the related methods. Theoretical analysis and numerical results show that the new scheme not only improves the accuracy by one order higher than the traditional fifth-order WENO schemes at transition point, but also maintains the fifth-order accuracy in smooth regions even at critical point. The new scheme is simple, which not only improves the computational efficiency of the original multistep WENO scheme, but also avoids the problem of reducing the computational accuracy of other multistep WENO schemes near discontinuities.
3) This paper proposed a low dissipation and low dispersion third-order WENO scheme.
Compared with higher order WENO scheme, the third-order WENO scheme has the advantages of good robustness and easy boundary treatment. However, the existing thirdorder WENO schemes are still too dissipative. In order to further develop the third-order WENO scheme with better performance, first, the functional relationship between the weights of different third-order WENO schemes and the smoothness indicators of substencils is analyzed. And then, a new low dispersion and low dissipation third order WENO scheme is developed by introducing a new function. The new scheme greatly improves the accuracy of the nonlinear weights approximating the optimal weights. And the low dispersion and low dissipation properties of the new scheme are demonstrated by spectral analysis and numerical results.
4) The application of parabolized Navier Stokes (PNS) equation to direct numerical simulation of supersonic boundary layer transition is explored. The PNS equation not only can reflect the characteristics of boundary layer, but also consider the interaction between viscous and inviscid. On the other hand, the PNS equation eliminates the viscous terms related to the approximate mainstream direction, hence it brings great convenience to numerical discretization. Considering that flow transition usually occurs in the boundary layer, it is undoubtedly of great significance to study whether PNS can correctly simulate supersonic boundary layer transition. In this paper, two different computational conditions, i.e., smooth plate with blowing-suction disturbance and plate with roughness elements, are designed. Then, the direct numerical simulation of PNS equation is carried out by using the high order scheme developed in this paper.The numerical results, which compared with the numerical results of the full NS equation, show that the PNS equation can simulate the transition phenomenon of supersonic flat-plate boundary layer.