In numerical simulations of complex fluid dynamical problems, unphysical negative density or pressure may appear, causing blow-up of the computation. With the aim of obtaining positivity-preserving solutions with multi-scale resolution for solving strongly compressible flows, including hypersonic flows and stiff detonation waves, we present a positivity-preserving hybrid discontinuous Galerkin/finite volume (DG/FV) method. The approach is based on a priori and a posteriori computational methodology. The a priori computation utilizes relaxed boundary variation diminishing (BVD) algorithm to find troubled cells where the DG operators are replaced by the FV operators. The FV operators then deploy a hyperbolic tangent function in the reconstruction procedure to prevent unphysical values appearing in the flux evaluation. The a posteriori computation detects unphysical negative values in a simplified version of multidimensional optimal order detection (MOOD) paradigm and in the worst case applies the first-order FV Godunov scheme to guarantee the positivity of the solution. The a priori computation produces fewer oscillatory solutions, so that the a posteriori computation triggers less first-order evaluation. A technique of utilizing the tangent of hyperbola for interface capturing (THINC) with anti-diffusion effect, which is also referred to as the technique of adaptive reconstruction in this work, is suggested to reduce the numerical dissipation of the shock-capturing scheme. The current approaches retain the capability of the DG scheme to resolve small scales and the capability of the FV scheme to capture sharp discontinuities at the subgrid level.
A thermodynamically consistent and robust four-equation model for multi-phase multi-component compressible flows using ENO-type schemes including interface regularization
Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows
An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers
本文发展了一个$ P $ -多重网格求解器，用隐式高阶方法模拟了低马赫数下局部预处理的非定常可压缩纳维-斯托克斯方程。具体而言，高阶通量重建/校正程序通过重建（FR/CPR）方法用于空间离散和高阶时间积分进行的显式第一阶段，单对角隐式龙格库塔（ESDIRK）方法。采用局部预处理来降低可压缩纳维-斯托克斯方程在低马赫数下的刚度，并仅在伪瞬态延拓中进行，以保证ESDIRK方法的高阶精度。我们采用元素雅可比平滑更新解决方案在不同的$ P $ -层次的$ P $ -多重网格求解器。通过自由流马赫数为0.005时的等熵涡传播验证了新求解器在低马赫数流动模拟中的高阶时空精度。通过数值实验研究了多项式阶数对$ P $ -多重网格法收敛速度的影响，包括$\text{Ma} = 0.001$下NACA 0012翼型的二维无粘和粘性流场，以及$\text{Ma} = 0.001$下球体的三维无粘流场。然后应用$ P $ -多重网格求解器对SD 7003机翼在$ 8^\circ $攻角下，雷诺数为60000，马赫数为0.1或0.01时的转捩流场进行了粗分辨率模拟。
A pressure-based method for weakly compressible two-phase flows under a Baer-Nunziato type model with generic equations of state and pressure and velocity disequilibrium
