In this work we present a novel exact Riemann solver for an artificial Equation of State (EoS) based modification of the incompressible Euler equations. Differently from the well known artificial compressibility method, this new approach overcomes the lack of the pressure-velocity coupling by using a suitably designed artificial EoS. The modified set of equations fits into the framework of first order hyperbolic conservation laws. Accordingly, an exact Riemann solver is derived. This new solver has the advantage to avoid a wave pattern violation issue which can affect the exact Riemann problem solution obtained by the standard artificial compressibility approach. The new artificial EoS based Riemann solver can be used as a tool for the definition of the advective Godunov fluxes in a Finite Volume or a discontinuous Galerkin discretization of the incompressible Navier–Stokes (INS) equations. We assess and analyse the new exact Riemann solver on 1D Riemann problems. Its capability and effectiveness are then shown in the context of a high-order accurate discontinuous Galerkin discretization of the INS equations. In particular, we verify the convergence properties, both in space and time, considering two test cases in 2D, namely, the Kovasznay test case and a damped travelling waves test case. Finally, we perform an implicit large eddy simulation of the incompressible turbulent flow over periodic hills where the classic forcing term formulation is modified to deal with variable time steps. Solution comparisons with respect to numerical and experimental results from the literature are given.

在这项工作中，我们提出了一种新颖的精确黎曼求解器，用于基于不可压缩欧拉方程的人工状态方程 (EoS) 的修改。与众所周知的人工可压缩性方法不同，这种新方法通过使用适当设计的人工 EoS 克服了压力-速度耦合的不足。修改后的方程组符合一阶双曲守恒定律的框架。因此，导出了精确的黎曼求解器。这种新的求解器具有避免波形违规问题的优势，该问题会影响通过标准人工可压缩性方法获得的精确黎曼问题解决方案。新的基于人工 EoS 的黎曼求解器可用作定义有限体积中的对流 Godunov 通量或不可压缩 Navier-Stokes (INS) 方程的不连续 Galerkin 离散化的工具。我们评估和分析一维黎曼问题的新精确黎曼求解器。然后在 INS 方程的高阶精确不连续 Galerkin 离散化的上下文中显示其能力和有效性。特别是，我们考虑了 2D 中的两个测试用例，即 Kovasznay 测试用例和阻尼行波测试用例，验证了空间和时间的收敛特性。最后，我们对周期性山丘上的不可压缩湍流进行隐式大涡模拟，其中修改了经典的强迫项公式以处理可变时间步长。