Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high accuracy before the simulation reaches the statistically stationary stage is computationally intensive. The goal of this work is to employ an efficient class of high order finite difference nonlinear filter methods for subsonic turbulence simulation with stochastic forcing. The 3D Euler equations for subsonic turbulence with temporally varying stochastic forcing at rms Mach numbers up to 0.6 are numerically solved using the Strang operator splitting of the homogeneous part of the Euler equations and the forcing terms. It was shown in Yee et al. (2013) that the Strang operator splitting is more stable than solving the full Euler equation with the source term included. The spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen (2007, 2011) are used to solve the homogeneous system and an ODE solver is used to solve the forcing source terms. The nonlinear filter method includes a full time step of a spatially eighth-order central base method, using a third-order TVD Runge-Kutta time integration. The solution computed with the central base method is then nonlinearly filtered by an adaptive flow sensor and the dissipative portion of a seventh-order WENO with the Roe Riemann solver. In order to improve nonlinear stability of the base method without added numerical dissipation, the central base method discretizes the skew-symmetric split form of the inviscid flux derivatives. Both Ducros et al. and Kennedy-Gruber skew-symmetric split forms are tested. Numerical stability, computational efficiency, and effective spectral bandwidth of the nonlinear filter schemes are compared with those of second-order TVD and fifth- and seventh-order WENO methods. It is shown that the nonlinear filter method for this application is substantially more efficient, accurate and yields a superior spectral bandwidth compared to the standard TVD and WENO methods. The nonlinear filter method also demonstrates robust long-time integration for moderately compressible, statistically stationary turbulence with large-scale solenoidal forcing, including small-scale quantities such as enstrophy and mean-square dilatation.

可压缩流体中强迫湍流的数值模拟具有挑战性，原因是问题的多尺度性质以及对数值方法的要求相互矛盾，以精确地解决小尺度问题，同时处理冲击波和其他不连续性而不产生伪振荡。在仿真达到统计稳定阶段之前，要在保持高精度的同时将非线性不稳定性和混叠误差最小化是计算密集型的。这项工作的目标是采用一类有效的高阶有限差分非线性滤波方法，对具有随机强迫的亚音速湍流进行仿真。亚音速湍流的3D Euler方程，其时变随机强迫为rms马赫数（最高0）。使用Euler方程的齐次部分和强迫项的Strang算子分裂对6进行数值求解。它在Yee等人中得到了证明。（2013年），Strang算子拆分比包含源项的完整Euler方程求解更稳定。由Yee＆Sjögreen（2007，2011）开发的具有自适应耗散控制的空间七阶非线性滤波方法用于求解齐次系统，而ODE求解器用于求解强迫源项。非线性滤波方法包括使用三阶TVD Runge-Kutta时间积分的空间八阶中心基础方法的全时步骤。然后，通过自适应流量传感器和Roe Riemann求解器对七阶WENO的耗散部分进行非线性滤波，以中心基础法计算出的解。为了在不增加数值耗散的情况下提高基本方法的非线性稳定性，中心基本方法离散化了无粘性通量导数的斜对称分裂形式。都Ducros等。和Kennedy-Gruber倾斜对称分裂形式进行了测试。将非线性滤波器方案的数值稳定性，计算效率和有效频谱带宽与二阶TVD以及五阶和七阶WENO方法进行了比较。结果表明，与标准TVD和WENO方法相比，用于此应用程序的非线性滤波方法实质上更加有效，准确，并且产生了出色的频谱带宽。