尺度无关高阶WCNS格式

高速流场的数值模拟中, 既要保证对小尺度结构的高保真分辨, 又要实现对激波稳定、无振荡地捕捉.当前工程中广泛应用的高精度数值格式虽然都能一定程度地满足上述两种要求, 但仍与理想目标存在较大差距.例如, 模拟雷诺应力模型等小尺度问题时, 高精度格式在间断解附近易产生数值振荡.基于高精度格式所存在的上述问题, 本文引入去尺度函数, 探索了一种更加简单稳定的非线性权重构造方法, 并将其应用于7阶精度加权紧致非线性格式WCNS, 提出了一种尺度无关的7阶WCNS格式.该格式的性能与灵敏度参数和尺度因子的选择无关, 并且在小尺度下仍可以有效捕捉流场激波.同时, 该格式在间断处具有基本无振荡性质, 且在任意尺度函数下保持尺度无关, 并且在极值点处也能保持最优精度.本文还推导了7阶D权函数的形式.最后, 在一维线性对流方程中验证了新格式在流场光滑区能够达到设计精度, 并通过一系列数值实验证明了尺度无关的7阶WCNS格式在激波捕捉能力上具有良好表现, 为WCNS格式改进和解决可压缩湍流等非线性问题提供了一种新途径. 

A SCALE-INVARIANT HIGH-ORDER WCNS SCHEME

For numerical simulation of high-speed flows, it requires that small-scale structures are resolved with high-fidelity, and discontinuities are stably captured without spurious oscillation. These two aspects put forward almost contradictory requirements for numerical schemes. The widely used high-order schemes can satisfy the two demands required above to some extent. However, they all have advantages and disadvantages compared to each other and no one can be considered perfect. For example, high-order schemes are prone to generating numerical oscillations near discontinuities when a small-scale problem is discretized, such as the Reynolds-stress model. To solve this deficiency, a simple, effective and robust modification is introduced to the seventh-order weight compact nonlinear scheme (WCNS) by making use of the descaling function to formulate a scale-invariant WCNS scheme. The descaling function is devised using an average of the function values and introduced into the nonlinear weights of the WCNS7-JS/Z/D schemes to eliminate the scale dependency. The design idea of the scale-invariant WCNS scheme is to make weights independent of the scale factor and the sensitivity parameter. In addition, the shock-capturing ability of the new scheme performs well even for small-scale problems. The new schemes can achieve an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). We derive the seventh-order D-type weights. The one-dimensional linear advection equation is solved to verify that WCNS schemes can achieve the optimal (seventh) order of accuracy. We test a series of one- and two-dimensional numerical experiments governed by Euler equations to demonstrate that the scale-invariant WCNS schemes perform well in the shock-capturing ability. Overall, the scale-invariant WCNS schemes provide a new method for improving WCNS schemes and solving nonlinear problems.