适于可压缩多尺度流动的紧致型激波捕捉格式
Compact shock capturing scheme for compressible multiscale flow

摘要 针对可压缩多尺度流动数值模拟特点，研究一种五阶高分辨率紧致型激波捕捉格式——紧致重构加权基本无振荡（CRWENO）格式。该格式利用非线性权系数将低阶紧致格式加权组合以达到高阶精度。在光滑区域蜕化成具有高分辨率的五阶线性紧致格式，在间断附近则能保持计算稳定无振荡。对CRWENO格式、目前广泛使用的加权基本无振荡（WENO）格式及两格式对应的线性格式（即五阶线性迎风格式和五阶紧致格式）进行数值性能研究，评估非线性权系数对格式耗散及频谱特性的影响。使用一维、二维、三维典型算例进行数值试验，探讨线性/非线性、紧致/非紧致格式在可压缩多尺度流动模拟中的优势和不足。结果表明，CRWENO格式在强压缩性流场模拟中能够稳定地捕捉激波，其紧致特性则改善了非线性格式普遍存在的耗散过大、分辨率较差的问题，使其能够清晰捕捉多尺度流动结构。因此，该格式在可压缩多尺度流动模拟中具有较大优势。
Abstract：Aimed at compressible multiscale flow simulations, a fifth-order high-resolution compact shock capturing scheme, compact-reconstruction weighted essentially non-oscillatory (CRWENO), is studied. Nonlinear weights are used to combine lower-order compact schemes to approximate a higher-order compact scheme. The scheme becomes the fifth-order linear compact scheme in smooth regions, while preserves computational stability across discontinuities. Numerical properties were analyzed for CRWENO and weighted essentially non-oscillatory (WENO) which is widely used these days, as well as the linear schemes that they correspond to, i.e. the fifth-order upwind linear scheme and the fifth-order compact scheme. The impacts of nonlinear weights on dissipation and spectral properties are evaluated. The advancements and drawbacks of linear/nonlinear and compact/explicit schemes in compressible multiscale flow simulations are discussed by performing 1D, 2D and 3D typical numerical tests. It can be concluded that CRWENO scheme can obtain non-oscillatory results near strong discontinuous regions. Its compact characteristic improves the problems of over-dissipation and low resolution exiting in nonlinear schemes and makes it clearly resolve multiscale flow structures. In a word, CRWENO is a proper candidate for compressible multiscale flow simulations.