Direct numerical simulations, performed with a high-order spectral-element method, are used to study coherent structures in turbulent pipe flow at friction Reynolds numbers $Re_τ = 180$ and $550$. The database was analysed using spectral proper orthogonal decomposition (SPOD) to identify energetically dominant coherent structures, most of which turn out to be streaks and quasi-streamwise vortices. To understand how such structures can be modelled, the linear flow responses to harmonic forcing were computed using the singular value decomposition of the resolvent operator, using the mean field as a base flow. The SPOD and resolvent analysis were calculated for several combinations of frequencies and wavenumbers, allowing to map out the similarities between SPOD modes and optimal responses for a wide range of relevant scales in turbulent pipe flows. In order to explore physical reasons behind the agreement between both methods, an indicator of lift-up mechanism in the resolvent analysis was introduced, activated when optimal forcing represents quasi-streamwise vortices and associated response corresponds to streaks. Good agreement between leading SPOD and resolvent modes is observed in a large region of parameter space. In this region, a significant gain separation is found in resolvent analysis, which may be attributed to the strong amplification associated with the lift-up mechanism. For both Reynolds numbers, the observed concordances were generally for structures with large energy in the buffer layer. The results highlight resolvent analysis as a pertinent reduced-order model for coherent structures in wall-bounded turbulence, particularly for streamwise elongated structures corresponding to near-wall streamwise vortices and streaks.
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
我们考虑频域形式的本征正交分解（POD）称为谱本征正交分解（SPOD）。谱POD是来自统计平稳流的时空POD问题，并导致模式，每个振荡在一个单一的频率。这种形式的POD可以追溯到Lumley的原始工作（Stochastic tools in turbulence，Academic Press，1970），但自20世纪90年代以来，POD的空间形式已经黯然失色。我们澄清了这两种形式的POD之间的关系，并表明SPOD模式代表的结构，在空间和时间上的连贯发展，而空间的POD模式一般不。我们还建立了SPOD和动态模式分解（DMD）之间的关系，我们表明，SPOD模式实际上是最佳平均DMD模式获得的系综DMD问题的固定流。因此，SPOD模式代表与DMD模式相同意义上的动态结构，但也最佳地考虑了湍流的统计变化。最后，我们建立了SPOD和预解式分析之间的联系。关键的观察结果是，解析模式的膨胀系数必须被视为统计量，以确保收敛近似的流量统计。当展开系数是不相关的，我们表明，SPOD和预解模式是相同的。我们的理论结果和整体效用的SPOD证明使用两个例子的问题：复杂的金斯堡-朗道方程和湍流射流。
Modeling of Reaction Dynamics in a Turbulent Hydrogen-Air Slot Flame Using Resolvent Analysis
Institute of Fluid Dynamics and Technical Acoustics (ISTA), TU Berlin（柏林工业大学流体动力学与技术声学研究所）；Chair and Institute for Technical Combustion, RWTH Aachen（亚琛工业大学技术燃烧学讲座与研究所）
