Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows(Gomez-de-Segura & Garcia-Mayoral 2019). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier-Stokes equations are interpreted as a linear forcing-response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition of the forcing-response transfer function---the resolvent operator---identifies response modes (resolvent modes) that are known to reproduce important structural and statistical features of wall-bounded turbulent flows. The effect of permeable substrates is introduced in this framework using the Volume-Averaged Navier-Stokes equations and a generalized form of Darcy's law. Substrates with high streamwise permeability and low spanwise permeability are found to suppress the forcing-response gain for the resolvent mode that serves as a surrogate for the energetic near-wall cycle. This reduction in mode gain is shown to be consistent with the drag reduction trends predicted by theory and observed in numerical simulations. Simulation results indicate that drag reduction is limited by the emergence of spanwise rollers resembling Kelvin-Helmholtz vortices beyond a threshold value of wall-normal permeability. The resolvent framework also predicts the conditions in which such energetic spanwise-coherent rollers emerge. These findings suggest that a limited set of resolvent modes can serve as the building blocks for computationally-efficient models that enable the design and optimization of permeable substrates for passive turbulence control.
Pore-resolved direct numerical simulations of turbulent open channel flow are performed comparing the structure and dynamics of turbulence over impermeable rough and smooth walls to a porous sediment bed at permeability Reynolds number ($Re_K$) of 2.6, representative of aquatic beds. Four configurations are investigated; namely, (i) permeable bed with randomly packed sediment grains, (ii) an impermeable-wall with full layer of roughness elements matching the top layer of the sediment bed, (iii) an impermeable-wall with half layer of roughness elements , and (iv) a smooth wall. A double-averagi…
This work introduces a formulation of resolvent analysis that uses wavelet transforms rather than Fourier transforms in time. This allows resolvent analysis to be extended to turbulent flows with non-stationary means in addition to statistically-stationary flows. The optimal resolvent modes for this formulation correspond to the potentially time-transient structures that are most amplified by the linearized Navier-Stokes operator. We validate this methodology for turbulent channel flow and show that the wavelet-based and Fourier-based resolvent analyses are equivalent for statistically-station…
Response modes computed via linear resolvent analysis of a turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such resolvent models predictive, the nonlinear forcing term must be closed. Strategies to do so include imposing self-consistent sets of triadic interactions, proposing various source models, or through turbulence modelling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In th…
The predictions of resolvent analysis for turbulent channel flow are evaluated for a friction Reynolds number of Retau = 550. In addition to the standard resolvent operator with kinematic viscosity, a resolvent operator augmented with the Cess eddy viscosity is considered. Adding eddy viscosity significantly alters the low-rank behavior of the resolvent. Regardless of the wave speed selected, the eddy resolvent is low-rank for spanwise wavelengths of ly+ = 80 and ly/h = 3.5 in comparison to the standard resolvent whose low-rank behavior depends on the wave speed. The leading eddy modes have hi…
This work applies resolvent analysis to incompressible flow through a rectangular duct, in order to identify dominant linear energy-amplification mechanisms present in such flows. In particular, we formulate the resolvent operator from linearizing the Navier--Stokes equations about a two-dimensional base/mean flow. The laminar base flow only has a nonzero streamwise velocity component, while the turbulent case exhibits a secondary mean flow (Prandtl's secondary flow of the second kind). A singular value decomposition of the resolvent operator allows for the identification of structures corresp…
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）
We employ a resolvent-based methodology to estimate velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of approximately 180, 550 and 1000 using measurements of shear stress and pressure at the walls, taken from direct numerical simulation (DNS) databases. Martini et al. (J. Fluid Mech., vol. 900, 2021, A2) showed that the resolvent-based estimator is optimal when the true space-time forcing statistics are utilized, thus providing an upper bound for the accuracy of any linear estimator. We use this framework to determine the flow structures that can b…
Interaction of forced Orr-Sommerfeld and Squire modes in a low-order representation of turbulent channel flow
基于分解的降阶表示被用来捕捉湍流槽道流的时间平均二阶统计。最近提出的预解算子分解成两个不同的家庭有关的Orr-Sommerfeld和Squire算子[K。Rosenberg和B. J. McKeon，Efficient representation of exact coherent states of the纳维-斯托克斯方程using reservent analysis，流体动力学Research 51，011401（2019）]导致匹配能谱和$uv$共谱的所有分量的能力显著提高。新的表示法的成功依赖于乡绅模式的能力，竞争与Orr-Sommerfeld模式，这是通过分解的统计数据到每个家庭的贡献所产生的涡度。然后示出，这种竞争可以用来推断两组模式之间的相位关系。此外，对于Moarref等人提出的普遍的预解模式类，导出了两类预解权的相对雷诺数标度。Moarref，A. S. Sharma，J. A. Tropp，和B. J. McKeon，高雷诺数湍流通道中流向能量密度的基于模型的缩放，流体力学杂志734，275（2013）]。这些发展可以被看作是一个起点，为进一步的建模工作，以量化非线性相互作用的壁边界湍流。
Resolvent analysis is a powerful tool for modeling and analyzing turbulent flows and in particular provides an approximation of coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this two-part paper, we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction. In this first part of the paper, we derive a well-posed and convergent one-way equa…
The decomposition of hydrodynamic and acoustic components of cavity flows can aid the understanding of the flow-acoustic interaction, which produces a variety of adverse effects in applications. We apply the approach of combining the resolvent analysis and Doak's momentum potential theory to examine the input-output flow-acoustic characteristics of compressible flow over an open cavity at Re=10,000. The methodology can decompose hydrodynamic and acoustic components from an input-output framework. We further localize the forcing at the leading-edge and front wall of the cavity and filter with a…
Self-similarity of wall-attached coherent structures in a turbulent channel at $Re_τ=543$ is explored by means of resolvent analysis. In this modelling framework, coherent structures are understood to arise as a response of the linearised mean-flow operator to generalised, frequency-dependent Reynolds stresses, considered to act as an endogenous forcing. We assess the self-similarity of both the wall-attached flow structures and the associated forcing. The former are educed from direct numerical simulation data by finding the flow field correlated with the wall shear, whereas the latter is ide…
The turbulent flow around a squared-section cylinder at $Re=22000$ exhibits both a low-frequency Vortex-Shedding (VS), a common feature of all bluff-body flows, and high-frequency Kelvin-Helmholtz (KH) structures evolving on the shear layers formed at the top and bottom side of the cylinder. The VS motion triggers a periodic movement of these shear layers, inducing oscillations of the strength of their velocity gradients. The high-frequency KH structures respond to these low-frequency oscillations of the base-flow gradients in a quasi-steady manner. In this paper we will propose a general fram…
The conceptual picture underlying resolvent analysis(RA) is that the nonlinear term in the Navier-Stokes(NS) equations provides an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. This inversion obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems leads to significant computational cost and memory requirements. In this work we suggest a…
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。
The interaction between the flow above and below a permeable wall is a central topic in the study of porous media. While previous investigations have provided compelling evidence of the strong coupling between the two regions, few studies have quantitatively measured the directionality, i.e., causal-and-effect relations, of this interaction. To shed light on the problem, we use transfer entropy as a marker to evaluate the causal interaction between the free turbulent flow and the porous media using interface-resolved direct numerical simulation. Our results show that the porosity of the porous…
