Error-landscape-based multiobjective calibration of the Smagorinsky eddy-viscosity using high-Reynolds-number decaying turbulence data

AbstractWe investigate a hierarchy of eddy-viscosity terms in proper orthogonal decomposition (POD) Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to large eddy simulation (LES) data for a flow around a vehicle-like bluff body called an Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (J. Fluid Mech., vol. 260, 1994a, pp. 351–375; J. Fluid Mech. vol. 275, 1994b, pp. 257–283). Robustness of the model solution with respect to initial conditions, eddy-viscosity values and model order is achieved only for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528, 2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier–Stokes-based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

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A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

Computers & Fluids ◽

10.1016/0045-7930(94)00032-t ◽

1995 ◽

Vol 24 (3) ◽

pp. 227-238 ◽

Cited By ~ 2804

Author(s):

Tsan-Hsing Shih ◽

William W. Liou ◽

Aamir Shabbir ◽

Zhigang Yang ◽

Jiang Zhu

Keyword(s):

Reynolds Number ◽

Turbulent Flows ◽

High Reynolds Number ◽

Eddy Viscosity ◽

Viscosity Model ◽

Eddy Viscosity Model ◽

High Reynolds

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Eddy viscosity of cellular flows

Journal of Fluid Mechanics ◽

10.1017/s0022112001005730 ◽

2001 ◽

Vol 446 ◽

pp. 173-198 ◽

Cited By ~ 18

Author(s):

ALEXEI NOVIKOV ◽

GEORGE PAPANICOLAOU

Keyword(s):

Reynolds Number ◽

High Reynolds Number ◽

Large Scale ◽

Eddy Viscosity ◽

Variational Principles ◽

Stokes Equations ◽

Stationary Solutions ◽

Effective Coefficients ◽

Cellular Flows ◽

High Reynolds

We analyse modulational (large-scale) perturbations of stationary solutions of the two-dimensional incompressible Navier–Stokes equations. The stationary solutions are cellular flows with stream function ϕ = sin y1 sin y2 + δ cos y1 cos y2, 0 [les ] δ [les ] 1. Using multiscale techniques we derive effective coefficients, including the eddy viscosity tensor, for the (averaged) modulation equations. For cellular flows with closed streamlines we give rigorous asymptotic bounds at high Reynolds number for the tensor of eddy viscosity by means of saddle-point variational principles. These results allow us to compare the linear and nonlinear modulational stability of cellular flows with no channels and of shear flows at high Reynolds number. We find that the geometry of the underlying cellular flows plays an important role in the stability of the modulational perturbations. The predictions of the multiscale analysis are compared with direct numerical simulations.

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Implicit Turbulence Modeling for High Reynolds Number Flows

Journal of Fluids Engineering ◽

10.1115/1.1514210 ◽

2002 ◽

Vol 124 (4) ◽

pp. 862-867 ◽

Cited By ~ 47

Author(s):

L. G. Margolin ◽

P. K. Smolarkiewicz ◽

A. A. Wyszogrodzki

Keyword(s):

Numerical Simulation ◽

Reynolds Number ◽

High Reynolds Number ◽

Turbulence Modeling ◽

Inviscid Limit ◽

Scale Model ◽

Spectral Studies ◽

Decaying Turbulence ◽

High Reynolds ◽

Pseudo Spectral

Implicit turbulence modeling is the numerical simulation of high Reynolds fluid flow using nonoscillatory finite volume (NFV) schemes without any explicit subgrid scale model. Here we investigate the ability of a particular NFV scheme, MPDATA, to simulate decaying turbulence in a triply periodic cube for a variety of viscosities, comparing our results to analogous pseudo-spectral studies. In the regime of direct numerical simulation, MPDATA is shown to agree closely with the pseudo-spectral results. As viscosity is reduced, the two model results diverge. We study the MPDATA results in the inviscid limit, using a combination of mathematical analysis and computational experiment. We validate these results as representing the turbulent flow in the limit of very high Reynolds number.

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Revisiting Batchelor's theory of two-dimensional turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112007008427 ◽

2007 ◽

Vol 591 ◽

pp. 379-391 ◽

Cited By ~ 30

Author(s):

DAVID G. DRITSCHEL ◽

CHUONG V. TRAN ◽

RICHARD K. SCOTT

Keyword(s):

Reynolds Number ◽

Mathematical Analysis ◽

High Reynolds Number ◽

Reynolds Numbers ◽

Inertial Range ◽

Two Dimensional ◽

Simple Modification ◽

High Reynolds Numbers ◽

Decaying Turbulence ◽

High Reynolds

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3k−1 with 〈ω2〉 k−1 (ln Re)−1).

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A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance

Journal of Fluid Mechanics ◽

10.1017/s0022112090001239 ◽

1990 ◽

Vol 210 ◽

pp. 113-153 ◽

Cited By ~ 175

Author(s):

D. J. Thomson

Keyword(s):

Reynolds Number ◽

Stochastic Model ◽

High Reynolds Number ◽

Three Dimensional ◽

Inertial Subrange ◽

Concentration Variance ◽

Particle Pairs ◽

Stationary Conditions ◽

Decaying Turbulence ◽

High Reynolds

A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

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Spectral and hyper eddy viscosity in high-Reynolds-number turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112000001671 ◽

2000 ◽

Vol 421 ◽

pp. 307-338 ◽

Cited By ~ 49

Author(s):

STEFANO CERUTTI ◽

CHARLES MENEVEAU ◽

OMAR M. KNIO

Keyword(s):

Reynolds Number ◽

Energy Transfer ◽

Strain Rate ◽

High Reynolds Number ◽

Eddy Viscosity ◽

Isotropic Turbulence ◽

Reynolds Numbers ◽

Local Isotropy ◽

Viscous Term ◽

High Reynolds

For the purpose of studying the spectral properties of energy transfer between large and small scales in high-Reynolds-number turbulence, we measure the longitudinal subgrid-scale (SGS) dissipation spectrum, defined as the co-spectrum of the SGS stress and filtered strain-rate tensors. An array of four closely spaced X-wire probes enables us to approximate a two-dimensional box filter by averaging over different probe locations (cross-stream filtering) and in time (streamwise filtering using Taylor's hypothesis). We analyse data taken at the centreline of a cylinder wake at Reynolds numbers up to Rλ ∼ 450. Using the assumption of local isotropy, the longitudinal SGS stress and filtered strain-rate co-spectrum is transformed into a radial co-spectrum, which allows us to evaluate the spectral eddy viscosity, v(k, kΔ). In agreement with classical two-point closure predictions, for graded filters, the spectral eddy viscosity deduced from the box-filtered data decreases near the filter wavenumber kΔ. When using a spectral cutoff filter in the streamwise direction (with a box-filter in the cross-stream direction) a cusp behaviour near the filter scale is observed. In physical space, certain features of a wavenumber-dependent eddy viscosity can be approximated by a combination of a regular and a hyper-viscosity term. A hyper-viscous term is also suggested from considering equilibrium between production and SGS dissipation of resolved enstrophy. Assuming local isotropy, the dimensionless coefficient of the hyper-viscous term can be related to the skewness coefficient of filtered velocity gradients. The skewness is measured from the X-wire array and from direct numerical simulation of isotropic turbulence. The results show that the hyper-viscosity coefficient is negative for graded filters and positive for spectral filters. These trends are in agreement with the spectral eddy viscosity measured directly from the SGS stress–strain rate co-spectrum. The results provide significant support, now at high Reynolds numbers, for the ability of classical two-point closures to predict general trends of mean energy transfer in locally isotropic turbulence.

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