Dynamical model of wall-bounded turbulence

Sirovich, Lawrence; Zhou, Xiang

doi:10.1103/physrevlett.72.340

Cited by 25 publications

(16 citation statements)

References 21 publications

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“…In the so-called DM we form a spatial correlation matrix R nm ij (y; y ) whose associated spatial eigenfunctions, nmq j (y), are solved for directly by means of (8). On the other hand, in the MOS, the spatial eigenfunctions are determined indirectly by ÿrst determining temporal eigenfunctions, a nmq , via (16).…”

Section: Discussionmentioning

confidence: 99%

“…An alternative procedure called the MOS is derived by changing the order of integration in (8) and (9) to give…”

Section: The Methods Of Snapshots For Three-dimensional Owsmentioning

confidence: 99%

“…This choice for yields the desired range (−1 6 y= 6 1) for the argument of the Chebyshev polynomials. To form the discrete problem for the DM we ÿrst apply the transformation y = cos(Â) to (8). In addition, to preserve the Hermitian nature of the kernel we introduce weights given by…”

Section: The Discrete Problemmentioning

confidence: 99%

“…In the case of turbulence, the KL method has been used to explore the nature of coherent structures embedded in seemingly chaotic ows [2][3][4][5]. It has also found use in the development of low-order dynamical system models [6][7][8] for these ows and in exploring non-linear energy dynamics [9].…”

Section: Introductionmentioning

confidence: 99%

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Karhunen–Loeve representations of turbulent channel flows using the method of snapshots

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Housiadas

Beris

2006

Numerical Methods in Fluids

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SUMMARYFor three-dimensional ows with one inhomogeneous spatial coordinate and two periodic directions, the Karhunen-Loeve procedure is typically formulated as a spatial eigenvalue problem. This is normally referred to as the direct method (DM). Here we derive an equivalent formulation in which the eigenvalue problem is formulated in the temporal coordinate. It is shown that this so-called method of snapshots (MOS) has some numerical advantages when compared to the DM. In particular, the MOS can be formulated purely as a matrix composed of scalars, thus avoiding the need to construct a matrix of matrices as in the DM. In addition, the MOS avoids the need for so-called weight functions, which emerge in the DM as a result of the non-uniform grid typically employed in the inhomogeneous direction. The avoidance of such weight functions, which may exhibit singular behaviour, guarantees satisfaction of the boundary conditions. The MOS is applied to data sets recently obtained from the direct simulation of turbulence in a channel in which viscoelasticity is imparted to the uid using a Giesekus model. The analysis reveals a steep drop in the dimensionality of the turbulence as viscoelasticity is increased. This is consistent with the results that have been obtained with other viscoelastic models, thus revealing an essential generic feature of polymer-induced drag reduced turbulent ows. Published in

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Section: Discussionmentioning

confidence: 99%

“…An alternative procedure called the MOS is derived by changing the order of integration in (8) and (9) to give…”

Section: The Methods Of Snapshots For Three-dimensional Owsmentioning

confidence: 99%

Section: The Discrete Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Karhunen–Loeve representations of turbulent channel flows using the method of snapshots

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Housiadas

Beris

2006

Numerical Methods in Fluids

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“…We concentrate on truly low-dimensional models which include modes with streamwise variation. Noteworthy POD-Galerkin models for the turbulent boundary layer which include modes with streamwise variation were obtained by Sanghi & Aubry (1993) and Sirovich & Zhou (1994a), the latter constructed to be very low-dimensional and the former testing the effect of the inclusion of modes with streamwise variation on the streamwise-invariant low-dimensional model of Aubry et al (1988).…”

Section: Introductionmentioning

confidence: 99%

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

2005

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We model turbulent plane Couette flow in the minimal flow unit (MFU) -a domain whose spanwise and streamwise extent is just sufficient to maintain turbulenceby expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier-Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on 'uncoupled' eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier-Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the 'backbone' for MFU turbulence is a periodic orbit, and support the roll-streak-breakdown-roll reformation picture of shear-driven turbulence.

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Energy dynamics in a turbulent channel flow using the Karhunen‐Loéve approach

Webber

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Sirovich

2002

Numerical Methods in Fluids

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SUMMARYThe dynamical equations for the energy in a turbulent channel ow have been developed by using the Karhunen-LoÃ eve modes to represent the velocity ÿeld. The energy balance equations show that all the energy in the ow originates from the applied pressure gradient acting on the mean ow. Energy redistribution occurs through triad interactions, which is basic to understanding the dynamics. Each triad interaction determines the rate of energy transport between source and sink modes via a catalyst mode. The importance of the proposed method stems from the fact that it can be used to determine both the rate of energy transport between modes as well as the direction of energy ow. The e ectiveness of the method in determining the mechanisms by which the turbulence sustains itself is demonstrated by performing a detailed analysis of triad interactions occurring during a turbulent burst in a minimal channel ow. The impact on ow modiÿcation is discussed.

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Dynamical model of wall-bounded turbulence

Cited by 25 publications

References 21 publications

Karhunen–Loeve representations of turbulent channel flows using the method of snapshots

Karhunen–Loeve representations of turbulent channel flows using the method of snapshots

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

Energy dynamics in a turbulent channel flow using the Karhunen‐Loéve approach

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