The energetics and colour for linearised
models of wall turbulence
models of wall turbulence
File(s)
Author(s)
Holford, Jacob
Myoungkyu, Lee
Hwang, Yongyun
Type
Journal Article
Abstract
By comparing the budget of a data-driven quasi-linear approximation (DQLA) (Holford,
Lee & Hwang, J. Fluid Mech., 2024, 980:A12) and direct numerical simulation (DNS)
(Lee & Moser, J. Fluid Mech., 2019, 860:886-938), the energetics of linear models for wall bounded turbulence are assessed. The DQLA is implemented with the linearised Navier Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The
self-consistent nature of the DQLA allows for a global comparison across all wavenumbers
to assess the role of the various terms in the linear model in replicating the features
present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a
spectral budget equation is derived, connecting Lyapunov-like equations to the transport
budget equations obtained from DNS. It is found that the DQLA and DNS are in good
qualitative agreement for the streamwise elongated structures present in DNS, comparing
well for production, viscous transport and wall-normal turbulent transport. However, the
DQLA does not have an energy-conservative nonlinear term. This results in no dissipation
under molecular viscosity, with energy instead being dissipated locally through the eddy
viscosity model, which models the energy removal by the nonlinear term at integral length
scales. Comparison of the pressure-strain statistics also highlights the absence of the
streak instability, with production and forcing mainly being retained in the streamwise
and wall-normal components or shifted to the spanwise component. It is demonstrated
that the eddy viscosity diffusion term locally enforces a self-similar budget, making the
model for the nonlinear term self-consistent with a logarithmic mean profile. Implications
and recommendations to improve the current eddy viscosity enhanced linear models
are also discussed concerning the comparison with DNS, as well as considerations with
regard to pressure statistics to mimic the role of the streak instability through colour of
turbulence models.
Lee & Hwang, J. Fluid Mech., 2024, 980:A12) and direct numerical simulation (DNS)
(Lee & Moser, J. Fluid Mech., 2019, 860:886-938), the energetics of linear models for wall bounded turbulence are assessed. The DQLA is implemented with the linearised Navier Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The
self-consistent nature of the DQLA allows for a global comparison across all wavenumbers
to assess the role of the various terms in the linear model in replicating the features
present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a
spectral budget equation is derived, connecting Lyapunov-like equations to the transport
budget equations obtained from DNS. It is found that the DQLA and DNS are in good
qualitative agreement for the streamwise elongated structures present in DNS, comparing
well for production, viscous transport and wall-normal turbulent transport. However, the
DQLA does not have an energy-conservative nonlinear term. This results in no dissipation
under molecular viscosity, with energy instead being dissipated locally through the eddy
viscosity model, which models the energy removal by the nonlinear term at integral length
scales. Comparison of the pressure-strain statistics also highlights the absence of the
streak instability, with production and forcing mainly being retained in the streamwise
and wall-normal components or shifted to the spanwise component. It is demonstrated
that the eddy viscosity diffusion term locally enforces a self-similar budget, making the
model for the nonlinear term self-consistent with a logarithmic mean profile. Implications
and recommendations to improve the current eddy viscosity enhanced linear models
are also discussed concerning the comparison with DNS, as well as considerations with
regard to pressure statistics to mimic the role of the streak instability through colour of
turbulence models.
Date Issued
2024-12-10
Date Acceptance
2024-09-06
Citation
Journal of Fluid Mechanics, 2024, 1000
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
1000
Copyright Statement
© The Author(s), 2024. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Identifier
https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/energetics-and-colour-for-linearised-models-of-wall-turbulence/E6BAE11A5C68A4AD7C6F7EAAA7CB87BF
Publication Status
Published
Article Number
A42
Date Publish Online
2024-11-26