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Generalised higher-order Kolmogorov scales

Published online by Cambridge University Press:  30 March 2016

Jonas Boschung ,
Fabian Hennig ,
Michael Gauding ,
Heinz Pitsch  and
Norbert Peters
Jonas Boschung*
Affiliation:
Institute for Combustion Technology, RWTH Aachen University, Templergraben 64, Aachen, Germany
Fabian Hennig
Affiliation:
Institute for Combustion Technology, RWTH Aachen University, Templergraben 64, Aachen, Germany
Michael Gauding
Affiliation:
Chair of Numerical Thermo-Fluid Dynamics, TU Bergakademie, Fuchsmühlenweg 9, Freiberg, Germany
Heinz Pitsch
Affiliation:
Institute for Combustion Technology, RWTH Aachen University, Templergraben 64, Aachen, Germany
Norbert Peters
Affiliation:
Institute for Combustion Technology, RWTH Aachen University, Templergraben 64, Aachen, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$, namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$. However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$. We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$, i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$. The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$, this leads to a modification of the estimate of grid points required for DNS.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 794 , 10 May 2016 , pp. 233 - 251
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© 2016 Cambridge University Press 

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