Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows |
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Authors: | Philip Schaefer Markus Gampert Jens Henrik Goebbert Lipo Wang Norbert Peters |
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Institution: | 1.Institut für Technische Verbrennung,RWTH Aachen University,Aachen,Germany |
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Abstract: | Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re
λ
between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)3. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then
obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation
ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other.
The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is
found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does.
In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles
of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138,
2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the
length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which
with increasing Reynolds number approach the empirical values. |
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