Technical Briefs

Analytical Solutions for the Modeled k Equation

[+] Author and Article Information
Rafik Absi

 EBI, 32 Boulevard du Port, 95094 Cergy-Pontoise Cedex, Francer.absi@ebi-edu.com

J. Appl. Mech 75(4), 044501 (May 13, 2008) (4 pages) doi:10.1115/1.2912722 History: Received February 20, 2006; Revised March 27, 2007; Published May 13, 2008

The semitheoretical function of Nezu and Nakagawa (1993, Turbulence in Open-Channel Flows, A. A. Balkema, ed., Rotterdam, The Netherlands) for the turbulent kinetic energy k is valid only where local equilibrium is a good approximation. From an estimation of the difference between the energy production and its dissipation Gϵ, we present in this study an analytical solution for the modeled k equation. Comparisons with direct numerical simulation data of turbulent channel flows show good agreement. A universal function for k+ is deducted for y+<20.

Copyright © 2008 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Turbulent kinetic energy. ○, DNS data (4) for Reτ=642; curve, Eq. 17 with A1+=360 and D1=4.2.

Grahic Jump Location
Figure 2

Dimensionless turbulence generation and dissipation rate versus dimensionless distance. (a) Turbulence generation. ○, measurements (5) (Profile 3). Curves, approximation 18; dashed line, with CG=1; solid line, with CG=0.6. (b) Turbulence dissipation rate. ×, measurements (5) (Profile 3). Curves, approximation 19; dashed line, with Cϵ=1; solid line, with Cϵ=0.5.

Grahic Jump Location
Figure 3

Turbulent kinetic energy. ○, DNS data (4) for Reτ=642; curves, proposed analytical solution 26 with C=1, A+=8 and B=0.14 for y+⩽20.

Grahic Jump Location
Figure 4

Turbulent kinetic energy for y+⩽20. ○, DNS data of Iwamoto (4); ×, DNS data of Kim (7); curves, proposed analytical solution 27. (a) Reτ=642; curve, A+=8 and B=0.14; (b) Reτ=395; curve, A+=8 and B=0.132; (c) Reτ=298; curve, A+=8 and B=0.127; (d) Reτ=150; A+=8 and B=0.116; (e) Reτ=109; A+=8 and B=0.11.

Grahic Jump Location
Figure 5

Dependency of the coefficient B on the Reynolds number Reτ. ○, values obtained from DNS data (4); curve, proposed function 28.



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