Nut-92 Model

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The Nut-92 Turbulence Model

This web page gives detailed information on the equations for various forms of the Nut-92 one-equation turbulence model. If any particular variant has been overlooked, please report it to the page curator.

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Nut-92 (1995) Model (Nut-92)

The Nut-92 model evolved from a model originally proposed by Kovasznay in 1967. The model was improved over the years, including one termed Nut-90. None of these earlier variants is described here (see the reference below for more details). The reference for the Nut-92 one-equation model is:

Shur, M., Strelets, M., Zaikov, L., Gulyaev, A., Kozlov, V., Secundov, A., "Comparative Numerical Testing of One- and Two-Equation Turbulence Models for Flows with Separation and Reattachment," AIAA Paper 95-0863, January 1995.

The model (written in conservation form) is given by the following:

$\frac{\partial (\rho \nu_t)}{\partial t} + \frac{\partial (\rho u_j \nu_t)}{\partial x_j} = P_{\nu} - D_{\nu} + \frac{\partial}{\partial x_j} \left[\rho \left(\nu + C_0 \nu_t \right)\frac{\partial \nu_t}{\partial x_j}\right] + \frac{\partial}{\partial x_j} \left[\rho \left(-\nu + (C_1 - C_0)\nu_t \right)\right] \frac{\partial \nu_t}{\partial x_j}$

where

$P_{\nu} = \rho C_2 F_2 \left( \nu_t \Gamma_1 + A_1 \nu_t^{4/3} \Gamma_2^{2/3} \right) + \rho C_2 F_2 A_2 N_1 \sqrt{(\nu + \nu_t)\Gamma_1} + \rho C_3 \nu_t \left( \frac{\partial^2 \nu_t}{\partial x_j \partial x_j} + N_2 \right)$

$D_{\nu} = \rho C_5 \nu_t^2 \Gamma_1^2 / a^2 + \rho C_4 \nu_t \left( \frac{\partial \langle u_j \rangle}{\partial x_j} + \left| \frac{\partial \langle u_j \rangle}{\partial x_j} \right| \right) + \rho \left[ C_6 \nu_t ( N_1 d_w + \nu_{t,w}) + C_7 F_1 \nu \nu_t \right] / d^2$

Here, a is the speed of sound and the angle braces < > represent a long-time average. The turbulent eddy viscosity is

$\mu_t = \rho \nu_t$

Other term appearing in the above equations are given by:

$F_1 = \frac{N_1 d_w + 0.4 C_8 \nu}{\nu_t + C_8 \nu + \nu_{t,w}}$

$F_2 = \frac{\chi^2 + 1.3 \chi + 0.2}{\chi^2 - 1.3 \chi + 1.0}$

$\chi = \frac{\nu_t}{7 \nu}$

$\Gamma_1 = \sqrt{ \frac{\partial u_i}{\partial x_j} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) }$

$\Gamma_2 = \sqrt{ \sum_i \left( \frac{\partial^2 u_i}{\partial x_j \partial x_j} \right)^2 } = \sqrt{ \left( \frac{\partial^2 u_i}{\partial x_n \partial x_n} \right) \times \left( \frac{\partial^2 u_i}{\partial x_m \partial x_m} \right) }$

$N_1 = \sqrt{ \frac{\partial \nu_t}{\partial x_j} \frac{\partial \nu_t}{\partial x_j} }$

$N_2 = \sqrt{ \frac{\partial N_1}{\partial x_j} \frac{\partial N_1}{\partial x_j} }$

The term $d_w$ is the distance to the nearest wall, and

$d = d_w + 0.01 k_s$

and $k_s$ is the Nikuradse roughness scale height (0 for smooth walls).

The constants are:

$A_1 =-0.5$ $A_2 = 4.0$

$C_0 = 0.8$ $C_1 = 1.6$ $C_2 = 0.1$

$C_3 = 4.0$ $C_4 = 0.35$ $C_5 = 3.5$

$C_6 = 2.9$ $C_7 = 31.5$ $C_8 = 0.1$

There are no specific farfield boundary conditions recommended for this model. At solid smooth walls:

$\nu_{t,w} = 0$

At solid rough walls:

$\nu_{t,w} = 0.02 k_s \sqrt{\frac{\tau_w}{\rho}}$

where $\tau_w$ is the wall shear stress and $\sqrt{\tau_w / \rho}$ is the friction velocity $u_{\tau}$ .

Nut-92 (1993) Model (Nut-92-FD)

The reference for this model is:

Gulyaev, A. N., Kozlov, V. E., Secundov, A. N., "A Universal One-Equation Model for Turbulent Viscosity," Fluid Dynamics, Vol. 28, No. 4, 1993, pp. 485-494 (translated from Russian, Consultants Bureau, NY).

The model is an earlier version of (Nut-92). It is given by the same main equation (written in conservation form). However, the following expressions are slightly different:

$F_1 = \frac{N_1 d + 0.4 C_8 \nu}{\nu_t + C_8 \nu}$

$d^2 = d_w^2 + 0.4 k_s d_w + 0.004 k_s^2$

For smooth walls, (Nut-92-FD) and (Nut-92) are identical.

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