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Turbulence Modeling Resource

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. All forms of the model given on this page are linear eddy viscosity models. Linear models use the Boussinesq assumption for the constitutive relation:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

where the last term is generally ignored for one-equation models like this one because k is not readily available (the term is sometimes ignored for non-supersonic speed flows for other models as well).

Unless otherwise stated, for compressible flow with heat transfer this model is implemented as described on the page Implementing Turbulence Models into the Compressible RANS Equations, with perfect gas assumed and Pr = 0.72, Pr_t = 0.90, and Sutherland's law for dynamic viscosity.

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The first version listed (Nut-92) is the latest "standard" version of this model.

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, https://doi.org/10.2514/6.1995-863.

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 (see also Sekundov, A. N., Fluid Dynamics 47(1):20-25, 2012, https://doi.org/10.1134/S0015462812010036). 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), https://doi.org/10.1007/BF01342683.

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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Recent significant updates:
6/30/2015 - mention Pr, Pr_t, and Sutherland's law

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Last Updated: 11/08/2021