Question

Expand the Reynolds stress term \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}}\) for the Spalart-Allmaras model.

(a) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_i}+\frac{\partial U_j}{\partial x_j})\)

(b) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\)

(c) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = 2\rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_i}+\frac{\partial U_j}{\partial x_j})\)

(d) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = 2\rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}) \)

The question was posed to me in an online quiz.

The query is from Turbulence Modelling topic in chapter Turbulence Modelling of Computational Fluid Dynamics

Answer 1

Right choice is (b) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\)

Easy explanation: The Reynolds stress term is given as

\(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho_t (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\)

Converting to Spalart-Allmaras terms,

\(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\) .