Reynolds Stress Model Based on the RDT Equations and Turbulence Dynamics in the Aerodynamic Nozzle

The Paradigm of the Turbulence Model

The well known challenge of turbulence modeling is the inclusion of the effects of the large-scale structure in the one-point model equations. We proposed an approach [1,2] to the Reynolds stresses modeling that is based on an unclosed equation in terms of the spectral tensorr $\Phi_{i,j}(\vec{k})$ deduced in [3]:

$\frac{d\Phi_{ij}}{dt} - E_{\alpha \beta} \frac{\partial k_{\alpha} \Phi_{ij}}{\partial k_{\beta}} + E_{i \alpha} \Phi_{\alpha j } + E_{j \alpha} \Phi_{i \alpha} - E_{j \alpha} \overline{u_{i} u_{\alpha}}$

$-2E_{\alpha \beta} \frac{k_\alpha}{k^2} (k_i \Phi_{\beta j} + k_j \Phi_{i \beta}) + \dots = 0 \,, (1)$

where $E_{i j}(t) = \partial U_i / \partial x_j$ and dots denote missed nonlinear and dissipation terms approximated in our model.