Reynolds Stress Transport Equations
\[
\underbrace{c_{ij}}_{\substack{\text{convection term}}}
=
\underbrace{d_{ij}}_{\substack{\text{diffusion term}}}
+ \underbrace{P_{ij}}_{\substack{\text{production term}}}
+ \underbrace{\phi_{ij}}_{\substack{\text{redistribution term}}}
- \underbrace{\varepsilon_{ij}}_{\substack{\text{dissipation term}}}
\]
The Reynolds stress transport equation \(
\frac{D \, \overline{u_i u_j}}{Dt}
= \frac{\partial}{\partial x_k} \left(
\nu \frac{\partial \overline{u_i u_j}}{\partial x_k}
- \overline{u_i u_j u_k}
- \frac{1}{\rho} \overline{u_i p} \, \delta_{jk}
+ \frac{1}{\rho} \overline{u_j p} \, \delta_{ik}
\right)
- \left(
\overline{u_k u_i} \frac{\partial U_j}{\partial x_k}
+ \overline{u_k u_j} \frac{\partial U_i}{\partial x_k}
\right)
+ \frac{2}{\rho} \, \overline{p s_{ij}}
- 2\nu \, \overline{\frac{\partial u_i}{\partial x_k} \frac{\partial u_j}{\partial x_k}}
\) that derived from the N–S equations contains all the information of turbulent motion under the sense of statistical averaging. Conceptually, this equation is composed of the following physical mechanisms:
- $c_{ij}$ — convection term, describing the convection of Reynolds stresses, i.e., the total rate of change of Reynolds stresses within a fluid control volume. This term represents the imbalance of all the terms on the right-hand side of the equation and also reflects the historical effect of turbulence transport
- $d_{ij}$ — diffusion term, representing the diffusivity of turbulence. For example, in a thin shear layer with $\partial U / \partial x$ as the dominant velocity gradient, when integrating across any control surface, its integral value is zero. Thus, it does not generate turbulence but promotes the spatial redistribution of turbulence. The diffusion of Reynolds stresses consists of three parts: diffusion caused by velocity fluctuations, diffusion caused by pressure fluctuations, and molecular viscosity diffusion, namely:
$
\overline{u_i u_j u_k}, \quad \overline{p u_i}/\rho, \quad \nu \overline{u_i u_j}/\partial x_k
$. Generally, diffusion due to velocity fluctuations (i.e., the triple velocity correlation term) is the main component in turbulence diffusion, while the latter two terms are significant only in near-wall regions
- $P_{ij}$ — production term, describing the rate of change of Reynolds stresses resulting from the interaction between the mean strain rate of the flow and the Reynolds stresses
- $\phi_{ij}$ — pressure–strain redistribution term, describing the redistribution of turbulent kinetic energy among different components. Since $\phi_{ii} = 0$ (for incompressible flows, $s_{ii} = 0$), this term does not appear in the turbulent kinetic energy equation, but it has the special meaning in the Reynolds stress transport process
- $\varepsilon_{ij}$ — dissipation term, describing the mechanism by which turbulent kinetic energy is dissipated through viscous action and converted into internal energy of the fluid
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