Reynolds stress

Remarks

Many authors call $R_{ij} = \overline{u_i' u_j'}$ itself the Reynolds stress tensor, without the factor $-\rho$. The matrix $(R_{ij})$ is symmetric and positive semidefinite, since $\sum_{i,j} a_i a_j R_{ij} = \overline{(a \cdot \mathbf{u}')^2} \ge 0$ for every vector $a$. Half its trace is the turbulent kinetic energy per unit mass, $k = \frac12 \overline{u_i' u_i'}$. The diagonal terms are normal stresses and the off-diagonal terms are shear stresses.

Example: take a turbulent channel flow with mean velocity $\bar u_1(x_2)$ increasing away from the lower wall. Eddies moving up ($u_2' > 0$) tend to carry slow fluid ($u_1' < 0$), so $\overline{u_1' u_2'} < 0$. Thus the shear stress $\tau_{12}$ is positive and acts like an extra friction. Non-example: in a laminar steady flow the fluctuations vanish, so all $\tau_{ij}$ are zero; the viscous stress $\mu (\partial_j u_i + \partial_i u_j)$ is not a Reynolds stress.