Reynolds stress ($R_{ij}=\rho \overline{u_i^{\prime }{u}_j^{\prime }}$)

Velocity-fluctuation covariances transport mean momentum. For positive wall shear, $-\overline{u^{\prime }{w}^{\prime }}$ is the downward momentum-transport magnitude and contributes to the total shear stress. Some stress conventions include the minus sign in the definition of the tensor itself; the momentum equation must use the same convention.

Rotating anisotropic convection can produce an azimuthal-meridional velocity correlation even without a gradient of the mean rotation rate. Its Reynolds stress transports angular momentum and can drive differential rotation, whereas a purely viscous closure only erases shear. Energy comes from the thermally driven convection; the Coriolis force changes the correlations but does no work on a parcel.

For shear ${\mathbf{u}}_0=-2Ax{\mathbf{e}}_y$, the spatially averaged perturbation kinetic energy obeys $dE/dt=2A\rho ⟨u_x^{\prime }{u}_y^{\prime }⟩-\rho \nu ⟨|\nabla {\mathbf{u}}^{\prime }{|}^2⟩$. Pressure and the Coriolis force do no net work. Thus positive radial-azimuthal velocity correlation extracts mean-flow energy. A leading two-dimensional shearing wave has $\mathrm{Re}(v_x{v}_y^{\ast })=-k_x{k}_y|Z{|}^2/k^4>0$.

// Target: astrophysics.bigb

For a stationary, horizontally homogeneous mean flow without a horizontal pressure gradient, the averaged horizontal Navier-Stokes equation is $\nu U_{zz}-(\overline{u^{\prime }{w}^{\prime }}{)}_z=0$. Integrating gives the displayed physical shear stress. The derivative of the Reynolds covariance belongs in the stress-divergence equation, not inside the stress itself. The kinematic stress is ${\tau }_d/\rho$, which must be used when an eddy viscosity with units of kinematic viscosity multiplies the mean gradient.

An eddy viscosity parametrizes turbulent momentum transport by $\overline{u^{\prime }{v}^{\prime }}=-{\nu }_T{U}_y$. The mixing length model gives ${\nu }_T=l^2|U_y|\ge 0$. This coefficient depends on the mean flow and position; it is distinct from molecular kinematic viscosity. It describes momentum transport rather than molecular energy dissipation.

For an incompressible mean flow, an eddy viscosity can model the non-isotropic part of the Reynolds stress through the strain-rate tensor $S_{ij}=(U_{i,j}+U_{j,i})/2$. The isotropic part is fixed separately by the turbulent kinetic energy $e_t=\overline{u_k^{\prime }{u}_k^{\prime }}/2$ in $d$ fluctuating dimensions. Taking the trace proves that the isotropic term is necessary: $S_{ii}=0$, whereas $-\rho \overline{u_i^{\prime }{u}_i^{\prime }}=-2\rho e_t$. In the mean momentum equation the isotropic term can be absorbed into $P_{\mathrm{eff}}=P+2\rho e_t/d$.

An Eulerian time average of an incompressible constant-density Navier-Stokes equation gives the displayed mean momentum equation. Decompose velocity into its mean and a zero-mean fluctuation. In conservative form, the averaged momentum flux is $\overline{u_i{u}_j}=U_i{U}_j+\overline{u_i^{\prime }{u}_j^{\prime }}$; mean incompressibility then turns its first term into mean advection. The fluctuation covariance is the Reynolds stress divided by density, using the positive covariance convention. Closing this extra unknown requires a turbulence model.