Turbulence Kinetic Energy Transport Equation

Equation

For $k = \tfrac12\overline{u_i' u_i'}$ (turbulent kinetic energy per unit mass):

\[\frac{\partial k}{\partial t} + \bar u_j\frac{\partial k}{\partial x_j} = -\frac{1}{\rho}\frac{\partial}{\partial x_j}\overline{u_j' p'} - \frac{1}{2}\frac{\partial}{\partial x_j}\overline{u_i' u_j' u_i'} + \nu\frac{\partial^2 k}{\partial x_j^2} - \overline{u_i' u_j'}\frac{\partial \bar u_i}{\partial x_j} - \nu\,\overline{\frac{\partial u_i'}{\partial x_j}\frac{\partial u_i'}{\partial x_j}}\]

Physical meaning — the energy budget of turbulence

Term	Name	Role
$\partial k/\partial t$	rate of change	local storage
$\bar u_j\,\partial_j k$	mean-flow convection	carried by the mean stream
$-\frac{1}{\rho}\partial_j\overline{u_j'p'}$	pressure diffusion	redistribution by pressure fluctuations
$-\frac12\partial_j\overline{u_i'u_j'u_i'}$	turbulent transport	eddies carrying their own energy
$\nu\,\partial_j^2 k$	viscous transport	molecular diffusion of $k$
$-\overline{u_i'u_j'}\,\partial_j\bar u_i$	production $\mathcal{P}$	extraction from mean shear via Reynolds stress
$-\nu\overline{(\partial_j u_i')^2}$	dissipation $\varepsilon$	conversion to heat at the Kolmogorov scale

In steady homogeneous shear, the budget collapses to $\mathcal{P} \approx \varepsilon$: what the mean flow feeds in, viscosity burns — through the energy cascade in between.

Variables

$k$ — TKE (m² s⁻²) · $\bar u_i$ — mean velocity · $u_i', p'$ — fluctuations · $\nu$ — kinematic viscosity · overbars — ensemble/time averages.

Assumptions

Incompressible · Newtonian · Reynolds decomposition meaningful (statistically steady or slowly varying flow).

Derivation

Multiply the fluctuating momentum equation (full Navier–Stokes minus its Reynolds average) by $u_i'$ and average. Each nonlinearity contributes one budget term; the algebra is bookkeeping, the physics is in the two sign-definite terms: production (usually $> 0$, since $\overline{u'v'}$ opposes mean shear) and dissipation (always $> 0$).

Applications

The backbone of the $k$–$\varepsilon$ and $k$–$\omega$ turbulence models (they carry modeled versions of exactly these terms)
Diagnosing simulations/experiments: where turbulence is born (production peaks near walls) and where it dies
Atmospheric boundary-layer meteorology (add buoyancy production)

Limitations

Averaged description only — no phase information, no coherent-structure detail; the higher-order correlations inside transport terms are unclosed (the closure problem again).

Navier–Stokes equation — parent
Kolmogorov spectrum — where $\varepsilon$ goes

Quiz

Q1 (conceptual). Why is the production term usually positive in a shear flow?

Answer

In shear $\partial\bar u/\partial y > 0$, eddies moving up ($v' > 0$) carry slower fluid ($u' < 0$) and vice versa, so $\overline{u'v'} < 0$; then $-\overline{u'v'}\,\partial\bar u/\partial y > 0$. Turbulence taxes the mean flow.

Q2 (multiple choice). In steady, homogeneous turbulence with no mean shear, the TKE equation reduces to:

(a) $\mathcal{P} = \varepsilon$ (b) $dk/dt = -\varepsilon$ (decay) (c) $k = $ const (d) $\varepsilon = 0$