Turbulence Kinetic Energy

In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations.

In Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a turbulence model. Generally, the TKE can be quantified by the mean of the turbulence normal stresses:

TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddie scales(integral scale). Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the Kolmogorov scale. This process of production, transport and dissipation can be expressed as:

where:

  • is the mean-flow material derivative of TKE;
  • is the turbulence transport of TKE;
  • is the production of TKE, and
  • is the TKE dissipation.

The full form of the TKE equation is

\underbrace{ \frac{\partial k}{\partial t}}_{ \begin{smallmatrix}\text{Local}\\\text{derivative}\end{smallmatrix}} + \underbrace{\overline{u}_j \frac{\partial k}{\partial x_j}}_{ \begin{smallmatrix}\text{Advection}\end{smallmatrix}} = - \underbrace{ \frac{1}{\rho_o} \frac{\partial \overline{u'_i p'}}{\partial x_i} } _{ \begin{smallmatrix}\text{Pressure}\\\text{diffusion}\end{smallmatrix}} - \underbrace{ \frac{\partial \overline{k u_i}}{\partial x_j} }_{ \begin{smallmatrix} \text{Turbulent}\\ \text{transport} \\ \mathcal{T} \end{smallmatrix}} + \underbrace{ \nu\frac{\partial^2 k}{\partial x^2_j} }_{\begin{smallmatrix} \text{Molecular}\\ \text{viscous}\\ \text{transport} \end{smallmatrix}} \underbrace{ - \overline{u'_i u'_j}\frac{\partial \overline{u_i}}{\partial x_j} }_{\begin{smallmatrix} \text{Production}\\ \mathcal{P} \end{smallmatrix}} - \underbrace{ \nu \overline{\frac{\partial u'_i}{\partial x_j}\frac{\partial u'_i}{\partial x_j}} }_{\begin{smallmatrix} \text{Dissipation}\\ \epsilon_k \end{smallmatrix}} - \underbrace{ \frac{g}{\rho_o} \overline{\rho' u'_i}\delta_{i3} }_{\begin{smallmatrix} \text{Buoyancy flux}\\ b \end{smallmatrix}}

By examining these phenomena, the turbulence kinetic energy budget for a particular flow can be found.

Read more about Turbulence Kinetic Energy:  Computational Fluid Dynamics

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