In fluid dynamics, turbulence kinetic energy 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 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 is defined to be half the sum of the variances of the velocity components: where the turbulent velocity component is the difference between the instantaneous and the average velocity, whose mean and variance are and, respectively. TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddy scales. 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:
Assuming density and viscosity both constant, the full form of the TKE equation is: By examining these phenomena, the turbulence kinetic energy budget for a particular flow can be found.
Computational fluid dynamics
In computational fluid dynamics, it is impossible to numerically simulate turbulence without discretizing the flow-field as far as the Kolmogorov microscales, which is called direct numerical simulation. Because DNS simulations are exorbitantly expensive due to memory, computational and storage overheads, turbulence models are used to simulate the effects of turbulence. A variety of models are used, but generally TKE is a fundamental flow property which must be calculated in order for fluid turbulence to be modelled.
simulations use the Boussinesq eddy viscosity hypothesis to calculate the Reynolds stress that results from the averaging procedure: where The exact method of resolving TKE depends upon the turbulence model used; – models assume isotropy of turbulence whereby the normal stresses are equal: This assumption makes modelling of turbulence quantities simpler, but will not be accurate in scenarios where anisotropic behaviour of turbulence stresses dominates, and the implications of this in the production of turbulence also leads to over-prediction since the production depends on the mean rate of strain, and not the difference between the normal stresses. Reynolds-stress models use a different method to close the Reynolds stresses, whereby the normal stresses are not assumed isotropic, so the issue with TKE production is avoided.
Initial conditions
Accurate prescription of TKE as initial conditions in CFD simulations are important to accurately predict flows, especially in high Reynolds-number simulations. A smooth duct example is given below. where is the initial turbulence intensity given below, and is the initial velocity magnitude; Here is the turbulence or eddy length scale, given below, and is a – model parameter whose value is typically given as 0.09; The turbulent length scale can be estimated as with a characteristic length. For internal flows this may take the value of the inlet duct width or the hydraulic diameter.
