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What is Kinetic Energy of Turbulence?

transition from laminar to turbulent flow in a plume of hot gas

The kinetic energy of turbulence is the energy associated with turbulent eddies in a fluid flow. It is a property that can be conserved in mathematical models of turbulent flow in computational fluid dynamics. It is expressed as the kinetic energy per unit mass, with units of velocity squared (m2/s2) and can be defined as:

k = 1/2 (mean(u'2) + mean(v'2) + mean(w'2) )

where u', v' and w' are the three fluctuating components of velocity.

The Need for Turbulence Models

Steady laminar flows, where viscous forces dominate, are relatively easy to model using a numerical solution of the Navier-Stokes equations. For turbulent flows, which cover the majority of practical fluid flow problems, the Navier-Stokes equations still apply, but the situation is much more complicated. This is because the equations must be solved on the scale of the smallest turbulent eddies and the flow is also time-dependent. The direct solution of the equations is impractical in all but very specific cases where Large Eddy Simulations are used, and even here there is a point where a model is used for the smallest eddies.

In practical turbulent flow modelling a turbulence model is required. The key assumption is of a mean flow with a statistical distribution of a fluctuating component of velocity; the Reynolds-averaged equations. By including this additional velocity component in the Navier-Stokes equations and rearranging, additional terms can be found that resemble the viscous stresses, and these are referred to as the Reynolds stresses. If the stresses can be assumed to be anisotropic then a single "effective viscosity" can be derived that includes the real molecular viscosity (which is usually negligible in fully developed turbulent flows) and an additional and usually much larger "eddy viscosity" (the Boussinesq hypothesis). This eddy viscosity is analogous to the molecular viscosity and represents the work of the eddies in transporting momentum in the same way that viscosity transports momentum at the molecular level.

Kinetic Energy in Turbulence Models

In order to achieve closure of the set of equations for turbulent fluid flow, some way of computing the unknown eddy viscosity is required. A number of turbulence models exist that ultimately include some degree of empirical input. The kinetic energy of the turbulence is usually involved directly as a conserved property of the flow. A source term involves a rate of production of kinetic energy derived from the mean rate of strain - the shear being the mechanism that causes the largest eddies to "roll up" in the first place as the inertial forces overcome the viscous drag. These eddies then decay into smaller and smaller eddies which carry less kinetic energy, until eventually the viscous forces dissipate them.

The kinetic energy of turbulence and its rate of production can be derived from the Reynolds averaged equations. But the rate of dissipation is a little more problematic and we must rely on some empirically based model. The various turbulence models differ mainly in the way this dissipation rate is derived. In some cases the rate of dissipation is derived from other properties of the flow, for example a length scale. In the widely used k-ε turbulence model the dissipation rate (ε) is modelled as if it were itself a conserved property.

Measuring Kinetic Energy of Turbulence

Kinetic Energy of Turbulence can be measured experimentally using hot-wire or hot-film anemometers, or using laser-Doppler or acoustic Doppler techniques. These techniques are used almost exclusively in fluid dynamics research.

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