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In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, turbulence kinetic energy (TKE) is the mean
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...
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 the Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a
turbulence model Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system, the airflow ...
. Generally, the TKE is defined to be half the sum of the variances (square of standard deviations) of the velocity components: k = \frac12 \left(\, \overline + \overline + \overline \,\right), where the turbulent velocity component is the difference between the instantaneous and the average velocity u' = u - \overline, whose
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...
and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
are \overline = \frac \int_0^T (u(t) - \overline) \, dt = 0 and \overline = \frac\int_0^T (u(t) - \overline)^2 \, dt \geq 0 , respectively. TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddy 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: \frac + \nabla \cdot T' = P - \varepsilon, 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. Assuming that molecular viscosity is constant, and making the Boussinesq approximation, the TKE equation is: \underbrace_ + \underbrace_ = - \underbrace _ - \underbrace_ + \underbrace_ \underbrace_ - \underbrace_ - \underbrace_ By examining these phenomena, the turbulence kinetic energy budget for a particular flow can be found.


Computational fluid dynamics

In
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate t ...
(CFD), it is impossible to numerically simulate turbulence without discretizing the flow-field as far as the Kolmogorov microscales, which is called direct numerical simulation (DNS). 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.


Reynolds-averaged Navier–Stokes equations

Reynolds-averaged Navier–Stokes (RANS) simulations use the Boussinesq eddy viscosity hypothesis to calculate the Reynolds stress that results from the averaging procedure: \overline = \frac23 k \delta_ - \nu_t \left( \frac + \frac \right), where \nu_t = c \cdot \sqrt \cdot l_m. The exact method of resolving TKE depends upon the turbulence model used; ''
The dash is a punctuation mark consisting of a long horizontal line. It is similar in appearance to the hyphen but is longer and sometimes higher from the baseline. The most common versions are the endash , generally longer than the hyphen b ...
'' (k–epsilon) models assume isotropy of turbulence whereby the normal stresses are equal: \overline = \overline = \overline. This assumption makes modelling of turbulence quantities ( and ) 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 (as they are, by assumption, equal). Reynolds-stress models (RSM) 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. k = \frac32 ( U I )^2, where is the initial turbulence intensity given below, and is the initial velocity magnitude; I = 0.16 Re^. Here is the turbulence or eddy length scale, given below, and is a – model parameter whose value is typically given as 0.09; \varepsilon = ^\frac34 k^\frac32 l^. The turbulent length scale can be ''estimated'' as l = 0.07L, with a characteristic length. For internal flows this may take the value of the inlet duct (or pipe) width (or diameter) or the hydraulic diameter.


References


External links


Turbulence kinetic energy
at CFD Online. *{{cite journal, last=Absi, first=R., title=Analytical solutions for the modeled {{mvar, k-equation , journal= Journal of Applied Mechanics, volume=75, date=2008, issue=44501, pages=044501, doi=10.1115/1.2912722, bibcode=2008JAM....75d4501A Computational fluid dynamics Turbulence Energy (physics)