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Reynold's Stress

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A closure problem arises in the RANS equation because of the non-linear term from the convective acceleration, known as the Reynolds stress,

Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant , the turbulent eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models or EVM's.

The Boussinesq hypothesis is used for the Spalart-Allmaras, k-ε, and k-ω models and offers a relatively low cost computation for the turbulent viscosity . The S-A model uses only one additional equation to model turbulent viscosity transport. Closing the RANS equation requires modeling the Reynold's stress . In 1887 Boussinesq proposed relating the turbulent stresses to the mean flow to close the system of equations. Thus, Boussinesq proposed a method using an eddy viscosity to solve for the Reynold's stress,

where is the mean rate of strain tensor.


Standard k-epsilon Model

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A two-equation k- turbulence model commonly used in modern engineering applications [1] uses the following two additional differential equations to close the RANS equation,

where K is the turbulent kinetic energy and ε is the turbulent kinetic energy dissipation rate. The buoyancy and compressibility production terms are ignored and not shown. K represents the kinetic energy contained in the fluctuating velocity term, from the decomposition,

Assuming that the Boussinesq hypothesis proposed holds, then it follows that

More to come, just need to copy and paste the equations from my thesis

The standard K- two-equation model commonly used in modern engineering applications uses the following two additional differential equations to close the RANS equation,

\nomenclature[ge]{}{turbulent kinetic energy dissipation rate [ms], Equation(\ref{eqn:k_decomp})}%

where is the turbulent kinetic energy and is the turbulent kinetic energy dissipation rate. The buoyancy and compressibility production terms are ignored and not shown. represents the kinetic energy contained in the fluctuating velocity term, from the decomposition,

\nomenclature[aK]{}{turbulent kinetic energy [ms], Equation(\ref{eqn:k_decomp})}%

For simplicity the is not written as , but it is indeed the energy of the fluctuating velocity component of the decomposition. Note that neither the mass nor density are factors in this kinetic energy. In Equation (\ref{eqn:K_transport}), is the turbulent kinetic energy production rate term defined as

Assuming that the Boussinesq hypothesis proposed in Equation (\ref{eqn:incompressible_boussinesq}) holds, then Equation (\ref{eqn:K_production}) becomes

Since the K- model uses a separate closed differential (transport) equation for , the only remaining term requiring modeling in the K transport equation is the last term. It is traditional to assume the term follows a gradient transport law~\cite{Bernard} such as,

where and are turbulent Prandtl Numbers. Finally, substituting this closure model into the K transport Equation (\ref{eqn:K_transport}), it reduces to

The left-hand side represents convection which is balanced by the right-hand side with production, dissipation, and transport. Lastly, the eddy viscosity is defined algebraically by choosing appropriate velocity and length scales such that . Since K provides a measure of the turbulence, it has been pursued as measure of the velocity scale

Single-equation turbulence models such as S-A, must assume a value for the length scale as part of a mixing-length assumption. Thus, single-equation models suffer because they may only work in certain classes of flows and may require externally supplied input for the length scale. A length scale is required for the following dimensional analysis relation to calculate the kinetic energy dissipation rate ,

However, a model that automatically selects the length scale is desirable, and this is where two-equation models offer a solution. Since the two-equation models use a separate transport equation to determine , Equation (\ref{eqn:eddy_visc_e}) can then be rearranged instead to estimate the length scale,

Multiplying Equations (\ref{eqn:eddy_visc_U}) and (\ref{eqn:eddy_visc_L}) yields Equation (\ref{eqn:eddy_visc}).

where is a constant in the standard k- model. This gives the two-equation models more robustness because the length scale is not required to determine the eddy viscosity.

Next, the energy dissipation rate is governed by a differential equation defined in Equation (\ref{eqn:epsilon_transport}). It is modeled with a combination of terms more challenging to model than the K transport equation (\ref{eqn:K_transport_2}). The equation comes completely from empiricism and uses arbitrary coefficients acquired during lab experiments, but behaves reasonably well for a wide variety of flows. Some of the terms are combined








Except for the last diffusion term , all of these equations contain velocity fluctuations () and must be modeled. is not modeled explicitly, but is considered to be contained within one of the other production terms. First, a model for the first two production terms is considered and relies on the formal assumption that the deviatoric parts of and are related to the anisotropy of turbulence and results in

The stretching () and dissipation () terms were derived for isotropic and homogeneous turbulence, but are applied to the general case because there are no means yet of determining the effects of anisotropy on the correlations.

where . Traditionally is assumed, in which case no contribution to the dissipation rate balance occurs from vortex stretching. Lastly, the transport terms are treated as a gradient law,

After substitution of all the models in Equations (\ref{eqn:epsilon_P_1})-(\ref{eqn:epsilon_T}), Equation (\ref{eqn:epsilon_transport}) transforms into

Similar to all the differential equations used in the K- model, the left-hand side represents convection which is balanced by the right-hand side with production, dissipation, and transport model terms. The default constants for all k- models are ~\cite[p. 316]{Bernard},

Realizable k-epsilon

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The realizable k- model was used during the present work for MESH II, and it differs from the standard and RNG k- models in that is no longer a constant. FLUENT computes this by %

%\nomenclature[ac]{}{wall normal distance, Equation (\ref{eqn:y_star})}% \nomenclature[gO]{}{rate of rotation tensor [], Equation (\ref{eqn:rateofrottensor})}% \nomenclature[aS]{}{mean rate of strain tensor [], Equation (\ref{eqn:mean_strain_rate})}%

where

and

Here, is the Levi-Civita symbol and is the mean rate-of-rotation tensor. Note that the term is ignored in the calculation of because it is an extra rotation term that is incompatible for meshes involving rotating reference frames, as in the present work. The model constants and are

where

and is the mean rate of strain tensor defined in Equation (\ref{eqn:mean_strain_rate}).



Table Test

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Col1 Col2 Col3 Col4 Col5
Name1 123456 3.1415926 Another Name Their Name\'s
Name2 123456 3.1415926 Another Name Their Name\'s
Name3 123456 3.1415926 Another Name Their Name\'s

References and notes

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  1. ^ Bernard, Peter S., Wallace, James M., Turbulent Flow Analysis, Measurement, and Prediction. 2002. p 312.