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Reynold's Stress

A closure problem arises in the RANS equation because of the non-linear term ${\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}$ from the convective acceleration, known as the Reynolds stress,

$R_{ij}={\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}$

Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant $\nu _{t}>0$, the turbulent eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models or EVM's.

$-{\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}=\nu _{t}\left({\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {\upsilon }}_{j}}{\partial x_{i}}}\right)-{\frac {2}{3}}\left(K+\nu _{t}{\frac {\partial {\bar {\upsilon }}_{k}}{\partial x_{k}}}\right)\delta _{ij}$

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 $\nu _{t}$. The S-A model uses only one additional equation to model turbulent viscosity transport. Closing the RANS equation requires modeling the Reynold's stress $R_{ij}$. 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,

$-R_{ij}=-{\overline {\upsilon _{i}^{\prime }\upsilon _{j}^{\prime }}}=2\nu _{t}S_{ij}-{\frac {2}{3}}K\delta _{ij}$

where $S_{ij}$ is the mean rate of strain tensor.

Standard k-epsilon Model

A two-equation k-$\epsilon$ turbulence model commonly used in modern engineering applications uses the following two additional differential equations to close the RANS equation,

${\frac {\partial K}{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial K}{\partial x_{j}}}={\mathcal {P}}-\epsilon +\nu \nabla ^{2}K-{\frac {\partial }{\partial x_{i}}}\left({\frac {\overline {p\upsilon _{i}^{\prime }}}{\rho }}+{\overline {\upsilon _{i}^{\prime }(\upsilon _{j}^{\prime \,2}/2)}}\right)$

${\frac {\partial \epsilon }{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial \epsilon }{\partial x_{j}}}=P_{\epsilon }^{1}+P_{\epsilon }^{2}+P_{\epsilon }^{3}+P_{\epsilon }^{4}+\Pi _{\epsilon }+T_{\epsilon }+D_{\epsilon }-\Upsilon _{\epsilon }$

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,

${\frac {1}{2}}{\overline {\upsilon _{i}^{2}}}={\overline {K}}+K={\frac {1}{2}}{\overline {\upsilon _{i}}}^{2}+{\frac {1}{2}}{\overline {\upsilon _{i}^{\prime \,2}}}$

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

${\mathcal {P}}=\nu _{t}{\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}\left({\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {\upsilon }}_{j}}{\partial x_{i}}}\right)$

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

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

${\frac {\partial K}{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial K}{\partial x_{j}}}={\mathcal {P}}-\epsilon +\nu \nabla ^{2}K-{\frac {\partial }{\partial x_{i}}}\left({\frac {\overline {p\upsilon _{i}^{\prime }}}{\rho }}+{\overline {\upsilon _{i}^{\prime }(\upsilon _{j}^{\prime \,2}/2)}}\right)$

${\frac {\partial \epsilon }{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial \epsilon }{\partial x_{j}}}=P_{\epsilon }^{1}+P_{\epsilon }^{2}+P_{\epsilon }^{3}+P_{\epsilon }^{4}+\Pi _{\epsilon }+T_{\epsilon }+D_{\epsilon }-\Upsilon _{\epsilon }$

\nomenclature[ge]{$\epsilon$}{turbulent kinetic energy dissipation rate [m$^{2}$s$^{-3}$], Equation(\ref{eqn:k_decomp})}%

where $K$ is the turbulent kinetic energy and $\epsilon$ 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,

${\frac {1}{2}}{\overline {\upsilon _{i}^{2}}}={\overline {K}}+K={\frac {1}{2}}{\overline {\upsilon _{i}}}^{2}+{\frac {1}{2}}{\overline {\upsilon _{i}^{\prime \,2}}}$

\nomenclature[aK]{$K$}{turbulent kinetic energy [m$^{2}$s$^{-2}$], Equation(\ref{eqn:k_decomp})}%

For simplicity the $K$ is not written as $K^{\prime }$, 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}), ${\mathcal {P}}$ is the turbulent kinetic energy production rate term defined as

${\mathcal {P}}=R_{ij}{\frac {\partial {\overline {\upsilon _{i}}}}{\partial x_{j}}}$

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

${\mathcal {P}}=\nu _{t}{\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}\left({\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {\upsilon }}_{j}}{\partial x_{i}}}\right)$

Since the K-$\epsilon$ model uses a separate closed differential (transport) equation for $\epsilon$, 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,

${\frac {1}{\rho }}{\overline {p\upsilon _{i}^{\prime }}}+{\overline {\upsilon _{i}^{\prime }(\upsilon _{j}^{\prime \,2}/2)}}=-{\frac {\nu _{t}}{\sigma _{K}}}{\frac {\partial K}{\partial x_{i}}},$

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

${\frac {\partial K}{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial K}{\partial x_{j}}}={\mathcal {P}}-\epsilon +{\frac {\partial }{\partial x_{i}}}\left[\left(\nu +{\frac {\nu _{t}}{\sigma _{K}}}\right){\frac {\partial K}{\partial x_{i}}}\right]$

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 $\nu _{t}={{\mathcal {U}}L}$. Since K provides a measure of the turbulence, it has been pursued as measure of the velocity scale

${\mathcal {U}}\sim {\sqrt {K}}$

Single-equation turbulence models such as S-A, must assume a value for the length scale ${\mathcal {L}}$ 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 $\epsilon$,

$\epsilon \sim {\frac {K^{3/2}}{\mathcal {L}}}$

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 $\epsilon$, Equation (\ref{eqn:eddy_visc_e}) can then be rearranged instead to estimate the length scale,

${\mathcal {L}}\sim {\frac {K^{3/2}}{\epsilon }}$

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

$\nu _{t}={{\mathcal {U}}L}=C_{\mu }{\frac {K^{2}}{\epsilon }}$

where $C_{\mu }=0.09$ is a constant in the standard k-$\epsilon$ 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 $\epsilon$ 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

$\epsilon _{ij}=2\nu {\overline {{\frac {\partial \upsilon _{i}}{\partial x_{k}}}{\frac {\partial \upsilon _{j}}{\partial x_{k}}}}},\qquad \epsilon _{ij}^{c}=2\nu {\overline {{\frac {\partial \upsilon _{k}}{\partial x_{i}}}{\frac {\partial \upsilon _{k}}{\partial x_{j}}}}}$

$P_{\epsilon }^{1}=-\epsilon _{ij}^{c}{\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}$

$P_{\epsilon }^{2}=-\epsilon _{ij}{\frac {\partial {\bar {\upsilon }}_{i}}{\partial x_{j}}}$

$P_{\epsilon }^{3}=-2\nu {\overline {\upsilon _{k}^{\prime }{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{j}}}}}{\frac {\partial {^{2}{\bar {\upsilon }}_{i}}}{\partial {x_{k}}\partial x_{j}}}$

$P_{\epsilon }^{4}=-2\nu {\overline {{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{k}}}{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{j}}}{\frac {\partial \upsilon _{k}^{\prime }}{\partial x_{j}}}}}$

$\Pi _{\epsilon }=-2\nu {\frac {\partial }{\partial x_{i}}}\left({\overline {{\frac {\partial p}{\partial x_{j}}}{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{j}}}}}\right)$

$T_{\epsilon }=-\nu {\frac {\partial }{\partial x_{k}}}\left(\upsilon _{k}^{\prime }{\overline {{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{j}}}{\frac {\partial \upsilon _{i}^{\prime }}{\partial x_{j}}}}}\right)$

$D_{\epsilon }=\nu \nabla ^{2}\epsilon$

$\Upsilon _{\epsilon }=2\nu ^{2}{\overline {\left({\frac {\partial ^{2}\upsilon _{i}^{\prime }}{\partial x_{j}\partial x_{k}}}\right)}}$

Except for the last diffusion term $D_{\epsilon }$, all of these equations contain velocity fluctuations ($\upsilon ^{\prime }$) and must be modeled. $P_{\epsilon }^{3}$ 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 $\epsilon _{ij}$ and $\epsilon _{ij}^{c}$ are related to the anisotropy of turbulence and results in

$P_{\epsilon }^{1}+P_{\epsilon }^{2}=C_{\epsilon 1}{\frac {\epsilon }{K}}{\mathcal {P}}$

The stretching ($P_{\epsilon }^{4}$) and dissipation ($\Upsilon _{\epsilon }$) 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.

$P_{\epsilon }^{4}-\Upsilon _{\epsilon }=C_{\epsilon 3}R_{T}^{1/2}{\frac {\epsilon ^{2}}{K}}-C_{\epsilon 2}{\frac {\epsilon ^{2}}{K}}$

where $R_{T}=K^{2}/\nu \epsilon$. Traditionally $C_{\epsilon 3}=0$ 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,

$T_{\epsilon }+\Pi _{\epsilon }={\frac {\partial }{\partial x_{i}}}\left({\frac {\nu _{t}}{\sigma _{\epsilon }}}{\frac {\partial \epsilon }{\partial x_{i}}}\right)$

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

${\frac {\partial \epsilon }{\partial t}}+{\bar {\upsilon }}_{j}{\frac {\partial \epsilon }{\partial x_{j}}}=C_{\epsilon 1}{\frac {\epsilon }{K}}{\mathcal {P}}-C_{\epsilon 2}{\frac {\epsilon ^{2}}{K}}+{\frac {\partial }{\partial x_{i}}}\left(\nu +{\frac {\nu _{t}}{\sigma _{\epsilon }}}{\frac {\partial \epsilon }{\partial x_{i}}}\right)$

Similar to all the differential equations used in the K-$\epsilon$ 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-$\epsilon$ models are ~\cite[p. 316]{Bernard},

$C_{1\epsilon }=1.44,\qquad C_{2\epsilon }=1.92,\qquad C_{\mu }=0.09,\qquad \sigma _{k}=1.0,\qquad \sigma _{\epsilon }=1.3,$

Realizable k-epsilon

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

$C_{\mu }={\frac {1}{A_{0}+A_{s}*{\frac {KU^{*}}{\epsilon }}}}$

%\nomenclature[ac]{$C_{\mu }$}{wall normal distance, Equation (\ref{eqn:y_star})}% \nomenclature[gO]{$\Omega _{ij}$}{rate of rotation tensor [$s^{-1}$], Equation (\ref{eqn:rateofrottensor})}% \nomenclature[aS]{$S_{ij}$}{mean rate of strain tensor [$s^{-1}$], Equation (\ref{eqn:mean_strain_rate})}%

where

$U^{*}\equiv {\sqrt {S_{ij}S_{ij}+{\tilde {\Omega }}_{ij}{\tilde {\Omega }}_{ij}}}$

and

${\tilde {\Omega }}_{ij}=\Omega _{ij}-2\epsilon _{ijk}\omega _{k}$

$\Omega _{ij}={\overline {\Omega _{ij}}}-\epsilon _{ijk}\omega _{k}$

Here, $\epsilon _{ijk}$ is the Levi-Civita symbol and ${\overline {\Omega _{ij}}}$ is the mean rate-of-rotation tensor. Note that the $2\epsilon _{ijk}\omega _{k}$ term is ignored in the calculation of ${\tilde {\Omega }}_{ij}$ because it is an extra rotation term that is incompatible for meshes involving rotating reference frames, as in the present work. The model constants $A_{0}$ and $A_{s}$ are

$A_{0}=4.04,A_{s}={\sqrt {6}}cos\phi$

where

$\phi ={\frac {1}{3}}cos^{-1}({\sqrt {6}}W),\;W={\frac {S_{ij}S_{jk}S_{ki}}{{\tilde {S}}^{3}}},\;{\tilde {S}}={\sqrt {{S_{ij}}^{2}}}$

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

Table Test

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