User:MasterHD/Sandbox

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_tS_{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^2K-\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 $$\nu_t={\mathcal UL}$$. 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 $$\epsilon$$,


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

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}).


 * $$ \nu_t={\mathcal UL}=C_\mu\frac{K^2}{\epsilon}$$

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_{\epsilon2}\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_{\epsilon2}\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}).