# Wall functions The Spalart-Allmaras turbulence model can be integrated throughout the inner layer of the turbulent boundary layer due to its built-in low Reynolds number damping functions. However, the model usually requires extremely fine near-wall resolutions on the order of $y ^ { + } < 2$ to accurately predict the eddy viscosity in the entire boundary layer. In general, the $y ^ { + } < 2$ near-wall resolution requirement is a very stringent constraint to enforce in complex high Reynolds number flow problems. Consequently, a wall-function approach is implemented to relax the near-wall resolution required by the Spalart-Allmaras model. The conventional wall-function approach is based on the law-of-the-wall, which is a semi-empirical universal velocity profile obtained in equilibrium wall-bounded flows when the flow velocity, $V$ , and wall-normal distance, $y ,$ are normalized with the kinematic viscosity, $\nu ,$ and friction velocity, $v _ { \tau }$ (known as viscous units or wall units): The friction velocity is computed as follows: $y ^ { * }$ is computed as follows: $$ y _ {(\chi)} ^ {*} = \frac {\chi}{\kappa}. $$ The Law-of-the-wall is computed viscous and logarithmic layer relations $V ^ { + } = V _ { ( y ^ { * } ) } ^ { + }$ . The friction velocity can be computed from the $$ v _ {\tau} = g v _ {\tau V i s} + (1 - g) v _ {\tau L o g}. $$ $$ v _ {\tau V i s} = \frac {\nu y _ {(\chi)} ^ {*}}{y}, $$ $$ v _ {\tau L o g} = \frac {V}{V _ {(y ^ {*} (\chi))} ^ {+}}. $$ The blending function is defined as $$ g = e ^ {- R e y / y _ {c} ^ {+}}, $$ $$ R e y = C _ {\mu} ^ {1 / 4} y ^ {*} (\chi). $$ The wall function is defined as $$ V _ {(y ^ {+})} ^ {+} = \left\{ \begin{array}{l l} y ^ {+} & \mathrm{if} y ^ {+} \leq y _ {c} ^ {+} \\ \frac {1}{\kappa} \ln (E y ^ {+}) & \mathrm{if} y ^ {+} > y _ {c} ^ {+}, \end{array} \right. $$ $$ V ^ {+} = \frac {V}{v _ {\tau}}, $$ $$ y ^ {+} = \frac {y v _ {\tau}}{\nu}, $$ $$ v _ {\tau} = \sqrt {\frac {\tau_ {w a l l}}{\rho}}. $$ In the above equations $\rho$ is the density, $\tau _ { w a l l }$ is the shear stress at the wall, $y _ { c } ^ { + }$ is the intersection point of the linear and logarithmic velocity profile, and $\kappa = 0 . 4 1$ and $E = 8 .$ are constants. The conventional wall-function approach has been adapted within Abaqus/CFD to a form that asymptotes to a standard wall function for coarse meshes, yet will also give results identical to a wall-function-free approach for fine meshes. It is termed a hybrid wall function. The key aspect in the implementation of the hybrid wall-function method within the Spalart-Allmaras model is to obtain the friction velocity as a function of $\widetilde \nu$ and the local velocity field, which was shown before. Having computed the friction velocity, the wall shear can be obtained as $$ \tau_ {w a l l} = \rho v _ {\tau} \frac {V}{V ^ {+} (y ^ {+})}, $$ $$ y ^ {+} = \frac {y v _ {\tau} (\chi)}{\nu}. $$ The hybrid wall-function approach is independent of the near-wall resolution; therefore, the law-ofthe-wall $V ^ { + } ( y ^ { + } )$ implemented needs to accurately predict the viscous-sublayer, the logarithmic-layer, and the buffer layer (the region that connects the viscous and logarithmic zones) since the cell-center adjacent to the wall can be located anywhere within the inner layer. Therefore, a single smooth correlation that reproduces the entire law-of-the-wall proposed by Reichardt (1951) is implemented. $$ V ^ {+} (y ^ {+}) = \frac {1}{\kappa} \mathrm{ln} (1 + \kappa y ^ {+}) + C \Big [ 1 - e ^ {- \frac {y ^ {+}}{y _ {c} ^ {+}}} - \frac {y ^ {+}}{y _ {c} ^ {+}} e ^ {- b y ^ {+}} \Big ], $$ where $$ C = \frac {1}{\kappa} \mathrm{ln} \left(\frac {E}{\kappa}\right), $$ $$ b = \frac {1}{2} \bigg (\frac {y _ {c} ^ {+} \kappa}{C} + \frac {1}{D} \bigg). $$ Implementation in the momentum equation For cases where the mesh resolution is not enough to capture the near-wall gradients, a near-wall model is required to provide the correct wall-shear stress in coarse meshes. The wall shear is obtained from the wall-function approach through an effective edge viscosity: $$ \mu_ {\mathrm{eff}} = \frac {\tau_ {w a l l} y}{V} = \left(\frac {\rho v _ {\tau} V}{V ^ {+}}\right) \frac {y}{V}. $$ Energy wall functions The wall-function approach can be extended to the energy equation by using the temperature law-of-thewall, which is a semi-empirical universal temperature profile obtained in equilibrium wall-bounded flows when the temperature, , and wall-normal distance, , are normalized with wall units. The standard temperature wall function is defined as $$ T ^ {+} (y ^ {+}) = \frac {(T _ {w} - T) \rho c _ {p} u _ {\tau}}{\dot {q}} = \left\{ \begin{array}{l l} P _ {r} y ^ {+} & \mathrm{if} y ^ {+} \leq y _ {T c} ^ {+} \\ P _ {r T} \Big (\frac {1}{\kappa} \ln (E y ^ {+}) + P (P _ {r}, P _ {r T}) \Big) & \mathrm{if} y ^ {+} > y _ {T c} ^ {+}, \end{array} \right. $$ where $y _ { T c } ^ { + }$ is the intersection point of the viscous-sublayer and the logarithmic layer in the temperature wall function, $T _ { w }$ is the wall temperature, $P _ { r }$ is the Prandtl number, $P _ { r T }$ is the turbulent Prandtl number, $\dot { q }$ is the wall-heat flux, $c _ { p }$ is the specific heat coefficient at constant pressure, and $P$ is computed using the Jayatilleke (1969) expression: $$ P (P _ {r}, P _ {r T}) = 9. 2 4 \biggl [ \biggl (\frac {P _ {r}}{P _ {r T}} \biggr) ^ {3 / 4} - 1 \biggr ] \biggl (1 + 0. 2 8 e ^ {- 0. 0 0 7 P _ {r} / P _ {r T}} \biggr). $$ For the hybrid wall-function approach a continuous temperature wall function proposed by Kader (1981) is implemented: $$ T ^ {+} (y ^ {+}) = e ^ {- \Gamma} P _ {r} y ^ {+} + e ^ {- 1 / \Gamma} P _ {r T} \left(\frac {1}{\kappa} \ln (E y ^ {+}) + P (P _ {r}, P _ {r T})\right) $$ with the blending function defined as $$ \Gamma = \frac {0 . 0 1 (P _ {r} y ^ {+}) ^ {4}}{1 + 5 P _ {r} ^ {3} y ^ {+}}. $$ Finally, the heat flux is obtained from the precomputed $v _ { \tau }$ flow properties temperature field and the continuous temperature wall function: $$ \dot {q} = \frac {(T _ {w} - T) \rho c _ {p} v _ {\tau}}{T ^ {+}}. $$ Implementation in the energy equation For cases where the mesh resolution is not enough to capture the near-wall gradients, a near-wall model is required to provide the correct wall-heat flux in coarse meshes. The wall-heat flux is obtained from the wall-function approach through an effective edge heat conductivity. $$ \kappa_ {\mathrm{eff}} = \dot {q} \frac {y}{(T _ {w} - T)}. $$ # k–epsilon RNG turbulence model The k– RNG model is a two-equation turbulence model that evolves an equation for the turbulent kinetic energy, k, and the energy dissipation rate, . The model equations are developed from fundamental physical principles and dimensional analysis; the equation for k is derived using first principles, and the equation for is postulated using physical insight. The main advantage of the RNG version is that the model coefficients are obtained using a mathematical approach called Renormalization Group theory, commonly used in physics. This calibration method removes most of the uncertainty introduced when the model coefficients are calibrated using a finite set of canonical flows as is done in most turbulence RANS models. In addition, it changes the epsilon equation (Yakhot et al., 1992). The model equations are as follows: $$ \frac {d}{d t} \int_ {V} \rho k d V + \int_ {S} \rho k (\mathbf {v} - \mathbf {v} _ {m}) \cdot \mathbf {n} d S $$ $$ = \int_ {S} \frac {(\mu + \mu_ {T})}{\sigma_ {k}} \nabla k \cdot \mathbf {n} d S + \int_ {V} \left(\tau_ {i j} S _ {i j} + G _ {b}\right) d V - \int_ {V} \rho \varepsilon d V, $$ $$ \frac {d}{d t} \int_ {V} \rho \varepsilon d V + \int_ {S} \rho \varepsilon (\mathbf {v} - \mathbf {v} _ {m}) \cdot \mathbf {n} d S $$ $$ = \int_ {S} \frac {(\mu + \mu_ {T})}{\sigma_ {\varepsilon}} \nabla \varepsilon \cdot \mathbf {n} d S + \int_ {V} C _ {\varepsilon_ {1}} \frac {\varepsilon}{k} (\tau_ {i j} S _ {i j} + C _ {\varepsilon 3} G _ {b}) d V - \int_ {V} C _ {\varepsilon_ {2}} \frac {\rho \varepsilon^ {2}}{k} d V, $$ where $\tau _ { i j }$ is the Reynolds stress tensor and is closed as follows: $$ \tau_ {i j} = 2 \mu_ {T} S _ {i j}, $$ $$ S _ {i j} = \frac {\partial u _ {i}}{\partial x _ {j}} + \frac {\partial u _ {j}}{\partial x _ {i}}. $$ The turbulent viscosity, $\mu _ { T }$ is $$ \mu_ {T} = \frac {\rho C _ {\mu} k ^ {2}}{\varepsilon}, $$ $$ C _ {\varepsilon_ {2}} = \widetilde {C} _ {\varepsilon_ {2}} + \frac {C _ {\mu} \eta^ {3} (1 - \eta / \eta_ {0})}{1 + \beta \eta^ {3}}, $$ and $$ \eta = \frac {k}{\varepsilon} \sqrt {2 S _ {i j} S _ {i j}}. $$ The second and third terms on the right-hand-side of the k– transport equations above represent the production and dissipation of k and , respectively. The k– RNG model coefficients are shown in Table 6.6.2–2. In addition, a turbulent Prandtl number $( P r _ { t } )$ can be specified. Table 6.6.2–2 k– RNG model coefficients.
$C_{\mu}$ $C_{\varepsilon_1}$ $\widetilde{C}_{\varepsilon_2}$ $\sigma_k$ $\sigma_\varepsilon$ $\beta$ $\eta_0$
0.0851.421.680.720.720.0124.38
Input File Usage: Use both of the following options: \*CFD \*TURBULENCE MODEL, TYPE=RNG KEPSILON Abaqus/CAE Usage: Step module: Create Step: General: Flow; Turbulence tabbed page: k-epsilon renormalization group (RNG) # Wall functions It is well known that the k– model has limitations, especially on wall-bounded flows where high values of eddy viscosity in the near-wall region are usually reproduced. For high Reynolds number flows often encountered in many industrial applications, a full resolution of the thin viscous sub-layer that occurs near a wall using a fine mesh may not be economical. Consequently, for meshes that cannot resolve the viscous sub-layer, wall functions are used to represent the effects of the viscous sub-layer on the transport processes. In Abaqus/CFD wall functions are used to avoid the need for highly resolved boundary layer meshes. This approach relies on the law of the wall to obtain the wall-shear stress. The law of the wall is a universal velocity profile that wall-bounded flows develop in the absence of pressure gradients. The law of the wall is $$ V ^ {+} = \left\{ \begin{array}{l l} y ^ {+} & \mathrm{if} y ^ {+} \leq y _ {c} ^ {+} \\ \frac {1}{\kappa} \ln (E y ^ {+}) & \mathrm{if} y ^ {+} > y _ {c} ^ {+}, \end{array} \right. $$ where $$ V ^ {+} = \frac {V}{v _ {\tau}}, $$ $$ y ^ {+} = \frac {y v _ {\tau}}{\nu}, $$ $$ v _ {\tau} = \sqrt {\frac {\tau_ {w a l l}}{\rho}}. $$ is the wall tangent velocity, is the kinematic viscosity, is the density, $\tau _ { w a l l }$ is the shear stress at the wall, $y _ { c } ^ { + }$ is the intersection point of the linear and logarithmic velocity profile, and $\kappa = 0 . 4 1$ and $E = 8 . 4$ are constants. The standard law of the wall profile is limited in its usage. For example, in recirculating flows the turbulent kinetic energy k becomes zero at separation and reattachment points, where, by definition, $v _ { \tau }$ is zero. This singular behavior causes the predicted results to be erroneous. To overcome this, the standard law of the wall is modified based on a new scale for the friction velocity following the method proposed by Launder and Spalding (1974). The modified friction velocity is given by $$ v ^ {*} = C _ {\mu} ^ {1 / 4} k ^ {1 / 2}, $$ which does not suffer from a singular behavior at flow reattachment, separation, and at points of flow impingement. Correspondingly, the wall distances are re-scaled as follows: $$ y ^ {*} = \frac {y v ^ {*}}{\nu} = \frac {y C _ {\mu} ^ {1 / 4} k ^ {1 / 2}}{\nu}. $$ The modified law of the wall reduces to the standard law of the wall under the conditions of uniform wall-shear stress and when the generation and dissipation of turbulent kinetic energy are in balance (i.e., when the turbulence structure is in equilibrium). Under such conditions, $v ^ { * } \approx v ^ { + }$ and, thus, $y ^ { * } \approx y ^ { + }$ . The wall-shear stress for the modified law of the wall can be evaluated as (Albets-Chico, et al., 2008) $$ \tau_ {w a l l} = \left\{ \begin{array}{l l} \frac {\mu V _ {p}}{y _ {p}} & \text {if} y ^ {*} \leq y _ {c} ^ {+} \\ \frac {\kappa \rho V _ {p} v ^ {*}}{\ln \left[ E y ^ {*} \right]} & \text {if} y ^ {*} > y _ {c} ^ {+}, \end{array} \right. $$ where the subscript p denotes the wall element center at which all the quantities of interest are evaluated. The use of the wall function requires the modification of the transport equations for k and for the wall layer of elements. Specifically, the production and dissipation terms in the governing transport equation for the turbulent kinetic energy k are modified to account for the presence of the wall. Following the procedure outlined in Craft et al. (2002), an average value of the production of k as given below is used in the transport equation. Such an average is obtained based on a two-layer model of the wall element (i.e., the wall element is divided into a partly viscous sub-layer region and a partly turbulent log-layer or inertial layer region). $$ \text {Average Production} (k) = \left\{ \begin{array}{l l} 0 & \text {if} y ^ {*} \leq y _ {c} ^ {+} \\ \frac {\tau_ {w a l l} ^ {2}}{\rho \kappa C _ {\mu} ^ {1 / 4} k ^ {1 / 2} y _ {n}} \ln \left(\frac {y _ {n}}{y _ {v}}\right) & \text {if} y ^ {*} > y _ {c} ^ {+}, \end{array} \right. $$ where $y _ { n }$ is the maximum of the wall normal distances of all the vertices of a given wall element, and $y _ { v }$ is the wall normal distance of the edge of the viscous sub-layer, where $$ y _ {v} ^ {*} = \frac {C _ {\mu} ^ {1 / 4} k ^ {1 / 2}}{\nu} y _ {v} = y _ {c} ^ {+}. $$ Similarly, an average value of the dissipation rate for k is also prescribed for the wall elements based on a two-layer integration and is given by $$ \text {Average Dissipation} (k) = \left\{ \begin{array}{l l} \frac {2 \mu k}{y _ {v} ^ {2}} & \text {if} y ^ {*} \leq y _ {c} ^ {+} \\ \frac {2 \mu k}{y _ {n} y _ {v}} + \frac {\rho C _ {\mu} ^ {3 / 4} k ^ {3 / 2}}{\kappa y _ {n}} \ln \left(\frac {y _ {n}}{y _ {v}}\right) & \text {if} y ^ {*} > y _ {c} ^ {+}. \end{array} \right. $$ The transport equation for is not solved for the wall layer elements. Instead, the value of is directly prescribed at the point p as follows: $$ \varepsilon = \left\{ \begin{array}{l l} \frac {2 \nu k}{y _ {v} ^ {2}} & \mathrm{if} y ^ {*} \leq y _ {c} ^ {+} \\ \frac {C _ {\mu} ^ {3 / 4} k ^ {3 / 2}}{\kappa y _ {p}} & \mathrm{if} y ^ {*} > y _ {c} ^ {+}. \end{array} \right. $$ Therefore, integration of the k and transport equations is performed with a zero flux (i.e., homogeneous Neumann boundary conditions) at the walls. # Guidelines on wall functions The main advantage of wall functions is the relaxed requirement on mesh resolution at walls. However, the main disadvantage of using wall functions is the dependence on the near-wall mesh resolution. Wall functions based on the law of the wall approach usually work best for wall elements whose centers lie in the fully turbulent layer (inertial or log layer) for which such functions are designed. This effectively imposes a lower limit on the value of the scaled wall coordinate, $y ^ { * }$ . For complex geometries, ensuring that all the near wall cells are outside the viscous sublayer is difficult. The precise location of the logarithmic region is solution dependent and may vary with time. To accommodate a more flexible mesh, a resolution-insensitive wall function (Durbin, 2009) has been implemented. Briefly, this wall function is based on limiting the minimum value of $y ^ { * }$ such that the value of the velocity gradient at the first wall-attached element is the same as if it was located on the edge of the viscous sub-layer. A best practice for wall-bounded flows is to have at least 8–10 points in the boundary layer region where $y ^ { \ast } \leq 3 0 0$ (see Casey and Wintergerste, 2000). # Modification of the momentum equation Using wall functions, the calculation of the wall-shear stress or the viscous flux of momentum based on the molecular viscosity $\mu$ and the numerically estimated velocity gradient can produce large errors. Hence, proper modifications to the momentum equations need to be introduced to account for the poorly resolved excess wall friction. The necessary modification is done through a modified viscosity for the wall elements that corrects for the erroneous estimate of the velocity gradient (Bredberg, 2000). This is implemented for the wall elements as: $$ \mu_ {\mathrm{eff}} = \left\{ \begin{array}{l l} \mu & \mathrm{if} y ^ {*} \leq y _ {c} ^ {+} \\ \frac {\rho \kappa C _ {\mu} ^ {1 / 4} k _ {p} ^ {1 / 2} y _ {p}}{\ln \left(E y _ {p} ^ {*}\right)} & \mathrm{if} y ^ {*} > y _ {c} ^ {+}. \end{array} \right. $$ Energy wall functions The wall-function approach can be extended to the energy equation by using the temperature law-of-thewall, which is a semi-empirical universal temperature profile obtained in equilibrium wall-bounded flows when the temperature, , and wall-normal distance, , are normalized with wall units. The standard temperature wall function is defined as $$ T ^ {+} (y ^ {+}) = \frac {(T _ {w} - T) \rho c _ {p} u _ {\tau}}{\dot {q}} = \left\{ \begin{array}{l l} P _ {r} y ^ {+} & \mathrm{if} y ^ {+} \leq y _ {T c} ^ {+} \\ P _ {r T} \Big (\frac {1}{\kappa} \ln (E y ^ {+}) + P (P _ {r}, P _ {r T}) \Big) & \mathrm{if} y ^ {+} > y _ {T c} ^ {+}, \end{array} \right. $$ where $y _ { T c } ^ { + }$ is the intersection point of the viscous-sublayer and logarithmic layer in the temperature wall function, $T _ { w }$ is the wall temperature, $P _ { r }$ is the Prandtl number, $P _ { r T }$ is the turbulent Prandtl number, $\dot { q }$ is the wall-heat flux, $c _ { p }$ is the specific heat coefficient at constant pressure, and is computed using the Jayatilleke (1969) expression: $$ P (P _ {r}, P _ {r T}) = 9. 2 4 \biggl [ \biggl (\frac {P _ {r}}{P _ {r T}} \biggr) ^ {3 / 4} - 1 \biggr ] \biggl (1 + 0. 2 8 e ^ {- 0. 0 0 7 P _ {r} / P _ {r T}} \biggr). $$ Finally, the heat flux is obtained from the precomputed $v _ { \tau }$ flow properties temperature field and the continuous temperature wall function: $$ \dot {q} = \frac {(T _ {w} - T) \rho c _ {p} v _ {\tau}}{T ^ {+}}. $$ Implementation in the energy equation For cases where the mesh resolution is not enough to capture the near-wall gradients, a near-wall model is required to provide the correct wall-heat flux in coarse meshes. The wall-heat flux is obtained from the wall-function approach through an effective edge heat conductivity: $$ \kappa_ {\mathrm{eff}} = \dot {q} \frac {y}{(T _ {w} - T)}. $$ # k–epsilon realizable turbulence model The k– realizable model is a two-equation turbulence model that evolves an equation for the turbulent kinetic energy, k, and the energy dissipation rate, . The model equations are developed from fundamental physical principles and dimensional analysis; the equation for k is derived using first principles, and the equation for is postulated using physical insight. This particular version uses realizability constraints, which imposes mathematical consistency in the Reynolds stresses (such as enforcing the positivity of the normal stresses and the Cauchy-Schwarz inequality) to modify the model coefficients and the epsilon equation. These modifications guarantee the physical consistency in the predicted Reynolds stresses, thus improving the accuracy of the predictions (Shih et al., 1995). The model equations are as follows: $$ \frac {d}{d t} \int_ {V} \rho k d V + \int_ {S} \rho k (\mathbf {v} - \mathbf {v} _ {m}) \cdot \mathbf {n} d S $$ $$ = \int_ {S} \Big (\mu + \frac {\mu_ {T}}{\sigma_ {k}} \Big) \nabla k \cdot \mathbf {n} d S + \int_ {V} \big (\tau_ {i j} S _ {i j} + G _ {b} \big) d V - \int_ {V} \rho \varepsilon d V, $$ $$ \frac {d}{d t} \int_ {V} \rho \varepsilon d V + \int_ {S} \rho \varepsilon (\mathbf {v} - \mathbf {v} _ {m}) \cdot \mathbf {n} d S $$ $$ = \int_ {S} \Big (\mu + \frac {\mu_ {T}}{\sigma_ {\varepsilon}} \Big) \nabla \varepsilon \cdot \mathbf {n} d S + \int_ {V} C _ {\varepsilon 1} (\rho S \varepsilon + \frac {\varepsilon}{k} C _ {\varepsilon 3} G _ {b}) d V - \int_ {V} C _ {\varepsilon 2} \frac {\rho \varepsilon^ {2}}{(k + \sqrt {\nu \varepsilon})} d V, $$ where the Reynolds stress tensor is defined as $$ \tau_ {i j} = 2 \mu_ {T} S _ {i j}, $$ with the eddy viscosity, $\mu _ { T }$ defined as $$ \mu_ {T} = C _ {\mu} \frac {\rho k ^ {2}}{\varepsilon}. $$ The turbulence viscosity coefficient is computed using realizability constraints: $$ C _ {\mu} = \frac {1}{A _ {0} + A _ {s} U ^ {(*)} \frac {k}{\varepsilon}}, $$ where $$ U ^ {(*)} = \sqrt {S _ {i j} S _ {i j} + R _ {i j} R _ {i j}}, $$ $$ A _ {s} = \sqrt {6} \cos (\phi), \quad \phi = \frac {1}{3} \arccos (\sqrt {6} W), $$ $$ W = \frac {S _ {i j} S _ {j k} S _ {k i}}{\tilde {S} ^ {3}}, \quad S = \sqrt {2 S _ {i j} S _ {i j}}, \quad \tilde {S} = \frac {S}{\sqrt {2}}, $$ $$ S _ {i j} = \frac {\partial u _ {i}}{\partial x _ {j}} + \frac {\partial u _ {j}}{\partial x _ {i}}, $$ $$ R _ {i j} = \frac {\partial u _ {i}}{\partial x _ {j}} - \frac {\partial u _ {j}}{\partial x _ {i}}. $$ Realizability conditions are imposed in the the production coefficient of the equation, which yields $$ C _ {\varepsilon 1} = \max \big (C _ {\varepsilon 1 m a x}, \frac {\eta}{5 + \eta} \big), $$ $$ \eta = \frac {S k}{\varepsilon}. $$ The $\pmb { k } \widetilde { \mathcal { E } } \delta$ realizable model coefficients are shown in Table 6.6.2–2. In addition, a turbulent Prandtl number, $P r _ { t }$ , can be specified. Table 6.6.2–3 k– realizable model coefficients.
$A_0$ $C_{\varepsilon 2}$ $\sigma_k$ $\sigma_\varepsilon$ $C_{\varepsilon 1max}$
4.01.91.01.20.43
Input File Usage: Use both of the following options: \*CFD \*TURBULENCE MODEL, TYPE=KEPSILON REALIZABLE Abaqus/CAE Usage: The $\pmb { k } { \mathcal { E } }$ realizable turbulence model is not supported in Abaqus/CAE. # Two-layer model To increase the accuracy of the $\pmb { k } \widetilde { \varrho }$ realizable model in cases where the mesh is not fine enough to resolve the inner layer in wall-bounded flows, a near-wall model, known as the two-layer model, is implemented. Following the work of Chen and Patel (1988), algebraic equations for and $\mu _ { T }$ are used in the inner layer to replace their turbulence model equations: $$ \varepsilon_ {\mathrm{2layer}} = \frac {k ^ {3 / 2}}{l _ {\varepsilon}}, $$ $$ l _ {\varepsilon} = C _ {l} y \big (1 - e ^ {(- R e _ {y} / A _ {\varepsilon})} \big), $$ where $$ A _ {\varepsilon} = 2 C _ {l}, \quad C _ {l} = \kappa C _ {\mu} ^ {- 3 / 4}. $$ The algebraic equation $\mu _ { T }$ is $$ \mu_ {\mathrm{T2layer}} = \mu R e _ {y} C _ {\mu} ^ {1 / 4} \kappa \big [ 1 - e ^ {(- R e _ {y} / A _ {\mu})} \big ], $$