/MAT/K-EPS

Format

Definitions

Example (Gas)

#RADIOSS STARTER
/UNIT/1
unit for mat
                  kg                   m                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/K-EPS/4/1
GAS
#              RHO_I               RHO_0
               .3828                   0
#                KNU                Pmin
             1.05E-4                   0
#            RHO0_K0                 SSL
                  20                   0    
#               C_MU               SIG_k             SIG_EPS         P_R_ON_P_RT
                   0                   0                   0
#             C_1eps              C_2eps              C_3eps
                   0                   0                   0              
#              KAPPA                   E              ALPHA                GSI_T
                   0                   0                   0                   0
/EOS/POLYNOMIAL/4/1
GAS
#                 C0                  C1                  C2                  C3
                   0                   0                   0                   0
#                 C4                  C5                  E0                Pmin               RHO_0
                 0.4                 0.4              253300                   0                1.22
/ALE/MAT/4
#     Modif. factor.
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

1. (1)
   $$S_{\mathit{ij}}=2\rho {\nu }_{\mathit{eq}}{\dot{e}}_{\mathit{ij}}$$
   Where,

   $S_{ij}$

   Deviatoric stress tensor

   $e_{ij}$

   Deviatoric strain tensor

   • If the element is connected to a boundary condition, a turbulent boundary layer model is used:
     (2)

     $$v_{\mathit{eq}}=\max \left(v,v\cdot \kappa \left(\frac{y^{+}}{\ln \left(E\cdot y^{+}\right)}\right)\right)$$

     (3)

     $$y^{+}=\frac{c_{\mu }{k}^2}{\epsilon \alpha \kappa \nu }$$

     (4)

     $$\chi =\left(1-{\chi }_t\right)+{\chi }_t\alpha \ln \left(E\cdot y^{+}\right)$$

     Where, $\kappa$ is the turbulent kinetic energy.

   • If the ratio between the laminar and the turbulent Prantl numbers is higher than $P_r/P_{rt}$ , then:
     • For laminar flow:(5)
       $${\nu }_{\mathit{eq}}=\nu$$
     • For turbulent flow:(6)
       $${\nu }_{\mathit{eq}}=\nu +\frac{c_{\mu }{k}^2}{\epsilon }$$
     Where, $\dot{\epsilon }$ is the turbulent dissipation and it is calculated using the following equations:(7)

     $$\frac{d}{dt}{\displaystyle \underset{V}{\int }\rho \epsilon dV}={\displaystyle \underset{S}{\int }\rho \epsilon \left(v-w\right)ndS}+{\displaystyle \underset{S}{\int }\frac{{\mu }_t}{{\sigma }_{\epsilon }}grad\left(\epsilon \right)ndS}+{\displaystyle \underset{V}{\int }{S}_{\epsilon }dV}$$

     With,

     ${\mu }_t=\frac{C_{\mu }\cdot {\kappa }^2}{\epsilon }$ (turbulent viscosity)(8)

     $$G=\frac{\partial v_i}{\partial x_j}\left[{\mu }_t\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}-\frac{2}{3}\rho \kappa {\delta }_{ij}\right)-\frac{2}{3}\rho \kappa {\delta }_{ij}\right]$$
     (9)
     $$S_{\epsilon }=\frac{\epsilon }{\kappa }\left(C_{1\epsilon }G-C_{2\epsilon }\rho \epsilon +C_{3\epsilon }\rho \kappa \frac{\partial v_j}{\partial x_j}\right)$$
     Where,

     $v$

     Material velocity

     $w$

     Grid velocity

2. Equation of state for hydrodynamic pressure has to be prescribed via the /EOS card.

   If $P=cst=0$ , then $C_1\mu +{\alpha }_{\nu }T=0$ , so $\mu =-\frac{{\alpha }_{\nu }T}{C_1}$

   Where,

   $\mu$

   Dilatation coefficient

   $\mu <0$

   Dilatation

   In this case the parameters C₂ and C₃ will not be taken into account.
3. If using LAW6 coupled with /MAT/LAW37 (BIPHAS) for liquid phase (without gas phase), the compatibility of the liquid EOS is:

   $\text{Δ}{P}_1=C_1\mu$ for /MAT/LAW37 (BIPHAS)

   $p=C_0+C_1\mu +C_2{\mu }^2+C_3{\mu }^3+\left(C_4+C_5\mu \right)E$ for LAW6, via a polynomial EOS defined in the example above,

   then, $p=C_1\mu$

4. If using LAW6 coupled with /MAT/LAW37 (BIPHAS) for gas phase (without liquid phase), the compatibility of the gas EOS is:

   $PV\gamma =const.$ for /MAT/LAW37 (BIPHAS)

   $p=\left(\gamma -1\right)\left(\mu +1\right)E$ , for LAW6, via the /EOS/IDEAL-GAS equation of state.

   Where, $E$ is the energy per unit volume.

5. All thermal data ( ${\rho }_0{C}_p,T_0,A,andB$ ) can be defined with keyword /HEAT/MAT.