CMPS Theory Manual › Transport Properties and Material Closures

Transport Properties and Material Closures

Source alignment. This page documents the active fluid-property evaluation paths used by the current coupled material state. It complements the thermochemistry chapter by separating thermodynamic state functions from viscosity, conductivity and diffusion closures.

Heat capacity and sensible thermodynamics

Fluid heat capacity is evaluated as either a constant or a piecewise temperature polynomial. In polynomial range \(r\),

\[C_p^{(r)}(T)=\sum_{d=0}^{m_r}a_{r,d}T^d.\]

The evaluator uses Horner form,

\[C_p^{(r)}(T)=a_{r,0}+T\left(a_{r,1}+T\left(a_{r,2}+\cdots\right)\right),\]

and clamps temperature to the nearest endpoint outside the complete range. The polynomial enthalpy is integrated analytically,

\[h^{(r)}(T)=\sum_{d=0}^{m_r}\frac{a_{r,d}}{d+1}T^{d+1}-h_0.\]

For constant heat capacity, \(h=C_pT\). The EOS determines the corresponding \(C_v\):

\[C_v=\begin{cases}C_p-R,&\text{ideal gas},\\ C_p/\gamma,&\text{stiffened gas},\\ C_p,&\text{constant density}.\end{cases}\]

The complete EOS relations are documented in Equations of State and Caloric Models.

Dynamic viscosity

The active fluid-material state accepts constant, piecewise-polynomial, Gupta-type and Sutherland viscosity closures. A polynomial range uses

\[\mu^{(r)}(T)=\sum_{d=0}^{m_r}b_{r,d}T^d.\]

The Sutherland form is

\[\boxed{\mu(T)=\mu_0\left(\frac{T}{T_0}\right)^{3/2}\frac{T_0+S}{T+S}}.\]

The implemented Gupta-type viscosity correlation is

\[\boxed{\mu(T)=\exp(C)\,T^{A\ln T+B}}.\]

Polynomial viscosity is clamped to the nearest valid endpoint when \(T\) lies outside the total property range.

Thermal conductivity

Thermal conductivity may be constant, piecewise polynomial, kinetic-theory based, or Gupta-type. The polynomial form is

\[k^{(r)}(T)=\sum_{d=0}^{m_r}c_{r,d}T^d.\]

The kinetic-theory option used by the fluid material class is

\[\boxed{k=\mu\left(C_p+\frac54R\right)}.\]

The Gupta-type conductivity law is

\[\boxed{k(T)=\exp(C_4)\,T^{C_0(\ln T)^3+C_1(\ln T)^2+C_2\ln T+C_3}}.\]

For GE \(k\!\! -\!\!\omega\) turbulence, turbulent conductivity is added as

\[k_t=\frac{\mu_t C_p}{Pr_t},\qquad k_{eff}=k_l+k_t.\]

Species diffusion

The laminar material/mixture diffusion coefficient is augmented by turbulent species diffusion when GE turbulence is active. The turbulent diffusivity is

\[D_t=\frac{\mu_t}{Sc_t\rho},\qquad Sc_t=0.9.\]

At an internal face, the laminar and turbulent diffusivities are independently harmonically interpolated. For a generic coefficient \(D\) and \(g=V_P/(V_P+V_F)\),

\[D_f=\frac{D_PD_F}{(1-g)D_P+gD_F}.\]

The independent-species face flux is the sum of the orthogonal and non-orthogonal contributions,

\[F_{Y_i}=-\rho_fD_{eff,f}\left[\frac{A_f}{d_{PF}}(Y_{i,F}-Y_{i,P})+\frac12(\nabla Y_{i,P}+\nabla Y_{i,F})\cdot\mathbf x_{dif}\right].\]

The density multiplying the diffusive coefficient is selected from the carrier upwind side. Species enthalpy diffusion contributes to the thermal/energy row through

\[F_{h,Y}=\sum_{i=1}^{N_s-1}h_i^{upw}F_{Y_i}.\]

For the constant-density \(N_s-1\) system, the dependent-species enthalpy is subtracted in this energy coupling so that the eliminated composition constraint is respected.

Species-mixture and VOF mixture rules

For a reacting ideal-gas species mixture with mass fractions \(Y_i\),

\[C_p=\sum_iY_iC_{p,i},\qquad R_m=\sum_iY_iR_i,\qquad C_v=C_p-R_m,\qquad \gamma=\frac{C_p}{C_v}.\]

The mixture molecular weight follows

\[\frac1{W_m}=\sum_i\frac{Y_i}{W_i},\qquad R_m=\frac{R_u}{W_m}.\]

For an ideal-gas mixture,

\[\rho=\frac{p}{R_mT},\qquad a=\sqrt{\gamma R_mT}.\]

The homogeneous VOF material state is different: each constituent is evaluated at common pressure and temperature and volumetric quantities are volume-fraction weighted,

\[\rho=\sum_k\alpha_k\rho_k,\qquad \mu=\sum_k\alpha_k\mu_k,\qquad k=\sum_k\alpha_k k_k.\]

Mass-specific mixture enthalpy is obtained from the volumetric sum,

\[h_m=\frac{\sum_k\alpha_k\rho_k h_k}{\sum_k\alpha_k\rho_k}.\]

Property admissibility

Before solving, the material state validates the property method and the resulting state. The basic admissibility set includes

\[\rho>0,\qquad C_p>0,\qquad C_v>0,\qquad \mu\ge0,\qquad k\ge0,\]

together with finite EOS derivatives. Stiffened-gas materials additionally require \(\gamma>1\), \(p_\infty\ge0\) and \(p+p_\infty>0\). The property evaluator does not silently convert an unsupported generic property method into an active fluid closure.