CMPS Theory Manual › Equations of State and Caloric Models

Equations of State and Caloric Models

Implementation convention. CMPS evaluates thermodynamic material states primarily from pressure and temperature. The equations below reproduce the active ideal-gas, stiffened-gas and constant-density closures used by the current carrier and homogeneous-VOF material-state paths.

Thermodynamic state and notation

For a fluid material, the primitive thermodynamic state is described by static pressure \(p\) and temperature \(T\). Density and caloric quantities are derived from the selected equation of state (EOS):

\[\rho=\rho(p,T),\qquad h=h(T),\qquad e=e(p,T),\qquad a=a(p,T).\]

The total specific energy stored by the compressible finite-volume formulation is

\[E=e+\frac12\mathbf u\cdot\mathbf u,\qquad \rho E=\rho e+\frac12\rho\lvert\mathbf u\rvert^2.\]

For pressure-based derivatives used in the coupled Jacobian, CMPS requires the thermodynamic sensitivities

\[\rho_p=\left(\frac{\partial\rho}{\partial p}\right)_T,\qquad \rho_T=\left(\frac{\partial\rho}{\partial T}\right)_p,\qquad e_p=\left(\frac{\partial e}{\partial p}\right)_T,\qquad e_T=\left(\frac{\partial e}{\partial T}\right)_p.\]

Ideal-gas EOS

For a single ideal-gas material with molecular weight \(W\),

\[R=\frac{R_u}{W},\qquad p=\rho R T,\qquad \rho=\frac{p}{RT}.\]

The density derivatives are therefore

\[\rho_p=\frac{1}{RT}=\frac{\rho}{p},\qquad \rho_T=-\frac{p}{RT^2}=-\frac{\rho}{T}.\]

The caloric relation is evaluated through the configured heat capacity. In the ideal-gas path,

\[C_v(T)=C_p(T)-R,\qquad \gamma(T)=\frac{C_p(T)}{C_v(T)},\qquad e(T)=h(T)-RT.\]

For a calorically perfect ideal gas with constant heat capacities and a zero sensible-energy reference, this reduces to

\[h=C_pT,\qquad e=C_vT,\qquad p=(\gamma-1)\rho e.\]

The acoustic speed used by the material closure is

\[a^2=\gamma\frac{p}{\rho}=\gamma RT.\]

Stiffened-gas EOS

The stiffened-gas model introduces a pressure offset \(p_\infty\). Define the thermodynamic pressure

\[\Pi=p+p_\infty.\]

The currently validated CMPS stiffened-gas material state requires constant \(C_p\) and constant \(\gamma>1\). Its effective gas constant and constant-volume heat capacity are

\[C_v=\frac{C_p}{\gamma},\qquad R_s=C_p-C_v=C_p\frac{\gamma-1}{\gamma}.\]

The pressure-temperature form of the EOS is

\[\boxed{\rho=\frac{p+p_\infty}{R_sT}}\qquad\Longleftrightarrow\qquad p=\rho R_sT-p_\infty.\]

CMPS also exposes the equivalent density/internal-energy form

\[\boxed{p=(\gamma-1)\rho e-\gamma p_\infty}.\]

With the current constant-\(C_p\) enthalpy path,

\[h=C_pT,\qquad e=h-\frac{p}{\rho}=C_vT+\frac{p_\infty}{\rho}.\]

This identity is important for liquids modeled as weakly compressible stiffened gases: the pressure offset contributes to internal energy even when \(C_p\) and \(C_v\) are constant.

The exact density derivatives used by the material-state Jacobian are

\[\rho_p=\frac{1}{R_sT},\qquad \rho_T=-\frac{\rho}{T}.\]

Because CMPS defines \(e=h-p/\rho\), the corresponding derivatives are

\[e_p=-\frac1\rho+\frac{p\rho_p}{\rho^2},\qquad e_T=C_p+\frac{p\rho_T}{\rho^2}.\]

The stiffened-gas sound speed follows from the shifted thermodynamic pressure,

\[\boxed{a^2=\gamma\frac{p+p_\infty}{\rho}=\gamma R_sT}.\]

The admissible state must satisfy

\[T>0,\qquad C_p>0,\qquad \gamma>1,\qquad p_\infty\ge0,\qquad p+p_\infty>0.\]

Constant-density incompressible closure

For the constant-density material model, density is a material constant rather than a thermodynamic unknown:

\[\rho=\rho_0,\qquad \rho_p=0,\qquad \rho_T=0.\]

The material caloric path sets

\[C_v=C_p,\qquad e=h.\]

No finite physical acoustic speed is manufactured from this closure. CMPS therefore uses the artificial-compressibility formulation for a carrier or homogeneous mixture that is entirely constant density.

Heat-capacity and enthalpy models

For a constant heat capacity, the active material evaluator uses

\[C_p(T)=C_p,\qquad h(T)=C_pT.\]

For a polynomial range \(r\),

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

The corresponding integrated enthalpy evaluator is

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

Outside the complete tabulated temperature interval, the current property path clamps evaluation to the nearest endpoint rather than extrapolating the polynomial indefinitely.

Total-state relations

For ideal and stiffened gases the total temperature relation used by the material helper is

\[T_0=T+\frac{\lvert\mathbf u\rvert^2}{2C_p}.\]

For an ideal gas, the isentropic total-pressure relation is

\[p_0=p\left(1+\frac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}.\]

For a stiffened gas the same relation is applied to the shifted pressure:

\[\boxed{p_0+p_\infty=(p+p_\infty)\left(1+\frac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}}.\]

The entropy-like isentropic invariant used by the material class has the common form

\[S=\frac{T^{\gamma/(\gamma-1)}}{p+p_\infty}\]

for a stiffened gas, with \(p_\infty=0\) recovering the ideal-gas expression.

Primitive-to-conservative thermodynamic Jacobian

For a compact compressible carrier state \(\mathbf V=[u_1,\ldots,u_d,p,T]^T\) and \(\mathbf W=[\rho u_1,\ldots,\rho u_d,\rho,\rho E]^T\), the EOS derivatives generate the pressure and temperature columns. For momentum,

\[\frac{\partial(\rho u_i)}{\partial p}=u_i\rho_p,\qquad \frac{\partial(\rho u_i)}{\partial T}=u_i\rho_T.\]

For the total-energy density,

\[\frac{\partial(\rho E)}{\partial p}=\rho_p\left(e+\frac12\lvert\mathbf u\rvert^2\right)+\rho e_p,\]

\[\frac{\partial(\rho E)}{\partial T}=\rho_T\left(e+\frac12\lvert\mathbf u\rvert^2\right)+\rho e_T,\qquad \frac{\partial(\rho E)}{\partial u_j}=\rho u_j.\]

These relations explain why a thermodynamic model in CMPS is more than a density formula: consistent derivatives propagate directly into the fully coupled implicit block.

Embedding the EOS in homogeneous VOF

Every VOF constituent is evaluated at the common \((p,T)\) using one of the supported material closures above. Mixture density is

\[\rho=\sum_k\alpha_k\rho_k(p,T).\]

At fixed phase mass fractions, the mixture density sensitivities are

\[\rho_p^Y=\rho\sum_k\alpha_k\frac{\rho_{k,p}}{\rho_k},\qquad \rho_T^Y=\rho\sum_k\alpha_k\frac{\rho_{k,T}}{\rho_k}.\]

With \(Y_k=\alpha_k\rho_k/\rho\) and \(C_p^Y=\sum_kY_kC_{p,k}\), the frozen-composition acoustic closure is

\[\frac1{a_m^2}=\rho_p^Y+\frac{\rho_T^Y}{\rho C_p^Y}.\]

This uses the constituent EOS derivatives directly; CMPS does not obtain \(a_m\) by averaging constituent sound speeds.

Model selection and admissibility

The compressible material-state path accepts ideal-gas and stiffened-gas closures; constant-density materials use the artificial-compressibility branch when the complete active carrier/VOF composition is incompressible. The dilute Eulerian particle phase has a separate pressureless/regularized formulation and should not be interpreted as another carrier thermodynamic EOS.

Before a material state is accepted, density, heat capacities and thermodynamic derivatives are required to be finite and physically admissible. This validation is especially important for stiffened-gas states because a numerically positive static pressure alone does not guarantee \(p+p_\infty>0\).