Constitutive equations

The conservation laws are complemented by empirical constitutive equa­tions. For simplicity we assume that the fluid is locally in a state close to thermodynamic equilibrium, so that we can express the internal energy in terms of two other state variables:

e = ^, s)

where s is the entropy per unit of mass. Using the thermodynamic equation:

we get the equations of state:

The speed of sound c is defined by:

In most applications we will consider an ideal gas for which:

de = cv dT

with cv the specific heat capacity at constant volume. For an ideal gas this is a function of the temperature only. This further implies:

and

c =. [YP (20)

V P

with R = cp — cv the specific gas constant, 7 = cp/cv the Poisson ratio and cp is the specific heat capacity at constant pressure. By definition:

where the specific enthalpy is defined by:

Assuming local thermodynamic equilibrium, fluxes are linear functions of the flow variables. For the heat flux we use the law of Fourier:

q = —KVT,

where K is the heat conductivity. The viscous stress tensor is defined by:

with Sij the Kronecker delta, equal to unity for i = j and otherwise zero. The viscous stress tensor is described for a so-called Newtonian fluid in terms of the dynamic viscosity n and the bulk viscosity ц:

Tij 2n (Dij 3Dkk &ij^ + PDkk$ij (26)

1.2 Boundary conditions

The boundary conditions corresponding to the continuum assumption and the local thermodynamic equilibrium are, for a solid impermeable wall with velocity vw and temperature Tw : v = vw and T = Tw.