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:

Подпись: (12)e = ^, s)

Подпись: de Подпись: Tds — pd Подпись: (13)

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

Constitutive equations Constitutive equations Подпись: ds . p Constitutive equations

we get the equations of state:

Подпись: c Подпись: Y dp [d~p Подпись: s Подпись: (17)

The speed of sound c is defined by:

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

Подпись: (18)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

Подпись: and Constitutive equations Constitutive equations Constitutive equations

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:

Подпись: i Подпись: . P e + - . P Подпись: (23)

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:

Подпись: (24)q = —KVT,

Подпись: Tij Подпись: pSij Рц Подпись: (25)

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:

Подпись: with Подпись: Dij Подпись: 1 ( dvi dvj 2 у dxj + дх,. Подпись: (27)

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.

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