The linearized Euler equations are derived from Euler’s equations, that is Equation 5-5 with no thermal conduction and no viscous losses. The fluid in the linearized Euler physics interface is assumed to be an ideal gas. The energy equation is also often written in terms of the pressure. This manipulation is possible using the thermodynamic differential for the entropy valid for an ideal gas. This is the common approach in literature. A review of the linearized Euler equations is found in, for example,
Ref. 12 and
Ref. 13.
The (time averaged) intensity vector I is given in the frequency domain by
In the frequency domain the Ingard-Myers equation (Ref. 1) gives an expression for the normal velocity at a boundary with a normal impedance condition. It is a so-called low-frequency approximation condition in the limit of very thin flow boundary layers (compared to the wavelength). Such conditions are used, for example, for porous lining conditions in ducts (
Ref. 2). The condition is given by:
where the surface normal n here points out of the domain and is the surface normal impedance.
If the flow is parallel to the impedance boundary condition u0·
n = 0, for example, slip flow over a mechanical impedance boundary condition (the same is true for the moving wall boundary condition described below), one can use a formulation with the tangential derivative (
) for the second term on the right-hand side:
Again these terms reduce significantly for the case where n⋅u0 = 0. If the boundary does not have curvature (planar boundary) then it is equal to zero. If the boundary is planar and the impedance condition is used inside the flow, for example at an outflow condition, then it reduces to the normal gradient of the velocity normal to the surface.
Myers’ equation (Ref. 1) gives the expressions used for a boundary condition at a moving wall. In the frequency domain it is given by
where un is the inward normal velocity is.