Governing Equations of the Convected Wave Equation
The governing equations solved in the Convected Wave Equation (CWE) interface describe the propagation of linear acoustic waves (assuming an adiabatic equation of state) in the presence of a background flow. These equation are derived by Pierce in Ref. 1 (section 8.6) and Ref. 2 and are the equations solved by this interface,
(7-2)
Pierce argues for the use of the adiabatic assumption for the acoustic processes (perturbations in the entropy, s = 0) but also for not retaining the 0th order entropy variable s0 (background mean flow entropy). The argument is that the entropy s only varies because of variations in the background fields (it is zero in a homogeneous medium). This leads to a term in the momentum equation that is second order in gradients of the background field, for example, (∂u0/∂x)2 and so forth. These terms are disregarded. This also means that the equations are not valid when these terms are large, meaning when gradients in the background fields are large.
In order to fit into the discontinuous Galerkin (dG) formulation, the governing equations need to be put on a general conservative form of the type
(7-3)
where U is the vector containing the dependent variables (p, u), da is the mass matrix of the system, Γ is the flux matrix, and S is the right-hand-side (RHS) source vector. The conservative form of Equation 7-2 is derived as follows. Start with Euler’s equations on a conservative form (omitting the RHS). For now, the dependent variables represent the full fields (not the acoustic perturbations). The continuity, momentum, and equation of state can be written
(7-4)
The equations describe the conservation of mass ρ and momentum flux ρu. Linearize these equations according to the usual scheme, using
Now, insert these into Equation 7-4 and retain only 1st order terms (the acoustic perturbations)
(7-5)
This is the conservative form of the equations implemented in the CWE interface. In the remaining of this section, the subscript 1 will be omitted from the acoustic fields. The subscript 0 is kept on the variables that represent the background mean properties.
Equation 7-5 can now be put on the form given in Equation 7-3 yielding the following components
(7-6)
with = [u, v, w]T and u0 = [u0, v0, w0]T, and the flux components are
(7-7)