Solution of low Mach number aeroacoustic flows using a Variational Multi-Scale finite element formulation of the compressible Navier–Stokes equations written in primitive variables

Abstract

In this work we solve the compressible Navier-Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic flows. We develop a Variational Multi-Scale formulation to stabilize the finite element discretization by including the orthogonal, dynamic and non-linear subscales, together with an implicit scheme for advancing in time. Three additional features define the proposed numerical scheme: the splitting of the pressure and temperature variables into a relative and a reference part, the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility, and the approximation of the dynamic stabilization matrix for the time dependent subscales. We also include a weak imposition of implicit non-reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes. The order of accuracy of the method is verified for two-and three-dimensional linear and quadratic elements using steady manufactured solutions. Several benchmark flow problems are studied, including transient examples and aeroacoustic applications. c

Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations

“…In this section, the finite element formulation of the compressible problem is described. The VMS formulation of the compressible problem that has been presented in [26,29] is recalled in the section. As it will be explained, the VMS framework is used for stabilizing the finite element approximation and allows us to use arbitrary interpolation spaces for the different variables of the problem.…”

“…, such that an approximation of the nonlinear operator applied to the subscales is made in each element. The definition for the diagonal matrix of stabilization parameters arises from the perspective of a Fourier analysis, as demonstrated in [26,29]. These components are defined for each formulation as…”

“…We also define the kinematic viscosity as ν = µ/ρ, and the thermal diffusivity as α = λ/ρc p . For the primitive variables formulation, we follow the definition in [29] for the matrix of stabilization parameters, in which u is a modified velocity that is calculated with the Gauss error function, u = |u| + erf(φ)c, where φ is a normalized compressibility that is defined as φ = 2 − 2( − M )/ . In the previous expression is a parameter that determines a certain transition from the compressible to the incompressible regime, which we assume as = 0.1 in the numerical examples presented in this work.…”

“…In the case of the primitive variables formulation, we avoid the off-diagonal terms that appear in matrix A 0 U n+1 by approximating (36) as described in [29]. Hence, for the primitive variables formulation we take the following diagonal definition of the dynamic operator:…”

“…The refined meshes driven by the subscales-based error and calculated with the entropy measure are plotted at the right side of the figure. It has been reported in [29] that the resolution of flow boundary layers and small perturbations of temperature (producing buoyancy) is enhanced with the inclusion of the dynamic subscales. We also observe that the error distribution, which is below the prescribed tolerance, and the mesh refinement are enhanced mostly at the lateral walls: the subscales-based error estimator is able to describe the boundary layer generated by the buoyancy of the flow.…”