A physics-based shock capturing method for unsteady laminar and turbulent flows

Abstract

We present a shock capturing method for unsteady laminar and turbulent flows. The proposed approach relies on physical principles to increase selected transport coefficients and resolve unstable sharp features, such as shock waves and strong thermal and shear gradients, over the smallest distance allowed by the discretization. In particular, we devise various sensors to detect when the shear viscosity, bulk viscosity and thermal conductivity of the fluid do not suffice to stabilize the numerical solution. In such cases, the transport coefficients are increased as necessary to optimally resolve these features with the available resolution. The performance of the method is illustrated through numerical simulation of external and internal flows in transonic, supersonic, and hypersonic regimes.

Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

“…In this approach, shock waves are stabilised by correcting the diffusive flux in equation ( 1) using the physicsbased approach proposed in [51]. This methodology, stemming from the work of Von Neumann and Richtmyer [132] and later considered in [35,75,76], relies on defining the diffusive flux as a combination of the physical flux G with an additional numerical contribution G * .…”

“…First, a dilatation-based shock sensor [51,87], which identifies the regions of high compression, is defined as…”

“…where Ψ denotes a smoothing operator consisting of a C 0 reconstruction, see [102], ε 0 is a user-defined positive constant and f β (s β ) = min {s max , max{s min , s β − s 0 }}. Following [51] the values ε 0 = 1.5, s 0 = 0.01, s min = 0 and s max = 2/ γ 2 − 1 and the artificial Prandtl number Pr β = 0.9 are employed in the numerical simulations of section 7.…”

“…The second alternative for the shock capturing detailed in this section consists of a discretised Laplace operator, applied in HDG discretisations [71,73] following standard approaches in the context of DG and SUPG methods [4,12,17,124]. Given the artificial viscosity ε, it relies on adding the term (∇W , ε∇U ) Ωe (51) to the left-hand side of the local equation (19c), or (24) for the Euler case. This approach is especially suited for the inviscid case, where the second-order term G vanishes and the mixed variables in (19a) and (19b) are neglected.…”

Hybridized Discontinuous Galerkin Methods for Wave Propagation

“…Third-order HDG and DIRK (3,3) schemes are used for the discretization. The details of the simulation setup are presented in [50]. Figure 9 shows 2D slices of the time-averaged (left) and instantaneous (right) pressure, temperature and Mach number fields.…”