Robust Spectral Solver for High-Fidelity Investigations of Aerospike Nozzle Flow Dynamics

TL;DR

This work delivers a kinetic-energy-preserving discontinuous Galerkin spectral element solver that hybridizes Entropy Viscosity (EV) with Finite Volume Subcell Element (FVSE) shock-capturing to simulate unsteady aerospike nozzle flows at high fidelity. The EV component smooths shocks while the FVSE component provides localized, robust dissipation in strongly shocked regions, with adaptive, per-element blending guided by an EV shock indicator; time integration is explicit RK4 and fluxes are designed to preserve kinetic energy. Across 2D perfectly expanded and underexpanded flows and a 3D Mach 2 case at $Re = 95{,}000$, the solver resolves unsteady wake dynamics, captures shocks, and exhibits implicit subgrid-scale dissipation consistent with ILES, while incurring modest computational overhead. These results demonstrate the method’s potential to study aerospike wake instabilities and their impact on flight performance, with stable behavior in hypersonic regimes and the ability to quantify energy transfer across resolved and subgrid scales.

Abstract

A spectral element solver is developed for the high-fidelity simulation of the unsteady flow over an aerospike nozzle. The Navier-Stokes solver is a kinetic-energy-preserving, discontinuous Galerkin spectral element method (DGSEM) combined with a hybridization of an entropy viscosity (EV) and a finite-volume subcell element (FVSE) shock-capturing scheme. The diffusive FVSE method is locally called only at locations where the EV method cannot sufficiently smooth the sharp solution gradients that suddenly appear in the supersonic, vortex-dominated jet generated by the aerospike nozzle. Two-dimensional tests of a perfectly expanded and an underexpanded nozzle flow demonstrate that the method is high-order accurate and captures unsteady flow phenomena at supersonic and hypersonic conditions. A resolved three-dimensional simulation at a Reynolds number of 95,000 shows that the solver implicitly models turbulent dissipation at the subgrid scales. To the authors' knowledge, these simulations represent the first DGSEM computations of the resolved, unsteady flow over an aerospike nozzle.

Figures (23)

• Figure 1: Schematic of the (a) open wake and (b) closed wake.
• Figure 2: A 1D spectral element (solid line) and corresponding finite volume subcell elements (dashed lines). $\Tilde{\mathbf{Q}}^N$ is the subcell average and $\Tilde{\mathbf{F}}^*$ is the interface flux.
• Figure 3: The FVSE blending coefficient $\Tilde{\alpha}$ as a function of the shock sensor $\Psi$.
• Figure 4: The computational domain of the two-dimensional aerospike test cases. The domain size is not to scale.
• Figure 5: The two-dimensional, computational grid. The domain and element sizes are not to scale.