Modeling of shock boundary layer interactions and stability analysis using particle approaches

Hypersonic flow separation and laminar shock wave boundary layer interactions (SWBLIs) have received considerable attention since these interactions can lead to laminar to turbulent transition, unsteadiness, and localized high pressure and heating regions. Accurate predictions of these phenomena, particularly when thermochemical nonequilibrium is present, play a crucial role for design purposes. In this regard, many experimental, theoretical, and numerical works have been conducted over the decades. In this work, numerical investigations of SWBLIs for hypersonic flows over a double wedge and cone and ”tick-shaped” model configurations have been conducted to investigate the origin of SWBLIs and to compare with measurements in the Hypervelocity Expansion Tube (HET), Calspan-University at Buffalo Research Center (CUBRC), and T-ADFA free-piston shock tunnel facilities using particle approaches to model the Boltzmann equation. The Boltzmann transport equation is the most general formulation of binary gas flows for a wide Knudsen number spectrum including rarefied, slip, continuum regimes. The direct simulation Monte Carlo (DSMC) method, a well-known stochastic approach to solving the Boltzmann equation, provides high-fidelity molecular transport and thermal nonequilibrium, commonly seen in strong shock-shock interactions and inherently captures rarefaction effects such as velocity slip and temperature jump without a priori specific model. Therefore, the DSMC method has been applied to SWBLIs in order to take all these effects into account. However, DSMC becomes prohibitively expensive for calculations in the continuum regime. In order to potentially reduce the computational costs related to DSMC computations for low Knudsen number flows, the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) model of the Boltzmann equation was developed and applied to shock dominated flows. The DSMC method has been used for modeling shock dominated separated hypersonic flows at Mach 7 for a unit Reynolds number of 4.15 × 105 m−1 previously studied in the HET over a double wedge configuration to investigate the impacts of thermochemical effects on SWBLIs by changing the chemical composition. The DSMC simulations are found to reproduce many of the classical features of Edney Type IV strong shock interactions. A comparison of simulated heat flux with measurements reveals that the calculated surface heating profiles were found to be time-dependent and in disagreement with experiments at later flow times, especially for the 2D wedge model. Further investigations using a three-dimensional model, taking the pressure relief into account, indicate that the simulated 3D heat fluxes, shock structure, and triple point movement were found to be in fair agreement with the experimental heat flux values, especially in the aft part of the wedge, and the shock tracking measurements. Nonetheless, both the 2D and 3D cases do not reach steady state for the duration of the experiment. To reduce 3D effects and to investigate time-dependency more closely, shock-dominated hypersonic lam- inar flows over a double cone are investigated using time-accurate DSMC combined with the residuals algorithm (RA) for unit Reynolds numbers gradually increasing from 9.35×104 to 3.74×105 m−1 at a Mach number of about 16. The main flow features, such as the strong bow-shock, location of the separation shock, the triple point, and the entire laminar separated region show a time-dependent behavior. As the Reynolds number is increased, larger pressure values in the under-expanded jet region due to strong shock interactions form more prominent λ-shocklets in the supersonic region between two contact surfaces. A Kelvin-Helmholtz instability arising at the shear layer results in an unsteady flow for the highest Reynolds number. These findings suggest that consideration of experimental measurement times is important when it comes to deter- mining the steady state surface parameters even for a relatively simple double cone geometry at moderately large Reynolds numbers. Further studies have been conducted to analyze the unsteadiness of the double cone flows using a com- bination of DSMC calculations, linear global instability analysis and momentum potential theory (MPT). Close to steady state linear analysis reveals the spatial structure of the underlying temporally stable global modes. Application of the MPT (valid for both linear and nonlinear signals) to the highest Reynolds number DSMC results shows that large acoustic and thermal potential variations exist in the vicinity of the sepa- ration shock, the λ-shock patterns, and the shear layers. It is further shown that the motion of the bow shock system is highly affected by non-uniformities in the acoustic field. At the highest Reynolds number considered here, the unsteadiness is characterized by Strouhal numbers in the shear layer and bow-shock regions. Lastly, a modal analysis with window proper orthogonal decomposition (WPOD) has been applied to hypersonic separated flows with different chemical composition over the double wedge near steady state in order to correlate POD modes with global modes, to predict future states without running computationally demanding simulations, and to eliminate statistical noise inherent to the DSMC method. Thermochemical nonequilibrium effects are found to change the shock structures, the size of the separation region, and the required time to reach steady state. The temporal analysis of POD modes shows that the decay rate of the least damped eigenmode for the chemically reacting air case is found to be smaller in comparison to the non-reacting air case. For the first time, steady state solutions for an unsteady, chemically reacting hypersonic flow are predicted using the WPOD method

Modeling of near-continuum laminar boundary layer shocks using DSMC

The hypersonic flow of nitrogen gas over a double wedge was simulated by the DSMC method using two-dimensional and three-dimensional geometries. The numerical results were compared with experiments conducted in the HET facility for a high-enthalpy pure nitrogen flow corresponding to a free stream Mach number of approximately seven. Since the conditions for the double wedge case are near-continuum and surface heat flux and size of the separation are sensitive to DSMC numerical parameters, special attention was paid to the convergence of these parameters for both geometries. At the beginning of the simulation, the separation zone was predicted to be small and the heat flux values for the 2-D model were comparable to the experimental data. However, for increasing time, it was observed that the heat flux values and shock profile strongly deviated from the experiment. Investigation of a three-dimensional model showed that the flow is truly three-dimensional and the side edge pressure relief provides good agreement between simulations and the experiment

Factors influencing flow steadiness in laminar boundary layer shock interactions

The Direct Simulation Monte Carlo method has been used to model laminar shock wave boundary interactions of hypersonic flow over a 30/55-deg double-wedge and “tick-shaped” model configurations studied in the Hypervelocity Expansion Tube facility and T-ADFA free-piston shock tunnel, respectively. The impact of thermochemical effects on these interactions by changing the chemical composition from nitrogen to air as well as argon for a stagnation enthalpy of 8.0 MJ/kg flow are investigated using the 2-D wedge model. The simulations are found to reproduce many of the classic features related to Edney Type V strong shock interactions that include the attached, oblique shock formed over the first wedge, the detached bow shock from the second wedge, the separation zone, and the separation and reattachment shocks that cause complex features such as the triple point for both cases. However, results of a reacting air flow case indicate that the size of the separation length, and the movement of the triple point toward to the leading edge is much less than the nitrogen case

Linear Instability of Shock-Dominated Laminar Hypersonic Separated Flows

The self-excited spanwise homogeneous perturbations arising in shock-wave/boundary-layer interaction (SWBLI) system formed in a hypersonic flow of molecular nitrogen over a double wedge are investigated using the kinetic Direct Simulation Monte Carlo (DSMC) method. The flow has transitional Knudsen and unit Reynolds numbers of 3.4 x 10$^{-3}$ and 5.2 x 10$^4$ m$^{-1}$, respectively. Strong thermal nonequilibrium exists downstream of the Mach 7 detached (bow) shock generated due to the upper wedge surface. A linear instability mechanism is expected to make the pre-computed 2-D base flow potentially unstable under spanwise perturbations. The specific intent is to assess the growth rates of unstable modes, the wavelength, location, and origin of spanwise periodic flow structures, and the characteristic frequencies present in this interaction.Comment: 10 pages, 6 figures. To appear in the proceedings of the IUTAM Transition 201

Implementation of a Monte Carlo method to a two-dimensional particle-in-cell solver using algebraic meshes

Particle-in-cell (PIC) technique is a widely used computational method in the simulation of low density collisionless plasma flows. In this study, a new two-dimensional (2-D) electrostatic particle-in-cell solver is developed that can be applied to non-rectangular configurations

A Monte Carlo-based Poisson's equation solver parallelized with Coarray Fortran

Poisson's equation is found in many scientific problems, such as heat transfer and electric field calculations. Although many different techniques are involved in solving Poisson's equation, we focused on the Monte Carlo method (MCM). We preferred the MCM not only because of its simple algorithm but also for its excellent parallel efficiency. Parallelization is one of the most effective techniques for reducing computation time. Among many parallelization paradigms, such as OpenMP (open multiprocessing), MPI (message passing interface), and PGAS (partitioned global address space), we adopted the PGAS-based Coarray Fortran (CAF). In this paper, we demonstrated that parallelization of Poisson's equation solver with CAF was quite painless. After parallelization, we solved Poisson's equation for a nonrectangular domain. We started with a workstation that consisted of 8 cores and we continued with a Cray supercomputer of up to 512 cores. The results of the parallel solvers were validated using exact solutions. We demonstrated that the error was less than 1.6%. Additionally, solution times and speedups of the CAF-based solver were compared with a solver that utilized MPI or OpenMP. OpenMP was not able to compete against CAF and MPI because of the "while" loop restriction. The CAF-based solver performed slightly better (7.5%) than the MPI provided that core numbers were between 2 and 32. However, CAF and MPI performed similarly for higher numbers of cores