Compressible Flow Solver Research Articles

A Numerical Investigation on the Equivalence of Shock−Wave/ Boundary-Layer Interactions using a Two Equations RANS Model

A detailed numerical investigation of two different modes of shock wave-turbulent boundary layer interaction (SWBLI) is presented. Equivalence of ramp induced SWBLI (R-SWBLI), and impingement shock based SWBLI (I-SWBLI) is explored from the computational study using an in-house developed compressible flow solver. Multiple flow deflection angles and ramp angles are employed for this study. For all the investigated cases, a freestream Mach number of 2.96 and Reynolds number of 3.47×107m−1 are considered. The k−ε model with the improved wall function of present solver predicted wall pressure distributions and separation bubble sizes very close to the experimental measurements. However, the separation bubble size is slightly over overpredicted by the k−ω model in most of the cases. The effect of overall flow deflection angle and upstream boundary layer thickness on the SWBLI phenomenon is also studied. A nearly linear variation in separation bubble size is observed with changes in overall flow deflection angle and upstream boundary layer thickness. However, the equivalence of SWBLI is noted to be independent of these two parameters. The undisturbed boundary thickness at the beginning of the interaction is identified as the most adequate scaling parameter for the length of the separated region.

Local-Maximum-and-Minimum-Preserving Solution Remapping Technique to Accelerate Flow Convergence for Discontinuous Galerkin Methods in Shape Optimization Design

In this work, a solution remapping technique is developed to accelerate the flow convergence for the intermediate shapes when a high-order discontinuous Galerkin (DG) method is employed as a compressible Euler flow solver in the airfoil design problems. Once the shape is updated, the proposed technique is applied to initialize the flow simulation for the new shape via a solution remapping formula and a maximum-and-minimum-preserving limiter. First, the solution remapping formula is used to remap the solution of the current shape into a piecewise polynomial on the mesh of the new shape. Then the piecewise polynomial is constrained with the maximum-and-minimum-preserving limiter. The modified piecewise polynomial is used as the initial value for the new shape. Numerical experiments show that the proposed technique can attractively accelerate flow convergence and significantly reduce up to 80% of the computational time in the airfoil design problems with a high-order DG solver.

Alteration in Structure of Underexpanded Freejet Through Gas-Dynamic Perspective

Studying the underexpanded jet has been essential for many fields of research because it might occur intentionally or spontaneously wherever a high-speed gas flow is possible. Therefore, an in-house compressible flow solver is employed to simulate an underexpanded sonic freejet. The behavior of the jet is examined in the near-field area, immediately downstream of the nozzle exit plane, to determine the parameters influencing the characteristics of the jet. The Mach reflection phenomenon is thoroughly studied, and a criterion is defined for the transformation from regular reflection to Mach reflection. Furthermore, the location and height of the Mach disk are found to be governed by the exit pressure ratio, the stagnation temperature, and the type of gas used. Although the temperature does not have a significant impact, the pressure ratio and specific heat ratio are found to affect the jet structure. The results obtained from the solver are compared with existing experimental results, highlighting the utility of the solver in probing the compressible flowfields.

Effect of stand-off distance and spatial resolution on the pressure impact of near-wall vapor bubble collapses

We consider the collapse behavior of cavitation bubbles near walls under high ambient pressure conditions. Generic configurations with different stand-off distances are investigated by numerical simulation using a fully compressible two-phase flow solver including phase change. The results show that the stand-off distance has significant effects on collapse dynamics, micro-jet formation, rebound, and maximum wall pressure. A relation between cavitation induced material damage and corresponding collapse mechanisms is obtained from pressure-impact data at the wall. We analyze the resolution dependence of collapse and rebound and the observed maximum pressure distributions. The comparison of the results on six different grid resolutions shows that main collapse features are already captured on the coarsest resolution, while the peak pressures are strongly resolution dependent.

Effect of vortex and entropy sources in sound generation for compressible cavity flow

The present study reports the effect of different source terms on the near and far-field acoustic characteristics of compressible flow over a rectangular cavity using hybrid computational aeroacoustics methodology. We use a low dispersive and dissipative compressible fluid flow solver in conjunction with an acoustic perturbation equation solver based on the spectral/hp element method. The hybrid approach involves calculating the base fields and the acoustic sources from a fluid simulation in the first step. In the next step, the acoustic solver utilizes the variables to predict the acoustic propagation due to the given sources. The validation of the methodology against benchmark cases provides quite accurate results while compared against the existing literature. The study is then extended to assess the importance of the entropy source term for the flow over a rectangular cavity. The predictions of hybrid simulations with vortex and entropy source terms reproduce the perturbation pressure values very close to the existing direct numerical simulation results. Moreover, the results suggest that the use of just the vortex source terms over-predicts the perturbation pressure near the source region. Finally, we have carried out detailed simulations with all the source terms to investigate the noise sources for compressible flow over the cavity for different Mach number ranges (M=0.4,0.5,0.6,0.7,1.5). The obtained acoustic spectra and the sound directivity are in close agreement with the reference experiment.

Drag and Heat Flux Reduction using Counterflow Jet and Spike - Analysis of their Equivalence for a Blunt Cone Geometry at Mach 8

This study aims to explore equivalence between active and passive flow control techniques in reducing the wave drag and surface heat flux over a blunt cone model kept in Mach 8 stream. Computational investigations were carried out by using finite volume-based compressible flow solver. Throughout the study, the solution of governing equations is sought by assuming two dimensional-axisymmetric nature of the flowfield. Both counter flow-stagnation point injection and forward facing-physical spike are considered to mitigate the excess drag and heat flux experienced by a blunt body representing the nose cone section of a hypersonic vehicle. Eventually, based on identified drag reductions, the present study proposes equivalence cases between these two methods. It is shown that a pointed spike of L/D=1 provides almost the same drag reduction as the counterflow injection jet with a pressure ratio of 8.25. Similarly, other equivalence cases are identified and the physics behind them is explored. The identified equivalence is expected to help the designers in effectively replacing one technique with another according to the requirement. Equivalence matrix is presented for different spike cases in terms of injection ratios of counterflow injection.

One-Sided GRP Solver and Numerical Boundary Conditions for Compressible Fluid Flows

One-Sided GRP Solver and Numerical Boundary Conditions for Compressible Fluid Flows

Comparison of impulsive wave forces on a semi-submerged platform deck, with and without columns and considering air compressibility effects, under regular wave actions

The decks of oil-extracting platforms could be damaged by great wave impact loads under harsh ocean conditions. Our understanding of the pressure distribution characteristics under the deck, especially those influenced by columns, is incomplete. A series of experiments are carried out to study the spatial and instantaneous distribution of the impact pressure generated by regular waves acting on the deck, and two cases are considered: a flat plate (FP for short) and a plate with four vertical columns (CP for short). Three-dimensional numerical simulations using a modified wave generating tool are conducted and compared to experimental data. Both compressible and incompressible flow solvers are applied to further quantitatively analyse the effect of air compressibility on impacting forces. The results show marked changes in both the pressure distribution scatter under the deck and the magnitude of the impact pressure when comparing the two cases. It has been demonstrated that run-up along the columns can cause intense localized pressure on the deck, and a close relationship exists between this increased pressure and fluid velocity. Additionally, the preliminary results showed that the phase compressibility increases the peak pressure compared with that of an incompressible solver, and the former matches the experimental measurements better.

Numerical simulation of flashing using a pressure-based compressible multiphase approach and a thermodynamic cavitation model

A recently published pressure-based compressible multiphase flow model Labois and Narayanan (2017) is validated for applications relevant to safety of pressurized systems, such as flashing of high-pressure water through valves and nozzles. The pressure-based compressible multiphase flow solver is based on non-conservative discretization of the mixture continuity equation and has the advantage of being applicable to interface tracking and n-phase mixture formulations. For simulation of flashing, two well-known cavitation models Singhal et al.(2002), Yuan et al.(2001) have been implemented along with thermodynamic effects such as latent heat of phase change, variable saturation pressure, and heat capacities. The limitations of the well-established cavitation models in terms of their applicability to compressible flows have been clarified. The model was applied to the Super Moby Dick experiment Rousseau (1987). Good flow rates, and pressure and void fraction variations were obtained for three chosen conditions by tuning the cavitation model under incompressible conditions. It was found that the void fraction stops increasing beyond the throat due to thermodynamic effects where the latent heat and heat capacity are functions of pressure and temperature. The reduction of latent heat and rapid changes in liquid and vapour heat capacities close to the saturation line at high pressures was shown to be important. The accuracy, robustness, and efficiency of the pressure-based method has been proven for flashing under strong depressurization conditions. The cavitation model predictions are strongly sensitive to model parameters such as the nucleate density making the models not general enough to be applied to different problems without calibration. It was shown that there is significant room for development of improved cavitation models especially focussing on the constant number density constraint due to the strong sensitivity of the current models to this input parameter. The assumption of incompressibility and homogeneity of the phasic velocities is also identified as serious limitations requiring further study.

Novel Expression-Template-Based Automatic Differentiation of Fortran Codes for Aerodynamic Optimization

An elegant and robust approach for discrete adjoint-based automatic differentiation of the Fortran codes is proposed in this work. The technique aims at bridging the gap within the Fortran programming language that currently lacks certain metaprogramming paradigms that are readily available at compile time in C/C++ codes. More specifically, the present work uses an expression-based tape approach, which is the first-of-its-kind implementation in Fortran programming language, that can significantly reduce the memory footprint while improving the computational efficiency of the adjoint-based automatic differentiation (AD). The proposed expression-template-based approach is incorporated in our in-house AD toolbox, which currently is the only Fortran-based tool in the literature that uses a fixed-point-type operator-overloading adjoint sensitivity analysis. The improved version of the fast automatic differentiation using operator-overloading technique toolbox is then coupled with the in-house unstructured parallel compressible (UNPAC) flow solver for a robust design optimization framework (DOF), called UNPAC-DOF. The efficiency and robustness of the proposed technique and the resulting framework are tested for aerodynamic shape optimization problems applied to airfoil and wing geometries.

Meshfree lattice Boltzmann flux solver for compressible inviscid flows

AbstractIn this work, a meshfree Lattice Boltzmann Flux Solver (LBFS) is proposed to resolve compressible flow problems based on scattered points without mesh connections. The new method employs the Least Square‐based Finite Difference (LSFD) scheme to discretize the governing equations. In order to simulate discontinuous problems such as shock wave, the mid‐point between two adjacent nodes is regarded as a discontinuous interface over which the Riemann problem is established. The local fluxes at this interface point are reconstructed by LBFS using the local solution of the Lattice Boltzmann Equation (LBE) as well as its correlations to macroscopic variables and moment relations. The LBFS is constructed based on the non‐free parameter D1Q4 model: the normal component of the particle velocity on the interface is retained, while the tangential component is reconstructed by the macroscopic variables on both sides of the interface. The meshfree LBFS expects some intriguing merits. On one hand, it inherits the physical robustness of the LBFS: the local fluxes are reconstructed from the physical solutions instead of mathematical interpolations. On the other hand, it allows the implementation at arbitrarily distributed nodes, which credits to the flexibility of the method. Representative examples of compressible flows, including Sod shock tube, Osher‐Shu shock tube, flow around NACA0012 airfoil, flow around staggered NACA0012 biplane configuration and shock reflection problem, are simulated by the proposed method for comprehensive evaluation of the meshfree LBFS.

Code Verification of a Pressure-Based Solver for Subsonic Compressible Flows

Abstract Although most flows in maritime applications can be modeled as incompressible, for certain phenomena like sloshing, slamming, and cavitation, this approximation falls short. For these events, it is necessary to consider compressibility effects. This paper presents the first step toward a solver for multiphase compressible flows: a single-phase compressible flow solver for perfect gases. The main purpose of this work is code verification of the solver using the method of manufactured solutions. For the sake of completeness, the governing equations are described in detail including the changes to the SIMPLE algorithm used in the incompressible flow solver to ensure mass conservation and pressure–velocity–density coupling. A manufactured solution for laminar subsonic flow was therefore designed. With properly defined boundary conditions, the observed order of grid convergence matches the formal order, so it can be concluded that the flow solver is free of coding mistakes, to the extent tested by the method of manufactured solutions. The performance of the pressure-based SIMPLE solver is quantified by reporting iteration counts for all grids. Furthermore, the use of pressure–weighted interpolation (PWI), also known as Rhie–Chow interpolation, to avoid spurious pressure oscillations in incompressible flow, though not strictly necessary for compressible flow, does show some benefits in the low Mach number range.

A Comparison of Interface Growth Models Applied to Rayleigh–Taylor and Richtmyer–Meshkov Instabilities

Abstract Sophisticated numerical models that contain fluid interfaces rely upon interface evolution models to approximate the transition to turbulence near interfaces, in the presence of Rayleigh–Taylor (RTI) or Richtmyer–Meshkov (RMI) instability. Semi-analytical models have been developed in recent decades to predict the interface growth from an initial state into the nonlinear regime. Two of these models are considered in this study. They are the Goncharov and the z-models. Both of these interface models have strengths and weaknesses, which are examined here. Both of them have been implemented in the xRAGE compressible flow solver, which models fluid interfaces. The flow solver provides the fluids acceleration as a body force to the interface model. The purpose of such interface model is to evolve the early times of interface position as a subgrid model within a compressible flow simulation in order to then initialize a turbulence model. In this work, the interface models are assessed and compared for their evolution of RTI and RMI. The z-model performed better than the Goncharov model for all cases that were explored.

Two-dimensional laboratory-scale DNS for knocking experiment using n-heptane at engine-like condition

A two-dimensional laboratory-scale DNS of a knocking experiment using a stoichiometric n-C7H16/O2/Ar mixture at engine-like condition with the latest reduced SIP isooctane kinetics was conducted. A compressible reactive flow solver PeleC combined with an in-house efficient chemical kinetics solver MACKS was used. The results of DNS were compared with a “knocking experiment by Kyushu University” conducted in a 14 × 14 × 80 mm rectangular constant volume chamber for the same mixture at an initial temperature of 480 K and pressure of 0.33 MPa. The comparisons showed the present DNS with the latest, validated reduced chemical kinetics successfully reproduced the knock onset timing within an error of 2% as well as the overall features of the experimental observations seen in the Schlieren images such as flame shape transitions, pressure history, and the characteristic faint increase in pressure due to the flame front attachment with the chamber wall. In addition, the present DNS captured that the cool flame ignition occurred homogeneously but accompanied by the rapid formation of density gradients vertical to the side walls in the unburned gas region.

Eulerian-Lagrangian modelling of detonative combustion in two-phase gas-droplet mixtures with OpenFOAM: Validations and verifications

A hybrid Eulerian-Lagrangian solver RYrhoCentralFoam is developed based on OpenFOAM® to simulate detonative combustion in two-phase gas-liquid mixtures. For Eulerian gas phase, RYrhoCentralFoam enjoys second order of accuracy in time and space discretizations and is based on finite volume method on polyhedral cells. The following developments are made based on the standard compressible flow solver rhoCentralFoam in OpenFOAM®: (1) multi-component species transport, (2) detailed fuel chemistry for gas phase combustion, and (3) Lagrangian solver for gas-droplet two-phase flows and sub-models for liquid droplets. To extensively verify and validate the developments and implementations of the solver and models, a series of benchmark cases are studied, including non-reacting multi-component gaseous flows, purely gaseous detonations, and two-phase gas-droplet mixtures. The results show that the RYrhoCentralFoam solver can accurately predict the flow discontinuities (e.g. shock wave and expansion wave), molecular diffusion, auto-ignition and shock-induced ignition. Also, the RYrhoCentralFoam solver can accurately simulate gaseous detonation propagation for different fuels (e.g. hydrogen and methane), about propagation speed, detonation frontal structures and cell size. Sub-models related to the droplet phase are verified and/or validated against analytical and experimental data. It is also found that the RYrhoCentralFoam solver is able to capture the main quantities and features of the gas-droplet two-phase detonations, including detonation propagation speed, interphase interactions and detonation frontal structures. As our future work, RYrhoCentralFoam solver can also be extended for simulating two-phase detonations in dense droplet sprays.

Transported PDF modeling of compressible turbulent reactive flows by using the Eulerian Monte Carlo fields method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

A local correlation-based zero-equation transition model

In this work, the local correlation-based one-equation transition model (Menter, F.R., Smirnov, P.E., Liu, T. and Avancha, R., A one-equation local correlation-based transition model. Flow, Turbulence and Combustion, vol. 95, no. 4, pp. 583–619, 2015.) is transformed into a zero-equation transition model. The new model provides an attractive choice in terms of quick implementation of a transition model in existing turbulent flow solvers with Menter’s shear-stress transport (SST) turbulence model, as it only introduces three extra source terms in the transport equation of turbulent kinetic energy. The model is validated against a set of benchmark flat-plate test cases: T3 series and SK, and also subsonic flows past two different airfoils: Aerospatiale A-airfoil (Re=2.1 million) and E387 (Re=0.2 million), and finally applied to a transonic flow over 3D DLR-F5 wing (Re=1.5 million). Results show that the proposed model produces similar transition prediction as the one-equation transition model, with a reduced computational effort. The computations are performed with an in-house finite-volume solver for compressible turbulent flows on block-structured grids.

CT-based simulation of ultrasound propagation and cavitation in the human body for the assessment of acoustic therapy monitoring in high intensity focused ultrasound

In situ control of therapy parameters based on real time monitoring of ablation targets is a promising strategy to improve the safety and efficacy of extracorporeal high-intensity focused ultrasound (HIFU). In this work, we present an approach to evaluate the accuracy of acoustic monitoring techniques using computational fluid dynamics simulation. In the numerical setup, computed tomography images are used to model anatomical structures that can scatter and dissipate therapy waves in the body of the patient, including fat, bone, tissues, and organs. An in-house, compressible flow solver is used to simulate ultrasound generation from a transducer, nonlinear focusing of the wave toward therapy targets in the body, and backscattering toward the body surface. The simulation also captures excitation of cavitation bubbles in the focal region and their influence on the pressure fields in the body. Sensing and processing of the scattered signals provide for acoustic monitoring. For demonstration, we apply this approach to HIFU-based lithotripsy that uses burst waves with a focal pressure of O(1) MPa and frequency of O(100) kHz. Accuracies of acoustic monitoring are compared for various sensing and signal processing methods in quantifying the effects of anatomical structures and cavitation on ultrasound irradiation into targeted kidney stones. Finally, we discuss the use of these numerical experiments for control and optimization of therapy parameters. [Work supported by NIH P01-DK043881.]

Development of accurate and robust genuinely two-dimensional HLL-type Riemann solver for compressible flows

When using the conventional direction splitting method to calculate multidimensional high speed gasdynamical flows, Riemann solvers capable of resolving contact surface and shear wave accurately will suffer from different forms of shock instability, such as the notorious carbuncle phenomenon. The stability analysis shows that the lack of dissipation in the direction transverse to the shock front leads to the shock instability of low-dissipation HLLEM solver. To overcome this defect, an accurate and carbuncle-free genuinely two-dimensional HLL-type Riemann solver is proposed. Using Zha-Bilgen splitting method, the flux vector of two-dimensional Euler equations is split into the convective flux and pressure flux. An algorithm similar to AUSM+ scheme is adopted to calculate the convective flux and the pressure flux is calculated by the low-dissipation HLLEM scheme. Following Balsara’s idea, the genuinely two-dimensional properties of the new solver are achieved by solving the two-dimensional Riemann problem that considers transversal features of the flow at each vertex of the cell interface. Numerical results of benchmark tests demonstrate that the new solver has higher resolution and better robustness than the conventional HLLEM solver implemented in dimension by dimension.

Hybrid scheme for complex flows on staggered grids and application to multiphase flows

A hybrid scheme is developed that enables advanced simulations, such as multiphase flows, in complex flow setups with compressible flow solvers using a staggered arrangement of flow variables. For this purpose, a weighted essentially non-oscillatory (WENO) scheme is adapted for staggered grids and combined with a finite difference central scheme. Hybrid weights that maintain continuity in time are computed based on a limiter constructed from the deviation of the local weights from the optimal weights. A total variation diminishing (TVD) interpolation is introduced to avoid numerical oscillations from the flow field. Additionally, the finite difference operators are reformulated in a flux-based finite volume form for convective terms in order to avoid spurious oscillations for turbulent configurations. For demonstrating the accuracy of the newly developed scheme and its improvements compared to the existing schemes, a modified wavenumber analysis is shown. Furthermore, the new scheme is applied to single and multiphase test cases. Finally, it is used for simulating the needle-opening process of a gasoline direct injector (GDI) and predicting the amount of gas inside the nozzle.