Compressible Flow Problems Research Articles

The Matrix‐Free Macro‐Element Hybridized Discontinuous Galerkin Method for Steady and Unsteady Compressible Flows

ABSTRACTThe macro‐element variant of the hybridized discontinuous Galerkin (HDG) method combines advantages of continuous and discontinuous finite element discretization. In this paper, we investigate the performance of the macro‐element HDG method for the analysis of compressible flow problems at moderate Reynolds numbers. To efficiently handle the corresponding large systems of equations, we explore several strategies at the solver level. On the one hand, we utilize a second‐layer static condensation approach that reduces the size of the local system matrix in each macro‐element and hence the factorization time of the local solver. On the other hand, we employ a multi‐level preconditioner based on the FGMRES solver for the global system that integrates well within a matrix‐free implementation. In addition, we integrate a standard diagonally implicit Runge–Kutta scheme for time integration. We test the matrix‐free macro‐element HDG method for compressible flow benchmarks, including Couette flow, flow past a sphere, and the Taylor–Green vortex. Our results show that unlike standard HDG, the macro‐element HDG method can operate efficiently for moderate polynomial degrees, as the local computational load can be flexibly increased via mesh refinement within a macro‐element. Our results also show that due to the balance of local and global operations, the reduction in degrees of freedom, and the reduction of the global problem size and the number of iterations for its solution, the macro‐element HDG method can be a competitive option for the analysis of compressible flow problems.

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Smoothed particle hydrodynamics (SPH) has attracted significant attention in recent decades, and exhibits special advantages in modeling complex flows with multiphysics processes and complex phenomena. Its accuracy depends heavily on the distribution of particles, and will generally be lower if the particles are distributed non-uniformly. A high-order SPH scheme is proposed in the present work for simulating both compressible and incompressible flows. It uses a Gaussian quadrature rule to perform the particle approximation of SPH by introducing Gaussian nodes. Unfortunately, the Gaussian nodes hardly overlap with SPH particles due to the Lagrangian feature, and thus we use a high-order interpolation method to obtain the corresponding physical quantities at the Gaussian nodes. The accuracy and robustness of the proposed Gaussian SPH are demonstrated by several numerical tests, including the Sod problem, Poiseuille flow, Couette flow, cavity flow, Taylor–Green vortex and dam break flow, and a convergence analysis is also conducted to evaluate the effects of particle resolution and distribution for reconstructing a given function. The simulation results for each test case are in good agreements with the available analytical, experimental or numerical results, showing that the proposed Gaussian SPH method is accurate and reliable but expensive for simulating compressible and incompressible flow problems.

On immiscibility preservation conditions of material interfaces in the generic five-equation model

Interfaces separating pure materials and mixtures tend to be severely smeared with interface-capturing methods for compressible multi-material flows, necessitating the requirement of various interface-sharpening techniques. However, these techniques have various problems related to consistency, conservation, and thermodynamic compatibility. In this work, we derive a general theoretical formulation of interface-sharpening techniques for the generic five-equation model. This theoretical formulation is not only conservative in mass, momentum, and total energy but also asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed, and is independent of various specific numerical algorithms. We further propose a general numerical method to solve this theoretical formulation. The proposed method is consistent and conservative, and it prevents spurious errors at the interfaces. Examples of one- and two-dimensional multimaterial compressible flow problems, including shocks and interfaces, are considered to verify the analysis and demonstrate the efficiency of the method.

Implementation and Validation of an Original OpenFOAM Code for Fluid–Structure Interaction Problems in Compressible Flow

Implementation and Validation of an Original OpenFOAM Code for Fluid–Structure Interaction Problems in Compressible Flow

A new re-redistribution scheme for weighted state redistribution with adaptive mesh refinement

State redistribution (SRD) is a recently developed technique for stabilizing cut cells that result from finite-volume embedded boundary methods. SRD has been successfully applied to a variety of compressible and incompressible flow problems. When used in conjunction with adaptive mesh refinement (AMR), additional steps are needed to preserve the accuracy and conservation properties of the solution if the embedded boundary is not restricted to a single level of the mesh hierarchy. In this work, we extend the weighted state redistribution algorithm to cases where cut cells live at or near a coarse-fine interface within the domain. We present numerical results that demonstrate that the algorithm is conservative when the coarse-fine interface intersects the embedded boundary. Additionally we compare the numerical solution of the Sod shock tube problem in an inclined cylinder with the analytic solution, and we compare the simulation of a shock hitting a cylindrical obstacle with experimental data. Finally we demonstrate the methodology for simulation of the multicomponent compressible Navier-Stokes equations in a piston-bowl geometry, and discuss the computational efficiency gained by not requiring the entire embedded boundary to be defined at the finest level.

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

Fluid–structure interaction modeling with nonmatching interface discretizations for compressible flow problems: simulating aircraft tail buffeting

Fluid–structure interaction modeling with nonmatching interface discretizations for compressible flow problems: simulating aircraft tail buffeting

A New Fifth-Order Weighted Compact Nonlinear Scheme with Multi-Order Candidates Weighting for Hyperbolic Conservation Laws

ABSTRACT Based on one 5-point and two 2-point stencils, a nonlinear weighted scheme adaptive to the weighting of multi-order interpolation candidates for shock capturing is proposed. The JS weight function is adopted. The smoothness indicator of the 5-point stencil is specially designed with a higher-order leading term similar to the reference global smoothness indicator τ in the Z weighting. The design maintains that the nonlinear weights satisfy the sufficient condition for the scheme to avoid degradation at extreme points. The form of the weighting is also simplified, obviously reducing the weighting procedure's computational costs. Inviscid and viscous compressible flow problems are simulated to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show no obvious oscillations near the discontinuity, and the resolution of both discontinuities and smooth regions is better than JS, Z, and ZQ weighting.

A parameter-free staggered-grid Lagrangian scheme for two-dimensional compressible flow problems

The present study aims to develop a parameter-free staggered-grid Lagrangian scheme that avoids the empirical parameters needed for artificial viscosity and anti-hourglass force. The artificial viscosity is equal to the pressure jump, which is constructed by the relationship between pressure and velocity. The anti-hourglass force is established using pressure compensation, which is the difference in pressure between sub-cells belonging to the same primary cell. The scheme maintains the conservation of total mass, momentum, and energy. Numerical experiments show that the scheme is highly robust for extreme flow problems with different fine meshes such as Sedov, Noh, Saltzmann triple point and Rayleigh–Taylor instability. The robust parameter-free scheme is well suited for multi-physics problems and engineering applications.

Development of an implicit high-order Flux Reconstruction solver for high-speed flows on simplex elements

In this study, a significant emphasis is placed on the extension of high-order Flux Reconstruction (FR) schemes to cater to high-speed flows, specifically those involving high Mach numbers. The paper delves into the development and implementation of these FR schemes on both straight and curved-edged 2D and 3D simplex elements within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The proposed FR solver provides more accurate detection of complex flow features over relatively coarser meshes when compared to their low-order peers (particularly in presence of shock waves), is fully implicit and able to simulate compressible flow problems governed by the Euler or Navier-Stokes equations on triangular and tetrahedral meshes. In order to simulate cases with shocks, a shock-capturing scheme previously developed for quadrilateral elements, was extended to tackle supersonic and hypersonic simulations. Extensive verification of the resulting FR solver (up to 7th order, a.k.a P6, for the solution polynomial and 3rd order a.k.a Q2 for the geometrical representation) has been performed on benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 9.6. The obtained results have been favorably compared to those available in the literature.

An adaptive, training‐free reduced‐order model for convection‐dominated problems based on hybrid snapshots

AbstractThe vast majority of reduced‐order models (ROMs) first obtain a low dimensional representation of the problem from high‐dimensional model (HDM) training data which is afterwards used to obtain a system of reduced complexity. Unfortunately, convection‐dominated problems generally have a slowly decaying Kolmogorov ‐width, which makes obtaining an accurate ROM built solely from training data very challenging. The accuracy of a ROM can be improved through enrichment with HDM solutions; however, due to the large computational expense of HDM evaluations for complex problems, they can only be used parsimoniously to obtain relevant computational savings. In this work, we exploit the local spatial coherence often exhibited by these problems to derive an accurate, cost‐efficient approach that repeatedly combines HDM and ROM evaluations without a separate training phase. Our approach obtains solutions at a given time step by either fully solving the HDM or by combining partial HDM and ROM solves. A dynamic sampling procedure identifies regions that require the HDM solution for global accuracy and the reminder of the flow is reconstructed using the ROM. Moreover, solutions combining both HDM and ROM solves use spatial filtering to eliminate potential spurious oscillations that may develop. We test the proposed method on inviscid compressible flow problems and demonstrate speedups up to a factor of five.

Efficient solution of bimaterial Riemann problems for compressible multi-material flow simulations

When solving compressible multi-material flow problems, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular idea in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. The work presented in this paper aims to accelerate the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed, including (1) a change of integration variable through rarefaction fans, (2) storing and reusing integration trajectory data, (3) step size adaptation, and (4) constructing an R-tree on the fly to generate initial guesses. The performance of these acceleration methods is assessed using four example problems in underwater explosion, laser-induced cavitation, and hypervelocity impact. These problems exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the acceleration methods are able to significantly reduce the computational cost without affecting solver robustness or solution accuracy. In different cases, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times.

An improved hybridization strategy for the fifth-order unequal-sized weighted essentially non-oscillatory scheme

In this paper, a hybrid unequal-sized weighted essentially non-oscillatory (US-WENO) scheme is developed to reduce the computational cost caused by the computation of nonlinear weights and local characteristic decompositions in its procedures. First, an improved discontinuous sensor is designed based on the mathematical properties of the fourth-degree reconstruction polynomial in the US-WENO scheme. The proposed hybridization strategy can automatically and efficiently identify the troubled cells, and does not contain artificial parameters related to the problems. Second, the efficient linear scheme and the existing US-WENO scheme are combined to simulate the inviscid and viscous compressible flow problems. Finally, some benchmark numerical examples are used to verify the performance of the presented hybrid WENO scheme in various aspects, such as computational efficiency, low dissipation characteristics, shock capture ability, etc. The numerical experiments show that the proposed hybrid method can inherit all the features of the existing US-WENO scheme while improving its computational efficiency.

Development of three-dimensional rotated lattice Boltzmann flux solver for the simulation of high-speed compressible flows

A three-dimensional (3D) Rotated Lattice Boltzmann Flux Solver (RLBFS) is proposed and analyzed with matrix-based linear stability theory for simulating compressible flows in a wide range of Mach numbers. The 3D RLBFS applies the finite volume method to discrete the Navier-Stokes equations and evaluates its fluxes at each cell interface. To improve numerical stability, the convective fluxes are obtained in a hybrid way by using the D1Q4 lattice Boltzmann model in two perpendicular directions decomposed by the outer normal vector of each cell interface. The viscous fluxes are computed in a conventional way using the second-order central scheme. The stability performance and order accuracy of the proposed 3D RLBFS are examined by using the matrix-based stability theory and L2 errors on different meshes respectively. It is shown that the 3D RLBFS is stable even at high Mach number and has the second-order accuracy in space. The reliability and capability of the proposed method is further evaluated by simulating several challenging compressible flow problems, including subsonic flow over the DLR-F4, transonic flow over the ONERA M6 wing, supersonic flow around NACA0012 wing, hypersonic flow over the Viking lander capsule, hypersonic flow over a hemisphere and hypersonic flow around a blunt nose double cone. The obtained numerical results are in good agreement with experimental and/or numerical data published in the literature, indicating that the present method provides a reliable and effective tool for simulating practical three-dimensional compressible flow problems in aeronautics and astronautics.

An improved discontinuity sensor for high-order weighted essentially non-oscillatory scheme on triangular meshes

In this paper, a new type of hybrid weighted essentially non-oscillatory (WENO) schemes with third-order and fourth-order accuracy is developed to simulate compressible flow problems on unstructured meshes. First, an improved discontinuity sensor is designed on unstructured meshes, which can automatically and accurately identify the troubled cells without manually adjusting the parameters related to the problem. Second, a hybrid unequal-sized WENO (US-WENO) scheme with different accuracy is developed based on the new hybridization strategy and the existing US-WENO scheme to reduce the computational overhead caused by the computation of nonlinear weights and local characteristic decompositions in WENO procedures. Finally, some benchmark examples are employed to verify the performance of the proposed method in terms of numerical dissipation, shock capture ability, and computational efficiency. Numerical experiments show that the proposed hybrid US-WENO schemes can inherit the excellent characteristics of the original US-WENO schemes while improving computational efficiency.

Verification of a High-Order FEM-based CFD Code using the Method of Manufactured Solutions

A high-order computational fluid dynamics (CFD) code capable of solving compressible turbulent flow problems was developed. The CFD code employs the Flowfield Dependent Variation (FDV) scheme implemented in a Finite Element Method (FEM) framework. The FDV scheme is basically derived from the Lax-Wendroff Scheme (LWS) involving the replacement of LWS’s explicit time derivatives with a weighted combination of explicit and implicit time derivatives. The code utilizes linear, quadratic and cubic isoparametric quadrilateral and hexahedral Lagrange finite elements with corresponding piecewise shape functions that have formal spatial accuracy of second-order, third-order and fourth-order, respectively. In this paper, the results of observed order-of-accuracy of the implemented FDV FEM-based CFD code involving grid and polynomial order refinements on uniform Cartesian grids are reported. The Method of Manufactured Solutions (MMS) is applied to governing 2-D Euler and Navier-Stokes equations for flow cases spanning both subsonic and supersonic flow regimes. Global discretization error analyses using discrete 𝐿2 norm show that the spatial order-of-accuracy of the FDV FEM-based CFD code converges to the shape function polynomial order plus one, in excellent agreement with theory. Uniquely, this procedure establishes the wider applicability of MMS in verifying the spatial accuracy of a high-order CFD code.

A consistent and conservative Phase-Field method for compressible multiphase flows with shocks

A key challenge with interface-capturing methods for compressible multiphase flows is that the numerical diffusion of upwind schemes causes material interfaces to continuously diffuse over time, possibly in a non-uniform fashion. In the present study, a consistent and conservative Phase-Field model for compressible multiphase flows is derived to control mixing between different fluids at material interfaces due to numerical diffusion. The model is general as it admits an arbitrary number of phases and has no prior assumption of the formulation of the Phase-Field mechanism. Necessary physical constraints to formulate the Phase-Field mechanism are obtained by analyzing physical principles, including the second law of thermodynamics. To numerically solve the model, two consistency requirements and a general numerical strategy, called reduction consistent formulation, are proposed. The proposed approach is consistent, conservative, bound/positivity preserving for the volume fraction/mass, and prevents spurious errors at interfaces. Various two-phase compressible flow problems including shocks and interfaces are performed to verify the analysis and demonstrate the efficacy of the method. Finally, temperature equilibrium and the incompressible limit are also discussed.

Generic five-equation model for compressible multi-material flows and its corresponding high-fidelity numerical algorithms

The five-equation model is widely used to simulate compressible multi-material flows. However, disunity and anomalous phenomena exist in both the model and corresponding numerical simulations. In this study, we first derived a unified formulation for the five-equation model, establishing a generic five-equation model. This model can not only recover existing typical five-equation models and generate new models but also provide a flexible framework for numerically simulating compressible multi-material flow problems in a consistent manner. Based on these properties, we constructed a suite of high-fidelity numerical algorithms, including a consistent algorithm for equations of volume fractions, technique for avoiding potential Mach number oscillation, and technique for restoring the non-monotonic sound speed if it is indeed physically possible. The proposed model and its corresponding high-fidelity numerical algorithms are verified on one-, two-, and three-dimensional problems, including artificial mixtures for separating pure or nearly pure fluids and physical fluid mixtures, and materials modeled as an ideal gas, stiffened gas, and Mie-Grüneisen-type equation of state. The numerical results validate our conclusions and confirm the broad applicability of the proposed model and its corresponding algorithm.

High-order implicit RBF-based differential quadrature-finite volume method on unstructured grids: Application to inviscid and viscous compressible flows

This paper exploits the potential of a high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method for effective simulation of inviscid and viscous compressible flows using unstructured grids. The framework of IRBFDQ-FV method is based on finite volume discretization which can guarantee conservation of mass, momentum and energy. The flow field variables within each control cell are represented by a fourth-order approximation constructed by a Taylor series expansion to the cell center with spatial derivatives as the undetermined coefficients. All spatial derivatives are computed by the meshless and highly converged RBF-based differential quadrature (RBFDQ) technique. Regarding flux evaluation, the discrete gas-kinetic flux solver is applied to evaluate inviscid and viscous fluxes simultaneously for compressible viscous flows. Resultantly, the special mathematical treatment for the viscous flux, which is usually adopted in other high-order methods, can be avoided. A point extrema-based extended bounds (PEEB) shock-capturing limiter is introduced to eliminate spurious numerical oscillations near discontinuities. Due to the implicit nature of the proposed method, implicit time-stepping schemes compatible with the spatial accuracy are devised to efficiently solve the resulting discretized equations. Representative compressible flow problems are simulated to verify the high-order accuracy, excellent efficiency and robustness of the present method. Comparison with two other high-order finite volume methods further shows competitiveness of the present method in solving compressible flow problems on unstructured grids.