Compressible Flow Problems Research Articles

Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models

In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overla...

Reduced-order modelling of parameterised incompressible and compressible unsteady flow problems using deep neural networks

A non-intrusive reduced-order model for nonlinear parametric flow problems is developed. It is based on extracting a reduced-order basis from full-order snapshots via proper orthogonal decomposition and using both deep and shallow neural network architectures to learn the reduced-order coefficients variation in time and over the parameter space. Even though the focus of the paper lies in developing a reduced-order methodology for approximating fluid flow problems, the methodology is generic and can be used for the order reduction of arbitrary time-dependent parametric systems. Since it is non-intrusive, it is independent of the full-order computational method and can be used together with black-box commercial solvers. An adaptive sampling strategy is proposed to increase the quality of the neural network predictions while minimising the required number of parameter samples. Numerical studies are presented for two canonical test cases, namely unsteady incompressible laminar flow around a circular cylinder and transonic inviscid flow around a pitching NACA 0012 aerofoil. Results show that the proposed methodology can be used as a predictive tool for unsteady parameter-dependent flow problems.

Reduced-order modelling of parameterised incompressible and compressible unsteady flow problems using deep neural networks

Reduced-order modelling of parameterised incompressible and compressible unsteady flow problems using deep neural networks

Numerical modeling of multiphase compressible flows with the presence of shock waves using an interface-sharpening five-equation model

An accurate shock- and interface-capturing method is introduced for simulations of compressible multiphase flows. First, an associated Godunov-type numerical scheme is established for a five-equation two-phase model obtained from a seven-equation model by assuming a single velocity and a single pressure between two phases. The computational finite-volume Riemann solver using the scheme and computing algorithms is presented. Next, an interface-sharpening technique (IST) is extended for the compressible two-phase model to improve numerical simulations and correct diffusion errors. The modified IST was applied as postprocessing to correct the numerical diffusion error in the solution of the discretization scheme while maintaining a sharp interface with a desired thickness after each time step. A mixture-consistent interface regularization approach of all conservative variables is combined with the IST to obtain consistent thermodynamic laws for the mixture, ensuring the consistency of the variables in the correction process. Several examples of fluid interface simulations including shock-tube, shock-bubble interactions, and underwater explosions were performed to demonstrate the accuracy and capability of the proposed method. Those compressible multiphase flow problems are complicated by the presence of both shock waves and the dynamics of interfaces. Comparisons of the numerical method with theoretical results and experimental data indicate that the present method can simulate interface dynamics with the presence of shock waves and large density differences.

Compressible Euler equations on a sphere and elliptic–hyperbolic property

Abstract In this work, we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow field. Under this assumption, the compressible Euler equations reduce to a system defined on the surface of the unit sphere. This compressible flow problem has been successfully used to study the steady supersonic flow past cones of arbitrary cross section by reducing the number of spatial dimensions from three down to two while still capturing many of the relevant 3D effects. In this paper, the powerful machinery of tensor calculus is utilized to avoid reference to any particular coordinate system. With the flexibility to use any coordinate system on the surface of a sphere, the equations can be more readily solved numerically when a structured mesh is used by defining the mesh lines to be the coordinate lines. The type of the system of partial differential equations would be hyperbolic or elliptic based on whether the crossflow Mach number is supersonic or subsonic.

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.

Parallel discontinuous Galerkin finite element method for computing hyperbolic conservation law on unstructured meshes

PurposeThe discontinuous Galerkin finite element method (DGFEM) is very suited for realizing high order resolution approximations on unstructured grids for calculating the hyperbolic conservation law. However, it requires a significant amount of computing resources. Therefore, this paper aims to investigate how to solve the Euler equations in parallel systems and improve the parallel performance.Design/methodology/approachDiscontinuous Galerkin discretization is used for the compressible inviscid Euler equations. The multi-level domain decomposition strategy was used to deal with the computational grids and ensure the calculation load balancing. The total variation diminishing (TVD) Runge–Kutta (RK) scheme coupled with the multigrid strategy was employed to further improve parallel efficiency. Moreover, the Newton Block Gauss–Seidel (GS) method was adopted to accelerate convergence and improve the iteration efficiency.FindingsNumerical experiments were implemented for the compressible inviscid flow problems around NACA0012 airfoil, over M6 wing and DLR-F6 configuration. The parallel acceleration is near to a linear convergence. The results indicate that the present parallel algorithm can reduce computational time significantly and allocate memory reasonably, which has high parallel efficiency and speedup, and it is well-suited to large-scale scientific computational problems on multiple instruction stream multiple data stream model.Originality/valueThe parallel DGFEM coupled with TVD RK and the Newton Block GS methods was presented for hyperbolic conservation law on unstructured meshes.

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 fast transient solver for low-Mach number aerodynamics and aeroacoustics

The fractional step method is an efficient time advancement scheme for problems involving transient incompressible flow. The method is desirable over traditional SIMPLE-like algorithms for its non-iterative nature. This paper presents a technique that extends the fractional step method to low-Mach number compressible flow problems. The proposed pressure-velocity coupling and a transient SIMPLEC algorithm are validated and verified for a 1D acoustic wave propagation and a convective isentropic vortex to qualitatively evaluate their convergence properties. Next, both algorithms are used to predict aerodynamic and aeroacoustic behaviour of flow past a wall-mounted side mirror. The proposed scheme is stable at acoustic CFL number greater than one, and at convective Co number equal one. The L2 error norms of both schemes exhibit second order accuracy as grids are refined, while the new scheme requires significantly less computation effort for an equal error. The new fractional step method consumes only 20% to 35% of the computation effort that is required by the transient SIMPLEC algorithm for the same error, and is easily scalable to a greater number of CPU processors.

Global strong solutions to the one-dimensional full compressible liquid crystal equations with temperature-dependent heat conductivity

In this paper, we consider the initial-boundary value problem of compressible non-isothermal liquid crystal flow in one-dimensional space. The existence of the global strong solution with large initial data is proved. In the proof of the main results, we assume that heat conductivity obeys the power law of the form κ(θ)=θβ with β>0.

NUMERICAL SCHEME OF MULTI-MATERIAL COMPRESSIBLE FLOW WITH SHARP INTERFACE ON EULERIAN GRIDS

Numerical simulation of multi-material compressible flow problem is of great importance in both the national defense and industry areas, such as weapon design and blast wave defense. Due to the property of large deformation and high nonlinearity, the efficient simulation of multi-material compressible flow becomes a quite challenging problem. A numerical scheme is developed to carry out the simulation of an immiscible multi-material compressible flow with sharp phase interface on two dimensional and three dimensional unstructured Eulerian grids, which can handle the large deformation of compressible fluid and elastoplastic solid under the extreme conditions. We use the level set method to depict the phase interface numerically, and explicitly reconstruct the phase interface in a piecewise linear manifold. The topological structure of the phase interface is constructed explicitly, which can handle any number of media in the whole computational domain and three media in a single cell. The traditional finite volume method is used to calculate the edge numeircal flux between the same material in adjacent cells, while the phase interface flux is calculated by exactly solving a one dimensional multi-material Riemann problem on the normal direction of the phase interface. The above procedures can keep the conservation of the phase interface flux, and the interaction between two media across the phase interface can keep consistent with the real situation. A robust aggregation algorithm is adopted to build cell patches and adjust the conservation variables around the phase interface, which can effectively remove the numerical instability due to the breakdown of the CFL constraint by the cell fragments. Some classical examples and application problems, such as one-dimensional multi-material Riemann problem, gas-bubble interaction problem, intensive airblast problem, sub-surface blast problem, and blast wave propagation in three dimensional sap, which have a good agreement with the corresponding analytical and experimental results, are presented to validate our numerical scheme.

Global classical solutions to the one dimensional free boundary problem for compressible non-isothermal liquid crystal flow with large data

In this paper, we study the free boundary problem of compressible non-isothermal liquid crystal flow in dimension one. For the vacuum interface condition, that is, the density is assumed to vanish and connect to vacuum continuously, we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data.

Evaluation of a nonlinear variational multiscale method for fluid transport problems

Diverse transport problems, especially those based on fluid flow models, are intrinsically multiscale and nonlinear, characteristics that often lead to intricate dynamics such as the development of instabilities and turbulence. Computational simulations that resolve all scales in these problems are often unfeasible, prompting to coarse-grained simulation strategies in which small-scale features are modeled instead of resolved. Variational Multiscale (VMS) methods, and particularly residual-based Large-Eddy Simulation (LES) approaches, have proven effective and robust for the coarse-grained simulation of complex transport problems. VMS methods avoid the assumption of separable nonlinearity and the reliance on empirical small-scale models by using a variational decomposition of scales together with a residual-based approximation of the small-scales. Evaluation of a nonlinear VMS approach, denoted as VMSn, is presented for the coarse-grained simulation of transient-advective-diffusive-reactive (TADR) transport problems arising from fluid flow models. In contrast to classical VMS approaches that neglect the effect of the small scales on the transport operator, VMSn treats the inter-dependence between large- and small-scales upfront. The treatment of inter-scale coupling involves the solution of a local algebraic nonlinear system describing the evolution of the small-scales. The VMSn approach is complemented with two algebraic approximations of the small-scales: one based on the main diagonal of the transport matrices and another that preserves transport fluxes and is suitable for generic TADR systems. The suitability of the VMSn approach for handling general TADR problems and regimes is evaluated with benchmark incompressible, compressible, and magnetohydrodynamic laminar flow problems, the incompressible Taylor-Green vortex flow, the turbulent free jet, and the two-temperature arc in crossflow. Simulation results show that VMSn leads to minor improvements in accuracy with respect to the classical VMS for the laminar flow problems, but to significantly greater accuracy for the turbulent flows and the unsteady plasma flow problems, while using the same cohesive numerical formulation.

On the shock change equations

We revisit and derive the shock change equations relating the dynamics of a shock wave with the partial derivatives describing the motion of a reactive fluid with the general equation of state in a stream-tube with arbitrary area variation. We specialize these to a perfect gas in which we obtain all shock change equations in closed form. These are further simplified for strong shocks. We discuss the general usefulness of these equations in problems of reactive compressible flow and in the development of intrinsic evolution equations for the shock, such as the approximations made by Whitham and Sharma.

Efficient Convergence for a Higher-Order Unstructured Finite Volume Solver for Compressible Flows

This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the -exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.

A Simple Modification of HLLEM Approximate Riemann Solver Applied to the Compressible Euler System at Low Mach Number

Due to its attractive properties in computing compressible flows, such as sharp capturing of discontinuities, satisfying entropy condition and positivity preservation, the HLLEM approximate Riemann solver has been widely applied for simulations of many compressible flow problems. However, when it comes to weakly compressible or incompressible flows, the HLLEM scheme cannot give physically correct solutions. In the current study, a simple low Mach number fix is applied to improve the accuracy and stability of HLLEM approximate Riemann solver in the low Mach limit. As a result, a modified HLLEM scheme called LM-HLLEM scheme is proposed. Numerical results demonstrate that the proposed LM-HLLEM scheme is able to compute various flow problems accurately and robustly ranging from compressible to low Mach incompressible flows.

Central upwind scheme based immersed boundary method for compressible flows around complex geometries

The present work combines easy to implement high resolution central upwind scheme (CUS) with a ghost cell based discrete forcing immersed boundary method (IBM) and proposes a novel CUS-IBM finite volume solver to simulate high-speed compressible inviscid/viscous flows around complex geometries on Cartesian grids. Proposed CUS-IBM solver has been tested for a variety of inviscid and viscous compressible flow problems, ranging from transonic to supersonic flow speeds (0.625 < M < 3), involving flow over different geometries, i.e., bump, airfoil and cylinder. Results from our computations clearly show that solutions computed with the proposed method are in good agreement with the numerical data available in literature for the various inviscid/viscous and external/internal compressible flow problems. The present study demonstrates that the proposed CUS-IBM exhibits nominal second-order accuracy in space and is an easy to implement, robust and promising alternative to the methods involving Riemann solver and characteristics decomposition.

Global large solutions to the planar magnetohydrodynamics equations with constant heat conductivity

This paper is concerned with global existence of large solutions to the initial-boundary value problem of the planar magnetohydrodynamic compressible flow. Under the assumptions that viscosity and heat conductivity coefficients are constants, magnetic diffusion is a function of the specific volume, we obtain the global existence of strong solutions. Some new methods are developed to deal with the complex interaction between the hydrodynamic and magnetodynamics effects.

Numerical simulation for steady anisotropic-diffusion convection problems of compressible flow in exponentially graded media

A boundary element method is utilized to find numerical solutions to boundary value problems of exponentially graded media governed by a spatially varying coefficients anisotropic-diffusion convection equation. The variable coefficients equation is firstly transformed into a constant coefficients equation for which a boundary integral equation can be formulated. A boundary element method (BEM) is then derived from the boundary integral equation. Some problems are considered. The numerical solutions justify the validity of the analysis used to derive the boundary element method with accurate and consistent solutions. A FORTRAN script is developed for the computation of the solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, results obtained from the considered examples show the effect of anisotropy of the media on the solutions. An example of a layered material is presented as an illustration of the application.

On the derivation of a boundary element method for diffusion convection-reaction problems of compressible flow in exponentially inhomogeneous media

Boundary value problems (BVPs) of anisotropic and exponentially-graded media governed by a diffusion convection-reaction (DCR) equation are considered. The governing equation is of spatially varying coefficients (with an anisotropic diffusion coefficient). The variable coefficients equation is firstly transformed into a constant coefficients equation, from which we derive a boundary integral equation. A boundary element method (BEM) is then constructed to be used for finding numerical solutions to the BVPs. For the computation of the solutions a FORTRAN code is developed. Some examples of problems are solved. The numerical solutions obtained verify the validity of the analysis used to construct the boundary element method with accurate and consistent solutions. The results also show that the BEM procedure elapses very efficient time in producing the solutions. In addition, the results indicate the effect of anisotropy of the media on the solutions.