Compressible Flow Problems Research Articles

A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems

We present a new Lagrangian cell-centered scheme for two-dimensional compressible flows. The primary variables in this new scheme are cell-centered, i.e., density, momentum, and total energy are defined by their mean values in the cells. The vertex velocities and the numerical fluxes through the cell interfaces are not computed independently, contrary to standard approaches, but are evaluated in a consistent manner due to an original solver located at the nodes. The main new feature of the algorithm is the introduction of four pressures on each edge, two for each node on each side of the edge. This extra degree of freedom allows us to construct a nodal solver which fulfills two properties. First, the conservation of momentum and total energy is ensured. Second, a semidiscrete entropy inequality is provided. In the case of a one-dimensional flow, the solver reduces to the classical Godunov acoustic solver: it can be considered as its two-dimensional generalization. Many numerical tests are presented. They are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver.

Forchheimer law derived by homogenization of gas flow in turbomachines

A general formulation of the homogenization problem of compressible fluid flow through a periodic porous material in turbomachines is presented here. This formulation is able to derive a Forchheimer law with a mean velocity dependent permeability as equivalent macroscopic behavior. To specify this permeability, additional flow problems are defined on the unit cell and solved by a mixed stabilized finite element discretization. The application of the Galerkin least-square (GLS) method requires the introduction of two stabilization terms with appropriate parameters. The mixed finite element discretization of these unit cell problems is finally outlined.

A Mach‐uniform pressure‐correction algorithm with AUSM+ flux definitions

PurposeTo present conversion of the advection upwind splitting method (AUSM+) from the conventional density‐based and coupled formulation to the pressure‐based and segregated formulation.Design/methodology/approachThe spatial discretization is done by a finite volume method. A collocated grid cell‐center formulation is used. The pressure‐correction procedure is set up in the usual way for a compressible flow problem. The conventional Rhie‐Chow interpolation methodology for the determination of the transporting velocity, and the conventional central interpolation for the pressure at the control volume faces, are replaced by AUSM+ definitions.FindingsThe AUSM+ flux definitions are spontaneously well suited for use in a collocated pressure‐correction formulation. The formulation does not require extensions to these flux definitions. As a consequence, the results of a density‐based fully coupled method, are identical to the results of a pressure‐based segregated formulation. The advantage of the pressure‐correction method with respect to the density‐based method, is the higher efficiency for low Mach number applications. The advantage of the AUSM+ flux definition for the transporting velocity with respect to the conventional Rhie‐Chow interpolation, is the improved accuracy in high Mach number flows. As a consequence, the combination of AUSM+ with a pressure‐correction method leads to an algorithm with improved performance for flows at all Mach numbers.Originality/valueA new methodology, with obvious advantages, is composed by the combination of ingredients from an existing spatial discretization method (AUSM+) and an existing time stepping method (pressure‐correction).

The Characteristic‐Based Split (CBS) scheme—a unified approach to fluid dynamics

AbstractThis paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the original characteristic schemes are discussed. The important features of the CBS scheme are brought out by studying several problems of compressible and incompressible flows. All special consideration necessary for solving these problems are thoroughly discussed. The CBS scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided. Several invicid and viscous flow examples are also provided to demonstrate the unified CBS approach. For sample two‐dimensional codes, input files and instructions, the readers are referred to ‘www.nithiarasu.co.uk’ Copyright © 2006 John Wiley & Sons, Ltd.

Inverse design of supersonic diffuser with flexible walls using a Genetic Algorithm

An efficient algorithm for the design optimization of the compressible fluid flow problem through a flexible structure is presented. The methodology has three essential parts: first the behavior of compressible flow in a supersonic diffuser was studied numerically in quasi-one-dimensional form using a flux splitting method. Second, a fully coupled sequential iterative procedure was used to solve the steady state aeroelastic problem of a flexible wall diffuser. Finally, a robust Genetic Algorithm was implemented and used to calculate the optimum shape of the flexible wall diffuser for a prescribed pressure distribution.

Multiscale finite-volume method for compressible multiphase flow in porous media

The Multiscale Finite-Volume (MSFV) method has been recently developed and tested for multiphase-flow problems with simplified physics (i.e. incompressible flow without gravity and capillary effects) and proved robust, accurate and efficient. However, applications to practical problems necessitate extensions that enable the method to deal with more complex processes. In this paper we present a modified version of the MSFV algorithm that provides a suitable and natural framework to include additional physics. The algorithm consists of four main steps: computation of the local basis functions, which are used to extract the coarse-scale effective transmissibilities; solution of the coarse-scale pressure equation; reconstruction of conservative fine-scale fluxes; and solution of the transport equations. Within this framework, we develop a MSFV method for compressible multiphase flow. The basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure. The effects of compressibility are taken into account in the solution of the coarse-scale pressure equation and, if necessary, in the reconstruction of the fine-scale fluxes. We consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows (tracer transport in gas flow, imbibition and drainage of partially saturated reservoirs, depletion of gas–water reservoirs, and flooding of oil–gas reservoirs). We demonstrate that the MSFV method provides accurate solutions for compressible multiphase flow problems. Whereas slightly compressible flows can be treated with a very simple model, a more sophisticate flux reconstruction is needed to obtain accurate fine-scale saturation fields in highly compressible flows.

Application of AUSM schemes to multi-dimensional compressible two-phase flow problems

This paper dicusses the extension of AUSM-type schemes to the solution of multidimensional compressible two-phase flow problems. We consider their first and second order approximations and several multidimensional problem have been analysed in order to illustrate the behaviour and performance of the schemes. Both two-fluid mixtures of water and air and phase change processes between water and steam have been considered. Thermodynamic variables have been calculated taking into account either analytical equations of state (water and air mixtures) or tabulated equations of state (water and steam mixtures). Non-conservative terms in the two-fluid system of equations have been discretized centrally. In addition, gravity forces, interfacial friction, interfacial heat and mass transfer, etc., have been taken into account using a point-wise evaluation.

Aspects of one-dimensional channel gas dynamics with friction and heat transfer

In the paper the second form of analytical solution is obtained for the problem of one-dimensional steady compressible flow of perfect gas with constant heat flux and friction in the constant-area duct. The possibility is analyzed for obtaining analytical solutions under other boundary conditions. The future studies and application of the analytical solution in engineering practice is discussed.

A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions

We present a simple volume-of-fluid approach to interface tracking for inviscid compressible multicomponent flow problems in two space dimensions. The algorithm uses a uniform Cartesian grid with some grid cells subdivided by tracked interfaces, approximately aligned with the material interfaces in the flow field. A standard volume-moving procedure that consists of two basic steps: (1) the update of a discrete set of volume fractions from the current time to the next and (2) the reconstruction of interfaces from the resulting volume fractions, is employed to find the new location of the tracked interfaces in piecewise linear form at the end of a time step. As in the previous work by LeVeque and the author to front tracking based on a surface-moving procedure (R.J. LeVeque, K.-M. Shyue, Two-dimensional front tracking based on high-resolution wave propagation methods, J. Comput. Phys. 123 (1996) 354–368), a conservative high-resolution wave propagation method is applied on the resulting slightly non-uniform grid to update all the cell values, while the stability of the method is maintained by using a large time step idea even in the presence of small cells and the use of a time step with respect to the uniform grid cells. We validate our algorithm by performing the simulation of a Mach 1.22 shock wave in air over a circular R22 gas bubble, where sensible agreement of some key flow features of the computed solutions are observed when direct comparison of our results are made with the existing experimental and numerical ones that appear in the literature. Other computations are also presented that show the feasibility of the algorithm together with a mixture type of the model equations developed by the author (K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with Mie–Grüneisen equation of state, J. Comput. Phys. 171 (2001) 678–707) for practical multicomponent problems with general compressible materials characterized by a Mie–Grüneisen form of the equation of state.

On steady state computation of turbulent flows using k–ε models approximated by the time splitting method

AbstractThe time splitting method is frequently used in numerical integration of flow equations with source terms since it allows almost independent programming for the source part. In this paper we will consider the question of convergence to steady state of the time splitting method applied to k–ε turbulence models. This analysis is derived from a properly defined scalar study and is carried out with success for the coupled k–ε equations. It is found that the time splitting method does not allow convergence to steady state for any choice of finite values of the time step. Numerical experiments for some typical turbulent compressible flow problems support the fact that the time splitting method is always nonconvergent, while its nonsplitting counterpart is convergent. Copyright © 2005 John Wiley & Sons, Ltd.

Adaptive delaunay triangulation with multidimensional dissipation scheme for high-speed compressible flow analysis

Adaptive Delaunay triangulation is combined with the cell-centered upwinding algorithm to analyze in viscid high-speed compressible flow problems. The multidimensional dissipation scheme was developed and included in the upwinding algorithm for unstructured triangular meshes to improve the computed shock wave resolution. The solution accuracy is further improved by coupling an error estimation procedure to a remeshing algorithm that generates small elements in regions with large change of solution gradients, and at the same time, larger elements in other regions. The proposed scheme is further extended to achieve higher-order spatial and temporal solution accuracy. Efficiency of the combined procedure is evaluated by analyzing supersonic shocks and shock propagation behaviors for both the steady and unsteady high-speed compressible flows.

The Simulation of Airframe Noise Applying Euler-Perturbation and Acoustic Analogy Approaches

The capability of three different perturbation approaches to tackle airframe noise problems is studied. The three approaches represent different levels of complexity and are applied to trailing edge noise problems. In the Euler-perturbation approach the linearized Euler equations without sources are used as governing acoustic equations. The sound generation and propagation is studied for several trailing edge shapes (blunt, sharp, and round trailing edges) by injecting upstream of the trailing edge test vortices into the mean-flow field. The efficiency to generate noise is determined for the trailing edge shapes by comparing the different generated sound intensities due to an initial standard vortex. Mach number scaling laws are determined varying the mean-flow Mach number. In the second simulation approach an extended acoustic analogy based on acoustic perturbation equations (APEs) is applied to simulate trailing edge noise of a flat plate. The acoustic source terms are computed from a synthetic turbulent velocity model. Furthermore, the far field is computed via additional Kirchhoff extrapolation. In the third approach the sources of the extended acoustic analogy are computed from a Large Eddy Simulation (LES) of the compressible flow problem. The directivities due to a modeled and a LES based source, respectively, compare qualitatively well in the near field. In the far field the asymptotic directivities from the Kirchhoff extrapolation agree very well with the analytical solution of Howe. Furthermore, the sound pressure spectra can be shown to have similar shape and magnitude for the last two approaches.

Conservative Velocity Interpolation for PDF Methods

AbstractIn many particle methods the accurate interpolation of a velocity field represented on a computational grid to arbitrary positions is crucial [2, 5]. Here, the importance of mass conservation and order of the interpolation scheme were analyzed. Initially equally distributed particles were tracked in a stationary, incompressible 2d flow field using different interpolation schemes. It could be demonstrated that especially mass conservation is of great importance, in particular in the case of complex flow patterns. The ideas presented in this paper are more general and the methods can be extended for unsteady, compressible 3d flow problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modified Multidimensional Dissipation Scheme on Unstructured Meshes for High-speed Compressible Flow Analysis

A Roe's flux-difference splitting scheme, combining with the entropy fix method according to Van Leer et al., and the H-correction entropy fix method by Pandolfi and D'Ambrosio, is proposed. The presented scheme eliminates unphysical flow behaviors such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the expansion shock generated from flow over a forward facing step. The proposed scheme is further extended to obtain high-order spatial and temporal solution accuracy. The scheme is, in addition, combined with an adaptive meshing technique that generates unstructured triangular meshes to resemble the flow phenomena for reducing computational effort. The entire procedure is evaluated by solving several benchmarks as well as complex steady-state and transient high-speed compressible flow problems.

Two-dimensional adaptive mesh generation algorithm and its application with higher-order compressible flow solver

A combined procedure for two-dimensional Delaunay mesh generation algorithm and an adaptive remeshing technique with higher-order compressible flow solver is presented. A pseudo-code procedure is described for the adaptive remeshing technique. The flux-difference splitting scheme with a modified multidimensional dissipation for high-speed compressible flow analysis on unstructured meshes is proposed. The scheme eliminates nonphysical flow solutions such as the spurious bump of the carbuncle phenomenon observed from the bow shock of the flow over a blunt body and the oscillation in the odd-even grid perturbation in a straight duct for the Quirk's odd-even decoupling test. The proposed scheme is further extended to achieve higher-order spatial and temporal solution accuracy. The performance of the combined procedure is evaluated on unstructured triangular meshes by solving several steady-state and transient high-speed compressible flow problems.

Flux‐difference splitting scheme with modified multidimensional dissipation on unstructured meshes

A flux‐difference splitting scheme with a modified multidimensional dissipation for high‐speed compressible flow analysis on unstructured meshes is presented. The scheme eliminates unphysical flow behaviors such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the expansion shock generated from flow over a forward facing step. The switching function suggested by Quirk is implemented as a choice to detect the vicinity of strong shock. The proposed scheme is further extended to obtain higher‐order spatial and temporal solution accuracy. The scheme is, in addition, combined with an adaptive meshing technique that generates unstructured triangular meshes to resemble the flow phenomena for reducing computational effort. The entire procedure is evaluated by solving several benchmarks as well as steady‐state and transient high‐speed compressible flow problems.

Comparison of Homogenization and Direct Techniques for the Treatment of Roughness in Incompressible Lubrication

Homogenization is a formal mathematical two-scale averaging process that can be applied to roughness problems and can replace previous heuristic averaging procedures, which have sometimes led to ambiguous results. This procedure was previously mathematically developed and applied to compressible flow problems. The purpose of this paper is the development of a special form of Reynolds equation for such homogenized conditions applied to the incompressible Newtonian case. The equation allows the calculation of the operating characteristics of a contact by taking into account the local geometry of surfaces while making a substantial improvement in computing time. The method allows for the study of rough surfaces, but requires considerably fewer calculated points than for traditional deterministic discretization methods.

MIXED ENTROPY FIX METHOD FOR ROE S FLUX-DIFFERENCE SPLITTING SCHEME WITH AUTOMATIC MESH ADAPTATION

A combined procedure for two-dimensional Delaunay mesh generation algorithm and an adaptive remeshing technique with higher-order compressible flow solver is presented. The flux-difference splitting scheme with a mixed entropy fix method is introduced for high-speed compressible flow analysis on unstructured meshes. The scheme eliminates unphysical flow solutions such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the oscillation in the odd-even grid perturbation in a straight duct for the Quirk's odd-even decoupling test. The proposed scheme is further extended to achieve higher-order spatial and temporal solution accuracy. The performance of the combined procedure is evaluated by solving several high-speed compressible flow problems on unstructured triangular meshes.

Global existence and blow-up phenomena of classical solutions for the system of compressible adiabatic flow through porous media

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small’ solution.

On the simulation of trailing edge noise with a hybrid LES/APE method

A hybrid method is applied to predict trailing edge noise based on a large eddy simulation (LES) of the compressible flow problem and acoustic perturbation equations (APE) for the time-dependent simulation of the acoustic field. The acoustic simulation in general considers the mean flow convection and refraction effects such that the computational domain of the flow simulation has to comprise only the significant acoustic source region. Using a modified rescaling method for the prediction of the unsteady turbulent inflow boundary layer, the LES just resolves the flow field in the immediate vicinity of the trailing edge. The linearized APE completely prevent the unbounded growth of hydrodynamic instabilities in critical mean flows.