Method For Compressible Flows Research Articles

Gas Injection and Suction Effect on the Instability of a Supersonic Boundary Layer

We present the results of an investigation of the boundary layer stability on a flat plate having a region of gas injection/suction normal to the wall at a large supersonic freestream Mach number. The laminar flow past a flat plate having a region of distributed injection/suction of fixed intensity is modeled numerically using the integration of Navier—Stokes equations. The unstable disturbances in the boundary layer distorted by injection/suction are analyzed within the framework of the linear stability theory and the eN{{e}^{N}} method for compressible flows. High-frequency disturbances belonging to the plane second mode of the boundary layer, which are the most unstable at high velocities, are considered. It is shown that gas injection/suction leads to a nonmonotonic variation of disturbance growth rates with the appearance of stabilization/destabilization regions, as compared with the case in which injection/suction is absent.

Finite Volume Methods for Compressible Flows and Application to Japanese Rockets

Finite Volume Methods for Compressible Flows and Application to Japanese Rockets

Low-Dissipation Simulation Methods and Models for Turbulent Subsonic Flow

The simulation of turbulent flows by means of computational fluid dynamics is highly challenging. The costs of an accurate direct numerical simulation (DNS) are usually too high, and engineers typically resort to cheaper coarse-grained models of the flow, such as large-eddy simulation (LES). To be suitable for the computation of turbulence, methods should not numerically dissipate the turbulent flow structures. Therefore, energy-conserving discretizations are investigated, which do not dissipate energy and are inherently stable because the discrete convective terms cannot spuriously generate kinetic energy. They have been known for incompressible flow, but the development of such methods for compressible flow is more recent. This paper will focus on the latter: LES and DNS for turbulent subsonic flow. A new theoretical framework for the analysis of energy conservation in compressible flow is proposed, in a mathematical notation of square-root variables, inner products, and differential operator symmetries. As a result, the discrete equations exactly conserve not only the primary variables (mass, momentum and energy), but also the convective terms preserve (secondary) discrete kinetic and internal energy. Numerical experiments confirm that simulations are stable without the addition of artificial dissipation. Next, minimum-dissipation eddy-viscosity models are reviewed, which try to minimize the dissipation needed for preventing sub-grid scales from polluting the numerical solution. A new model suitable for anisotropic grids is proposed: the anisotropic minimum-dissipation model. This model appropriately switches off for laminar and transitional flow, and is consistent with the exact sub-filter tensor on anisotropic grids. The methods and models are first assessed on several academic test cases: channel flow, homogeneous decaying turbulence and the temporal mixing layer. As a practical application, accurate simulations of the transitional flow over a delta wing have been performed.

A hybrid flux splitting method for compressible flow

ABSTRACTA hybrid flux splitting scheme, called AUSM+–FVS((advection upstream splitting method)+–flux vector splitting), is proposed in this article to calculate the inviscid fluxes of Euler/Navier–Stokes equations. This new scheme is obtained by hybridizing the AUSM+ scheme with FVS method. When the local Mach number tends to zero, this scheme is similar to the AUSM+. Contrarily, this scheme is similar to the FVS at the shock region. Thus, this scheme has the accuracy of AUSM+ in boundary layer region and the robustness of FVS in shock region. Several numerical tests show that AUSM+–FVS can reduce the shock instability and has a good accuracy in boundary layer region.

A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods

Most high order computational fluid dynamics (CFD) methods for compressible flows are based on Riemann solver for the flux evaluation and Runge–Kutta (RK) time stepping technique for temporal accuracy. The advantage of this kind of space–time separation approach is the easy implementation and stability enhancement by introducing more middle stages. However, the nth-order time accuracy needs no less than n stages for the RK method, which can be very time and memory consuming due to the reconstruction at each stage for a high order method. On the other hand, the multi-stage multi-derivative (MSMD) method can be used to achieve the same order of time accuracy using less middle stages with the use of the time derivatives of the flux function. For traditional Riemann solver based CFD methods, the lack of time derivatives in the flux function prevents its direct implementation of the MSMD method. However, the gas kinetic scheme (GKS) provides such a time accurate evolution model. By combining the second-order or third-order GKS flux functions with the MSMD technique, a family of high order gas kinetic methods can be constructed. As an extension of the previous 2-stage 4th-order GKS, the 5th-order schemes with 2 and 3 stages will be developed in this paper. Based on the same 5th-order WENO reconstruction, the performance of gas kinetic schemes from the 2nd- to the 5th-order time accurate methods will be evaluated. The results show that the 5th-order scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier–Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation. In comparison with Riemann solver based 5th-order RK method, the high order GKS has advantages in terms of efficiency, accuracy, and robustness, for all test cases. The 4th- and 5th-order GKS have the same robustness as the 2nd-order scheme for the capturing of discontinuous solutions. The current high order MSMD GKS is a multi-dimensional scheme with incorporation of both normal and tangential spatial derivatives of flow variables at a cell interface in the flux evaluation. The scheme can be extended straightforwardly to viscous flow computation in unstructured mesh. It provides a promising direction for the development of high-order CFD methods for the computation of complex flows, such as turbulence and acoustics with shock interactions.

Development of the generalized MacCormack scheme and its extension to low Mach number flows

SummaryThe low Mach number performance of the MacCormack scheme is examined. The inherent dissipation in the scheme is found to suffer from the degradation in accuracy observed with traditional, density‐based methods for compressible flows. Two specific modifications are proposed, leading to the formation of the generalized MacCormack scheme within a dual‐time framework (called GMC‐PC). The first modification involves reformulating the flux by splitting it into particle convection and acoustic parts, with the former terms treated using the traditional MacCormack discretization and the latter terms augmented by the addition of a pressure‐based artificial dissipation. The second modification involves a reformulation of the traditional nonlinear fix introduced by MacCormack in 1971, which is found to be necessary to suppress pressure oscillations at low Mach numbers. The new scheme is demonstrated to have superior performance, independent of Mach number, compared with standard MacCormack implementations using several canonical test problems. Copyright © 2017 John Wiley & Sons, Ltd.

Local preconditioning and variational multiscale stabilization for Euler compressible steady flow

This paper introduces a preconditioned variational multiscale stabilization (P-VMS) method for compressible flows. In this introductory paper we focus on inviscid flow and steady state problems. The Euler equations are solved on fully unstructured grids and discretized using the finite element method. The P-VMS method can be decomposed into three parts. First, a local preconditioner is applied to the continuous equations to reduce the stiffness while covering a wide range of Mach numbers. Then, the resulting preconditioned system is discretized in space using finite elements and stabilized with a variational multiscale stabilization method adapted for the preconditioned equations. In this paper, the solution is advanced in time using a fully explicit time discretization, although P-VMS is general and can be applied to fully implicit solvers. The proposed method is assessed by comparing convergence and accuracy of the solutions between the non-preconditioned and preconditioned cases, in particular for van Leer–Lee–Roe’s (1991) and Choi-Merkle’s (1993) preconditioners, in some selected examples covering a large range of Mach numbers.

Direct numerical simulation of aeroacoustic sound by volume penalization method

The volume penalization (VP) method for compressible flows is investigated as a tool of direct numerical simulation of aeroacoustic sound in problems where not only acoustic pressure but also hydrodynamic pressure depends on time and position. First, it is shown that the method proposed by Liu and Vasilyev (2007) [30] is not Galilean invariant. It is corrected to satisfy Galilean invariance. Next, numerical accuracy of the corrected VP method is investigated in problems of simple geometry which can be simulated also by a standard method on a body-fitted coordinate system: sound generation in (i) flow past a fixed square/circular cylinder, (ii) flow past an oscillating square/circular cylinder, and (iii) flow past two square cylinders. The results confirm that the corrected VP method gives reasonably accurate results for sound pressure which is much smaller than hydrodynamic pressure within 5% error. Finally, the corrected method is applied to two examples of complex geometry, which cannot be simulated by standard methods using body-fitted coordinate systems without considerable difficulty: sound generation in (i) flow past an oscillating cylinder and a fixed cylinder behind it and (ii) flow past a bundle of cylinders. The results show that the present method is in principle applicable to aeroacoustic problems in any complex geometry including practical engineering ones.

ON THE PIECEWISE PARABOLIC METHOD FOR COMPRESSIBLE FLOW WITH STELLAR EQUATIONS OF STATE

The piecewise parabolic method and related schemes are widely used to model stellar flows. Several different methods for extending the validity of these methods to a general equation of state (EOS) have been proposed over time, but direct comparisons among one-another and exact solutions with stellar EOSs are not widely available. We introduce some simple test problems with exact solutions run with a popular stellar EOS and test how two existing codes with different approaches to incorporating general gases perform. The source code for generating the exact solutions is made available.

A symmetry-preserving discretisation and regularisation model for compressible flow with application to turbulent channel flow

Most simulation methods for compressible flow attain numerical stability at the cost of swamping the fine turbulent flow structures by artificial dissipation. This article demonstrates that numerical stability can also be attained by preserving conservation laws at the discrete level. A new mathematical explanation of conservation in compressible flow reveals that many conservation properties of convection are due to the skew-symmetry of the convection operator. By preserving this skew-symmetry at the discrete level, a fourth-order accurate collocated symmetry-preserving discretisation with excellent conservation properties is obtained. Also a new symmetry-preserving regularisation subgrid-scale model is proposed. The proposed techniques are assessed in simulations of compressible turbulent channel flow. The symmetry-preserving discretisation for compressible flow has good stability without artificial dissipation and yields acceptable results already on coarse grids. Regularisation does not consistently improve upon no-model results, but often compares favourably with eddy-viscosity models.

An Immersed Boundary Method for Compressible Flows with Complex Boundaries

An immersed boundary method based on the ghost-cell approach is presented in this paper. The compressible Navier-Stokes equations are discretized using a flux-splitting method for inviscid fluxes and second-order central-difference for the viscous components. High-order accuracy is achieved by using weighted essentially non-oscillatory (WENO) and Runge-Kutta schemes. Boundary conditions are reconstructed by a serial of linear interpolation and inverse distance weighting interpolation of flow variables in fluid domain. Two classic flow problems (flow over a circular cylinder, and a NACA 0012 airfoil) are simulated using the present immersed boundary method, and the predictions show good agreement with previous computational results.

Modified kinetic flux vector splitting (m-KFVS) method for compressible flows

To resolve many flow features accurately, like accurate capture of suction peak in subsonic flows and crisp shocks in flows with discontinuities, to minimise the loss in stagnation pressure in isentropic flows or even flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order kinetic flux vector splitting (KFVS) method has been found to be very robust but suffers from the problem of having much more numerical diffusion than required, resulting in inaccurate computation of the above flow features. However, numerical dissipation can be reduced by refining the grid or by using higher order kinetic schemes. In flows with strong shock waves, the higher order schemes require limiters, which reduce the local order of accuracy to first order, resulting in degradation of flow features in many cases. Further, these schemes require more points in the stencil and hence consume more computational time and memory. In this paper, we present a low dissipative modified KFVS (m-KFVS) method which leads to improved splitting of inviscid fluxes. The m-KFVS method captures the above flow features more accurately compared to first order KFVS and the results are comparable to second order accurate KFVS method, by still using the first order stencil.

Fast p-Multigrid Discontinuous Galerkin Methodfor Compressible Flows at All Speeds

Ap-multigrid (wherep is the polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature ofthisp-multigrid method is the application of an explicit smoother on the higher-level approximations (p > 0) and an implicit smoother on the lowest-level approximation (p = 0), resulting in a fast (and low) storage method that can be efficiently used to accelerate the convergence to a steady-state solution. Furthermore, this p-multigrid method can be naturally applied to compute the flows with discontinuities, in which a monotonic limiting procedure is usually required for discontinuous Galerkin methods. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries [Krivodonova, L., and Berger, M., High-Order Implementation of Solid Wall Boundary Conditions in Curved Geometries, Journal of Computational Physics, Vol. 211, No.2,2006, pp. 492-512; and Luo, H., Baum, J. D., and Lohner, R., On the Computation of Steady-State Compressible Flows Using a Discontinuous Galerkin Method, International Journal for Numerical Methods in Engineering, Vol. 73, No.5,2008, pp. 597-623]. A variety of compressible flow problems for a wide range of flow conditions from low Mach number to supersonic in both two-dimensional and three-dimensional configurations are computed to demonstrate the performance ofthisp-multigrid method. The numerical results obtained strongly indicate the order-independent property of this p-multigrid method and demonstrate that this method is orders-of-magnitude faster than its explicit counterpart The performance comparison with a finite volume method shows that using this p-multigrid method, the discontinuous Galerkin method, provides a viable, attractive, competitive, and probably even superior alternative to the finite volume method for computing compressible flows at all speeds.

Doyle D. Knight: Elements of numerical methods for compressible flows

Doyle D. Knight: Elements of numerical methods for compressible flows

The Characteristics-Based Matching (CBM) Method for Compressible Flow With Moving Boundaries and Interfaces

Recently, Eulerian methods for capturing interfaces in multi-fluid problems become increasingly popular. While these methods can effectively handle significant deformations of interface, the treatment of the boundary conditions in certain classes of compressible flows are known to produce nonphysical oscillations due to the radical change in equation of state across the material interface. One promising recent development to overcome these problems is the Ghost Fluid Method (GFM). The present study initiates a new methodology for boundary condition capturing in multifluid compressible flows. The method, named Characteristics-Based Matching (CBM), capitalizes on recent developments of the level set method and related techniques, i.e., PDE-based re-initialization and extrapolation, and the Ghost Fluid Method (GFM). Specifically, the CBM utilizes the level set function to capture interface position and a GFM-like strategy to tag computational nodes. In difference to the GFM method, which employs a boundary condition capturing in primitive variables, the CBM method implements boundary conditions based on a characteristic decomposition in the direction normal to the boundary. In this way overspecification of boundary conditions is avoided and we believe so will be spurious oscillations. In this paper, we treat (moving or stationary) fluid-solid interfaces and present numerical results for a select set of test cases. Extension to fluid-fluid interfaces will be presented in a subsequent paper.

A low-diffusion MUSCL scheme for LES on unstructured grids

We are interested in the development of large-eddy simulation (LES) methods for compressible flows in complex geometries. The starting point is a numerical scheme applicable to unstructured tetrahedrizations, conservative, upwind of MUSCL type and vertex centered. A low-diffusion version stabilized with sixth-order spatial derivatives is proposed. The obtained scheme is combined with two LES models, derived from the Smagorinsky model and its dynamic version. The basic test-case chosen is the flow around a square cylinder. Calculations around a forward-swept wing and a business jet are also presented.

Review of Flow visualization , edited by A. J. Smits and T. T. Lim

Flow visualization is perhaps the oldest investigative tool in fluid mechanics and hydraulics. Remarkable insights into fluid flow, such as those made by Leonardo da Vinci, were obtained over time using this straightforward method. The advent of the digital revolution in modern times transformed these essentially qualitative techniques to strong quantitative ones. Among the distinctive advantages of flow visualization over the other experimental approaches is that they are essentially nonintrusive. In recent years, twoor three-dimensional vector and associated scalar-flow quantities have become available through continued and sophisticated technological development. Currently, the modern flow visualization instruments gain continuously spatial and temporal capabilities such that they are increasingly complementing the data obtained by Computational Fluid Dynamics ~CFD!, another offspring of the digital revolution. The rapid, intensive, and extensive evolution of the flow visualization techniques in past decades, did not allow fast assimilation of the techniques in current laboratory practice. The need of evaluation, validation, and dissemination of the benefits of these techniques was continuously felt during these decades. From this respect, the publication of this book is valuable and timely. The book consists of twelve chapters covering a large range of flow visualization techniques including new methods not widely covered in the technical literature ~e.g., molecular tagging velocimetry, thermo-chromic liquid crystals, threeand four-dimensional ~inclusive of time! imaging!. The contributing authors are leading scientific authorities with extensive expertise in the presented techniques. Chapter 1 familiarizes the reader with the interpretation of flow patterns with a strong emphasis on the significance of the critical points. The relationships between tensor invariants and streamline patterns are explained in detail using comprehensive illustrations and theoretical derivations. It is also shown that a combination of invariants can be used to detect local vortices from the three-dimensional velocity information obtained by CFD. Chapter 2 discusses the well-known hydrogen bubble technique, initially developed for visualizing organized turbulent structures in turbulent boundary layers. The configuration of the bubble-generating systems, probes, lighting procedures, and the limitations of the method are briefly described. Chapter 3 describes another traditional experimental approach, namely visualization using dye and smoke. The use of the technique is separately described in accordance with the working fluid used in the experiments, i.e., water or air. Special attention is given to the capturing of the visualized information. Chapters 4 and 5 treat two newer techniques based on laser light interaction with atoms or molecules contained in the flows. Chapter 4, dealing with Molecular Tagging Velocimetry ~MTV!, first introduces the reader to the particularities of the photochromic materials. Similarly, Chapter 5 familiarizes the reader with the underlying principles of the planar-laser imaging. LaserInduced Fluorescence ~LIF! and Planar-Doppler Velocimetry ~PDV! are subsequently described in this chapter. Results obtained with the above mentioned methods are illustrated in the last parts of Chapters 4 and 5. Chapter 6 describes Digital Particle-Image Velocimetry ~DPIV!, one of the most innovative flow diagnostic technologies in contemporary fluid mechanics. Following a brief description of the measurement principle involved, post-processing methods, and a valuable section on the sources of errors associated with DPIV measurements are presented. Given the widespread use of the DPIV, even for largescale flows, such as Fig. 1 shows for a river cross section, the application section of this chapter could have been more generous. Chapters 7 and 8 are associated with surface measurement technologies. These methods have evolved considerably in recent years. Surface temperature sensing techniques using thermochromic Liquid Crystals ~LCs! are concisely described in Chapter 7. Examples of LC application to boundary layer heat transfer processes are illustrated at the end of the chapter. Surface pressure and surface shear stress measurements made with the aid of a special coating technology are detailed in Chapter 8. Pressuresensitive paint ~PSP!, shear-sensitive liquid crystal coating ~SSLCC!, and Fringe-Imaging Skin-Friction Interferometry ~FISF! are extensively documented. Practical means to achieve proper lighting, calibration, and data reduction methods are described for each of the methods. Well-illustrated examples support the importance of these methods to nonintrusively measure the pressure distribution and shear stresses in practical aerodynamics and experimental fluid mechanics. Chapter 9 introduces the reader to the visualization methods for compressible flows. A comprehensive introduction to basic optical concepts related to the subsequent described techniques makes this chapter selfcontained. Practical configurations and results obtained with Shadowgraph, Schlieren, Interferometers, and Holography are succinctly presented at the end of the chapter. Chapter 10 and 11 concern 3and 4D flow measurement methods. These quite complex techniques are the most recent developments of the experimental methods technologies and continue

ADAPTIVE FINITE ELEMENT METHODS FOR COMPRESSIBLE TWO-PHASE FLOW

We apply the adaptive streamline diffusion method for compressible flow in conservation variables using P1×P0 finite elements to a conservative model of two-phase flow. The adaptive algorithm is based on an a posteriori error estimate involving certain stability factors related to a linearized dual problem. For a model problem we prove that the stability factors are bounded. We compute the stability factors for some numerical examples in one- and two-space dimensions.

ADAPTIVE FINITE ELEMENT METHODS FOR REACTIVE COMPRESSIBLE FLOW

We extend the adaptive streamline diffusion finite element method for compressible flow in conservation variables using P1× P0 space–time elements to include chemical reactions. The adaptive error control is based on an a posteriori error estimate involving a stability factor, which is estimated numerically. We prove for a model problem that the stability factor is bounded by a moderate constant.

Modified finite element method for compressible flow

Modified finite element method for compressible flow