Compressible Flow Problems Research Articles

A GPU-accelerated implicit meshless method for compressible flows

This paper develops a recently proposed GPU based two-dimensional explicit meshless method (Ma et al., 2014) by devising and implementing an efficient parallel LU-SGS implicit algorithm to further improve the computational efficiency. The capability of the original 2D meshless code is extended to deal with 3D complex compressible flow problems. To resolve the inherent data dependency of the standard LU-SGS method, which causes thread-racing conditions destabilizing numerical computation, a generic rainbow coloring method is presented and applied to organize the computational points into different groups by painting neighboring points with different colors. The original LU-SGS method is modified and parallelized accordingly to perform calculations in a color-by-color manner. The CUDA Fortran programming model is employed to develop the key kernel functions to apply boundary conditions, calculate time steps, evaluate residuals as well as advance and update the solution in the temporal space. A series of two- and three-dimensional test cases including compressible flows over single- and multi-element airfoils and a M6 wing are carried out to verify the developed code. The obtained solutions agree well with experimental data and other computational results reported in the literature. Detailed analysis on the performance of the developed code reveals that the developed CPU based implicit meshless method is at least four to eight times faster than its explicit counterpart. The computational efficiency of the implicit method could be further improved by ten to fifteen times on the GPU.

Numerical study of the blockage length effect on the transient wave in pipe flows

ABSTRACTA pipeline with an extended blockage is modelled as a system of three pipes in series. A computational fluid dynamics approach using the ADINA software is adapted to solve the slightly compressible turbulent flow problem with complex geometry. Two-dimensional axisymmetric computational results are validated by several examples. In practice, the water hammer wave generated by rapid valve closure usually exhibits wave front smearing. The numerical analysis shows that for a blockage length larger than the wave front thickness, the magnitude of the incident wave may be significantly suppressed as a result of the interaction between the blockage and the wave front. By reflecting negative pressure waves toward the incident wave before its maximum pressure arrives at the blockage, the maximum transient pressure may be considerably reduced. Consequently, the blockage behaves as a discrete blockage. However, if the blockage length is large compared with the thickness of the wave front, the incident wave conserves its initial amplitude, and the blockage behaves as an extended blockage. Therefore, the occurrences of discrete and extended blockages depend on the thickness of the wave front and thus on the rapidity of the manoeuvre that generates the transient.

Numerical modeling of compressible fluid flow through elastic porous medium

A numerical technique for solving an initial-boundary value problem of the single compressible fluid flow through elastic porous medium is presented. The model of compressible fluid flow in elastic porous medium is derived within the theory of hyperbolic thermodynamically compatible systems of conservation laws that provides a formulation of mathematically and physically well-posed governing equations of physical processes. Numerical method is based on the WENO–Runge–Kutta algorithm. For the boundary flux evaluation the characteristic decomposition and linearized boundary conditions are used. A two-dimensional problem of the fluid inflow into the saturated porous rectangular domain from its left boundary is studied numerically.

Double-deck structure of the boundary layer in the problem of a compressible flow along a plate with small irregularities on the surface

We consider the problem of a viscous compressible subsonic fluid flow along a flat plate with small periodic irregularities on the surface for large Reynolds numbers. We obtain a formal asymptotic solution with double-deck structure of the boundary layer. We present the results of numerical simulation of flow in the thin boundary layer (i.e., in the near-boundary region).

Constrained Interpolation Profile Conservative Semi-Lagrangian Scheme Based on Third-Order Polynomial Functions and Essentially Non-Oscillatory (CIP-CSL3ENO) Scheme

AbstractWe propose a fully conservative and less oscillatory multi-moment scheme for the approximation of hyperbolic conservation laws. The proposed scheme (CIP-CSL3ENO) is based on two CIP-CSL3 schemes and the essentially non-oscillatory (ENO) scheme. In this paper, we also propose an ENO indicator for the multimoment framework, which intentionally selects non-smooth stencil but can efficiently minimize numerical oscillations. The proposed scheme is validated through various benchmark problems and a comparison with an experiment of two droplets collision/separation. The CIP-CSL3ENO scheme shows approximately fourth-order accuracy for smooth solution, and captures discontinuities and smooth solutions simultaneously without numerical oscillations for solutions which include discontinuities. The numerical results of two droplets collision/separation (3D free surface flow simulation) show that the CIP-CSL3ENO scheme can be applied to various types of fluid problems not only compressible flow problems but also incompressible and 3D free surface flow problems.

An hp‐adaptive unstructured finite volume solver for compressible flows

SummaryThe idea of hp‐adaptation, which has originally been developed for compact schemes (such as finite element methods), suggests an adaptation scheme using a mixture of mesh refinement and order enrichment based on the smoothness of the solution to obtain an accurate solution efficiently. In this paper, we develop an hp‐adaptation framework for unstructured finite volume methods using residual‐based and adjoint‐based error indicators. For the residual‐based error indicator, we use a higher‐order discrete operator to estimate the truncation error, whereas this estimate is weighted by the solution of the discrete adjoint problem for an output of interest to form the adaptation indicator for adjoint‐based adaptations. We perform our adaptation by local subdivision of cells with nonconforming interfaces allowed and local reconstruction of higher‐order polynomials for solution approximations. We present our results for two‐dimensional compressible flow problems including subsonic inviscid, transonic inviscid, and subsonic laminar flow around the NACA 0012 airfoil and also turbulent flow over a flat plate. Our numerical results suggest the efficiency and accuracy advantages of adjoint‐based hp‐adaptations over uniform refinement and also over residual‐based adaptation for flows with and without singularities.

A-posteriori error estimation for the finite point method with applications to compressible flow

An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented.

The low Mach number limit for the compressible flow of liquid crystals

In this paper, we are concerned with the compressible flow of liquid crystals. Based on the convergence–stability principle, it is shown that, for the Mach number sufficiently small, the Cauchy problem of compressible liquid crystal flow has a unique smooth solution on the (finite) time interval where the incompressible liquid crystal flow exists. Furthermore, it is justified that, as the Mach number tends to zero, the smooth solutions converge rigorously to those of the incompressible equations, and the sharp convergence orders are also obtained.

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level.This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.

A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

SummaryWe propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.

A new method of adjustment of inlet boundary for improving heat transfer of mixed convection in a vertical channel

This article is dedicated to a new method for improving heat transfer of mixed convection in a vertical channel from the numerical perspective. An adjustable inlet boundary is proposed in the new method to substitute an inlet that is completely occupied by the amount of fluid provided by forced convection included in mixed convection. According to this new method, the flow reversal occurring in the large Richardson number situation can be eliminated significantly, which results in an improvement in the heat transfer rate of mixed convection. In order to investigate a compressible flow problem, this study focuses on four methods, i.e., Roe scheme, preconditioning, dual time stepping, and the LUSGS (Lower–Upper Symmetric-Gauss–Seidel), to solve the governing equations. Except for the aperture occupied by the mount of fluid provided by forced convection, this study simulates the non-reflecting boundary conditions in other open boundaries. Based on our previous study which has successfully explored a phenomenon of flow reversal, this study intends to enhance the heat transfer rate under this circumstance, and the result has shown a significant improvement, by 29 percentage, comparing with the previous study. Also, a criterion of estimation of the necessity of forced convection included in mixed convection is proposed for enhancing heat transfer rate. Present results and existing results have good agreement.

A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids

In this work, a spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional inviscid compressible flows on structured grids. The compressible form of the discrete Boltzmann–BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is applied. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function. For time accurate solutions, the fourth-order Runge–Kutta scheme is used to discretize the temporal term in the LB equation. Note that the procedure of implementing the SD method to solve the LB equation is nearly the same as that procedure developed in the literature for the solution of the Euler equations. The accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems. A sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size/distribution on the accuracy and performance of the solution. At first, three inviscid compressible flow problems, namely, the stationary isentropic vortex, the shock tube and the shock–vortex interaction are solved by using the SDLBM to demonstrate the accuracy and robustness of the present solution methodology. Results computed for these test cases are in good agreement with the analytical and the available numerical solutions. To more assess the accuracy and robustness of the SDLBM, a third-order finite volume LBM (FVLBM) is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other. It is demonstrated that the SDLBM is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible LB-based flow solvers. As the final problem, the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the SDLBM in reaching to a steady-state solution.

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

Improving a conjugate-heat-transfer immersed-boundary method

Purpose– The purpose of this paper is to provide the current state of the art in the development of a computer code combining an immersed boundary method with a conjugate heat transfer (CHT) approach, including some new findings. In particular, various treatments of the fluid-solid-interface conditions are compared in order to determine the most accurate one. Most importantly, the method is capable of computing a challenging three dimensional compressible turbulent flow past an air cooled turbine vane.Design/methodology/approach– The unsteady Reynolds-averaged Navier–Stokes (URANS) equations are solved within the fluid domain, whereas the heat conduction equation is solved within the solid one, using the same spatial discretization and time-marching scheme. At the interface boundary, the temperatures and heat fluxes within the fluid and the solid are set to be equal using three different approximations.Findings– This work provides an accurate and efficient code for solving three dimensional CHT problems, such as the flow through an air cooled gas turbine cascade, using a coupled immersed boundary (IB) CHT methodology. A one-to-one comparison of three different interface-condition approximations has shown that the two multidimensional ones are slightly superior to the early treatment based on a single direction and that the one based on a least square reconstruction of the solution near the IB minimizes the oscillations caused by the Cartesian grid. This last reconstruction is then used to compute a compressible turbulent flow of industrial interest, namely, that through an air cooled gas turbine cascade. Another interesting finding is that the very promising approach based on wall functions does not combine favourably with the interface conditions for the temperature and the heat flux. Therefore, current and future work aims at developing and testing appropriate temperature wall functions, in order to further improve the accuracy – for a given grid – or the efficiency – for a given accuracy – of the proposed methodology.Originality/value– An accurate and efficient IB CHT method, using a state of the art URANS parallel solver, has been developed and tested. In particular, a detailed study has elucidated the influence of different interface treatments of the fluid-solid boundary upon the accuracy of the computations. Last but not least, the method has been applied with success to solve the well-known CHT problem of compressible turbulent flow past the C3X turbine guide vane.

A p-adaptation method for compressible flow problems using a goal-based error indicator

An accurate calculation of aerodynamic force coefficients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier–Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coefficients. To reduce error in the calculated force coefficients whilst keeping computational cost minimal, we propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, we use control theory to derive an adjoint problem which provides the sensitivity of the functional with respect to changes in the flow variables, and assume that these changes are represented by the local truncation error. In its final form, the goal-based error indicator represents the effect of truncation error on the objective functional, suitably weighted by the adjoint solution. Both flow governing and adjoint equations are solved by the same high-order method, where we allow the degree of the polynomial within an element to vary across the mesh.We initially calculate a steady-state solution to the governing equations using a low polynomial order and use the goal-based error indicator to identify parts of the computational domain that require improved solution accuracy which is achieved by increasing the approximation order. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coefficients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.

A hybrid reconstructed discontinuous Galerkin method for compressible flows on arbitrary grids

A class of reconstructed discontinuous Galerkin methods is described for solving compressible flow problems on arbitrary grids. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy, efficiency, and robustness I. Abstract The discontinuous Galerkin methods 1-25 (DGM) have recently become popular for the solution of systems of conservation laws. Nowadays, they are widely used in computational fluid dynamics, computational acoustics, and computational electromagnetics. The discontinuous Galerkin methods combine two advantageous features commonly associated to finite element and finite volume methods. As in classical finite element methods, accuracy is obtained by means of high-order polynomial approximation within an element rather than by wide stencils as in the case of finite volume methods. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces. In this respect, the methods are therefore similar to finite volume methods. The discontinuous Galerkin methods have many attractive features:1) They have several useful mathematical properties with respect to conservation, stability, and convergence; 2) The method can be easily extended to higher-order (>2 nd

Algebraic approximation of sub-grid scales for the variational multiscale modeling of transport problems

Variational Multiscale (VMS) Finite Element Methods (FEMs) are robust for the development of general formulations for the solution of multiphysics and multiscale transport problems. To obtain a tractable and computationally efficient model, VMS methods often rely on a residual-based algebraic approximation of the sub-grid scales (small or unresolved features of the solution field not captured by the discretization) using a so-called intrinsic time scales matrix, which depends on the problem’s overall differential operator and represents the main model parameter. A novel technique for approximating the intrinsic time scales matrix for generic transport problems in a relatively inexpensive manner (e.g., does not rely on eigenvalue computations) is presented. The method is denoted Transport-Equivalent Scaling (TES) and is based on the monolithic casting of the transport problem as a system of transient–advective–diffusive–reactive (TADR) equations and a subsequent scaling of the coefficient matrices such to preserve each type of transport flux. An algebraic VMS formulation incorporating the TES method is complemented with a discontinuity-capturing (DC) approach and implemented within a FEM solver for the solution of TADR problems. The solution of the global discrete system is accomplished using a generalized-alpha time-stepper together with a globalized inexact Newton–Krylov nonlinear solver. The effectiveness of the TES formulation is verified with the simulation of benchmark incompressible, compressible, and magnetohydrodynamic flow problems. The results demonstrate that the TES method seamlessly handles incompressible–compressible flows in a unified manner (e.g., without assessing the compressibility of the flow). The convergence process using the TES approach and a more standard approximation for the intrinsic time scales, as well as the effect of the DC approach, are also investigated. Analysis of the intrinsic time scales for a one-dimensional incompressible flow model reveals the similitudes and differences between the TES formulation and other conventional methods.

Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall

Phenomena of the flow downward penetration in vertical parallel plates natural convection with an asymmetrically heated wall are investigated numerically. For clarifying the occurrence of flow downward penetration, the width of parallel plates is further broadened and broader than that investigated in the previous study in which the flow reversal was studied in detail. In order to simulate realistically, the compressibility and the viscosity of the working fluid are simultaneously considered, and the non-reflecting boundary is adopted at apertures of plates. The methods of the Roe scheme, preconditioning, and dual time stepping matching LUSGS are coordinated to solve governing equations of a low-speed compressible flow problem. The flow downward penetration, in which fluids are sucked from surroundings outside the outlet to flow into the parallel plates and through the inlet flow out of parallel plates, is composed of a few fluids and firstly observed. Also, two flow reversal caused by the buoyancy force appear in the upper left region, and one flow reversal induced by the flow downward penetration appears in the low right region. The usage of the existing results and present study, an equivalent ratio of the width to the length of parallel plates, of which the heat transfer rate is the same as the single vertical heated plate, can be expediently obtained. In this study, the ratio is about 0.6.

On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

Application of compact schemes in the CUSP framework for strong shock–vortex interaction

We present a methodology for application of compact schemes for unsteady compressible flow problems involving strong shocks and vortices. A compact interpolation formula is combined with a solver based on the ‘convective upwind and split pressure’ (CUSP) scheme [32] in conjunction with a limiter to restrict the numerical fluxes at the cell faces. The interpolation scheme is derived in a manner so that its use in a first derivative calculation results in high accuracy in the wave number space. This is established by the modified wavenumber and the numerical group velocity of the scheme. Use of compact schemes in compressible flow with shocks has mostly been limited to shocks at lower Mach number range. We test our solver by computing a Mach 7 shock interacting with a strong vortex. Results from the lower resolution version of the solver match well with existing data from an ENO scheme based solver. The high resolution scheme shows the break-up of the vortex into smaller vortices in great details, including roll-up of Kelvin–Helmholtz instability generated small scale vortices from slip lines. The presented method can be seamlessly integrated into existing finite volume or finite difference compressible solvers when such high resolution capability is required.