Isentropic Vortex Research Articles

Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

The accuracy of gradient reconstruction methods on unstructured meshes is analyzed both mathematically and numerically. Mathematical derivations reveal that, for gradient reconstruction based on the Green-Gauss theorem (the GG methods), if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process, which rarely occurs, second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient. However, gradient reconstruction based on the least-squares approach (the LSQ methods) is at least first-order on arbitrary unstructured grids. Verifications are performed on typical isotropic grid stencils by analyzing the relationship between the discretization error of gradient reconstruction and the discretization error of the face midpoint value approximation of a given analytic function. Meanwhile, the numerical accuracy of gradient reconstruction methods is examined with grid convergence study on typical isotropic grids. Results verify the phenomenon of accuracy degradation for the GG methods when the face midpoint value condition is not satisfied. The LSQ methods are proved to be at least first-order on all tested isotropic grids. To study gradient accuracy effects on inviscid flow simulation, solution errors are quantified using the Method of Manufactured Solutions (MMS) which was validated before adoption by comparing with an exact solution case, i.e., the 2-dimensional (2D) inviscid isentropic vortex. Numerical results demonstrate that the order of accuracy (OOA) of gradient reconstruction is crucial in determining the OOA of numerical solutions. Solution accuracy deteriorates seriously if gradient reconstruction does not reach first-order.

Free-stream preserving linear-upwind and WENO schemes on curvilinear grids

Applying high-order finite-difference schemes, like the extensively used linear-upwind or WENO schemes, to curvilinear grids can be problematic. If the scheme doesn't satisfy the geometric conservation law, the geometrically induced error from grid Jacobian and metrics evaluation can pollute the flow field, and degrade the accuracy or cause the simulation failure even when uniform flow imposed, i.e. free-stream preserving problem. In order to address this issue, a method for general linear-upwind and WENO schemes preserving free-stream on stationary curvilinear grids is proposed. Following Lax-Friedrichs splitting, this method rewrites the numerical flux into a central term, which achieves free-stream preserving by using symmetrical conservative metric method, and a numerical dissipative term with a local difference form of conservative variables for neighboring grid-point pairs. In order to achieve free-stream preservation for the latter term, the local differences are modified to share the same Jacobian and metric terms evaluated by high order schemes. In addition, this method allows a simple hybridization switching between linear-upwind and WENO schemes is proposed for improving computational efficiency and reducing numerical dissipation. A number of testing cases including free-stream, isentropic vortex convection, double Mach reflection, flow past a cylinder and supersonic wind tunnel with a step are computed to verify the effectiveness of this method.

Hybrid recursive regularized thermal lattice Boltzmann model for high subsonic compressible flows

A thermal lattice Boltzmann model with a hybrid recursive regularization (HRR) collision operator is developed on standard lattices for simulation of subsonic and sonic compressible flows without shock. The approach is hybrid: mass and momentum conservation equations are solved using a lattice Boltzmann solver, while the energy conservation is solved under entropy form with a finite volume solver. The defect of Galilean invariance related to Mach number is corrected by the third order equilibrium distribution function, supplemented by an additional correcting term and hybrid recursive regularization. The proposed approach is assessed considering the simulation of i) an isentropic vortex convection, ii) a two dimensional acoustic pulse and iii) non-isothermal Gaussian pulse with Ma number in range of 0 to 1. Numerical simulations demonstrate that the flaw in Galilean invariance is effectively eliminated by the compressible HRR model. At last, the compressible laminar flows over flat plate at Ma number of 0.3 and 0.87, Reynolds number of 105 are considered to validate the capture of viscous and diffusive effects.

An analysis of the stability of the least square finite difference scheme and its shock capturing ability in inviscid flows

SummaryA meshless method – The Least Square Finite Difference scheme (LSFD) with diffusion is analyzed and applied to inviscid flows. The scheme is made second‐order by using a modified difference in the formulation of LSFD. Several numerical experiments, namely the Sod shock tube and the shallow water problems, are carried out and, in the limelight of the results obtained, the ability of the scheme to resolve shock wave, rarefaction wave, and contact discontinuity is discussed. The conditional stability of the LSFD scheme is established. The LSFD uses weights to diagonalize the least square matrix resulting in the spatial discretization in order to gain computational time. We prove that there exists a unique weight for the resulting optimization problem. The weighted version of LSFD is used to solve the isentropic vortex problem numerically and the results are used to discuss the dissipative nature of the scheme. Five configurations of the two‐dimensional Riemann problems are used in our numerical experiments. The capability of the scheme to capture the complex interaction of multiple planar waves is discussed in the limelight of the results on the Riemann problems. The result of the shock reflection problem shows that the scheme is minimally dissipative and leads to sharp and well‐resolved shocks.

Unsteady Discontinuous Galerkin Method of a High Order of Accuracy for Modeling Turbulent Flows

This paper presents a brief description of the Zhukovsky Central Aerohydrodynamic Institute (TsAGI) code based on the Galerkin method of a high order of accuracy with discontinuous basic functions. The functions is reconstructed for conservative variables. Gradients of the variables are determined using the Bassi–Rebay 2 method. For integrating, the Gaussian quadrature rules are used. The coordinates are transformed by serendipian elements. In computing by schemes of an order higher than the second order, the curvature of grid lines is taken into account. A comparison with finite volume methods is performed, including the WENO method with constant weights and a single quadrature point on a cell face. The classical tests are used, namely, a subsonic flow around a circular cylinder in an ideal gas, the diagonal convection of a two-dimensional isentropic vortex, and the decay of the Taylor–Green vortex.

Time integration of unsteady nonhydrostatic equations with dual time stepping and multigrid methods

A time integration method based on dual time stepping coupled with a multigrid and implicit iterative solver is discussed for overcoming the temporal stiffness associated with modeling unsteady atmospheric flows. Adopting the dual time stepping allows solving an unsteady flow problem as a steady-state one by introducing a pseudo-time derivative into the governing equations. Consequently, the model uses a large time step and leads to a reduction in computational time. In the framework of the dual time stepping, a full approximation storage multigrid and lower–upper symmetric Gauss–Seidel implicit solver are employed to accelerate convergence in a structured, quadrilateral grid. The discretization of a physical-time derivative is based on a second-order backward differentiation formula or a family of singly diagonally implicit Runge–Kutta methods. The spatial terms of nonhydrostatic equations are discretized with a second-order Roe scheme in the finite volume method. To assess the performance of the model in terms of accuracy and efficiency, which are indicated by the normalized error norms and wall-clock time, respectively, three idealized tests are performed with the unsteady flow problems of isentropic vortex, rising thermal bubble and inertia-gravity waves. The results of numerical experiments consistently demonstrate that the dual time stepping coupled with explicit-first, singly diagonally implicit Runge–Kutta and multigrid methods is more efficient than the three-stage, third-order explicit Runge–Kutta method with comparable accuracy. The numerical experiment also indicates that the dual time stepping is more efficient when a stretched grid is employed. The efficiency in the framework of the dual time stepping is dependent on various numerical factors, including a multigrid level and cycle, Butcher coefficients, an implicit solver and a physical-time step.

Toward vortex identification based on local pressure-minimum criterion in compressible and variable density flows

We propose a dynamical vortex definition (the ‘$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition’) for flows dominated by density variation, such as compressible and multi-phase flows. Based on the search of the pressure minimum in a plane,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$defines a vortex to be a connected region with two negative eigenvalues of the tensor$\unicode[STIX]{x1D64E}^{M}+\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$. Here,$\unicode[STIX]{x1D64E}^{M}$is the symmetric part of the tensor product of the momentum gradient tensor$\unicode[STIX]{x1D735}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$and the velocity gradient tensor$\unicode[STIX]{x1D735}\unicode[STIX]{x1D66A}$, with$\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$denoting the symmetric part of momentum-dilatation gradient tensor$\unicode[STIX]{x1D735}(\unicode[STIX]{x1D717}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$, and$\unicode[STIX]{x1D717}\equiv \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D66A}$, the dilatation rate scalar. The$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition is examined and compared with the$\unicode[STIX]{x1D706}_{2}$definition using the analytical isentropic Euler vortex and several other flows obtained by direct numerical simulation (DNS) – e.g. liquid jet breakup in a gas, a compressible wake, a compressible turbulent channel and a hypersonic turbulent boundary layer. For low Mach number ($M\lesssim 5$) compressible flows, the$\unicode[STIX]{x1D706}_{2}$and$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$structures are nearly identical, so that the$\unicode[STIX]{x1D706}_{2}$method is still valid for low$M$compressible flows. But, the$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition is needed for studying vortex dynamics in highly compressible and strongly varying density flows.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools

First- and second-order accurate numerical methods, implemented for CPUs, underpin the majority of industrial CFD solvers. Whilst this technology has proven very successful at solving steady-state problems via a Reynolds Averaged Navier–Stokes approach, its utility for undertaking scale-resolving simulations of unsteady flows is less clear. High-order methods for unstructured grids and GPU accelerators have been proposed as an enabling technology for unsteady scale-resolving simulations of flow over complex geometries. In this study we systematically compare accuracy and cost of the high-order Flux Reconstruction solver PyFR running on GPUs and the industry-standard solver STAR-CCM+ running on CPUs when applied to a range of unsteady flow problems. Specifically, we perform comparisons of accuracy and cost for isentropic vortex advection (EV), decay of the Taylor–Green vortex (TGV), turbulent flow over a circular cylinder, and turbulent flow over an SD7003 aerofoil. We consider two configurations of STAR-CCM+: a second-order configuration, and a third-order configuration, where the latter was recommended by CD-adapco for more effective computation of unsteady flow problems. Results from both PyFR and STAR-CCM+ demonstrate that third-order schemes can be more accurate than second-order schemes for a given cost e.g. going from second- to third-order, the PyFR simulations of the EV and TGV achieve 75× and 3× error reduction respectively for the same or reduced cost, and STAR-CCM+ simulations of the cylinder recovered wake statistics significantly more accurately for only twice the cost. Moreover, advancing to higher-order schemes on GPUs with PyFR was found to offer even further accuracy vs. cost benefits relative to industry-standard tools.

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.

On application of high-order compact finite-difference schemes to compressible vorticity confinement method

The main goal of this study is to assess the application of high-order compact finite-difference schemes for the solution of the Euler equations in conjunction with the compressible vorticity confinement method on both uniform Cartesian and curvilinear grids. Here, the spatial discretization of the governing equations is performed by the fourth-order compact finite-difference scheme and the temporal term is discretized by the fourth-order Runge–Kutta method. To stabilize the numerical solution, appropriate dissipation terms are applied and a detail assessment is performed to study the effects of the values of confinement and dissipation coefficients on the solution to reasonably preserve the structure of vortical flows. The accuracy and performance of the proposed solution procedure by applying the fourth-order compact finite-difference scheme are also examined by comparison with the solution obtained by the second-order central finite-difference scheme. Low-pass high-order filters are also applied to the fourth-order compact finite-difference scheme to investigate the performance of the vorticity confinement with the filtering scheme. The advection of a 2D isentropic vortex and a shock–vortex interaction problem are two test cases simulated for the assessment of the present solution methodology. The study shows that the high-order compact finite-difference schemes in conjunction with the vorticity confinement method can reasonably preserve the structure of vortical flows in coarse uniform Cartesian and curvilinear grids. Indications are that the present solution methodology is accurate and effective for solving compressible flow problems with vortical structures when coarse grids are used.

Improvement of weighted compact scheme with multi-step strategy for supersonic compressible flow

With shock capturing capability and spectral like resolution, weighted compact schemes are good choices for supersonic turbulent flow simulation. However, because of the implicit nature of compact schemes, accuracy decrease at one point would contaminate the overall solution. The recently proposed compact-reconstructed WENO (CRWENO) scheme (Ghosh and Baeder, 2012) is an efficient weighted compact scheme. Analysis shows that the accuracy of the CRWENO scheme is only second order for points near discontinuities. In this paper, a multi-step strategy is proposed to increase the accuracy of the CRWENO scheme near discontinuities. Instead of constructing the fifth order compact scheme directly from three third order compact schemes, two fourth order compact schemes are first constructed. The final fifth order scheme is then obtained by these two fourth order schemes. In this way, the accuracy of the CRWENO scheme is increased to third order for points near discontinuities. The dissipation and dispersion properties of the new scheme are analyzed by employing the modified wave number method. Several examples, including linear advection cases, three one-dimensional shock tube problems, two-dimensional isentropic vortex convection, double Mach reflection, and shock/mixing layer interaction problems, are considered to assess the present method. Numerical results show that the present multi-step CRWENO scheme has lower dissipation and higher resolution in both smooth and discontinuous regions compared with the original scheme.

Comparison of discontinuous Galerkin time integration schemes for the solution of flow problems with deformable domains

The computational efficiency of numerical solvers is known to strongly depend on the chosen time integration scheme. Thus, when solving viscous flow problems on time-varying domains, an efficient and reliable solver is one of the prerequisites for a successful solution. For this reason, we provide a comparison of two time integration schemes in terms of numerical error and CPU time usage.The contribution of the paper lies in the mutual comparison of two different approaches for time integration of the compressible Navier–Stokes equations in arbitrary Lagrangian–Eulerian (ALE) description that are spatially discretised by the discontinuous Galerkin finite element method. The computational ability of implicit Crank–Nicolson scheme and explicit three-step Runge–Kutta scheme, computational performance of which is improved by the local time-stepping technique, are tested on two flow problems: the propagation of an isentropic vortex, which has a known analytical solution, and the viscous flow around an oscillating NACA 0012 airfoil. All numerical simulations are carried out on unstructured triangular meshes by using a continuous mapping between the reference and time-varying domains.

Theoretical and numerical analysis of nonconforming grid interface for unsteady flows

A framework is proposed for the spectral analysis of the numerical scheme applied on a nonconforming grid interface between two structured blocks for the two-dimensional advection equation, in a cell-centered finite-volume formalism. The conservative flux computation on the nonconforming grid interface is based on the sum of partial fluxes computed with the same numerical scheme used for a standard cell interface. This framework is used to analyze the effect of grid refinement or coarsening on the stability of the second-order centered scheme. New theoretical results are given and compared to numerical results. Considering the convection of a two-dimensional isentropic vortex, the refinement/coarsening are shown to be the cause of instabilities, poor accuracy and reflection of high-frequency waves. A new approach to compute partial fluxes, which is based both on a high-order extrapolation that accounts for the local metric and on a Riemann solver, is then proposed to reduce spurious modes.

On the Geometric Conservation Law for Unsteady Flow Simulations on Moving Mesh

When using the dynamic mesh method to deal with moving boundary problems, the Geometric Conservation Law (GCL) must be considered carefully. This paper reviews the study on GCL problem of dynamic grid in finite volume framework based on Arbitrary Lagrangian-Eulerian (ALE) methods. Several common approaches to satisfy discretized geometric conservation law (DGCL) are studied and simplified to a uniform form. The uniform flow and the isentropic vortex are tested to validate the geometric conservation property and the temporal-accuracy. Numerical results illustrate that although the violating of GCL does not pollute the original time-accuracy of numerical schemes for the governing equations, it may introduce extra artificial errors. When the “Adding Source Term” methods are adopted, the temporal accuracy order of face velocity scheme must match the temporal accuracy order of numerical schemes for the governing equations, otherwise it will pollute the original temporal accuracy.

Adaptive IMEX schemes for high-order unstructured methods

We present an adaptive implicit–explicit (IMEX) method for use with high-order unstructured schemes. The proposed method makes use of the Gerschgorin theorem to conservatively estimate the influence of each individual degree of freedom on the spectral radius of the discretization. This information is used to split the system into implicit and explicit regions, adapting to unsteady features in the flow. We dynamically repartition the domain to balance the number of implicit and explicit elements per core. As a consequence, we are able to achieve an even load balance for each implicit/explicit stage of the IMEX scheme. We investigate linear advection–diffusion, isentropic vortex advection, unsteady laminar flow over an SD7003 airfoil, and turbulent flow over a circular cylinder. Results show that the proposed method consistently yields a stable discretization, and maintains the theoretical order of accuracy of the high-order spatial schemes.

Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

SUMMARYIn this paper, a high‐order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second‐order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG‐MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.

Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite-difference schemes on moving and deforming grids

The role of a geometric conservation law (GCL) on a finite-difference scheme is revisited for conservation laws, and the conservative forms of coordinate-transformation metrics are introduced in general dimensions. The sufficient condition of a linear high-order finite-difference scheme is arranged in detail, for which the discretized conservative coordinate-transformation metrics and Jacobian satisfy the GCL identities on three-dimensional moving and deforming grids. Subsequently, the geometric interpretation of the metrics and Jacobian discretized by a linear high-order finite-difference scheme is discussed, and only the symmetric conservative forms of the discretized metrics and Jacobian are shown to have the appropriate geometric structures. The symmetric and asymmetric conservative forms of the metrics and Jacobian are examined by the computation of an inviscid compressible fluid on highly-skewed stationary and deforming grids using sixth-order compact and fourth-order explicit central-difference schemes, respectively. The resolution of the isentropic vortex and the robustness of the computation are improved by employing symmetric conservative forms on the coordinate-transformation metrics and Jacobian that have an appropriate geometry background. An integrated conservation of conservative quantities is also attained on the deforming grid when symmetric conservative forms are adopted to the time metrics and Jacobian.

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.

A High-Order Moving Mesh Kinetic Scheme Based on WENO Reconstruction for Compressible Flows on Unstructured Meshes

In this paper, we present a high-order moving mesh (HMM) kinetic scheme for compressible flow computations on unstructured meshes. To construct the scheme, we employ the frame of the remapping-free ALE-type kinetic method (Ni et al. in J Comput Phys 228:3154---3171, 2009) to get the discretization of compressible system. For the space accuracy, we use the weighted essential non-oscillatory reconstruction on the adaptive moving mesh from Tang and Tang (SIAM J Numer Anal 41:487---515 2003) to achieve time accuracy,we make use of the kinetic flux which includes time accurate integral, and thus obtain a HMM scheme. A number of numerical examples are given, especially an isentropic vortex problem to show the convergence order of the scheme. Numerical results demonstrate the accuracy and robustness of the scheme.