A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

Coupling the lattice‐Boltzmann and spectrin‐link methods for the direct numerical simulation of cellular blood flow

The implementation of a spectrin‐link (SL) red blood cell (RBC) membrane method coupled with a lattice‐Boltzmann (LB) fluid solver is discussed. Details of the methodology are included along with subtleties associated with its integration into a massively parallel hybrid LB finite element (FE) suspension flow solver. A comparison of the computational performance of the coupled LB–SL method with that of the previously implemented LB–FE is given for an isolated RBC and for a dense suspension in Hagen–Poiseuille flow. Validating results for RBCs isolated in shear and parachuting in microvessel flow are also presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled solution of the species conservation equations using unstructured finite‐volume method

A coupled solver was developed to solve the species conservation equations on an unstructured mesh with implicit spatial as well as species‐to‐species coupling. First, the computational domain was decomposed into sub‐domains comprised of geometrically contiguous cells—a process similar to additive Schwarz decomposition. This was done using the binary spatial partitioning algorithm. Following this step, for each sub‐domain, the discretized equations were developed using the finite‐volume method, and solved using an iterative solver based on Krylov sub‐space iterations, that is, the pre‐conditioned generalized minimum residual solver. Overall (outer) iterations were then performed to treat explicitness at sub‐domain interfaces and nonlinearities in the governing equations. The solver is demonstrated for both two‐dimensional and three‐dimensional geometries for laminar methane–air flame calculations with 6 species and 2 reaction steps, and for catalytic methane–air combustion with 19 species and 24 reaction steps. It was found that the best performance is manifested for sub‐domain size of 2000 cells or more, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block (coupled) Gauss–Seidel procedure. All calculations were performed on a single processor machine. The largest calculations were performed for about 355 000 cells (4.6 million unknowns) and required 900 MB of peak runtime memory and 19 h of CPU on a single processor. Copyright © 2009 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace methods used in solving two‐dimensional transient two‐phase flows

This paper investigates the performance of preconditioned Krylov subspace methods used in a previously presented two‐fluid model developed for the simulation of separated and intermittent gas–liquid flows. The two‐fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two‐step semi‐implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi‐CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, τ) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable. Copyright © 1999 John Wiley & Sons, Ltd.     

Monte Carlo method of radiative transfer applied to a turbulent flame modeling with LES

Radiative transfer plays an important role in the numerical simulation of turbulent combustion. However, for the reason that combustion and radiation are characterized by different time scales and different spatial and chemical treatments, the radiation effect is often neglected or roughly modelled. The coupling of a large eddy simulation combustion solver and a radiation solver through a dedicated language, CORBA, is investigated. Two formulations of Monte Carlo method (Forward Method and Emission Reciprocity Method) employed to resolve RTE have been compared in a one-dimensional flame test case using three-dimensional calculation grids with absorbing and emitting media in order to validate the Monte Carlo radiative solver and to choose the most efficient model for coupling. Then the results obtained using two different RTE solvers (Reciprocity Monte Carlo method and Discrete Ordinate Method) applied on a three-dimensional flame holder set-up with a correlated-k distribution model describing the real gas medium spectral radiative properties are compared not only in terms of the physical behavior of the flame, but also in computational performance (storage requirement, CPU time and parallelization efficiency). To cite this article: J. Zhang et al., C. R. Mecanique 337 (2009).     

Execution of a parallel edge-based Navier–Stokes solver on commodity graphics processor units

The implementation of an edge-based three-dimensional Reynolds Average Navier–Stokes solver for unstructured grids able to run on multiple graphics processing units (GPUs) is presented. Loops over edges, which are the most time-consuming part of the solver, have been written to exploit the massively parallel capabilities of GPUs. Non-blocking communications between parallel processes and between the GPU and the central processor unit (CPU) have been used to enhance code scalability. The code is written using a mixture of C++ and OpenCL, to allow the execution of the source code on GPUs. The Message Passage Interface (MPI) library is used to allow the parallel execution of the solver on multiple GPUs. A comparative study of the solver parallel performance is carried out using a cluster of CPUs and another of GPUs. It is shown that a single GPU is up to 64 times faster than a single CPU core. The parallel scalability of the solver is mainly degraded due to the loss of computing efficiency of the GPU when the size of the case decreases. However, for large enough grid sizes, the scalability is strongly improved. A cluster featuring commodity GPUs and a high bandwidth network is ten times less costly and consumes 33% less energy than a CPU-based cluster with an equivalent computational power.     

A high-order solver for simulating vortex-induced vibrations using the sliding-mesh spectral difference method and hybrid grids

We present a high-order solver for simulating vortex-induced vibrations (VIVs) at very challenging situations, for example, VIVs of a row of very closely placed objects with large relative displacements. This solver works on unstructured hybrid grids by employing the high-order tensor-product spectral difference method for quadrilateral grids and the Raviart-Thomas spectral difference method for triangular grids. To deal with the challenging situations where a traditional conforming moving mesh is incapable, we split a computational domain into nonoverlapping subdomains, where each interior subdomain encloses an object and moves freely with respect to its neighbors. A nonuniform sliding-mesh method that ensures high-order accuracy is developed to deal with sliding interfaces between subdomains. A monolithic approach is adopted to seamlessly couple the fluid and solid vibration equations. Moreover, the solver is parallelized to further improve its efficiency on distributed-memory computers. Through a series of numerical tests, we demonstrate that this solver is high-order accurate for both inviscid and viscous flows and has good parallel efficiency, making it ideal for VIV studies.     

Multi‐resolution analysis for high accuracy and efficiency of Euler computation

A multi‐resolution analysis (MRA) is proposed for efficient flow computation with preserving the high‐order numerical accuracy of a conventional solver. In the MRA process, the smoothness of a flow pattern is assessed by the difference between original flow property values, and the values approximated by high‐order interpolating polynomial in decomposition. Insignificant data in smooth region are discarded, and flux computation is performed only where crucial features of a solution exist. The reduction of expensive flow computation improves the overall computational efficiency. In order to maintain the high‐order accuracy, modified thresholding procedure restricts the additional error introduced by the thresholding below the order of accuracy of a conventional solver. The practical applicability of the MRA method is validated in various continuous and discontinuous flow problems. The MRA stably computes the Euler equations for continuous and discontinuous flow problems and maintains the accuracy of a conventional solver. Overall, it substantially enhances the computational efficiency of the conventional third‐order accurate solver. Copyright © 2014 John Wiley & Sons, Ltd.     

A high-order discontinuous Galerkin method for simulating incompressible fluid-thermal-structural interaction problems

This paper presents a framework for the application of the discontinuous Galerkin(DG) finite element method to the multi-physics simulation of the solid thermal deformation interacting with incompressible flow problems in two-dimensions. Recent applications of the DG method are primarily for thermoelastic problems in a solid domain or fluid-structure interaction problems without heat transfer. Based on a recently published conjugate heat transfer solver, the incompressible Navier-Stokes equation, the fluid advection-diffusion equation, the Boussinesq term, the solid heat equation and the solid linear elastic equation are solved using an explicit DG formulation. A Dirichlet-Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux computed at interface quadrature points in the fluid-solid interface. Formal hp convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved for each solver. The algorithm is then further validated against several existing benchmark cases including the in-plane loaded square, the Timoshenko Beam, the laminated beam subject to thermal-loads and the lid-driven cavity with a flexible bottom wall. The computational effort demonstrates that for all cases the highest order accurate algorithm has several magnitudes lower error than the second-order schemes for a given computational effort. It is a strong justification for the development of such high order discretisations. The solver can be employed to predict thermal deformation of a structure due to convective and conductive heat transfer at low Mach, such as chip deformation on a printed circuit board, wave-guide structure optimization, thermoelectric cooler simulation, and optics mounting method verification.     

A bore-capturing finite volume method for open-channel flows

A high-resolution finite volume hydrodynamic solver is presented for open-channel flows based on the 2D shallow water equations. This Godunov-type upwind scheme uses an efficient Harten–Lax–van Leer (HLL) approximate Riemann solver capable of capturing bore waves and simulating supercritical flows. Second-order accuracy is achieved by means of MUSCL reconstruction in conjunction with a Hancock two-stage scheme for the time integration. By using a finite volume approach, the computational grid can be irregular which allows for easy boundary fitting. The method can be applied directly to model 1D flows in an open channel with a rectangular cross-section without the need to modify the scheme. Such a modification is normally required for solving the 1D St Venant equations to take account of the variation of channel width. The numerical scheme and results of three test problems are presented in this paper. © 1998 John Wiley & Sons, Ltd.     

The complementary RANS equations for the simulation of viscous flows

A complementary set of Reynolds‐averaged Navier–Stokes (RANS) equations has been developed for steady incompressible, turbulent flows. The method is based on the Helmholtz decomposition of the velocity vector field into a viscous and a potential components. In the complementary RANS solver a potential solution coexists with a viscous solution with the purpose of contributing to a fastest decay of the viscous solution in the far field. The proposed complementary RANS equations have been validated for steady laminar and turbulent flows. The computational results show that the complementary RANS solver is able to produce less grid‐dependent solutions than a conventional RANS solver. Copyright © 2005 John Wiley & Sons, Ltd.     

Clustering solver for displacement-based numerical homogenization

Based on strain-clustering via k-means, we decompose computational domain into clusters of possibly disjoint cells. Assuming cells in each cluster take the same strain, we reconstruct displacement field. We further propose a new way to condensate the governing equations of displacement-based finite element method to reduce the complexity while maintain the accuracy. Numerical examples are presented to illustrate the efficiency of the clustering solver. Numerical convergence studies are performed for the examples. Avoiding complexities which is common in existing clustering analysis methods, the proposed clustering solver is easy to implement, particularly for numerical homogenization using commercial softwares.     

Solution of incompressible flows with or without a free surface using the finite volume method on unstructured triangular meshes

An incompressible Navier–Stokes solver based on a cell‐centre finite volume formulation for unstructured triangular meshes is developed and tested. The solution methodology makes use of pseudocompressibility, whereby the convective terms are computed using a Godunov‐type second‐order upwind finite volume formulation. The evolution of the solution in time is obtained by subiterating the equations in pseudotime for each physical time step, with the pseudotime step set equal to infinity. For flows with a free surface the computational mesh is fitted to the free surface boundary at each time step, with the free surface elevation satisfying a kinematic boundary condition. A ‘leakage coefficient’, ε, is introduced for the calculation of flows with a free surface in order to control the leakage of flow through the free surface. This allows the assumption of stationarity of mesh points to be made during the course of pseudotime iteration. The solver is tested by comparing the output with a wide range of documented published results, both for flows with and without a free surface. The presented results show that the solver is robust. © 1999 John Wiley & Sons, Ltd.     

Fluid–structure interaction analysis of flexible turbomachinery

A method for the performance computation of an expandable-impeller pump is developed and validated. Large deformations of the highly flexible pump impellers result in a strong coupling between the impeller and fluid flow. The computational method therefore requires simultaneous solution of fluid flow and structural response. OpenFOAM provides the flow and mesh motion solvers and is coupled to an author-developed structural solver in a tightly coupled approach using a fixed-point iteration. The structural deformations are time-dependent because the material exhibits stress relaxation. The time-constant of the relaxation, however, is very large, thereby allowing quasi-steady simulations. A water-tunnel test of a viscoelastic hydrofoil is employed to validate the solver. Simulations of the test problem show good agreement with the experimental results and demonstrate the need for several sub-iterations of the solver even for the quasi-steady simulations.     

On the role of (weak) compressibility for fluid-structure interaction solvers

In the present study, a weakly compressible formulation of the Navier-Stokes equations is developed and examined for the solution of fluid-structure interaction (FSI) problems. Newtonian viscous fluids under isothermal conditions are considered, and the Murnaghan-Tait equation of state is employed for the evaluation of mass density changes with pressure. A pressure-based approach is adopted to handle the low Mach number regime, ie, the pressure is chosen as primary variable, and the divergence-free condition of the velocity field for incompressible flows is replaced by the continuity equation for compressible flows. The approach is then embedded into a partitioned FSI solver based on a Dirichlet-Neumann coupling scheme. It is analytically demonstrated how this formulation alleviates the constraints of the instability condition of the artificial added mass effect, due to the reduction of the maximal eigenvalue of the so-called added mass operator. The numerical performance is examined on a selection of benchmark problems. In comparison to a fully incompressible solver, a significant reduction of the coupling iterations and the computational time and a notable increase in the relaxation parameter evaluated according to Aitken's Δ² method are observed.     

Aeroelastic prediction of the limit cycle oscillations of a cropped delta wing

The flutter and limit cycle oscillation (LCO) behavior of a cropped delta wing are investigated using a newly developed computational aeroelastic solver. This computational model includes a well-validated Euler finite difference solver coupled to a high-fidelity finite element structural solver. The nonlinear structural model includes geometric nonlinearities which are modelled using a co-rotational formulation. The LCOs of the cropped delta wing are computed and the results are compared to previous computations and to experiment. Over the range of dynamic pressures for which experimental results are reported, the LCO magnitudes computed using the current model are comparable to those from a previous computation which used a lower-order von Karman structural model. However, for larger dynamic pressures, the current computational model and the model which used the von Karman theory start to differ significantly, with the current model predicting larger deflections for a given dynamic pressure. This results in a LCO curve which is in better qualitative agreement with experiment. Flow features which were present in the previous computational model such as a leading edge vortex and a shock wave are enhanced in the current model due to the prediction of larger deflections and rotations at the higher dynamic pressures.     

A finite volume method for viscous incompressible flows using a consistent flux reconstruction scheme

An incompressible Navier–Stokes solver using curvilinear body‐fitted collocated grid has been developed to solve unconfined flow past arbitrary two‐dimensional body geometries. In this solver, the full Navier–Stokes equations have been solved numerically in the physical plane itself without using any transformation to the computational plane. For the proper coupling of pressure and velocity field on collocated grid, a new scheme, designated ‘consistent flux reconstruction’ (CFR) scheme, has been developed. In this scheme, the cell face centre velocities are obtained explicitly by solving the momentum equations at the centre of the cell faces. The velocities at the cell centres are also updated explicitly by solving the momentum equations at the cell centres. By resorting to such a fully explicit treatment considerable simplification has been achieved compared to earlier approaches. In the present investigation the solver has been applied to unconfined flow past a square cylinder at zero and non‐zero incidence at low and moderate Reynolds numbers and reasonably good agreement has been obtained with results available from literature. Copyright © 2006 John Wiley & Sons, Ltd.     

An approximate‐state Riemann solver for the two‐dimensional shallow water equations with porosity

PorAS, a new approximate‐state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd.     

The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation

Godunov‐type algorithms are very attractive for the numerical solution of discontinuous flows. The reconstruction of the profile inside the cells is crucial to scheme performance. The non‐linear generalization of the discontinuous profile method (DPM) presented here for the modelling of two‐phase flow in pipes uses a discontinuous reconstruction in order to capture shocks more efficiently than schemes using continuous functions. The reconstructed profile is used to define the Riemann problem at cell interfaces by averaging of the components of the variable in the base of eigenvectors over their domain of dependence. Intercell fluxes are computed by solving the Riemann problem with an approximate‐state solver. The adapted treatment of boundary conditions is essential to ensure the quality of the computational results and a specific procedure using virtual cells at both extremities of the computational domain is required. Internal boundary conditions can be treated in the same way as external ones. Application of the DPM to test cases is shown to improve the quality of computational results significantly. Copyright © 2001 John Wiley & Sons, Ltd.     

A harmonic balance technique for the reduced-order computation of vortex-induced vibration

We present a harmonic balance (HB) method to model frequency lock-in effect during vortex-induced vibration (VIV) of elastically mounted circular cylinder and a flexible riser section in a freestream uniform flow. The fluid flow and structure are coupled by a fixed-point iteration process through a frequency updating algorithm. By minimizing the structural residual in the standard least-square norm, the convergence of HB-based fixed-point algorithm is achieved for a range of reduced velocity. To begin with, the HB solver is first assessed for a periodic unsteady flow around a stationary circular cylinder. A freely vibrating circular cylinder is then adopted for the reduced-order computation of VIV at low Reynolds numbers of Re=100 and 180 with one- and two-degrees-of-freedom. The coupled VIV dynamics and the frequency lock-in phenomenon are accurately captured. The results show that the HB solver is able to predict the amplitude of vibration, frequency and forces comparable to its time domain counterpart, while providing a significant reduction with regard to overall computational cost. The proposed new scheme is then demonstrated for a fully-coupled three dimensional (3D) analysis of a linear-elastic riser section undergoing vortex-induced vibration in the lock-in range. The results reveal the 3D effects through isosurfaces of streamwise vorticity blobs distributed over the span of flexible riser section. In comparison to time domain results, the 3D flow-structure interactions are accurately predicted while providing a similar speed up rate that of 2D simulations. This further corroborates that the HB solver can be extended to 3D flow-structure dynamics without compromising efficiency and accuracy.