Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility

Transitions in large eddy simulation of box turbulence

One promising decomposition of turbulent dynamics is that into building blocks such as equilibrium and periodic solutions and orbits connecting these. While the numerical approximation of such building blocks is feasible for flows in small domains and at low Reynolds numbers, computations in developed turbulence are currently out of reach because of the large number of degrees of freedom necessary to represent Navier-Stokes flow on all relevant spatial scales. We mitigate this problem by applying large eddy simulation (LES), which aims to model, rather than resolve, motion on scales below the filter length, which is fixed by a model parameter. By considering a periodic spatial domain, we avoid complications that arise in LES modelling in the presence of boundary layers. We consider the motion of an LES fluid subject to a constant body force of the Taylor-Green type as the separation between the forcing length scale and the filter length is increased. In particular, we discuss the transition from laminar to weakly turbulent motion, regulated by simple invariant solution, on a grid of $32^3$ points

Convergence Acceleration of Runge-Kutta Schemes for Solving the Navier-Stokes Equations

The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 can be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. This RK/implicit scheme is used as a smoother for multigrid. Fourier analysis is applied to determine damping properties. Numerical dissipation operators based on the Roe scheme, a matrix dissipation, and the CUSP scheme are considered in evaluating the RK/implicit scheme. In addition, the effect of the number of RK stages is examined. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. Turbulent flows over an airfoil and wing at subsonic and transonic conditions are computed. The effects of the cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 x 10(exp 6) and 100 x 10(exp 6). It is demonstrated that the implicit preconditioner can reduce the computational time of a well-tuned standard RK scheme by a factor between four and ten

Convergence Acceleration for Multistage Time-Stepping Schemes

The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 could be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. Numerical dissipation operators (based on the Roe scheme, a matrix formulation, and the CUSP scheme) as well as the number of RK stages are considered in evaluating the RK/implicit scheme. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. In two dimensions, turbulent flows over an airfoil at subsonic and transonic conditions are computed. The effects of mesh cell aspect ratio on convergence are investigated for Reynolds numbers between 5.7 x 10(exp 6) and 100.0 x 10(exp 6). Results are also obtained for a transonic wing flow. For both 2-D and 3-D problems, the computational time of a well-tuned standard RK scheme is reduced at least a factor of four

Towards Efficient and Scalable Discontinuous Galerkin Methods for Unsteady Flows

openNegli ultimi anni, la crescente disponibilit`a di risorse computazionali ha contribuito alla diffusione della fluidodinamica computazionale per la ricerca e per la progettazione industriale. Uno degli approcci pi promettenti si basa sul metodo agli elementi finiti discontinui di Galerkin (dG). Nell’ambito di queste metodologie, il contributo della tesi e' triplice. Innanzi- tutto, il lavoro introduce un algoritmo di parallelizzazione ibrida MPI/OpenMP per l’utilizzo efficiente di risorse di super calcolo. In secondo luogo, propone strategie di soluzione efficienti, scalabili e con limitata allocazione di memoria per la soluzione di problemi complessi. Infine, confronta le strategie di soluzione introdotte con nuove tecniche di discretizzazione dette “ibridizzabili”, su problemi riguardanti la soluzione delle equazioni di Navier–Stokes non stazionarie. L’efficienza computazionale e' stata valutata su casi di crescente complessita' riguardanti la simulazione della turbolenza. In primo luogo, e' stata considerata la convezione naturale di Rayleigh-Benard e il flusso turbolento in un canale a numeri di Reynolds moderatamente alti. Le strategie di soluzione proposte sono risultate fino a cinque volte piu` veloci rispetto ai metodi standard allocando solamente il 7% della memoria. In secondo luogo, e' stato analizzato il flusso attorno ad una piastra piana con bordo arrotondato sottoposta a diversi livelli di turbolenza in ingresso. Nonostante la maggiore complessità' dovuta all’uso di elementi curvi ed anisotropi, l’algoritmo proposto e' risultato oltre tre volte piu` veloce allocando il 15% della memoria rispetto ad un metodo standard. Concludendo, viene riportata la simulazione del “Boeing Rudimentary Landing Gear” a Re = 10^6. In tutti i casi i risultati ottenuti sono in ottimo accordo con i dati sperimentali e con precedenti simulazioni numeriche pubblicate in letteratura.In recent years the increasing availability of High Performance Computing (HPC) resources strongly promoted the widespread of high fidelity simulations, such as the Large Eddy Simulation (LES), for industrial research and design. One of the most promising approaches to those kind of simulations is based on the discontinuous Galerkin (dG) discretization method. The contribution of the thesis towards this research area is three-fold. First, the work introduces an efficient hybrid MPI/OpenMP parallelisation paradigm to fruitfully exploit large HPC facilities. Second, it reports efficient, scalable and memory saving solution strategies for stiff dG discretisations. Third, it compares those solution strategies, for the first time using the same numerical framework, to hybridizable discontinuous Galerkin (HDG) methods, including a novel implementation of a p-multigrid preconditioning approach, on unsteady flow problems involving the solution of the NavierStokes equations. The improvements in computational efficiency have been evaluated on cases of growing complexity involving large eddy simulations of turbulent flows. First, the Rayleigh-Benard convection problem and the turbulent channel flow at moderately high Reynolds numbers is presented. The solution strategies proposed resulted up to five times faster than standard matrix-based methods while al- locating the 7% of the memory. A second family of test cases involve the LES simulation of a rounded leading edge flat plate under different levels of free-stream turbulence. Although the increased stiffness of the iteration matrix due to the use of curved and stretched elements, the solver resulted more than three times faster while allocating the 15% of the memory if compared to standard methods. Finally, the large eddy simulation of the Boeing Rudimentary Landing Gear at Re = 10^6 is reported. In all the cases, a remarkable agreement with experimental data as well as previous numerical simulations is documented.INGEGNERIA INDUSTRIALEopenFranciolini, Matte

Multigrid for hypersonic viscous two- and three-dimensional flows

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that removes the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional flow over a blunt biconic

An Initial Investigation of the Effects of Turbulence Models on the Convergence of the RK/Implicit Scheme

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. This scheme has been applied to both the compressible and essentially incompressible Reynolds-averaged Navier-Stokes (RANS) equations using the algebraic turbulence model of Baldwin and Lomax (BL). In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulence are represented by either the Spalart-Allmaras model or the Wilcox k-! model, which are frequently used models in practical fluid dynamic applications. Convergence behavior of the scheme with these turbulence models and the BL model are directly compared. For this initial investigation we solve the flow equations and the partial differential equations of the turbulence models indirectly coupled. With this approach we examine the convergence behavior of each system. Both point and line symmetric Gauss-Seidel are considered for approximating the inverse of the implicit operator of the flow solver. To solve the turbulence equations we use a diagonally dominant alternating direction implicit (DDADI) scheme. Computational results are presented for three airfoil flow cases and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and transport-type equations for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses the RK/implicit scheme for the flow equations

Multigrid calculation of three-dimensional viscous cascade flows

A 3-D code for viscous cascade flow prediction was developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full multigrid method. The Baldwin-Lomax eddy viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency