3,750 research outputs found
Unsteady residual distribution schemes for transition prediction
In this work, the unsteady simulation of the Navier–Stokes equations is carried out by using a Residual Distribution Schemes (RDS) methodology. This algorithm has a compact stencil (cell-based computations) and uses a finite element like method to compute the residual over the cell. The RDS method has been successfully proven in steady Navier–Stokes computation but its application to fully unsteady configurations is still not closed, because some of the properties of the steady counterpart can be lost. Here, we proposed a numerical solution for unsteady problems that is fully compatible with the original approach. In order to check the method, we chose a very demanding test case, namely the numerical simulation of a Tollmien–Schlichting (TS) wave in a 2D boundary layer. The evolution of this numerical perturbation is accurately computed and checked against theoretical results
A bounded upwinding scheme for computing convection-dominated transport problems
A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data
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
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
Multigrid solution of the Navier-Stokes equations on triangular meshes
A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms
Implicit time integration for high-order compressible flow solvers
The application of high-order spectral/hp element discontinuous Galerkin (DG)
methods to unsteady compressible flow simulations has gained increasing popularity.
However, the time step is seriously restricted when high-order methods are applied
to an explicit solver. To eliminate this restriction, an implicit high-order compressible flow solver is developed using DG methods for spatial discretization, diagonally
implicit Runge-Kutta methods for temporal discretization, and the Jacobian-free
Newton-Krylov method as its nonlinear solver. To accelerate convergence, a block
relaxed Jacobi preconditioner is partially matrix-free implementation with a hybrid
calculation of analytical and numerical Jacobian.The problem of too many user-defined parameters within the implicit solver is
then studied. A systematic framework of adaptive strategies is designed to relax the
difficulty of parameter choices. The adaptive time-stepping strategy is based on the
observation that in a fixed mesh simulation, when the total error is dominated by the
spatial error, further decreasing of temporal error through decreasing the time step
cannot help increase accuracy but only slow down the solver. Based on a similar
error analysis, an adaptive Newton tolerance is proposed based on the idea that
the iterative error should be smaller than the temporal error to guarantee temporal
accuracy. An adaptive strategy to update the preconditioner based on the Krylov
solver’s convergence state is also discussed. Finally, an adaptive implicit solver is
developed that eliminates the need for repeated tests of tunning parameters, whose
accuracy and efficiency are verified in various steady/unsteady simulations. An improved shock-capturing strategy is also proposed when the implicit solver
is applied to high-speed simulations. Through comparisons among the forms of
three popular artificial viscosities, we identify the importance of the density term
and add density-related terms on the original bulk-stress based artificial viscosity.
To stabilize the simulations involving strong shear layers, we design an extra shearstress based artificial viscosity. The new shock-capturing strategy helps dissipate
oscillations at shocks but has negligible dissipation in smooth regions.Open Acces
A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
We present a comparison between hybridized and non-hybridized discontinuous
Galerkin methods in the context of target-based hp-adaptation for compressible
flow problems. The aim is to provide a critical assessment of the computational
efficiency of hybridized DG methods. Hybridization of finite element
discretizations has the main advantage, that the resulting set of algebraic
equations has globally coupled degrees of freedom only on the skeleton of the
computational mesh. Consequently, solving for these degrees of freedom involves
the solution of a potentially much smaller system. This not only reduces
storage requirements, but also allows for a faster solution with iterative
solvers. Using a discrete-adjoint approach, sensitivities with respect to
output functionals are computed to drive the adaptation. From the error
distribution given by the adjoint-based error estimator, h- or p-refinement is
chosen based on the smoothness of the solution which can be quantified by
properly-chosen smoothness indicators. Numerical results are shown for
subsonic, transonic, and supersonic flow around the NACA0012 airfoil.
hp-adaptation proves to be superior to pure h-adaptation if discontinuous or
singular flow features are involved. In all cases, a higher polynomial degree
turns out to be beneficial. We show that for polynomial degree of approximation
p=2 and higher, and for a broad range of test cases, HDG performs better than
DG in terms of runtime and memory requirements
High-Order Hyperbolic Residual-Distribution Schemes on Arbitrary Triangular Grids
In this paper, we construct high-order hyperbolic residual-distribution schemes for general advection-diffusion problems on arbitrary triangular grids. We demonstrate that the second-order accuracy of the hyperbolic schemes can be greatly improved by requiring the scheme to preserve exact quadratic solutions. We also show that the improved second-order scheme can be easily extended to third-order by further requiring the exactness for cubic solutions. We construct these schemes based on the LDA and the SUPG methodology formulated in the framework of the residual-distribution method. For both second- and third-order-schemes, we construct a fully implicit solver by the exact residual Jacobian of the second-order scheme, and demonstrate rapid convergence of 10-15 iterations to reduce the residuals by 10 orders of magnitude. We demonstrate also that these schemes can be constructed based on a separate treatment of the advective and diffusive terms, which paves the way for the construction of hyperbolic residual-distribution schemes for the compressible Navier-Stokes equations. Numerical results show that these schemes produce exceptionally accurate and smooth solution gradients on highly skewed and anisotropic triangular grids, including curved boundary problems, using linear elements. We also present Fourier analysis performed on the constructed linear system and show that an under-relaxation parameter is needed for stabilization of Gauss-Seidel relaxation
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
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