Implicit methods for viscous incompressible flows

Numerical methods and simulation tools for incompressible flows have been advanced largely as a subset of the computational fluid dynamics (CFD) discipline. Especially within the aerospace community, simulation of compressible flows has driven most of the development of computational algorithms and tools. This is due to the high level of accuracy desired for predicting aerodynamic performance of flight vehicles. Conversely, low-speed incompressible flow encountered in a wide range of fluid engineering problems has not typically required the same level of numerical accuracy. This practice of tolerating relatively low-fidelity solutions in engineering applications for incompressible flow has changed. As the design of flow devices becomes more sophisticated, a narrower margin of error is required. Accurate and robust CFD tools have become increasingly important in fluid engineering for incompressible and low-speed flow. Accuracy depends not only on numerical methods but also on flow physics and geometry modeling. For high-accuracy solutions, geometry modeling has to be very inclusive to capture the elliptic nature of incompressible flow resulting in large grid sizes. Therefore, in this article, implicit schemes or efficient time integration schemes for incompressible flow are reviewed from a CFD tool development point of view. Extension of the efficient solution procedures to arbitrary Mach number flows through a unified time-derivative preconditioning approach is also discussed. The unified implicit solution procedure is capable of solving low-speed compressible flows, transonic, as well as supersonic flows accurately and efficiently. Test cases demonstrating Mach-independent convergence are presented.     

Hybrid spectral-element-low-order methods for incompressible flows

In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforcesC ¹ continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near rough wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors.     

Spectral domain decomposition techniques for viscous incompressible flows

Spectral domain decomposition methods are described for solving the equations governing the flow of viscous incompressible fluids in rectangularly decomposable domains. The domain of interest is divided into a number of rectangular subdomains on each of which a spectral approximation of the flow variables is sought. For Newtonian flows a stream function formulation is used whereas for nonNewtonian flows the components of the extra-stress tensor are also used. Efficient direct methods for the solution of the algebraic systems are discussed. Numerical results are presented for laminar flow through a channel contraction and for the stick-slip problem.     

The dissipative structure of variational multiscale methods for incompressible flows

In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.     

Numerical methods of higher order of accuracy for incompressible flows

The work deals with numerical solution of the Navier–Stokes equations for incompressible fluid using finite volume and finite difference methods. The first method is based on artificial compressibility where continuity equation is changed by adding pressure time derivative. The second method is based on solving momentum equations and the Poisson equation for pressure instead of continuity equation. The numerical solution using both methods is compared for backward facing step flows. The equations are discretized on orthogonal grids with second, fourth and sixth orders of accuracy as well as third order accurate upwind approximation for convective terms. Not only laminar but also turbulent regimes using two-equation turbulence models are presented.     

Least-squares p-r finite element methods for incompressible non-Newtonian flows

A least-squares mixed p-type finite element method is developed for numerical solution of incompressible non-Newtonian flows. Hierarchical piecewise polynomials are introduced as element basis functions while singularities and boundary layers are treated by a combination of mesh redistribution and polynomial refinement. Scaling of the original differential equations is found to be important for the least-squares minimization process. We discuss both nonlinear algebraic and differential constitutive models and present numerical examples to illustrate benefits and shortcomings of the present approach.     

Boundary conditions for incompressible flows

A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.     

A variational multiscale method for incompressible turbulent flows: Bubble functions and fine scale fields

This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re_T = 395 and 590, and non-equilibrium turbulent flow around a cylinder at Re = 3,900.     

Spectral method of decoupling the vorticity and stream function for the incompressible fluid flows

A spectral method is proposed for the vorticity-stream function equations of the incompressible fluid flows. It is effective to overcome the lack of vorticity boundary condition. This method decouples the vorticity and stream function. At each time step, first, the vorticity is explicitly solved and the stream function is evaluated by a Poisson-like equation; then the vorticity is determined by a Poisson-like equation again. The numerical experiments show that this method is of efficiency and high accuracy.     

High order finite difference methods for unsteady incompressible flows in multi-connected domains

Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the “holes”. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger.     

Three time integration methods for incompressible flows with discontinuous Galerkin Boltzmann method

This paper presents three time integration methods for incompressible flows with finite element method in solving the lattice-BGK Boltzmann equation. The space discretization is performed using nodal discontinuous Galerkin method, which employs unstructured meshes with triangular elements and high order approximation degrees. The time discretization is performed using three different kinds of time integration methods, namely, direct, decoupling and splitting. From the storage cost, temporal accuracy, numerical stability and time consumption, we systematically compare three time integration methods. Then benchmark fluid flow simulations are performed to highlight efficient time integration methods. Numerical results are in good agreement with others or exact solutions.     

Application of single-level,pointwise algebraic,and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.     

A local mesh-refinement technique for incompressible flows

As part of a Multi-Grid scheme for the solution of the Navier-Stokes equations in primitive variables, we introduce a local mesh refinement procedure. New cartesian sub-grids are introduced into regions where the estimated truncation errors are too large. Through the Multi-Grid processing, informations is transferred among the grids in a stable and efficient manner. A simple pointer system allows the storage of the dependent variables, without increasing in the required computer memory. Two computed examples of incompressible flow problems are discussed.     

CFD for incompressible flows at NASA Ames

Over the past 30 years, numerical methods and simulation tools for incompressible flows have been advanced as a subset of the computational fluid dynamics (CFD) discipline. Although incompressible flows are encountered in many areas of engineering, the simulation of compressible flows has driven most of the development of computational algorithms and tools at NASA Ames Research Center. This is due to the stringent requirements for predicting aerodynamic performances of flight vehicles. Conversely, low-speed incompressible flow through or past flow devices did not require the same numerical accuracy. This practice of tolerating relatively low-fidelity solutions in engineering applications has changed, as the design of low-speed flow devices have become more sophisticated, along with more strict efficiency requirements. Accurate and robust CFD tools have become increasingly important in fluid engineering for incompressible and low-speed flow. This paper reviews advances in computational technologies for incompressible flow simulation developed at Ames, and some engineering successes brought about by these advances made during the same period. Additionally, some of the current challenges faced in computing incompressible flows are presented.     

Computational method for turbulent supersonic flows

The calculation features of the turbulent flows described by the Reynolds equations and the two-equation model of turbulence are examined for an explicit high-order accurate Godunov method. Under these features, a new version of a high order of Godunov’s method is developed for calculating the compressible turbulent flows. To illustrate the capability of the new method, some results of the calculation are shown for a supersonic turbulent jet with a complex shock-wave structure and for a separate flow in a plane nozzle.     

Domain decomposition for near-wall turbulent flows

Modeling near-wall high-Reynolds-number turbulent flows is a time-consuming problem. A domain decomposition approach is developed to overcome the problem. The original computational domain is split into a near-wall (inner) subdomain and an outer subdomain. The developed approach is applied to a model 2D equation simulating major peculiarities of near-wall high-Reynolds-number flows. On the base of the Calderon–Ryaben’kii potential theory it is possible to consider the near-wall (inner) problem independently on the outer problem. The influence of the inner problem can exactly be represented by a pseudo-differential equation formulated on the intermediate boundary. In a 1D case, it leads to the wall functions represented by Robin boundary conditions, which can be determined either analytically or numerically. It is important that the wall functions (or boundary conditions) are mesh independent and can be realized in a separate routine. Thus, the original problem can only be solved in the outer domain with some specific nonlocal boundary conditions called nonlocal wall functions. The technique can be extended to 3D problems straightforward.     

Large-eddy simulation of multi-component compressible turbulent flows using high resolution methods

The ability of a finite volume Godunov and a semi-Lagrangian large-eddy simulation (LES) method to predict shock induced turbulent mixing has been examined through simulations of the half-height experiment [Holder and Barton. In: Proceedings of the international workshop on the physics of compressible turbulent mixing, 2004]. Very good agreement is gained in qualitative comparisons with experimental results for combined Richtmyer-Meshkov and Kelvin-Helmholtz instabilities in compressible turbulent multi-component flows. It is shown that both numerical methods can capture the size, location and temporal growth of the main flow features. In comparing the methods, there is variability in the amount of resolved turbulent kinetic energy. The semi-Lagrangian method has constant dissipation at low Mach number, thus allowing the initially small perturbations to develop into Kelvin-Helmholtz instabilities. These are suppressed at the low Mach stage in the Godunov method. However, there is an excellent agreement in the final amount of fluid mixing when comparing both numerical methods at different grid resolutions.     

A class of high-resolution algorithms for incompressible flows

We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed by the incompressible Navier-Stokes equations. The algorithms use high-resolution finite volume methods developed in LeVeque (SIAM J Numer Anal 1996;33:627-665) for the advective terms and finite difference methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advects the cell-centered velocities using the divergence-free cell-edge velocities. The resulting cell-centered velocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robust for constant-density flows at high Reynolds numbers via an example of lid-driven cavity flow. With a slight modification for the projection operator in the constant-density solvers, the algorithms also solve incompressible flows with finite-amplitude density variation. The strength of such algorithms is illustrated through problems like Rayleigh-Taylor instability and the Boussinesq equations for Rayleigh-Bénard convection. Numerical studies of the convergence and order of accuracy for the velocity field are provided. While simulations for two-dimensional regular-geometry problems are presented in this study, in principle, extension of the algorithms to three dimensions with complex geometry is feasible.