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.     

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 element-Fourier methods for incompressible turbulent flows

A mixed spectral element-(Fourier) spectral method is proposed for solution of the incompressible Navier-Stokes equations in general, curvilinear domains. The formulation is appropriate for simulations of turbulent flows in complex geometries with only one homogeneous flow direction. The governing equations are written in a form suitable for both direct (DNS) and large-eddy (LES) simulations allowing a unified implementation. The method is based on skew-symmetric convective operators that induce minimal aliasing errors and fast Helmholtz solvers that employ efficient iterative algorithms (e.g. multigrid). Direct numerical simulations of channel flow verified that the proposed method can sustain turbulent fluctuations even at ‘marginal’ Reynolds numbers. The flexibility of the method to efficiently simulate complex-geometry flows is demonstrated through an example of transitional flow in a grooved channel and an example of transitional-turbulent flow over rough wall surfaces.     

A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

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.     

Numerical studies of finite element variational multiscale methods for turbulent flow simulations

Different realizations of variational multiscale (VMS) methods within the framework of finite element methods are studied in turbulent channel flow simulations. One class of VMS methods uses bubble functions to model resolved small scales whereas the other class contains the definition of the resolved small scales by an explicit projection in its set of equations. All methods are employed with eddy viscosity models of Smagorinsky type. The simulations are performed on grids for which a Direct Numerical Simulation blows up in finite time.     

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.     

Dynamical geometry for multiscale dissipative particle dynamics

In this paper, we review the computational aspects of a multiscale dissipative particle dynamics model for complex fluid simulations based on the feature-rich geometry of the Voronoi tessellation. The geometrical features of the model are critical since the mesh is directly connected to the physics by the interpretation of the Voronoi volumes of the tessellation as coarse-grained fluid clusters. The Voronoi tessellation is maintained dynamically in time to model the fluid in the Lagrangian frame of reference, including imposition of periodic boundary conditions. Several algorithms to construct and maintain the periodic Voronoi tessellation are reviewed in two and three spatial dimensions and their parallel performance discussed. The insertion of polymers and colloidal particles in the fluctuating hydrodynamic solvent is described using surface boundaries.     

The variational multiscale method for laminar and turbulent flow

Summary  The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.     

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.     

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.     

Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes

In this work, we combine (i) NURBS-based isogeometric analysis, (ii) residual-driven turbulence modeling and iii) weak imposition of no-slip and no-penetration Dirichlet boundary conditions on unstretched meshes to compute wall-bounded turbulent flows. While the first two ingredients were shown to be successful for turbulence computations at medium-to-high Reynolds number [I. Akkerman, Y. Bazilevs, V. M. Calo, T. J. R. Hughes, S. Hulshoff, The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech. 41 (2008) 371–378; Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007) 173–201], it is the weak imposition of no-slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher Reynolds number [Y. Bazilevs, T.J.R. Hughes. Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput. Fluids 36 (2007) 12–26; Y. Bazilevs, C. Michler, V.M. Calo, T.J.R. Hughes, Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4853–4862]. These three ingredients form a basis of a possible practical strategy for computing engineering flows, somewhere between RANS and LES in complexity. We demonstrate this by solving two challenging incompressible turbulent benchmark problems: channel flow at friction-velocity Reynolds number 2003 and flow in a planar asymmetric diffuser. We observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data. This lends some credence to the proposed approach, which we believe may become a viable engineering tool.     

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.     

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.     

EULAG, a computational model for multiscale flows

EULAG (Eulerian/semi-Lagrangian fluid solver) is an established computational model for simulating thermo-fluid flows across a wide range of scales and physical scenarios. It is noteworthy for its nonoscillatory integration algorithms, robust elliptic solver, and generalized coordinate formulation enabling grid adaptivity technology. In this paper we highlight the key model ingredients, demonstrate its capabilities with a select subset of recent applications, and show its performance both in terms of accuracy and scalability on massively parallel processor architectures. A comprehensive list of references is provided to facilitate more detailed study.