Variational Multiscale Research Articles

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

We propose in this work an adaptive variational multiscale method for two-fluid flows with surface tension. A level set function is used to provide a precise position of the interfaces. The implementation of the surface tension in the context of the Continuum Surface Force is proposed to circumvent the capillary time step restriction. The obtained system is then solved using a new derived Variational Multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios. Combined with an a posteriori error estimator, we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two-phase incompressible isothermal flows. Stable and accurate results are obtained for all two- and three-dimensional numerical examples. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation.

Algebraic approximation of sub-grid scales for the variational multiscale modeling of transport problems

Variational Multiscale (VMS) Finite Element Methods (FEMs) are robust for the development of general formulations for the solution of multiphysics and multiscale transport problems. To obtain a tractable and computationally efficient model, VMS methods often rely on a residual-based algebraic approximation of the sub-grid scales (small or unresolved features of the solution field not captured by the discretization) using a so-called intrinsic time scales matrix, which depends on the problem’s overall differential operator and represents the main model parameter. A novel technique for approximating the intrinsic time scales matrix for generic transport problems in a relatively inexpensive manner (e.g., does not rely on eigenvalue computations) is presented. The method is denoted Transport-Equivalent Scaling (TES) and is based on the monolithic casting of the transport problem as a system of transient–advective–diffusive–reactive (TADR) equations and a subsequent scaling of the coefficient matrices such to preserve each type of transport flux. An algebraic VMS formulation incorporating the TES method is complemented with a discontinuity-capturing (DC) approach and implemented within a FEM solver for the solution of TADR problems. The solution of the global discrete system is accomplished using a generalized-alpha time-stepper together with a globalized inexact Newton–Krylov nonlinear solver. The effectiveness of the TES formulation is verified with the simulation of benchmark incompressible, compressible, and magnetohydrodynamic flow problems. The results demonstrate that the TES method seamlessly handles incompressible–compressible flows in a unified manner (e.g., without assessing the compressibility of the flow). The convergence process using the TES approach and a more standard approximation for the intrinsic time scales, as well as the effect of the DC approach, are also investigated. Analysis of the intrinsic time scales for a one-dimensional incompressible flow model reveals the similitudes and differences between the TES formulation and other conventional methods.

Fluid–Structure Interaction Modeling for Fatigue-Damage Prediction in Full-Scale Wind-Turbine Blades

This work presents a collection of advanced computational methods, and their coupling, that enable prediction of fatigue-damage evolution in full-scale composite blades of wind turbines operating at realistic wind and rotor speeds. The numerical methodology involves: (1) a recently developed and validated fatigue-damage model for multilayer fiber-reinforced composites; (2) a validated coupled fluid–structure interaction (FSI) framework, wherein the 3D time-dependent aerodynamics based on the Navier–Stokes equations of incompressible flows is computed using a finite-element-based arbitrary Lagrangian–Eulerian–variational multiscale (ALE–VMS) technique, and the blade structures are modeled as rotation-free isogeometric shells; and (3) coupling of the FSI and fatigue-damage models. The coupled FSI and fatigue-damage formulations are deployed on the Micon 13M wind turbine equipped with the Sandia CX-100 blades. Damage initiation, damage progression, and eventual failure of the blades are reported.

A variational multiscale finite element method for monolithic ALE computations of shock hydrodynamics using nodal elements

We present a monolithic arbitrary Lagrangian–Eulerian (ALE) finite element method for computing highly transient flows with strong shocks. We use a variational multiscale (VMS) approach to stabilize a piecewise-linear Galerkin formulation of the equations of compressible flows, and an entropy artificial viscosity to capture strong solution discontinuities. Our work demonstrates the feasibility of VMS methods for highly transient shock flows, an area of research for which the VMS literature is extremely scarce. In addition, the proposed monolithic ALE method is an alternative to the more commonly used Lagrangian+remap methods, in which, at each time step, a Lagrangian computation is followed by mesh smoothing and remap (conservative solution interpolation). Lagrangian+remap methods are the methods of choice in shock hydrodynamics computations because they provide nearly optimal mesh resolution in proximity of shock fronts. However, Lagrangian+remap methods are not well suited for imposing inflow and outflow boundary conditions. These issues offer an additional motivation for the proposed approach, in which we first perform the mesh motion, and then the flow computations using the monolithic ALE framework. The proposed method is second-order accurate and stable, as demonstrated by extensive numerical examples in two and three space dimensions.

Computational analysis of flow-driven string dynamics in turbomachinery

Abstract We focus on computational analysis of flow-driven string dynamics. The objective is to understand how the strings carried by a fluid interact with the solid surfaces present and get stuck on or around those surfaces. Our target application is turbomachinery, such as understanding how strings get stuck on or around the blades of a fan. The components of the method we developed for this purpose are the Space–Time Variational Multiscale (ST-VMS) and ST Slip Interface (ST-SI) methods for the fluid dynamics, and a one-way-dependence model and the Isogeometric Analysis (IGA) for the string dynamics. The ST-VMS method is the core computational technology and it also has the features of a turbulence model. The ST-SI method allows in a consistent fashion slip at the interface between the mesh covering a spinning solid surface and the mesh covering the rest of the domain, and with this, we maintain high-resolution representation of the boundary layers near spinning solid surfaces such as fan blades. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small. The IGA for the string dynamics gives us not only a higher-order method and smoothness in the structure shape, but also smoothness in the fluid dynamics forces calculated on the string. To demonstrate how the method can be used in computational analysis of flow-driven string dynamics, we present the pilot computations we carried out, for a duct with cylindrical obstacles and for a ventilating fan.

A palette of fine-scale eddy viscosity and residual-based models for variational multiscale formulations of turbulence

We explore a general family of eddy viscosity models for the large-eddy simulation of turbulence within the framework of the Variational Multiscale Method. Our investigation encompasses various fine-scale eddy viscosities and coarse-scale residual-based constructs. We delineate the domain of parameter space in which physically and mathematically suitable models exist, and identify several sub-families of potentially useful models that are either entirely new or extend previously proposed ones. We also combine classical modeling ideas, that lead to turbulent kinetic energy evolution equations, with the residual-based approach to derive a new residual-driven, one-equation dynamic model.

Computational thermo-fluid analysis of a disk brake

We present computational thermo-fluid analysis of a disk brake, including thermo-fluid analysis of the flow around the brake and heat conduction analysis of the disk. The computational challenges include proper representation of the small-scale thermo-fluid behavior, high-resolution representation of the thermo-fluid boundary layers near the spinning solid surfaces, and bringing the heat transfer coefficient (HTC) calculated in the thermo-fluid analysis of the flow to the heat conduction analysis of the spinning disk. The disk brake model used in the analysis closely represents the actual configuration, and this adds to the computational challenges. The components of the method we have developed for computational analysis of the class of problems with these types of challenges include the Space---Time Variational Multiscale method for coupled incompressible flow and thermal transport, ST Slip Interface method for high-resolution representation of the thermo-fluid boundary layers near spinning solid surfaces, and a set of projection methods for different parts of the disk to bring the HTC calculated in the thermo-fluid analysis. With the HTC coming from the thermo-fluid analysis of the flow around the brake, we do the heat conduction analysis of the disk, from the start of the breaking until the disk spinning stops, demonstrating how the method developed works in computational analysis of this complex and challenging problem.

Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG

The computational solution of the governing balance equations for mass, momentum, heat transfer and magnetic induction for resistive magnetohydrodynamics (MHD) systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that the interactions of these physical mechanisms produce. This paper explores the development of a scalable, fully-implicit stabilized unstructured finite element (FE) capability for 3D incompressible resistive MHD. The discussion considers the development of a stabilized FE formulation in context of the variational multiscale (VMS) method, and describes the scalable implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton–Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, that include MHD duct flows, an unstable hydromagnetic Kelvin–Helmholtz shear layer, and a 3D island coalescence problem used to model magnetic reconnection. Initial results that explore the scaling of the solution methods are also presented on up to 128K processors for problems with up to 1.8B unknowns on a CrayXK7.

On the Influence of Polynomial De-aliasing on Subgrid Scale Models

In this work we investigate the interplay of polynomial de-aliasing and sub-grid scale models for large eddy simulations based on discontinuous Galerkin discretizations. It is known that stability is a major concern when simulating underresolved turbulent flows with high order nodal collocation type discretizations. By changing the interpolatory character of the nodal collocation type discretization to a projection based discretization by increasing the number of quadrature points (polynomial de-aliasing), one is able to remove the aliasing induced stability problems. We focus on this effect and on the consequence for large eddy simulations with explicit subgrid scale models. Often, subgrid scale models have to achieve two possibly conflicting tasks in a single simulation: firstly stabilizing the numerics and secondly modeling the physical effect of the missing scales. Within a discontinuous Galerkin approach, it is possible to use either a fast (but potentially aliasing afflicted) nodal collocation discretization or a projection-based (but computationally costly) variant in combination with an explicit subgrid scale model. We use this framework to investigate the effect on the appropriate model parameter of a standard Smagorinsky subgrid scale model and of a Variational Multiscale Smagorinsky formulation. For this we first consider the 3-D viscous Taylor-Green vortex example to investigate the impact on the stability of the method and second the turbulent flow past a circular cylinder to investigate and compare the accuracy of the results. We show that the aliasing instabilities of collocative discretizations severely limit the choice of the model constant, in particular for high order schemes, while for de-aliased DG schemes, the closure model parameters can be chosen independently from the numerical scheme. For the cylinder flow, we also find that for the same model settings, the projection-based results are in better agreement with the reference DNS than those of the collocative scheme.

Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: they are based on the variation al formulation of the incompressible Navier–Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used math ematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

MODIFIED METHOD OF CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME DEPENDENT NAVIER-STOKES PROBLEMS

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

Dynamic models for Large Eddy Simulation of compressible flows with a high order DG method

The impact of dynamic models for applications to LES of compressible flows is assessed in the framework of a numerical model based on high order discontinuous finite elements. The projections onto lower dimensional subspaces associated with lower degree basis functions are used as LES filter, along the lines proposed in Variational Multiscale templates. Comparisons with DNS results available in the literature for plane and constricted channel flows at Mach numbers 0.2, 0.7 and 1.5 show clearly that the dynamic models are able to improve the prediction of most key features of the flow with respect to the Smagorinsky models employed so far in a VMS-DG context.

The characteristic variational multiscale method for time dependent conduction–convection problems

In this paper, the characteristic variational multiscale (C-VMS) method is proposed to solve the nonstationary conduction–convection problems. The stability analysis is carried out using the energy estimate method. Compared with the standard variational multiscale (VMS) method, the C-VMS method does not need nonlinear iteration. Finally, some numerical examples are given, which show that the C-VMS method is efficient, reliable and can save a lot of CPU time for this problem, besides, it can deal with the high Rayleigh number.

New directions and challenging computations in fluid dynamics modeling with stabilized and multiscale methods

In this paper, we provide a brief overview of the development of stabilized and multiscale methods in fluid dynamics. We mainly focus on recent developments and new directions in the variational multiscale (VMS) methods. We also discuss applications of the VMS techniques to fluid dynamics problems involving computational challenges associated with high-Reynolds-number flows, wall-bounded turbulent flows, flows with moving domains including subdomains in relative motion, and free-surface flows.

Multiscale space–time methods for thermo-fluid analysis of a ground vehicle and its tires

We present the core and special multiscale space–time (ST) methods we developed for thermo-fluid analysis of a ground vehicle and its tires. We also present application of these methods to thermo-fluid analysis of a freight truck and its rear set of tires. The core multiscale ST method is the ST variational multiscale (ST-VMS) formulation of the Navier–Stokes equations of incompressible flows with thermal coupling, which is multiscale in the way the small-scale thermo-fluid behavior is represented in the computations. The special multiscale ST method is spatially multiscale, where the thermo-fluid computation over the global domain with a reasonable mesh refinement is followed by a higher-resolution computation over the local domain containing the rear set of tires, with the boundary and initial conditions coming from the data computed over the global domain. The large amount of time-history data from the global computation is stored using the ST computation technique with continuous representation in time (ST-C), which serves as a data compression technique in this context. In our thermo-fluid analysis, we use a road-surface temperature higher than the free-stream temperature, and a tire-surface temperature that is even higher. We also include in the analysis the heat from the engine and exhaust system, with a reasonably realistic representation of the rate by which that heat transfer takes place as well as the surface geometry of the engine and exhaust system over which the heat transfer occurs. We take into account the heave motion of the truck body. We demonstrate how the spatially multiscale ST method, with higher-refinement mesh in the local domain, substantially increases the accuracy of the computed heat transfer rates from the tires.

Space–time VMS method for flow computations with slip interfaces (ST-SI)

We present the space–time variational multiscale (ST-VMS) method for flow computations with slip interfaces (ST-SI). The method is intended for fluid–structure interaction (FSI) analysis where one or more of the subdomains contain spinning structures, such as the rotor of a wind turbine, and the subdomains are covered by meshes that do not match at the interface and have slip between them. The mesh covering a subdomain with the spinning structure spins with it, thus maintaining the high-resolution representation of the boundary layers near the structure. The starting point in the development of the method is the version of the arbitrary Lagrangian–Eulerian VMS (ALE-VMS) method designed for computations with "sliding interfaces". Interface terms similar to those in the ALE-VMS version are added to the ST-VMS formulation to account for the compatibility conditions for the velocity and stress. In addition to having a high-resolution representation of the boundary layers, because the ST framework allows NURBS functions in temporal representation of the structure motion, we have exact representation of the circular paths associated with the spinning. The ST-SI method includes versions for cases where the SI is between fluid and solid domains with weakly-imposed Dirichlet conditions for the fluid and for cases where the SI is between a thin porous structure and the fluid on its two sides. Test computations with 2D and 3D models of a vertical-axis wind turbine show the effectiveness of the ST-SI method.

Finite element dynamical subgrid-scale model for low Mach number flows with radiative heat transfer

Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields. Design/methodology/approach – The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations. Findings – As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show...

Simulation of laminar flow through eccentric annuli using isogeometric variational multiscale method

We present an application of the Residual-based variational multiscale (RBVMS) methodology to the computation of laminar eccentric annular pipe flow with eccentricities of 0.5 and 1. Isogeometric analysis is utilized for higher order approximation of the geometry and solution using Non-uniform rational B-splines (NURBS) functions. The ability of NURBS exactly representing curved circular geometries makes NURBS-based isogeometric analysis attractive for the application to the flow through the eccentric annuli. By using the exact representation of circular boundary, the limiting case of the eccentricity, i.e. the inner circular wall actually touches the outer circular wall, is successfully demonstrated.

Two-grid variational multiscale method with bubble stabilization for convection diffusion equation

A two-grid variational multiscale (VMS) method based on two local Gauss integrations for the convection dominated nonlinear convection diffusion equation is investigated. This method combines the two-grid strategy with the variational multiscale method which chooses polynomial bubble functions as subgrid scale. Two local Gauss integrations are applied to replace the projection operator without adding any extra storage. Moreover, the error estimates for the algorithms of the two-grid method are obtained. Numerical examples validate the theoretical results of the presented methods.