Residual-based Variational Multiscale Method Research Articles

Space–time computational flow analysis: Unconventional methods and first-ever solutions

The Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method was introduced in 1990 as a moving-mesh method for computational analysis of flows with moving boundaries and interfaces (MBI), which is a wide class of problems that includes fluid–particle and fluid–structure interactions and free-surface and multi-fluid flows. The method was inspired by Thomas Hughes’s 1987 work on space–time finite element methods for elastodynamics. The original DSD/SST method is now called “ST-SUPS”, reflecting its stabilization components, which are the Streamline-Upwind/Petrov–Galerkin (SUPG) method, pioneered by Hughes, and the Pressure-Stabilizing/Petrov–Galerkin (PSPG) method, inspired by Hughes’s work on a Stokes-flow Petrov–Galerkin formulation allowing equal-order interpolations for velocity and pressure. Hughes’s work on the residual-based variational multiscale (RBVMS) method inspired the ST-VMS method, which is the VMS version of the DSD/SST. A number of special methods were introduced in connection with the core methods ST-SUPS and ST-VMS. Hughes’s work on isogeometric analysis (IGA) inspired one of those special methods, the “ST-IGA”, with IGA basis functions not only in space but also in time. The core and special ST methods enabled first-ever solutions in some of the most challenging classes of MBI problems, including particle-laden flows, spacecraft parachute fluid–structure interactions, and car and tire aerodynamics. We provide an overview of the ST methods inspired by Hughes’s work and highlight some of the first-ever solutions in these three classes of MBI problems.

Phase-field Navier–Stokes model for vesicle doublets hydrodynamics in incompressible fluid flow

In this paper, we study the hydrodynamics of a vesicle doublet suspended in an external viscous fluid flow. Vesicles in this study are modeled using the phase-field model. The bending energy and energies associated with enforcing the global volume and area are considered. In addition, the local inextensibility condition is ensured by introducing an additional equation to the system. To prevent the vesicles from overlapping, we deploy an interaction energy definition to maintain a short-range repulsion between the vesicles. The fluid flow is modeled using Navier–Stokes equations under the assumption of incompressible flows, and the vesicle evolution in time is modeled using two advection equations describing the process of advecting each vesicle by the fluid flow. Rather than solving the velocity–pressure saddle point system, we apply the Residual-Based Variational MultiScale (RBVMS) method to Navier–Stokes equations and solve the coupled systems monolithically using isogeometric analysis. We study vesicle doublet hydrodynamics in shear flow, planar extensional flow, and parabolic flow under various configurations and boundary conditions. Based on the fluid flow profile and domain configuration, we have observed various dynamics like tank-threading, locking, doublet separation, sliding, and shape transformation in tubular channels.

Isogeometric variational multiscale modeling of stably stratified flow over complex terrains

Numerical modeling of stratified boundary layer over complex terrain has been an ongoing challenge in the field of environmental fluid dynamics. In this work, we present a computational framework aiming to tackle that challenge. The key components of the framework are residual-based variational multiscale method, isogeometric analysis, and weak imposition of Dirichlet boundary condition. The framework is validated against a laboratory experiment on strongly stratified flow past a three-dimensional bell-shaped hill. Good agreement is observed for qualitative flow physics, with the predicted occurrences of flow separation, recirculation, and hydraulic jump closely matching those in the experiment. In addition, the dividing-streamline height and wavelength of lee wave computed from the present framework compare well to the theoretical predictions. We show that the present framework is able to tackle various degrees of stratifications. The effect of weak imposition of Dirichlet boundary condition on the performance of the framework is also examined. This paper is concluded with an outlook toward applying the present framework to modeling microscale stratified flow past real-world terrains by simulating stratified flow past a two-dimensional environmental terrain.

Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach

In this paper, we present an isogeometric analysis for studying the dynamical behavior of inextensible vesicles under an external fluid flow with inertial forces. We consider a phase-field model Aland, et al. (2014) for the coupled fluid–vesicle problem which enforces global area and volume constraints using a Lagrange multiplier method and employs an extra equation for enforcing local inextensibility condition. Full Navier–Stokes equations are considered and their finite element formulation is presented based on a residual-based variational multiscale method while a standard Galerkin finite element framework is employed for the rest of partial differential equations in the model. We solve the system of PDEs using an implicit, monolithic scheme based on the generalized-α time integration method. Compared to the system of equations considered in Aland, et al. (2014), we reduce the number of equations to be solved by leveraging high continuity of NURBS functions. We also extend the algorithm of the phase-field method to three-dimensional problems. A number of two-dimensional numerical examples which model the dynamics of a vesicle in a quiescent fluid, in a shear flow, and in plane Poiseuille flow with and without obstructions are studied. The resistive immersed surface method is employed for dealing with obstructions. We also consider a 3D example where we study the dynamics of a vesicle in a constricted channel which resembles the situation that a vesicle experiences in a stenosed microchannel.

Computational study of natural ventilation in a sustainable building with complex geometry

We deploy a fully coupled thermo-fluidic finite element approach to simulating natural ventilation in a sustainably designed building with complex geometry. The ‘interlock house’ uses building design for climate control instead of mechanical means (such as air conditioning). Therefore, accurately modeling the natural ventilation flows is crucial to assess thermal comfort in such designs. A residual-based variational multiscale method (VMS) is employed, which is a Large Eddy Simulation (LES) type approach to turbulence modeling. The VMS formulation is further augmented with a weakly enforced boundary condition method to efficiently resolve the effect of boundary layers. We validate the framework using a canonical Rayleigh Bénard convection problem across different flow regimes. We deploy the framework to analyze thermal flows in the house under two natural ventilation configurations characterized by window opening strategies. Mesh convergence study using one of the configurations is performed to verify the framework. Comparisons of the flow fields and temperature distributions between the two scenarios are discussed. Air diffusion performance index (ADPI) and predicted mean vote (PMV) are computed to investigate thermal comfort in both configurations. This work illustrates the ability of the framework to comprehensively model and predict natural ventilation under various operating scenarios.

The Pseudo-Direct Numerical Simulation method for multi-scale problems in mechanics

We present an overview of the Pseudo-Direct Numerical Simulation (P-DNS in short) method for the solution of multi-scale phenomena. The method can be seen as an adaptation of the variational multi-scale (VMS) method, where the fine solution is solved numerically instead of analytically. Also, from the point of view of homogenization methods it can be seen as an evolution of Finite Element square (FE2) methods, where the most expensive part of the computations is performed offline. The name of P-DNS arises from the premise that in all multi-scale simulations the numerical result obtained with a very fine discretization is correct without the need to introduce any additional model (i.e. turbulence models) or stabilization procedures for transport equation terms (as in residual-based VMS methods). What is intended here is that the P-DNS solution tends to the DNS solution of the problem, accepting as a premise that the DNS solution is a reliable result. In this paper we present first an overview of the P-DNS methodology in the context of an abstract Dirichlet problem involving a second order differential operator that could be non-symmetric and non-necessarily positive definite. Next, the P-DNS approach is applied to the fluid mechanics equations accounting for turbulent phenomena. Examples showing the applicability of the P-DNS method for solving transport problems involving different scales are then presented.

An octree-based immersogeometric approach for modeling inertial migration of particles in channels

In this paper, we develop a scalable, adaptively refined, octree-based finite element approach with immersogeometric analysis to track inertial migration of particles in microchannels. Fluid physics is modeled using a residual-based variational multiscale method, and the particle movement is modeled as rigid body motion. A parallel, hierarchically refined octree mesh is employed as the background mesh, on which a variationally consistent immersogeometric formulation is adopted for tracking the particle motion in the fluid. Adaptations of immersogeometric analysis on an octree background mesh are developed to enable efficient searching of background element for a given surface quadrature point, as well as a distribution of surface quadrature points over processors to reduce memory overhead and better parallelize the surface assembly. An octree-based adaptive mesh refinement algorithm adapted to in-out test in the immersogeometric approach is also developed. The validation of our octree-based immersogeometric approach is carried out using a benchmark case of a sphere settling in quiescent fluid, with good agreement presented. In addition, good strong (and weak) scalability on supercomputing resources for this benchmark case up to 16,384 processes is demonstrated. The proposed method is further deployed for exploring particle migration in straight and converging-diverging channels. This example illustrates the potential of the octree-based immersogeometric approach for efficiently tracking particle motion in complex channel flows – a problem with a diverse array of applications.

Simulation of stably stratified turbulent channel flow using residual-based variational multiscale method and isogeometric analysis

Stably stratified turbulent channel flow (STCF) is a canonical flow that is representative of shear-driven boundary mixing processes in geophysical settings. We examine the performance of a turbulence modeling framework composed of residual-based variational multiscale method (RBVMS) and isogeometric analysis (IGA) applied to STCF at friction Reynolds number Reτ=180. The framework is implemented with two augmentation companion features, namely weak imposition of Dirichlet boundary conditions (WD) and a new subgrid-scale (SGS) model. The performance of the modeling framework, as well as its interaction with the two companion features, are assessed in both weakly and strongly stratified regimes. In comparison to existing direct numerical simulation (DNS) data, our study reveals that RBVMS–IGA framework is able to faithfully capture the flow structures and one-point statistics in STCF simulation with relatively coarse grid resolution. The framework also demonstrates its capability of replicating intermittent flow dynamics under strong stratification. Such characteristics dynamics are reproduced robustly when the modeling framework is enhanced with WD and the new SGS model, features that are shown to generally improve numerical accuracy of simulations for the cases tested. Our results confirm the computational efficiency as well as the robustness of RBVMS–IGA framework in modeling wall-bounded stratified flows.

Computational analysis of particle-laden-airflow erosion and experimental verification

Computational analysis of particle-laden-airflow erosion can help engineers have a better understanding of the erosion process, maintenance and protection of turbomachinery components. We present an integrated method for this class of computational analysis. The main components of the method are the residual-based Variational Multiscale (VMS) method, a finite element particle-cloud tracking (PCT) method with ellipsoidal clouds, an erosion model based on two time scales, and the Solid-Extension Mesh Moving Technique (SEMMT). The turbulent-flow nature of the analysis is addressed with the VMS, the particle-cloud trajectories are calculated based on the time-averaged computed flow field and closure models defined for the turbulent dispersion of particles, and one-way dependence is assumed between the flow and particle dynamics. Because the target-geometry update due to the erosion has a very long time scale compared to the fluid–particle dynamics, the update takes place in a sequence of “evolution steps” representing the impact of the erosion. A scale-up factor, calculated based on the update threshold criterion, relates the erosions and particle counts in the evolution steps to those in the PCT computation. As the target geometry evolves, the mesh is updated with the SEMMT. We present a computation designed to match the sand-erosion experiment we conducted with an aluminum-alloy target. We show that, despite the problem complexities and model assumptions involved, we have a reasonably good agreement between the computed and experimental data.

An immersogeometric formulation for free-surface flows with application to marine engineering problems

An immersogeometric formulation is proposed to simulate free-surface flows around structures with complex geometry. The fluid–fluid interface (air–water interface) is handled by the level set method, while the fluid–structure interface is handled through an immersogeometric approach by immersing structures into non-boundary-fitted meshes and enforcing Dirichlet boundary conditions weakly. Residual-based variational multiscale method (RBVMS) is employed to stabilize the coupled Navier–Stokes equations of incompressible flows and level set convection equation. Other level set techniques, including re-distancing and mass balancing, are also incorporated into the immersed formulation. Adaptive quadrature rule is used to better capture the geometry of the immersed structure boundary by accurately integrating the intersected background elements. Generalized-α method is adopted for time integration, which results in a two-stage predictor multi-corrector algorithm. GMRES solver preconditioned with block Jacobian matrices of individual fluid and level set subproblems is used for solving the coupled linear systems arising from the multi-corrector stage. The capability and accuracy of the proposed method are assessed by simulating three challenging marine engineering problems, which are a solitary wave impacting a stationary platform, dam break with an obstacle, and planing of a DTMB 5415 ship model. A refinement study is performed. The predictions of key quantities of interest by the proposed formulation are in good agreement with experimental results and boundary-fitted simulation results from others. The proposed formulation has great potential for wide applications in marine engineering problems.

Immersogeometric analysis of moving objects in incompressible flows

We deploy the immersogeometric approach for tracking moving objects. The method immerses objects into non-boundary-fitted meshes and weakly enforces Dirichlet boundary conditions on the object boundaries. The object motion is driven by the integrated surface force and external body forces. A residual-based variational multiscale method is employed to stabilize the finite element formulation for incompressible flows. Adaptively refined quadrature rules are used to better capture the geometry of the immersed boundaries by accurately integrating the intersected background elements. Treatment for the freshly-cleared nodes (i.e. background mesh nodes that are inside the object at one time step, but are in the fluid domain at the next time step) is considered. We assess the accuracy of the method by analyzing object motion in different flow structures including objects freely dropping in viscous fluids and particle focusing in unobstructed and obstructed micro-channels. We show that key quantities of interest are in very good agreements with analytical, numerical and experimental solutions. We also show a much better computational efficiency of this framework than current commercial codes using adaptive boundary-fitted approaches. We anticipate deploying this framework for applications of particle inertial migration in microfluidic channels.

Finite element simulation of complex dense granular flows using a well-posed regularization of the μ(I)-rheology

Dense granular flows play an important role in industry and nature. Numerical simulation is a helpful tool to understand their behavior. The high particle concentration in a particle-laden flow leads to non-Newtonian fluid properties. The μ(I)-rheology is a constitutive model to describe a complex, unconventional viscoplastic, shear-thinning flow behavior that agrees well with experiments of dense granular flow. This constitutive model is based on the Coulomb [1] friction and assumes that the pressure is proportional to the tangential stress and related to it by a friction parameter. The friction parameter depends on a dimensionless inertial number I which describes the macroscopic and microscopic timescales of the particle rearrangement. Barker et al. [2] showed that the constitutive equations of the μ(I)-rheology are ill-posed for low and high inertial number regimes. This often results in instabilities, oscillations and high sensitivity of the solution to input parameters. We apply a partial regularization proposed by Barker and Gray [3] for a finite element simulation of a 2D and 3D column collapse with the intention to get stable results and show the disadvantageous outcome of ill-posed constitutive equations. Here, we use a Navier-Stokes solver and the volume-of-fluid method to describe the interface between a dense granular column and a light surrounding fluid. The implementation is done in libMesh using the residual-based variational multiscale method. The objective of this study is to analyze the effects of the partial regularization on the computational simulation of a column collapse using the finite element method. We observe that the viscosity field using the partially well-posed regularized formulation of the μ(I)-rheology by Barker and Gray [3] has a smoother distribution and no spurious oscillations or shear bands, even in simulations with high density and viscosity contrasts.

A residual-based variational multiscale method with weak imposition of boundary conditions for buoyancy-driven flows

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Bénard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers (Ra∼103 to 1010), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.

A staggered procedure for fluid–object interaction with free surfaces, large rotations and driven by adaptive time stepping

The coupling between a rigid body under large rotations and incompressible fluids is investigated within the arbitrary Lagrangian–Eulerian framework. We use here a staggered type of coupling with a predictor/corrector approach for the forces applied to the rigid body. Adaptive time stepping based on feedback control theory imposing a CFL condition on the mesh is investigated. The coupling scheme is first tested on a case illustrating vortex-induced vibrations around a rotating plate. We show the advantages of using the residual-based variational multiscale method for the fluid in the present context. Also, the time-step control and the role of the parameters introduced for the predictor/corrector approach are illustrated using the same test case. A reduced model FPSO ship is then studied, comparing its pitch decay with experimental results. A complex wave–rigid body interaction calculation is finally presented. Results demonstrated the robustness of the predictor/corrector staggered approach with adaptive time-step control for simulating complex interactions of a rigid body under large rotations and free-surface flows.

A new variational multiscale formulation for stratified incompressible turbulent flows

For geophysical and environmental flows in the high-Reynolds-number regime, stable density stratification strongly affects the turbulent fluid motions. While turbulent flows, due to the presence of a cascade of spatial and temporal scales, present several challenges to accurate numerical approximation, density stratification in these flows exacerbates the situation further by suppressing vertical velocity fluctuations thereby enhancing the anisotropy of the turbulence. In this paper, based on the framework of residual-based variational multiscale (RBVMS) methods, we design a new numerical formulation for incompressible stratified flows that introduces coupling between the velocity fine scales and densityequation residual, and gives improved numerical performance as evidenced by the results of two challenging turbulent-flow simulations.

Adjoint consistency analysis of residual-based variational multiscale methods

We investigate the conditions under which residual-based variational multiscale methods are adjoint, or dual, consistent for model hyperbolic and elliptic partial differential equations. In particular, while many residual-based variational multiscale stabilizations are adjoint consistent for hyperbolic problems and finite-element spaces, only a few are adjoint consistent for elliptic problems.

An isogeometric variational multiscale method for large-eddy simulation of coupled multi-ion transport in turbulent flow

Electrochemical processes, such as electroplating of large items in galvanic baths, are often coupled to turbulent flow. In this study, we propose an isogeometric residual-based variational multiscale finite element method for multi-ion transport in dilute electrolyte solutions under turbulent flow conditions. In other words, this means that the concepts of isogeometric discretization and variational multiscale methods are successfully combined for developing a method capable of simulating the challenging problem of coupled multi-ion transport in turbulent flow. We present a comprehensive three-dimensional computational method taking into account, among others, coupled convection–diffusion-migration equations subject to an electroneutrality constraint in combination with phenomenological electrode-kinetics modeling. The electrochemical subproblem is one-way coupled to turbulent incompressible flow via convection. Ionic mass transfer in turbulent Taylor–Couette flow is investigated, representing an important model problem for rotating-cylinder-electrode configurations. Multi-ion transport as considered here is an example for mass transport at high Schmidt number (Sc=1389). An isogeometric discretization is especially advantageous for the present problem, since (i) curved boundaries can be represented exactly, and (ii) it has been proven to provide very accurate solutions for flow quantities when being applied in combination with residual-based variational multiscale modeling. We demonstrate that the method is robust and provides results which are in good agreement with direct numerical simulation results as well as empirical mass-transfer correlations reported in literature.

SPACE–TIME VMS METHODS FOR MODELING OF INCOMPRESSIBLE FLOWS AT HIGH REYNOLDS NUMBERS

Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation was developed for flow problems with moving interfaces and has been successfully applied to some of the most complex problems in that category. A new version of the DSD/SST method for incompressible flows, which has additional subgrid-scale representation features, is the space–time version of the residual-based variational multiscale (VMS) method. This new version, called DSD/SST-VMST and also Space–Time VMS (ST-VMS), provides a more comprehensive framework for the VMS method. We describe the ST-VMS method, including the embedded stabilization parameters, and assess its performance in computation of flow problems at high Reynolds numbers by comparing the results to experimental data. The computations, which include those with 3D airfoil geometries and spacecraft configurations, signal a promising future for the ST-VMS method.

Multifractal subgrid-scale modeling within a variational multiscale method for large-eddy simulation of turbulent flow

Multifractal subgrid-scale modeling within a variational multiscale method is proposed for large-eddy simulation of turbulent flow. In the multifractal subgrid-scale modeling approach, the subgrid-scale velocity is evaluated from a multifractal description of the subgrid-scale vorticity, which is based on the multifractal scale similarity of gradient fields in turbulent flow. The multifractal subgrid-scale modeling approach is integrated into a variational multiscale formulation, which constitutes a new application of the variational multiscale concept. A focus of this study is on the application of the multifractal subgrid-scale modeling approach to wall-bounded turbulent flow. Therefore, a near-wall limit of the multifractal subgrid-scale modeling approach is derived in this work. The novel computational approach of multifractal subgrid-scale modeling within a variational multiscale formulation is applied to turbulent channel flow at various Reynolds numbers, turbulent flow over a backward-facing step and turbulent flow past a square-section cylinder, which are three of the most important and widely-used benchmark examples for wall-bounded turbulent flow. All results presented in this study confirm a very good performance of the proposed method. Compared to a dynamic Smagorinsky model and a residual-based variational multiscale method, improved results are obtained. Moreover, it is demonstrated that the subgrid-scale energy transfer incorporated by the proposed method very well approximates the expected energy transfer as obtained from appropriately filtered direct numerical simulation data. The computational cost is notably reduced compared to a dynamic Smagorinsky model and only marginally increased compared to a residual-based variational multiscale method.

Space–time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle

We present a detailed computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle (MAV). The computational techniques used include the Deforming-Spatial-Domain/Stabilized Space---Time (DSD/SST) formulation, which serves as the core computational technique. The DSD/SST formulation is a moving-mesh technique, and in the computations reported here we use the space---time version of the residual-based variational multiscale (VMS) method, which is called "DSD/ SST-VMST." The motion and deformation of the wings are based on data extracted from the high-speed, multi-camera video recordings of a locust in a wind tunnel. A set of special space---time techniques are also used in the computations in conjunction with the DSD/SST method. The special techniques are based on using, in the space---time flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the wings and for the mesh moving and remeshing. The computational analysis starts with the computation of the base case, and includes computations with increased temporal and spatial resolutions compared to the base case. In increasing the temporal resolution, we separately test increasing the temporal order, the number of temporal subdivisions, and the frequency of remeshing. In terms of the spatial resolution, we separately test increasing the wing-mesh refinement in the normal and tangential directions and changing the way node connectivities are handled at the wingtips. The computational analysis also includes using different combinations of wing configurations for the MAV and investigating the beneficial and disruptive interactions between the wings and the role of wing camber and twist.