Variational Multiscale Research Articles

Chemo-mechanical coupling and material evolution in finitely deforming solids with advancing fronts of reactive fluids

A new stabilized method is presented for coupled chemo-mechanical problems involving chemically reacting fluids flowing through deformable elastic solids. A mixture theory model is employed wherein kinematics is represented via an independent set of balance laws for each of the interacting constituents. A significant feature of the mixture model is the interactive force field in the momentum balance equations that couples the constituents implicitly at the level of the governing system of equations. The constitutive relations for the constituents in the mixture model are based on maximization of the rate of entropy production. Since each constituent is not discretely modeled and the interactive effects are mathematically coupled at the local continuum level, the resulting system serves as a physics-based reduced-order model for the complex microstructure of the material system. When constitutive equations are substituted into the balance laws, they give rise to a system of coupled nonlinear PDEs. Evolving nonlinearity and coupled chemo-mechanical effects give rise to spatially localized phenomena, namely boundary layers, shear bands, and steep gradients that appear at the reaction fronts. For large reaction rates, the balance of mass of the fluid becomes a singularly perturbed equation (reaction-dominated), which may exhibit boundary and/or internal layers. Likewise, for large reaction rates and/or low diffusivity, the balance of linear momentum for the fluid constituent also becomes a singularly perturbed PDE. Presence of these features in the solution requires stable numerical methods, and we present a variational multiscale (VMS)-based stabilized finite element method for the initial-boundary value problem. Mathematical attributes of the method are investigated via a range of numerical test cases that involve diffusion of chemically reacting fluids through nonlinear elastic solids. Enhanced stabilization features and higher spatial accuracy of the models and the methods are highlighted.

The actuator line method for wind turbine modelling applied in a variational multiscale framework

The actuator line method is implemented into a residual-based variational multiscale (VMS) fluid modelling framework to simulate airflow around a wind turbine. In the actuator line method, the turbine blades are represented as rotating lines of body force applied to the flow field. The NREL 5MW turbine is simulated first to validate the model for a large scale reference turbine and investigate the effect of model parameters such as the width of the body force projection and the mesh resolution. The wind turbine designed by Norwegian University of Science and Technology and the NREL Phase VI wind turbine are simulated next and results are compared with the corresponding wind tunnel experiments. Velocity deficit and turbulence kinetic energy in the wake region between three and five down-stream diameters show good agreement with experimental data. We conclude with a view towards using this framework in an integrated simulation of a wind farm in complex terrain by simulating a single turbine in an area of complex terrain.

Investigation of flow regimes in arc plasma–gas interactions using a two-temperature arc in crossflow model

The perpendicular impingement of a gas stream on an electric arc, a configuration known as the arc in crossflow, is of primary relevance in the study of plasma–gas interactions as well as in industrial applications such as circuit breakers and wire-arc spraying. The flow dynamics in the arc in crossflow are the result of coupled fluid-thermal-electromagnetic phenomena accompanied by large property gradients, which can produce significant deviations from Local Thermodynamic Equilibrium (LTE) among electrons and gas species. These characteristics can lead to the establishment of distinct flow regimes depending on the relative values of the controlling parameters of the system, such as inflow velocity, arc current, and inter-electrode spacing. A two-temperature non-LTE model is used to investigate the arc dynamics and the establishment of flow regimes in the arc in crossflow. The plasma flow model is implemented within a nonlinear Variational Multiscale (VMS) numerical discretization approach that is less dissipative and, hence, better suited to capture unstable behavior than traditional VMS methods commonly used in computational fluid dynamics simulations. The Reynolds and the Enthalpy dimensionless numbers, characterizing the relative flow strength and arc strength, respectively, are chosen as the controlling parameters of the system. Simulation results reveal the onset of dynamic behavior and the establishment of steady, periodic, quasi-periodic, and chaotic or potentially turbulent regimes, as identified by distinct spatiotemporal fluctuations. The computational results reveal the role of increasing the relative arc strength on enhancing flow stability by delaying the growth of fluctuating and unstable flow behavior.

FSI analysis and simulation of flexible blades in a Wells turbine for wave energy conversion

In this paper a preliminary design and a 2D computational fluidstructure interaction (FSI) simulation of a flexible blade for a Wells turbine is presented, by means of stabilized finite elements and a strongly coupled approaches for the multi-physics analysis. The main objective is to observe the behaviour of the flexible blades, and to evaluate the eventual occurrence of aeroelastic effects and unstable feedbacks in the coupled dynamics. A series of configurations for the same blade geometry, each one characterized by a different material and mechanical properties distribution will be compared. Results will be given in terms of total pressure difference, supported by a flow survey. The analysis is performed using an in-house build software, featured of parallel scalability and structured to easy implement coupled multiphysical systems. The adopted models for the FSI simulation are the Residual Based Variational MultiScale method for the Navier-Stokes equations, the Total Lagrangian formulation for the nonlinear elasticity problem, and the Solid Extension Mesh Moving technique for the moving mesh algorithm.

Anisotropic boundary layer mesh generation for reliable 3D unsteady RANS simulations

This paper proposes a Computational Fluid Dynamics (CFD) framework with the aim of combining consistency and efficiency for the numerical simulation of high Reynolds number flows encountered in engineering applications for aerodynamics. The novelty of the framework is the combination of a Reynolds-Averaged Navier–Stokes (RANS) model with an anisotropic mesh adaptation strategy handling arbitrary immersed geometries by building the corresponding boundary layer meshes. The numerical algorithm consists of robust and accurate solution of the unsteady incompressible Navier–Stokes equations supplemented with a Spalart–Allmaras turbulence model and boundary layer remeshing relying on a specifically designed metric. The flow solver is formulated as a Variational Multiscale (VMS) finite element method for the momentum balance and the incompressibility constraint, and as an upwind Petrov–Galerkin method for the nonlinear turbulent equation. The boundary layer remeshing strategy is flexible as it allows the adaptation of arbitrary coarse meshes by modifying the size and the orientation of elements along the immersed boundary to ensure a smooth gradation along the curvature of the body's geometry. The solver is capable of handling highly stretched anisotropic elements and is shown to successfully predict both mean and fluctuating drag/lift coefficients. Laminar and turbulent test cases in 2D and 3D are presented to assess the performance of this framework against experimental results relevant to external aerodynamics, including an airship and a flying drone.

The variational multiscale formulation for the fully-implicit log-morphology equation as a tensor-based blood damage model.

We derive a variational multiscale (VMS) finite element formulation for a viscoelastic, tensor-based blood damage model. The tensor equation is numerically stabilized by a logarithmic shape tensor description that prevents unphysical, negative eigenvalues. The resulting VMS stabilization terms for this so-called log-morph equation are presented together with their special numerical treatment. Results for a 2D rotating stirrer test case obtained from log-morph simulations with both SUPG and VMS stabilization show significantly improved numerical behavior if compared with Galerkin/least squares (GLS) stabilized untransformed morphology simulation results. The newly proposed method is also successfully applied to a state-of-the-art centrifugal ventricular assist device (VAD), and clear advantages of the VMS stabilization compared with the SUPG-stabilized formulation are presented.

Adaptive Eulerian framework for boiling and evaporation

We propose in this work an adaptive Eulerian framework for the simulation of both boiling and evaporation phenomena occurring at the interface of a heated 3D solid immersed in a liquid tank. It simultaneously takes into account the gas-liquid phase changes, the vapor formation and their dynamics, and consequently the 3D quenching or cooling of a heated solid. It uses a Level Set method to separate and to track each phase. The phase change is performed using the balance of heat fluxes at the interface without the use of conforming mesh. Instead, the use of an a posteriori error estimate leading to highly stretched anisotropic elements at the interface enables to drastically reduce errors on computed jumps. This avoids the need of interface reconstruction or interpolation procedure. Finally, a Variational Multiscale solver for the Navier-Stokes equations is extended with implicit treatment of the surface tension. A series of 2D and 3D problems are solved to highlight the efficiency and the accuracy of the proposed framework. The cooling of an immersed solid is also presented and shows good agreement with experimental data. To the best of our knowledge, direct numerical simulations of quenching using an Eulerian framework with boiling and evaporation have never been considered.

Anatomically realistic lumen motion representation in patient-specific space–time isogeometric flow analysis of coronary arteries with time-dependent medical-image data

Patient-specific computational flow analysis of coronary arteries with time-dependent medical-image data can provide valuable information to doctors making treatment decisions. Reliable computational analysis requires a good core method, high-fidelity space and time discretizations, and an anatomically realistic representation of the lumen motion. The space–time variational multiscale (ST-VMS) method has a good track record as a core method. The ST framework, in a general context, provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow in the artery. The moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces. The ST isogeometric analysis is a superior discretization method. With IGA basis functions in space, it enables more accurate representation of the lumen geometry and increased accuracy in the flow solution. With IGA basis functions in time, it enables a smoother representation of the lumen motion and a mesh motion consistent with that. With cubic NURBS in time, we obtain a continuous acceleration from the lumen-motion representation. Here we focus on making the lumen-motion representation anatomically realistic. We present a method to obtain from medical-image data in discrete form an anatomically realistic NURBS representation of the lumen motion, without sudden, unrealistic changes introduced by the higher-order representation. In the discrete projection from the medical-image data to the NURBS representation, we supplement the least-squares terms with two penalty terms, corresponding to the first and second time derivatives of the control-point trajectories. The penalty terms help us avoid the sudden unrealistic changes. The computation we present demonstrates the effectiveness of the method.

On the performance of a high-order multiscale DG approach to LES at increasing Reynolds number

The variational multiscale (VMS) approach based on a high-order discontinuous Galerkin (DG) method is used to perform LES of the sub-critical flow past a circular cylinder at Reynolds 3 900, 20 000 and 140 000. The effect of the numerical flux function on the quality of the LES solutions is also studied in the context of very coarse discretizations of the TGV configuration at Re=20000. The potential of using p-adaption in combination with DG-VMS is illustrated for the cylinder flow at Re=140000 by considering a non-uniform distribution of the polynomial degree based on a recently developed error estimation strategy [57]. The results from these tests demonstrate the robustness of the DG-VMS approach with increasing Reynolds number on a highly curved geometrical configuration.

Validation study of computational fluid dynamics models of hemodynamics in the human aorta

AbstractWe report on conducting a rigorous validation study where we attempt to compare simulated hemodynamics in a mock model of the human aorta with experimental bi‐plane particle image velocimetry (PIV) data. Within the cardiac modeling framework CARP (Cardiac Arrhythmia Research Package) [12] we implemented a finite element CFD solver based on the residual‐based variational multiscale (RBVMS) discretization of the incompressible Navier‐Stokes equations. CFD simulation results compare favorably with experimental PIV data suggesting that our CFD analysis tools are representative of blood flow in the human aorta.

A moving-domain CFD solver in FEniCS with applications to tidal turbine simulations in turbulent flows

A moving-domain computational fluid dynamics (CFD) solver is designed by deploying the Arbitrary Lagrangian Eulerian Variational Multi-scale Formulation (ALE-VMS) enhanced with weak enforcement of essential boundary conditions (weak BC) into the open-source finite element automation software FEniCS. The mathematical formulation of ALE-VMS, which is working as a Large Eddy Simulation (LES) model for turbulent flows, and weak BC, which is acting as a wall model, are presented with the implementation details in FEniCS. To validate the CFD solver, simulations of flow past a stationary sphere are performed with moving meshes first. Refinement study shows the results quickly converge to the reference results with fixed grids. Then, the solver is utilized to simulate a tidal turbine rotor with uniform and turbulent inflow conditions. Good agreement is achieved between the computational results and experimental measurements in terms of thrust and power coefficients for the uniform inflow case. The effect of the inflow turbulence intensity on the tidal turbine performance is quantified.

Variational multiscale interpolating element-free Galerkin method for the nonlinear Darcy–Forchheimer model

In this paper, the variational multiscale interpolating element-free Galerkin (VMIEFG) method is developed to obtain the numerical solution ofthenonlinearDarcy–Forchheimer model. We use the interpolating moving least squares method instead of the moving least squares approximation to construct meshless shape functions with delta function properties. Then the flux boundary condition of the Darcy–Forchheimer model can be handled easily. Hughes’ variational multiscale (HVM) method is applied to overcome the numerical oscillation caused by equal-order basis for the velocity and pressure. Moreover, the HVM ensures that the resultant formulation in the VMIEFG method is consistent and the stabilization parameter (or tensor) appears naturally. Consequently, the stabilization parameter is free of user-defined. The fixed point iteration method is used to deal with the nonlinear term. Some numerical examples are provided to illustrate the stability and performance of the proposed method for solving the nonlinear Darcy–Forchheimer model.

Pseudoplastic fluid flows for different Prandtl numbers: Steady and time-dependent solutions

In this work, a variational multiscale (VMS) finite element formulation is used to approximate numerically the natural convection in square cavity with differentially heated from sidewalls problem for Newtonian and power-law fluids. The problem is characterized for going through a Hopf bifurcation when reaching high enough Rayleigh numbers, which initiates the transition between steady and time dependent behavior, however, results found in the literature are only for air Prandtl number. The presented VMS formulation is validated using existing results, and is used to study highly convective cases, to determine the flow conditions at which it becomes time dependent, and to establish new benchmark solutions for non-Newtonian fluid flows for different Pr and power-law indexes n. The range of solutions were found in the range 0.6<n<1 and 0.01<Pr<1,000, and the critical Rayleigh number (Rac) where Hopf bifurcations appear were identified for all cases. Obtained results have good agreement with those previously reported in the specific literature, and new data related to the heat transfer capabilities of pseudoplastic fluids and its oscillatory behavior was identified. This non-Newtonian influence of the fluid is later checked in a 3D model of a simplified heat exchanger, where the capability of pseudoplastic fluids for energy transport proved to be enhanced when compared to the Newtonian case.

Space\u2013time computational analysis of tire aerodynamics with actual geometry, road contact, tire deformation, road roughness and fluid film

The space–time (ST) computational method “ST-SI-TC-IGA” has recently enabled computational analysis of tire aerodynamics with actual tire geometry, road contact and tire deformation. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). These ST methods played their parts in overcoming the computational challenges, including (i) the complexity of an actual tire geometry with longitudinal and transverse grooves, (ii) the spin of the tire, (iii) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flow-domain topology change created by the road contact, and (iv) the turbulent nature of the flow. The combination of the ST-VMS, ST-SI and the ST-IGA has also recently enabled solution of fluid film problems with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality. This was accomplished with the computational flexibility to go beyond the limitations of the Reynolds-equation model. Here we include and address the computational challenges associated with the road roughness and the fluid film between the tire and the road. The new methods we add to accomplish that include a remedy for the trapped fluid, a method for reducing the number of control points as a space occupied by the fluid shrinks down to a narrow gap, and a method for representing the road roughness. We present computations for a 2D test problem with a straight channel, a simple 2D model of the tire, and a 3D model with actual tire geometry and road roughness. The computations show the effectiveness of our integrated set of ST methods targeting tire aerodynamics.

Computational Cardiovascular Flow Analysis with the Variational Multiscale Methods

Computational cardiovascular flow analysis can provide valuable information to medical doctors in a wide range of patientspecific cases, including cerebral aneurysms, aortas and heart valves. The computational challenges faced in this class of flow analyses also have a wide range. They include unsteady flows, complex cardiovascular geometries, moving boundaries and interfaces, such as the motion of the heart valve leaflets, contact between moving solid surfaces, such as the contact between the leaflets, and the fluid–structure interaction between the blood and the cardiovascular structure. Many of these challenges have been or are being addressed by the Space–Time Variational Multiscale (ST-VMS) method, Arbitrary Lagrangian–Eulerian VMS (ALE-VMS) method, and the VMS-based Immersogeometric Analysis (IMGA-VMS), which serve as the core computational methods, and the special methods used in combination with them. We provide an overview of the core and special methods and present examples of challenging computations carried out with these methods, including aorta and heart valve flow analyses. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

A reduced order variational multiscale approach for turbulent flows

The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

ALE-VMS methods for wind-resistant design of long-span bridges

For unsteady aerodynamics, the Arbitrary Lagrangian-Eulerian Variational Multi-Scale (ALE-VMS) formulation for incompressible flows has proven an accurate and powerful method. In this paper we present an overview of some of its applications to wind engineering of long-span bridges, including flutter and buffeting analysis and vortex-induced vibrations. In general the numerical results compare well with the companion wind-tunnel experiments. Further, we use the method to address more special topics of bluff-body aerodynamic; the impact of inlet turbulence on the self-excited forces and the nonconforming span-wise coherence structures of turbulence and buffeting forces. The present work demonstrates the completeness and accuracy of ALE-VMS, and proves it to be a viable engineering tool for bridge aerodynamics.

Logarithmic conformation reformulation in viscoelastic flow problems approximated by a VMS-type stabilized finite element formulation

The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina (2014) for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids.

Space\u2013time VMS flow analysis of a turbocharger turbine with isogeometric discretization: computations with time-dependent and steady-inflow representations of the intake/exhaust cycle

Many of the computational challenges encountered in turbocharger-turbine flow analysis have been addressed by an integrated set of space–time (ST) computational methods. The core computational method is the ST variational multiscale (ST-VMS) method. The ST framework provides higher-order accuracy in general, and the VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow. The moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST slip interface (ST-SI) method enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST Isogeometric Analysis enables more accurate representation of the turbine geometry and increased accuracy in the flow solution. The ST/NURBS Mesh Update Method enables exact representation of the mesh rotation. A general-purpose NURBS mesh generation method makes it easier to deal with the complex geometries involved. An SI also provides mesh generation flexibility in a general context by accurately connecting the two sides of the solution computed over nonmatching meshes, and that is enabling the use of nonmatching NURBS meshes in the computations. The computational analysis needs to cover a full intake/exhaust cycle, which is much longer than the turbine rotation cycle because of high rotation speeds, and the long duration required is an additional computational challenge. As one way of addressing that challenge, we propose here to calculate the turbine efficiency for the intake/exhaust cycle by interpolation from the efficiencies associated with a set of steady-inflow computations at different flow rates. The efficiencies obtained from the computations with time-dependent and steady-inflow representations of the intake/exhaust cycle compare well. This demonstrates that predicting the turbine performance from a set of steady-inflow computations is a good way of addressing the challenge associated with the multiple time scales.

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.