Vaibhav Joshi

A positivity preserving variational method for multi-dimensional convectiondiffusionreaction equation

A new positivity preserving variational (PPV) procedure is proposed to solve the convectiondiffusionreaction (CDR) equation. Through the generalization of stabilized finite element methods, the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection, diffusion and reaction effects. By means of Fourier analysis, we first review the shortcomings of the Galerkin/Least-Squares (GLS) and the Subgrid Scale (SGS) methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined GLS-SGS methodology. Discrete upwind operator with a solution-dependent nonlinear term is then introduced in high gradient regions, which enables the positivity preserving property in the variational formulation. Direct extension to multi-dimensions is carried out by considering the principle streamline and crosswind directions. The efficacy of the method is demonstrated by the systematic accuracy and stability analyses in one- and two-dimensions. Results show the reduction of oscillations in the solution in one- and two-dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient. The proposed formulation provides a superior solution in the reaction-dominated as well as the convection-dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions.

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

Abstract We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio. The variational finite element technique relies on the Allen–Cahn phase-field equation for capturing the phase interface on a fixed Eulerian mesh with mass conservative and energy-stable discretization. The mass conservation is achieved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase-field equation. To make the scheme energy-stable in a variational sense, we discretize the spatial part of the Lagrange multiplier in the phase-field equation by the mid-point approximation. The proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second-order accuracy of both spatial and temporal discretizations. We integrate the Allen–Cahn phase-field equation with the incompressible Navier–Stokes equations for modeling a broad range of two-phase flow and fluid-fluid interface problems. The coupling of the implicit discretizations corresponding to the phase-field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure. Comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen–Cahn two-phase solver. We examine the convergence and accuracy properties of the coupled phase-field solver through the standard benchmarks of the Laplace–Young law and a sloshing tank problem. Two- and three-dimensional dam break problems are simulated to assess the capability of the phase-field solver for complex air-water interfaces involving topological changes on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical offshore engineering application of wave-structure interaction.

An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model

Abstract In this paper, we present an adaptive variational procedure for unstructured meshes to capture fluid–fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen–Cahn equation. For the adaptivity procedure, we consider the residual-based error estimates for the underlying differential equations of the two-phase system. In particular, the adaptive refinement/coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the Allen–Cahn equation. The coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh, thus providing the ease of numerical implementation. Furthermore, the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations. We investigate the adaptive phase-field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume-conserved interface motion driven by the mean curvature flow for two circles in a square domain. We then assess the accuracy and efficiency of the proposed procedure by modeling the free-surface motion in a sloshing tank. In contrast to the non-adaptive Eulerian grid counterpart, we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy. The mass loss in the Allen–Cahn equation via adaptivity process is also reduced by nearly three times compared to the non-adaptive mesh. Finally, we apply the adaptive numerical framework to solve the application of a dam-breaking problem with topological changes.

Flow-Induced Vibrations of Riser Array System

When a riser array system is subjected to a uniform flow, an unstable flow-induced vibration commonly occurs among cylinders, generally called fluid-elastic instability. It can cause long-term or short-term damage to the riser array system. A numerical investigation has been performed in the present study. Generally, flow-induced vibrations include vortex-induced vibration (VIV), wake-induced vibration (WIV), jet switching, turbulent buffeting and fluid-elastic instability. The dynamic interactions among the fluid-induced vibrations, wake interference and proximity interference pose difficulties in the design and operation of the riser array system. The dynamics of a riser array system is very different from that of basic canonical configurations such as side-by-side, tandem and staggered arrangements. In a riser array system, the interferences come from all possible nearby constituent risers. There is a synchronization phenomenon among the cylinders, which may lead to detrimental collisions and short-term failures. It is known that the vortex-induced vibration (VIV) of an isolated circular cylinder is self-limiting. An extensive vibration occurs in the lock-in region within which the frequency of the vortex shedding matches the structural frequency of the immersed structure. In a riser array system, there is a point at which the vibration of cylinder suddenly increases. The vibration of the constituent risers increases without bound with the increment of the free-stream velocity. This free-stream velocity is defined as the critical velocity. The interference not only comes from the inline and cross-flow directions, but also the wake interference from the diagonal upstream risers. In a riser array system, each riser vibrates independently. However, there is symmetry of frequency spectrum observed about the inline direction along the middle row of the risers.In this study, the dynamic response of the different risers in the array system is investigated with the help of the amplitude response results from the canonical arrangements (side-by-side and tandem) and wake flow structures. The long top-tensioned riser system can be idealized by two-dimensional elastically mounted cylinders to solve the complex fluid-structure interaction problem. The dynamic response of a typical riser array system has been analyzed at low and high Reynolds number. It is encouraging to see that the results reported in the present investigation can provide useful insight and suggestions in the design and optimization of riser systems to avoid collisions and various long-or short-term failures.Copyright

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

Abstract We present a three-dimensional (3D) common-refinement method for non-matching unstructured meshes between non-overlapping subdomains of incompressible turbulent fluid flow and nonlinear hyperelastic structure. The fluid flow is discretized using a stabilized Petrov–Galerkin method, and the large deformation structural formulation relies on a continuous Galerkin finite element method. An arbitrary Lagrangian–Eulerian formulation with a nonlinear iterative force correction (NIFC) coupling is achieved in a staggered partitioned manner by means of fully decoupled implicit procedures for the fluid and solid discretizations. To begin, we first investigate the accuracy of common-refinement method (CRM) to satisfy the traction equilibrium condition along the fluid-elastic interface with non-matching meshes. We systematically assess the accuracy of CRM against the matching grid solution by varying grid mismatch between the fluid and solid meshes over a tubular elastic body. We demonstrate the second-order accuracy of CRM through uniform refinements of fluid and solid meshes along the interface. We then extend the error analysis to transient data transfer across non-matching meshes between the fluid and solid solvers. We show that the common-refinement discretization across non-matching fluid–structure grids yields accurate transfer of the physical quantities across the fluid–solid interface. We next solve a 3D fluid–structure interaction (FSI) problem of a cantilevered hyperelastic plate behind a circular bluff body and verify the accuracy of coupled solutions with respect to the available solution in the literature. By varying the solid interface resolution, we generate various non-matching grid ratios and quantify the accuracy of CRM for the nonlinear structure interacting with a laminar flow. We illustrate that the CRM with the partitioned NIFC treatment is stable for low solid-to-fluid density ratio and non-matching meshes for the 3D FSI problem. Finally, we demonstrate the 3D parallel implementation of the common-refinement with the NIFC method for a realistic engineering problem of drilling riser undergoing complex vortex-induced vibration with strong added mass effects and turbulent wake flow.

A hybrid variational Allen-Cahn/ALE scheme for the coupled analysis of two-phase fluid-structure interaction: A stable variational Allen-Cahn-ALE scheme for two-phase FSI

We present a novel partitioned iterative formulation for modeling of fluid-structure interaction in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz., incompressible viscous fluid, a rigid or flexible structure and two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Chan phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner. We assess the accuracy and stability of the new phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free-surface. We consider the decay test problems of increasing complexity, namely free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Using three-dimensional unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.

Investigation on Standing and Traveling Wave Response Patterns in Long Flexible Risers

Due to the complexity involved in the vortex-induced vibration (VIV) of long offshore risers, the fundamental understanding of the coupled kinematics and dynamics of the standing and traveling waves is not well established. In the present contribution, a systematic numerical study on slender flexible riser immersed in a turbulent flow is performed on a flexible riser pinned at both the ends to investigate the standing and traveling wave responses. This wake-body resonance problem requires a stable coupling of the Navier-Stokes equation with the low mass flexible riser structure subjected to strong inertial effects from the surrounding fluid flow. A partitioned iterative scheme that relies on the nonlinear interface force corrections is employed for the modeling of coupled fluid-riser problem. The study here includes a flexible cylindrical riser considered as a long tensioned beam via linear modal analysis. Full three-dimensional simulations are performed on the flexible riser exposed to two different inflow conditions: uniform and linearly sheared. At first, the response characteristics of the riser model are validated with experimental measurements under pinned-pinned condition for uniform current. A detailed analysis is performed on the response characteristics and vorticity dynamics at various locations along the span of the flexible riser. Our simulations show that for uniform inflow condition, the flexible riser exhibits a standing wave-like phenomenon. On the other hand, for linearly sheared inflow, a traveling wave response is observed for both cross-flow and inline oscillations. These traveling waves travel from the top point to the bottom point.Copyright