Prescribed Velocity Field Research Articles

A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field

A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field

Isosurface extraction for piecewise-linear reconstruction of complex interfaces on arbitrary grids

In the field of volume of fluid type schemes, there is still great room for improvement in the accuracy of the methods used to reconstruct interfaces, especially when highly complex interfaces are involved. Isosurface extraction is exploited in a series of piecewise-linear interface reconstruction methods to simulate complex interface dynamic problems on arbitrary grids, and high-order surface patches are applied to achieve significantly improved accuracy. Isosurface extraction is also used to choose a convenient set of neighboring cells, hereafter called reconstruction stencil, for the reconstruction of highly curved interfaces when using well-known linearity-preserving methods. The relevance of the proper selection of the reconstruction stencil is highlighted, showing that the convergence of formal second-order methods can deteriorate when inappropriate reconstruction stencils are used. The accuracy, computational efficiency and performance of the implemented piecewise-linear interface reconstruction methods coupled with different reconstruction stencils are thoroughly analyzed for the reconstruction of interfaces in static conditions and dynamic simulations that produce highly deformed interfaces by using an accurate unsplit geometric advection method and imposing prescribed velocity fields.

An interface and geometry preserving phase-field method for fully Eulerian fluid-structure interaction

We present an interface and geometry preserving (IGP) method for modeling fully Eulerian fluid-structure interaction via phase-field formulation. While the time-dependent mobility model preserves the hyperbolic tangent interface profile, the proposed method maintains the geometry of the solid-fluid interface by minimizing the volume-conserved mean curvature flow. To reduce the curvature flow, we construct a gradient-minimizing velocity field (GMV) for the convection of the order parameter. The constructed velocity field retains the solid velocity in the solid domain while extending the velocity in the normal direction throughout the diffuse interface region. With this treatment, the GMV reduces the normal velocity difference of the isosurfaces of the order parameter, alleviating the undesired thickening or thinning of the diffuse interface region due to the convection. As a result, the time-dependent mobility coefficient is substantially reduced and there is a lesser volume-conserved mean curvature flow. Furthermore, the GMV ensures that the diffuse interface region moves with the solid bulk despite the fluid flow, such that the fluid-solid interface conforms to the geometry of the solid. Using the unified momentum and mass conservation equations via the phase-dependent interpolation, we integrate the IGP method into a fully Eulerian fluid-structure interaction solver based on the incompressible viscous fluid and the neo-Hookean solid models. The kinematics of the solid in an Eulerian frame of reference is resolved by evolving the left Cauchy-Green tensor. We first demonstrate the ability of the phase-field-based IGP method for the convection of circular and square interfaces in a prescribed velocity field. The variational fluid-structure interaction framework with the IGP method is then examined on the deformation of a solid block under cavity flow. We next explore the geometry-preserving effect of the proposed framework when the fluid flow passes over a fixed deformable block in a channel. Finally, the vibration of a flexible plate attached behind a stationary cylinder is considered to demonstrate the ability of the framework in solving unsteady fluid-structure interaction problems.

High-order level set reinitialization for multiphase flow simulations based on unstructured grids

Reinitialization is essential to maintaining the accuracy of the level set method at capturing interfaces. In this paper, a high-order reinitialization method for the level set function which has previously been developed for structured grids was extended to unstructured grids. The proposed method involves constructing a stencil of an unstructured grid to define a high-order polynomial that approximates a piecewise segment of the interface. Then, the closest point method is adopted to estimate the signed distance function at grid points. The proposed method was validated by solving some benchmark problems of static cases and then the evolution of moving interfacial problems subject to prescribed velocity fields for various grid resolutions consisting of triangular elements. The accuracy of the proposed method was evaluated for computing geometric quantities such as the normal vector and curvature field. The proposed method proved to have high-order accuracy for estimating not only the signed distance function but also geometric quantities for a smooth interface. Finally, a local correction procedure was developed to apply the proposed method to level set problems involving the interface with kinks. The proposed method can accurately estimate normal vector and curvature fields including singularities both for static cases and for dynamic cases where an interface experiences topological changes. • High-order reinitialization of level set method on unstructured grid is developed. • We conduct a high-order estimation of normal vector and curvature on unstructured grid. • We propose a local correction procedure to manage kinks in level set method. • We successfully solve level set problems involving topological changes with kinks.

Second-Order Finite Difference Approximations of the Upper-Convected Time Derivative

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions. The theoretical analysis of the truncation errors of the methods takes into account the linear and quadratic interpolation operators based on a Lagrangian framework. Numerical experiments illustrating the theoretical results for the model equation defined in one and two dimensions are included. Finally, the finite difference approximations of second-order in time are also applied for solving a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field at different Weissenberg numbers.

Contact line advection using the geometrical Volume-of-Fluid method

We consider the interface advection problem by a prescribed velocity field in the special case when the interface intersects the domain boundary, i.e. in the presence of a contact line. This problem emerges from the discretization of continuum models for dynamic wetting. The kinematic evolution equation for the dynamic contact angle (Fricke et al., 2019) expresses the fundamental relationship between the rate of change of the contact angle and the structure of the transporting velocity field. The goal of the present work is to develop an interface advection method that is consistent with the fundamental kinematics and transports the contact angle correctly with respect to a prescribed velocity field. In order to verify the advection method, the kinematic evolution equation is solved numerically and analytically (for special cases).We employ the geometrical Volume-of-Fluid (VOF) method on a structured Cartesian grid to solve the hyperbolic transport equation for the interface in two spatial dimensions. We introduce generalizations of the Youngs and ELVIRA methods to reconstruct the interface close to the domain boundary. Both methods deliver first-order convergent results for the motion of the contact line. However, the Boundary Youngs method shows strong oscillations in the numerical contact angle that do not converge with mesh refinement. In contrast to that, the Boundary ELVIRA method provides linear convergence of the numerical contact angle transport.

A semi-analytic accuracy benchmark for Stokes flow in 3-D spherical mantle convection codes

One of the main challenges in numerical mantle circulation simulations is the determination of accurate solutions to the Stokes equation in combination with an additional constraint arising from the continuity equation. Here we derive a semi-analytic solution to the Stokes equation in a spherical shell using scalar and vector spherical harmonics. For the simple case of an incompressible flow with uniform viscosity we show that a direct relation exists between the flow velocity and the driving force through a fourth-order ordinary differential equation. The latter can be exploited to derive the forcing term from a prescribed velocity field. This feature lends itself to the design of a self-contained benchmark set-up that can be performed without relying on the numerical output from other codes. We demonstrate the applicability of the benchmark by verifying the convergence behaviour of the Stokes solver in the prototype of a new mantle convection modelling framework for high performance computing based on Hierarchical Hybrid Grids (HHG).

Contact Line Advection using the Level Set Method

AbstractIn this work, we consider the geometrical problem of the numerical advection of a hypersurface by a prescribed velocity field in the special case when the hypersurface intersects the domain boundary. This problem emerges from the discretization of continuum models for dynamic wetting. The kinematic evolution equation [1], [2] expresses the fundamental relationship between the rate of change of the contact angle and the structure of the transporting velocity field. We employ a simple version of the Level Set method to numerically solve the hyperbolic transport equation for the interface in two dimensions. The results are validated against an analytic solution of the kinematic evolution equation. Full access to the data and C++‐implementations is provided via an open research data repository [3].

Penetration resistance and the critical cavitation velocity for an ogive-nosed rigid projectile penetrating into a semi-infinite metallic target

This paper studies penetration resistance and projectile-target separation for an ogive-nosed rigid projectile penetrating into a metallic target. A prescribed normal velocity field is introduced to determine the contact stress on the projectile nose, based on which the dynamic equation of a rigid projectile is formulated. Projectile-target separation occurs when the impact velocity of the projectile is greater than the critical cavitation velocity according to Hill's cavitation criterion. Target material is assumed incompressible in both elastic and rigid, perfectly plastic regions in order to simplify the analyses. It shows that the penetration resistance derived from the prescribed velocity field has a different form from that based on the dynamic cavity expansion model. When the impact velocity is greater than the critical cavitation velocity, the projectile-target separation point moves further to the nose tip, leading to the reduction of the contact area between the projectile nose and the target and a large increase of the inertia term (velocity square term) in penetration resistance. The predicted penetration depth and critical cavitation velocity agree reasonably with experimental and numerical results.

Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation

We consider the time‐dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal‐based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi‐discretized in space to get bounds on stabilization parameters where we distinguish equal‐order elements and Taylor‐Hood elements. Different numerical experiments are performed to illustrate our theoretical findings.

A 3D kinematic Babcock Leighton solar dynamo model sustained by dynamic magnetic buoyancy and flux transport processes

Context. Magnetohydrodynamic interactions between plasma flows and magnetic fields is fundamental to the origin and sustenance of the 11-year sunspot cycle. These processes are intrinsically three-dimensional (3D) in nature. Aims. Our goal is to construct a 3D solar dynamo model that on the one hand captures the buoyant emergence of tilted bipolar sunspot pairs, and on the other hand produces cyclic large-scale field reversals mediated via surface flux-transport processes – that is, the Babcock-Leighton mechanism. Furthermore, we seek to explore the relative roles of flux transport by buoyancy, advection by meridional circulation, and turbulent diffusion in this 3D dynamo model. Methods. We perform kinematic dynamo simulations where the prescribed velocity field is a combination of solar-like differential rotation and meridional circulation, along with a parametrized turbulent diffusivity. We use a novel methodology for modeling magnetic buoyancy through field-strength-dependent 3D helical up-flows that results in the formation of tilted bipolar sunspots. Results. The bipolar spots produced in our simulations participate in the process of poloidal-field generation through the Babcock-Leighton mechanism, resulting in self-sustained and periodic large-scale magnetic field reversal. Our parameter space study varying the amplitude of the meridional flow, the convection zone diffusivity, and parameters governing the efficiency of the magnetic buoyancy mechanism reveal their relative roles in determining properties of the sunspot cycle such as amplitude, period, and dynamical memory relevant to solar cycle prediction. We also derive a new dynamo number for the Babcock-Leighton solar dynamo mechanism which reasonably captures our model dynamics. Conclusions. This study elucidates the relative roles of different flux-transport processes in the Sun’s convection zone in determining the properties and physics of the sunspot cycle and could potentially lead to realistic, data-driven 3D dynamo models for solar-activity predictions and exploration of stellar magnetism and starspot formation in other stars.

An iterative interface reconstruction method for PLIC in general convex grids as part of a Coupled Level Set Volume of Fluid solver

Reconstructing the interface within a cell, based on volume fraction and normal direction, is a key part of multiphase flow solvers which make use of piecewise linear interface calculation (PLIC) such as the Coupled Level Set Volume of Fluid (CLSVOF) method. In this paper, we present an iterative method for interface reconstruction (IR) in general convex cells based on tetrahedral decomposition. By splitting the cell into tetrahedra prior to IR the volume of the truncated polyhedron can be calculated much more rapidly than using existing clipping and capping methods. In addition the root finding algorithm is designed to take advantage of the nature of the relationship between volume fraction and interface position by using a combination of Newton's and Muller's methods. In stand-alone tests of the IR algorithm on single cells with up to 20 vertices the proposed method was found to be 2 times faster than an implementation of an existing analytical method, while being easy to implement. It was also found to be 3.4–11.8 times faster than existing iterative methods using clipping and capping and combined with Brent's root finding method. Tests were then carried out of the IR method as part of a CLSVOF solver. For a sphere deformed by a prescribed velocity field the proposed method was found to be up to 33% faster than existing iterative methods. For simulations including the solution of the velocity field the maximum speed up was found to be approximately 52% for a case where 12% of cells lie on the interface. Analysis of the full simulation CPU time budget also indicates that while the proposed method has produced a considerable speed-up, further gains due to increasing the efficiency of the IR method are likely to be small as the IR step now represents only a small proportion of the run time.

Combined Effects of Grain Boundary Convection and Migration in Dynamic Phase Transformations

During large strain deformation of polycrystals, grain or interphase boundaries are driven by the material flow, which is a convection movement. By contrast, upon static recrystallization or grain growth, their motion takes place with respect to matter, which is referred to as grain boundary or interphase migration. During hot working, where dynamic phase transformations commonly occur, convection and migration operate simultaneously. According to local geometrical (e.g., prescribed velocity field, grain boundary curvature) and physical (e.g., grain boundary mobility, dislocation densities) conditions, they can reinforce or oppose each other, but generally combine in more complex ways. The aim of this work is to analyze such effects on the basis of simple analytical approaches. The results suggest that second phase particles or grains dynamically generated (i.e., during straining) exhibit approximately equiaxed shapes.

Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity

It is proved the existence of a unique, global strong solution to the two-dimensional Navier-Stokes initial-value problem in exterior domains in the case where the velocity field tends, at large spatial distance, to a prescribed velocity field that is allowed to grow linearly.

From large-scale to small-scale dynamos in a spherical shell

Kinematic dynamo action in a spherical shell is studied with a small-scale cellular prescribed velocity field. Three velocity fields are considered, all of which are axisymmetric and have a large-scale separation between the shell size and the dominant scale of the motion. The first flow is steady and strongly helical, so that a mean field dynamo might be expected. We find that indeed large-scale dynamo action is obtained at onset, but that the dynamo is of small-scale type at large magnetic Reynolds number Rm, where advection processes dominate and the magnetic field is generated on the scale of the cells in the flow. We study the transition between these two dynamo processes and find a gradual transition as Rm is increased, where the energy is slowly passed from the large scales to the small scales as the two dynamo processes morph into one another. The second flow, a time-dependent version of the first, produces almost identical results, but the transition appears to occur at smaller Rm than in the ste...

Passive scalar decay in chaotic flows with boundaries

This paper considers the long-time decay rate of a passive scalar in two-dimensional flow. The focus is on the effects of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain . The boundary conditions on the passive scalar are imposed on the boundary of by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar is exponential in time with a decay rate γ(κ), where κ is the molecular diffusivity. Scaling laws of the form γ(κ) ≃ Cκα for small κ are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, α ≃ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α ≃ 1/2 for a plane layer, but find α ≃ 2/3 in a square subdomain where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain , the exponent measuring the decay of the scalar field is α ≃ 1/2 for a slip flow condition and α ≃ 3/4 for a no-slip condition. The scaling law exponents α for chaotic time-periodic flows are compared with those for similarly constructed random flows.

Variational data assimilation for the initial-value dynamo problem

The secular variation of the geomagnetic field as observed at the Earth's surface results from the complex magnetohydrodynamics taking place in the fluid core of the Earth. One way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation, in much the same way as is employed in meteorology and oceanography. The aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations. Taking the Earth's core to be an electrically conducting fluid sphere in which convection takes place, we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method. These adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution. The initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere. For the magnetic field, conservation of the divergence-free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition. We thus find that solving the forward and adjoint dynamo system requires different numerical algorithms. In this paper, an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere: one is a kinematic problem with prescribed velocity field, and the second is associated with the Hall-effect dynamo, exhibiting considerable nonlinearity. The algorithm exhibits reliable numerical accuracy and stability. Using both the analytical and the numerical techniques of this paper, the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem. These numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries.

Explicit characteristic-based finite volume element method for level set equation

Accurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two-phase flow problems. This paper presents an explicit characteristic-based finite volume element method for solving the two-dimensional level set equation. The method is applicable for the case of non-divergence-free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time. Copyright © 2010 John Wiley & Sons, Ltd.

A conservative Fourier-finite-element method for solving partial differential equations on the whole sphere

Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude–longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost. The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourier-finite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N2 degrees of freedom, the computational cost is 𝒪(N2logN), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs 𝒪(N2), and less than with a spherical harmonics method, which costs 𝒪(N3). Differential operators with latitude-dependent coefficients are inverted at a cost of 𝒪(N2). We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier–Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described. Copyright © 2009 Royal Meteorological Society

A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves

A coupled level set and volume-of-fluid (CLSVOF) method is implemented for the numerical simulations of interfacial flows in ship hydrodynamics. The interface is reconstructed via a piecewise linear interface construction scheme and is advected using a Lagrangian method with a second-order Runge–Kutta scheme for time integration. The level set function is re-distanced based on the reconstructed interface with an efficient re-distance algorithm. This level set re-distance algorithm significantly simplifies the complicated geometric procedure and is especially efficient for three-dimensional (3D) cases. The CLSVOF scheme is incorporated into CFDShip-Iowa version 6, a sharp interface Cartesian grid solver for two-phase incompressible flows with the interface represented by the level set method and the interface jump conditions handled using a ghost fluid methodology. The performance of the CLSVOF method is first evaluated through the numerical benchmark tests with prescribed velocity fields, which shows superior mass conservation property over the level set method. With combination of the flow solver, a gas bubble rising in a viscous liquid and a water drop impact onto a deep water pool are modeled. The computed results are compared with the available numerical and experimental results, and good agreement is obtained. Wave breaking of a steep Stokes wave is also modeled and the results are very close to the available numerical results. Finally, plunging wave breaking over a submerged bump is simulated. The overall wave breaking process and major events are identified from the wave profiles of the simulations, which are qualitatively validated by the complementary experimental data. The flow structures are also compared with the experimental data, and similar flow trends have been observed.