428 research outputs found
Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions
We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodalbased finite elements may converge to the wrong solution due to a version of the Babuška paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms. Keywords: Meshfree Finite-differences; Navier-Stokes; Incompressible; Vector Poisson equation; Pressure Poisson equation; Reformulation; Manufactured solution; High-orderNational Science Foundation (U.S.) (Grant DMS–1318942
High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions
Pressure Poisson equation (PPE) reformulations of the incompressible
Navier-Stokes equations (NSE) replace the incompressibility constraint by a
Poisson equation for the pressure and a suitable choice of boundary conditions.
This yields a time-evolution equation for the velocity field only, with the
pressure gradient acting as a nonlocal operator. Thus, numerical methods based
on PPE reformulations, in principle, have no limitations in achieving high
order. In this paper, it is studied to what extent high-order methods for the
NSE can be obtained from a specific PPE reformulation with electric boundary
conditions (EBC). To that end, implicit-explicit (IMEX) time-stepping is used
to decouple the pressure solve from the velocity update, while avoiding a
parabolic time-step restriction; and mixed finite elements are used in space,
to capture the structure imposed by the EBC. Via numerical examples, it is
demonstrated that the methodology can yield at least third order accuracy in
space and time
Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions
We demonstrate how meshfree finite difference methods can be applied to solve
vector Poisson problems with electric boundary conditions. In these, the
tangential velocity and the incompressibility of the vector field are
prescribed at the boundary. Even on irregular domains with only convex corners,
canonical nodal-based finite elements may converge to the wrong solution due to
a version of the Babuska paradox. In turn, straightforward meshfree finite
differences converge to the true solution, and even high-order accuracy can be
achieved in a simple fashion. The methodology is then extended to a specific
pressure Poisson equation reformulation of the Navier-Stokes equations that
possesses the same type of boundary conditions. The resulting numerical
approach is second order accurate and allows for a simple switching between an
explicit and implicit treatment of the viscosity terms.Comment: 19 pages, 7 figure
An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field
This paper is concerned with the numerical approximation of the isothermal
Euler equations for charged particles subject to the Lorentz force. When the
magnetic field is large, the so-called drift-fluid approximation is obtained.
In this limit, the parallel motion relative to the magnetic field direction
splits from perpendicular motion and is given implicitly by the constraint of
zero total force along the magnetic field lines. In this paper, we provide a
well-posed elliptic equation for the parallel velocity which in turn allows us
to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system.
This scheme gives rise to both a consistent approximation of the Euler-Lorentz
model when epsilon is finite and a consistent approximation of the drift limit
when epsilon tends to 0. Above all, it does not require any constraint on the
space and time steps related to the small value of epsilon. Numerical results
are presented, which confirm the AP character of the scheme and its Asymptotic
Stability
Comparison and numerical treatment of generalized Nernst--Planck models
In its most widespread, classical formulation, the Nernst-Planck-Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson-Boltzmann system describing the thermodynamic equilibrium is a stationary state of the discretized time-dependent generalized Nernst-Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach
Models and numerical methods for electrolyte flows
The most common mathematical models for electrolyte flows are based on the dilute solution assumption, leading to a coupled system of the Nernst--Planck--Poisson drift-diffusion equations for ion transport and the Stokes resp. Navier--Stokes equations for fluid flow. This contribution discusses historical and recent model developments beyond the dilute solution assumption and focuses on the effects of finite ion sizes and solvation. A novel numerical solution approach is presented and verified here which aims at preserving on the discrete level consistency with basic thermodynamic principles and structural properties like independence of flow velocities from gradient contributions to external forces
Self-consistent modeling of laminar electrohydrodynamic plumes from ultrasharp needles in cyclohexane
This paper presents a self-consistent model of electrohydrodynamic (EHD) laminar plumes produced by electron injection
from ultra-sharp needle tips in cyclohexane. Since the density of electrons injected into the liquid is well described by the
Fowler-Nordheim field emission theory, the injection law is not assumed. Furthermore, the generation of electrons in
cyclohexane and their conversion into negative ions is included in the analysis. Detailed steady-state characteristics of EHD
plumes under weak injection and space-charge limited injection are studied. It is found that the plume characteristics far from
both electrodes and under weak injection can be accurately described with an asymptotic simplified solution proposed by
Vazquez et al. Physics of Fluids 12, 2809 (2000) when the correct longitudinal electric field distribution and liquid velocity
radial profile are used as input. However, this asymptotic solution deviates from the self-consistently calculated plume
parameters under space-charge limited injection since it neglects the radial variations of the electric field produced by a highdensity
charged core. In addition, no significant differences in the model estimates of the plume are found when the
simulations are obtained either with the Finite Element Method or with a diffusion-free particle method. It is shown that the
model also enables the calculation of the current-voltage (IV) characteristic of EHD laminar plumes produced by electron
field emission, with good agreement with measured values reported in the literature.Ministerio de Economía y Competitividad FIS2014-54539-P
Transient electrohydrodynamic flow with concentration dependent fluid properties: modelling and energy-stable numerical schemes
Transport of electrolytic solutions under influence of electric fields occurs
in phenomena ranging from biology to geophysics. Here, we present a continuum
model for single-phase electrohydrodynamic flow, which can be derived from
fundamental thermodynamic principles. This results in a generalized
Navier-Stokes-Poisson-Nernst-Planck system, where fluid properties such as
density and permittivity depend on the ion concentration fields. We propose
strategies for constructing numerical schemes for this set of equations, where
solving the electrochemical and the hydrodynamic subproblems are decoupled at
each time step. We provide time discretizations of the model that suffice to
satisfy the same energy dissipation law as the continuous model. In particular,
we propose both linear and non-linear discretizations of the electrochemical
subproblem, along with a projection scheme for the fluid flow. The efficiency
of the approach is demonstrated by numerical simulations using several of the
proposed schemes
Multiphysics simulation of corona discharge induced ionic wind
Ionic wind devices or electrostatic fluid accelerators are becoming of
increasing interest as tools for thermal management, in particular for
semiconductor devices. In this work, we present a numerical model for
predicting the performance of such devices, whose main benefit is the ability
to accurately predict the amount of charge injected at the corona electrode.
Our multiphysics numerical model consists of a highly nonlinear strongly
coupled set of PDEs including the Navier-Stokes equations for fluid flow,
Poisson's equation for electrostatic potential, charge continuity and heat
transfer equations. To solve this system we employ a staggered solution
algorithm that generalizes Gummel's algorithm for charge transport in
semiconductors. Predictions of our simulations are validated by comparison with
experimental measurements and are shown to closely match. Finally, our
simulation tool is used to estimate the effectiveness of the design of an
electrohydrodynamic cooling apparatus for power electronics applications.Comment: 24 pages, 17 figure
- …