2D Incompressible Fluid Research Articles

A local domain boundary element method for solving 2D incompressible fluid flow problems

The Boundary Element Method (BEM) is a well-established, accurate numerical method for solving engineering problems described by linear partial differential equations. However, the application of BEM to non-linear problems leads to the appearance of volume integrals in the integral formulation, and combined with the non-local character of the method the required computational effort is dramatically increased. In this work, a Local Domain BEM (LD-BEM) is proposed for solving incompressible fluid flow problems. The domain of interest is fragmented into small subdomains, in each of which the integral representation of the solution is considered separately. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and reduced computational complexity, overcoming some of the well-known disadvantages of the conventional BEM. The proposed method is applied to the solution of the well-known two-dimensional (2D) lid-driven cavity problem, for Reynolds numbers up to Re=125 × 103, with a discretization involving more than 10 million unknowns.

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equationsto explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equationsof motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.

Swift generator for three-dimensional magnetohydrodynamic turbulence.

Magnetohydrodynamic turbulence is central to laboratory and astrophysical plasmas, and is invoked for interpreting many observed scalings. Verifying predicted scaling law behavior requires extreme-resolution direct numerical simulations (DNS), with needed computing resources excluding systematic parameter surveys. We here present an analytic generator of realistically looking turbulent magnetic fields, that computes three-dimensional (3D) O(1000^{3}) solenoidal vector fields in minutes to hours on desktop computers. Our model is inspired by recent developments in 3D incompressible fluid turbulence theory, where a Gaussian white noise vector subjected to a nonlinear transformation results in an intermittent, multifractal random field. Our B×C model has only few parameters that have clear geometric interpretations. We directly compare a (costly) DNS with a swiftly B×C-generated realization, in terms of its (1) characteristic sheetlike structures of current density, (2) volume-filling aspects across current intensity, (3) power-spectral behavior, (4) probability distribution functions of increments for magnetic field and current density, structure functions, and spectra of exponents, and (5) partial variance of increments. The model even allows to mimic time-evolving magnetic and current density distributions and can be used for synthetic observations on 3D turbulent data cubes.

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2d Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2d Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

Performance of combined spectral collocation method and artificial compressibility method for 3D incompressible fluid flow and heat transfer

Purpose The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer. Design/methodology/approach The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method. Findings The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step. Research limitations/implications It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer. Practical implications The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes. Originality/value The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

Affine Motion of 2d Incompressible Fluids Surrounded by Vacuum and Flows in $$\mathrm{SL}(2,\mathbb {R})$$

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $$\mathrm{SL}(2,\mathbb {R})$$. In the case of perfect fluids, the motion is given by geodesic flow in $$\mathrm{SL}(2,\mathbb {R})$$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in $$\mathrm{SL}(2,\mathbb {R})$$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $$t\rightarrow \pm \infty $$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

Normal mode analysis of 3D incompressible viscous fluid flow models

In this paper, we study the normal mode solutions of 3D incompressible viscous fluid flow models. The obtained theoretical results are then applied to analyze several time-stepping schemes for the numerical solutions of the 3D incompressible fluid flow models.

The aggregation equation with Newtonian potential: The vanishing viscosity limit

The viscous and inviscid aggregation equation with Newtonian potential models a number of different physical systems and has close analogs in 2D incompressible fluid mechanics. We consider a slight generalization of these equations in the whole space establishing well-posedness of the viscous and inviscid equations, spatial decay of the viscous solutions, and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero.

A three-dimensional numerical approach on water entry of a horizontal circular cylinder using the volume of fluid technique

In this paper, complicated hydrodynamics of a horizontal circular cylinder entering water is investigated numerically for low Froude numbers. A numerical approach is used to model the solid-liquid interactions in presence of a free-surface. The governing equations for the 3D incompressible fluid flow are continuity and Navier-Stokes equations along with an equation for the free surface advection. To track the free-surface motion, the fast-fictitious-domain method is integrated into the volume-of-fluid (VOF) technique. The governing equations are solved everywhere in the computational domain including the horizontal cylinder. A rigid body motion is applied to the region occupied by the circular cylinder. The no-slip boundary condition on the solid-liquid interface is exerted implicitly via increasing the viscosity of the region occupied by the solid. To validate the numerical scheme, the results are compared with those of the experiments available in the literature. The effects of cylinder diameter, length, impact velocity, and cylinder-water density ratio on the non-dimensional depth are also investigated.

Global Regularity for Several Incompressible Fluid Models with Partial Dissipation

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

Performance analyses of the IDEAL algorithm combined with the fuzzy control method for 3D incompressible fluid flow and heat transfer problems

ABSTRACTIDEAL proposed by the present author is an efficient segregated algorithm for solving the incompressible fluid flow and heat transfer problems. However, its convergence rate is greatly influenced by the under-relaxation factor. The convergence rate under an optimum under-relaxation factor is dozens of times quicker than that under the most unfavorable under-relaxation factor. To lessen the influence, the IDEAL algorithm combined with a fuzz control method, called IDEAL+FC, is introduced to automatically regulate the values of the under-relaxation factor for accelerating the iteration convergence. Finally, it is demonstrated that IDEAL+FC is superior to IDEAL in terms of convergence rate and robustness. The rapid convergence rate can be achieved by IDEAL+FC even if the initial under-relaxation factor is the most unfavorable value.

Flows of vector fields with point singularities and the vortex-wave system

The vortex-wave system is a version of the vorticity equation governing the motion of 2D incompressible fluids in which vorticity is split into a finite sum of Diracs, evolvedthrough an ODE, plus an $L^p$ part, evolved through an active scalar transport equation. Existence of a weak solution for this systemwas recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for $p>2$, but their result left open the existence and basic propertiesof the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the)linear transport problem associated to the vortex-wave system. To this end, we study the flow associated toa two-dimensional vector field which is singular at a moving point. We first observe that existence and uniqueness of the regular Lagrangian flow are ensured by combining previous results by Ambrosio and by Lacave and Miot. In addition we prove that, generically, the Lagrangian trajectories do not collide with the point singularity. In the second part we present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector fields.

High performance computations of steady-state bifurcations in 3D incompressible fluid flows by Asymptotic Numerical Method

This paper presents a powerful numerical model that implements the Asymptotic Numerical Method to compute 3D steady-state incompressible fluid flow solutions. This continuation algorithm enables to explore branches of steady-state solutions, stable or unstable, to accurately determine any simple steady-state bifurcation points and their emanating bifurcated branches. The powerfulness of the model stands on an optimal step length continuation thanks to the combination of power series analysis in the framework of ANM along with an efficient parallel implementation of the resulting algorithm on high performance computers. The outcome of this approach is demonstrated throughout 3D incompressible fluid flows inside a sudden expansion channel (expansion ratio E=3, cross-section aspect ratio 10≤B≤20). We have computed for the first time up to four steady symmetry breaking (pitchfork) bifurcations together with their associated bifurcated branches. The main characteristic of this 3D symmetric expansion configuration is that for a given cross-section aspect ratio the first bifurcation mode induces a top–bottom asymmetry, as in the 2D case, whereas the subsequent ones modulate the former in the span-wise direction with increasing wave numbers.

Simulation of liquid sloshing in 2D containers using the volume of fluid method

In this paper, a two-dimensional numerical model is developed to study liquid sloshing in containers considering liquid free surface deformation, liquid viscosity and surface tension. The model is validated by a comparison between the computational and theoretical/experimental results for various sloshing scenarios with different time-dependent linear/angular accelerations. The governing equations for the 2D incompressible fluid flow are continuity and Navier–Stokes equations along with an equation for the free surface advection. The deformation of the liquid–gas interface is modeled using the volume-of-fluid (VOF) method. The fluid flow equations describing the fluid sloshing in the container and the dynamic equation which describes the movement of the container are solved separately in two coupled programs. In each time step of computations, the outputs of the fluid program (forces and torque) are obtained and used as inputs for the dynamic program. The forces and torque are applied to the body of the container resulting in translational and rotational accelerations which are then used as inputs to the fluid program in the next time step. The model is also used to simulate the movement of a liquid container in a general case where a complete interaction between the liquid and solid body of the container exists and the container has both linear and rotational accelerations.

3D incompressible fluids: Combinatorial models, Eigenspace models, and a conjecture about well-posedness of the 3D zero viscosity limit

3D incompressible fluids: Combinatorial models, Eigenspace models, and a conjecture about well-posedness of the 3D zero viscosity limit

A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible fluid flow

In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier–Stokes equations for a 2D incompressible fluid flow. We present a numerical test of our theoretical result, for an unsteady fluid flow in a channel around a circular cylinder.

The Navier-Stokes-Voight model for image inpainting

In this paper we investigate the use of the 2D Navier-Stokes-Voight (NSV) model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et. al. in 2001 on exploiting an analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady state solution of the 2D Navier-Stokes vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics (CFD). Recently, the three-dimensional (3D) Navier-Stokes-Voight (NSV) model of viscoelastic fluid, was suggested by Cao, et. al. as an inviscid regularization to the 3D Navier-Stokes equations. We give some background on the NSV mathematical model, describe why it is a good candidate sub-grid-scale turbulence model, and propose this model as an alternative for image inpainting. We describe an implementation of inpainting use the NSV model, and present numerical results comparing the resulting images when using the NSE and NSV for inpainting. Our results show that the NSV model allows for a larger time step to converge to the steady state solution, yielding a more efficient numerical process when automating the inpainting process. We compare quality of the resulting images using subjective measure (human evaluation) and objected measure (by calculating the peak signal-to-noise ratio (PSNR), also known as peak signal-to-reconstructed measure). We also present some new theoretical results based on energy methods comparing the sufficient conditions for stability of the discretization scheme for the two model equations. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem. CONTENTS

Fast single domain–subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer

AbstractIn this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity–vorticity formulation of the incompressible Navier–Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary‐domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved.The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non‐linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid‐driven cavity and the onset of natural convection in a differentially heated enclosure. Copyright © 2008 John Wiley & Sons, Ltd.

Large-scale numerical simulations: Convection in an annular channel rotating about a vertical axis with side-walls

AbstractFor the purpose of understanding the dynamics of planetary atmospheres, we use the annular convection model to simulate the dynamics of atmospheres of Jupiter and Saturn. The model (annular channel) rotates about a vertical axis with side-walls, and it is heated from below.We use the software NaSt3DGP (a parallel software package to solve the 3D incompressible fluid dynamic problems in Cartesian coordinates by using Finite Difference Method) for the computation. It's reliability is tested by our application to simulate fully three-dimensional nonlinear convection in a box with lateral stress-free side-walls, uniformly heated from below. We found that, at moderately large Rayleigh numbers, the complex formation of multiple-jet flows can be maintained by the traveling convective eddies; we also found that the type of the sidewall velocity condition does not play an essential role in determining the primary properties of strongly nonlinear convection.

Application of Local MQ-DQ Method to Solve 3D Incompressible Viscous Flows with Curved Boundary

The local multiquadric-based differ- ential quadrature (MQ-DQ) method proposed by (Shu, Ding, and Yeo (2003)) is a natural mesh- freeapproach forderivativeapproximation,which is easy to be implemented to solve problems with curved boundary. Previously, it has been well tested for the two-dimensional (2D) case. In this work, thismesh-free methodwasextendedtosim- ulate fluid flow problems with curved boundary in three-dimensional (3D) space. The main con- cern of this work is to numerically study the per- formance of the 3D local MQ-DQ method and demonstrate its capability and flexibility for sim- ulation of 3D incompressible fluid flows with curved boundary. Fractional step method was adopted for the solution of Navier-Stokes (N-S) equations in the primitive-variable form. Flow past a sphere with various Reynolds numbers was chosen as a test case to validate the 3D local MQ- DQ method. The computed solution was com- pared well with available data in the literature. The numerical solution shows that the local MQ- DQ method can be applied to solve incompress- ible viscous flow problems with curved boundary in 3D space effectively. Keyword: Local MQ-DQ method; Error esti- mate; Flow past a sphere