Terms Of Navier-Stokes Equations Research Articles

A Numerical Procedure Based on 1D-IRBFN and Local MLS-1D-IRBFN Methods for Fluid-Structure Interaction Analysis

The partition of unity method is employed to incorporate the moving least square (MLS) and one dimensional-integrated radial basis function (1D-IRBFN) techniques in a new approach, namely local MLS-1D-IRBFN or LMLS-1D-IRBFN. This approach leads to sparse system matrices and offers a high level of accuracy as in the case of 1D-IRBFN method. A new numerical procedure based on the 1D-IRBFN method and LMLS-1D-IRBFN approach is presented for a solution of fluid-structure interaction (FSI) problems. A combination of Chorin's method and pseudo-time subiterative technique is presented for a transient solution of 2-D incompressible viscous Navier-Stokes equations in terms of primitive variables. Fluid domains are discretised by using Cartesian grids. The fluid solver is first verified through a solution of mixed convection in a lid-driven cavity with a hot lid and a cold bottom wall. The structural solver is verified with an analytical solution of forced vibration of a beam. The Newmark's method is employed for the forced vibration analysis of the beam based on the Euler-Bernoulli theory. The FSI numerical procedure is then applied to simulate flows in a lid-driven open-cavity with a flexible bottom wall.

On possible isolated blow-up phenomena and regularity criterion of the 3D Navier-Stokes equation along the streamlines

Abstract: The first goal of our paper is to give a new type of regularity criterion for solutions u to Navier-Stokes equation in terms of some supercritical function space condition u ∈ L∞(Lα,∗) (with 3 4 (17 1 2 − 1) < α < 3) and some exponential control on the growth rate of div( u |u| ) along the streamlines of u. This regularity criterion greatly improves a previous result of the first author. However, we also point out that totally new idea which involves the use of the new supercritical function space condition is necessary for the success of our new regularity criterion in this paper. The second goal of our paper is to construct a divergence free vector field u within a flowinvariant tubular region with increasing twisting of streamlines towards one end of a bundle of streamlines. The increasing twisting of streamlines is controlled in such a way that the associated quantities ‖u‖Lp (2 < p < 3) and ‖div( u |u| )‖L6 blow up while preserving the finite energy property u ∈ L at the same time. We also briefly mention how this construction is related to the regularity criterion proved in our paper.

Viscous Potential Flow Analysis of Rayleigh-Taylor Instability of Cylindrical Interface

The present paper deals with the study of Rayleigh-Taylor instability at the cylindrical interface using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. A dispersion relation is derived and stability is discussed in terms of various parameters such as Ohnesorge number, density ratio etc. A condition for neutral stability is obtained and it is given in terms of critical value of the wave number. It is observed that the Ohnesorge number has destabilizing effect while inner fluid fraction has stabilizing effect on the stability of the system.

Viscous potential flow analysis of Kelvin-Helmholtz instability of cylindrical interface

A linear analysis of Kelvin-Helmholtz instability of cylindrical interface is carried out using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. Both asymmetric and axisymmetric disturbances are considered. A dispersion relation has been obtained and stability criteria are given in the terms of the relative velocity. A comparison between inviscid potential flow and viscous potential flow has been made. It has been observed that Reynolds number and inner fluid fraction both have destabilizing effect on the stability of the system.

Differential forms, fluids, and finite models

By rewriting the Navier-Stokes equation in terms of differential forms we give a formulation which is abstracted and reproduced in a finite dimensional setting. We give two examples of these finite models and, in the latter case, prove some approximation results. Some useful properties of these finite models are derived.

Response characteristics of a vortex-excited circular cylinder in laminar flow

This paper presents numerical simulation results for vortex-induced vibration of a circular cylinder in laminar flow. A vortex method is implemented to solve the two-dimensional Navier-Stokes equations in terms of vorticity. In order to validate the numerical code, the flow past a fixed cylinder is first investigated for which enough experimental and numerical results are available. Basic characteristics of the dynamic response and vortex shedding for an elastically mounted circular cylinder are then investigated for 70 < Re < 170. The lock-in phenomenon is captured at certain reduced velocities where the lift coefficient takes a considerable value associated with a high amplitude response. The wake structure exhibits the 2S or C (2S) modes of vortex shedding in this range of Reynolds numbers, as opposed to the 2P mode which is observed in the turbulent flow regime. The numerical results are in acceptable agreement with available experimental and numerical data.

Regularity criteria in weak spaces for $3$-dimensional Navier-Stokes equations in terms of the pressure

In this paper we consider the Cauchy problem for the $3$-dimensional Navier-Stokes equations and we establish some Serrin-type regularity criterion in weak spaces involving the summability of the pressure or the gradient of the pressure.

Smoothness Criterion for Navier-Stokes Equations in Terms of Regularity along the Streamlines

Smoothness Criterion for Navier-Stokes Equations in Terms of Regularity along the Streamlines

A stochastic representation for backward incompressible Navier-Stokes equations

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.

Regularity Criterion for Weak Solutions to the Navier-Stokes Equations in Terms of the Gradient of the Pressure

We prove a regularity criterion for weak solutions to the Navier-Stokes equations in three-space dimensions. This improves the available result with .

SOLVING BURGERS EQUATION BY A MESHLESS METHOD

Burgers equation is a fundamental partial differential equation of second order to describe the integrated process of convection-diffusion in physics. It occurs in various areas of applied mathematics and physics, such as modeling of turbulence, boundary layer behavior, shock wave formation, and mass transport. The convective and diffusive terms in Navier-Stokes equation are included in Burgers equation while the pressure term is neglected. A least-square point collocation meshless formula is proposed to discretize the Burgers equation. To verify the present meshless approach, the distributions of velocity for Burgers equation under different Reynolds numbers are investigated. Numerical results show that the proposed approach presents a good simulation of shock wave for Burgers equation with large Reynolds number.

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.

Three-Dimensional Model of Navier-Stokes Equations for Water Waves

A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.

Acoustics in viscous subsonic flow models with nonreflecting boundary conditions

The article investigates acoustic waves in a subsonic viscous thermally conducting gas. Two descriptions of the medium are compared: a description in terms of Navier-Stokes equations and a description as a quasi-gas-dynamic system based on a difference approximation of the Boltzmann equation. Reflection of acoustic waves from artificial boundaries in the numerical region is considered, and special boundary conditions are constructed to ensure damping of nonphysical modes. A difference approximation of such boundary conditions is investigated. The procedure is tested in application to viscous flow around plates.

Convergence of gauge method for incompressible flow

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable Φ, u = a + ⊇Φ, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field a are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint Δt/Δx < C. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.

Implementation of boundary conditions for meshless methods

Boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite-element-like Kronecker delta condition. Furthermore, it is frequently desirable, especially in fluid mechanics, to impose general, nonlinear boundary and interface constraints. This paper describes a computationally efficient algorithm based on d'Alembert's principle that can be used for general constraints both in meshless methods and finite elements. First, a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed. Subsequently, an analogous method is developed for nonlinear constraints. Special attention is given to imposing general boundary and fluid-structure interface conditions on the Navier-Stokes equations in terms of conservative variables. Numerical results using d'Alembert's principle with the Reproducing Kernel Particle Method (RKPM), including viscous, supersonic flow past a NACA 7012 airfoil, are shown.

Numerical treatment of the instability and the breakup of a liquid capillary column in a bounded immiscible phase

The capillary instability of an infinite axisymmetric viscous liquid column in an immiscible medium is investigated. The process of disintegration is simulated numerically using a (second-order) finite-difference method applied to the ‘vorticity-stream function’ formulation of the Navier-Stokes equations. These equations and corresponding boundary conditions are written in there detailed form including convective terms in Navier-Stokes equations and nonlinear terms in the mass and momentum conservation equations at the unknown interface. Then the evolution in time of a given cosinusoidal disturbance is studied subjected to the action of the nonlinear effects. In these conditions the formation of a satellite drop attached to the main drop is observed. In the case where the liquid column is submerged into a low density inviscid fluid, the basic characteristics of the column disintegration such as drops sizes and breakup time are in a good agreement with those calculated by previous authors. New results are obtained for the instability parameters of a liquid column surrounded by another viscous fluid.

A TVD Convective-Difference Scheme Using an Arbitrarily Shaped Grid for Analyzing 3-D Unsteady Incompressible Flow. 2nd Report. 3-D Navier-Stokes Equations.

A numerical method for analyzing the 3-D unsteady incompressible viscous flow with complicated boundaries or irregular shaped grids is proposed. In this method, the unsteady Navier-Stokes equations in terms of the primitive variables are solved using a time-splitting method and the convective-difference scheme which is stabilized by means of the TVD conception. Poisson's equation for pressure is integrated using Gaussian Integration in order to remove the spurious errors, and solved by the iteration scheme. For irregular grids the finite-difference operators derived by the least squares polynomial approximation are used. As a numerical example, a complex recirculating flow in a three-dimensional cavity was calculated.

On Approximate Inertial Manifolds for Stochastic Navier-Stokes Equations

The stochastic approximate inertial manifold is constructed for 2D Navier-Stokes equations with random initial data and excited by additive white noise. We estimate the size of the corresponding attracting neighborhood for this manifold. Our consideration relies on some additional information on the nonlinear term in Navier-Stokes equations.

Finite element solutions of nonsteady incompressible viscous flow between two rotating concentric spheres

Abstract The problem of unsteady, incompressible viscous flow between two rotating concentric spheres has been investigated here. The full Navier-Stokes equations in terms of the velocity components u, v w and the pressure p , using spherical coordinates for axially symmetric flow, were solved by means of the finite element method in the spatial dimension and the alternating-direction method in the time dimension using Glowinski's algorithm. The element used is an annular-sector-type element with a bilinear approximation for the velocity components and with constant pressure within the element. Reynolds numbers in the range from 1 to 1000, gap size 0.5 and different combinations of the angular velocity of the inner and outer spheres were studied. In some of these cases a steady-state solution was possible, while in others only a transient solution was possible. This method proved to be successful and powerful in predicting the behavior of the flow for these nonlinear-type problems.