Steady-state Navier-Stokes Equations Research Articles

Hydrodynamic force distribution on a fractal cluster.

The force distribution around a fractal aggregate in a shear flow is investigated. The steady-state Navier-Stokes equation is solved analytically on a model fractal boundary composed of a succession of small-amplitude sinusoidal buckles. It is found that the moments of the force distribution on the fractal scale as powers of the system size, yielding a nontrivial series of exponents, which implies an extremely broad, multifractal force profile. The forces normal and tangential to the boundary are both calculated. The experimental consequences for fractal structures in solution are discussed.

An adaptive multigrid technique for the incompressible Navier-Stokes equations

An automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady-state incompressible Navier-Stokes equations. Solutions have been obtained for the driven cavity and flow over a backward-facing step, for Reynolds numbers up to 5000 and 800, respectively. The refinement criterion is based on the local truncation error. The solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level. For driven cavity flow at Re = 1000, the adaptive refinement approach reduced the computer memory and CPU time to 20 and 40% of the requirements of the ''pure'' multigrid method. The primitive-variable formulation of the Navier-Stokes equations is used so the method can be extended easily to three dimensions. Application to other nonlinear elliptic problems is equally possible.

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Abstract Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.

Parallel multigrid solution of the Navier-Stokes equations on general 2D domains

Multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency. Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).

A finite element computation of moderate Reynolds fluid flow using a modified marquardt method

A modification of the Marquardt method is combined with a Galerkin finite element method to produce an efficient algorithm for the solution of the steady-state Navier-Stokes equation at moderate Reynolds numbers. The algorithm is shown to be 3 to 4 times faster than more conventional algorithms based on the modified Newton method with loading on the nonlinear term. Numerical results are presented for a 4:1 contraction at Reynolds numbers in the range 400–1000 and for problems varying in size from 1107 to 3576 equations.

Navier-Stokes Computation of Radial Inflow Turbine Distributor

A two-dimensional flow analysis of a radial inflow turbine distributor using full steady-state Reynolds-averaged Navier-Stokes equations is made. The numerical prediction of the total energy loss and the wicket gate torque is compared with experimental data. Also, a parametric study is carried out in order to evaluate the behavior of the numerical algorithm.

The steady state Navier–Stokes equations for incompressible flows with rotating boundaries

SynopsisWhen a rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity field and the pressure field of the fluid motion and the angular velocity of the rigid body.In this paper we obtain existence and uniqueness results for the steady state problem in which a rigid body rotates about an axis of symmetry in a viscous incompressible fluid.

Multiple discrete solutions of the incompressible steady-state Navier-Stokes equations

This paper discusses the computation of multiple solutions of various discretizations of the steady state incompressible Navier-Stokes equations. Solution paths (?,R) satisfying the discrete system of equationsH(?,R)=0, where ? represents the discrete flow field andR is the Reynolds number, are computed using a pseudo arc-length continuation procedure. For flows over the back end of an axially symmetric body with a cusped tail in a coaxial circular cylinder, the solution paths often exhibit "hairpin" turning points. Dependence of the paths on the mesh spacing and various selections for the discretizations of the convective and diffusive terms are presented.

Three-dimensional natural convection heat transfer of a liquid metal in a cavity

The paper describes a numerical algorithm for the solution of the steady-state Navier-Stokes equations in three dimensions for the problem of natural convection in a rectangular cavity as a result of differential side heating. Numerical results for two- and three-dimensional models are reported for a cavity filled with a low-Prandtl-number fluid. Supporting experiments using gallium as a working fluid are described. Measured temperatures are compared with predictions of the three-dimensional model. Agreement between data and predictions is only fair and reasons for the discrepancy are identified.

A finite-difference scheme for the solution of the steady Navier-Stokes equations

Abstract A new finite-difference scheme of second-order accuracy, free of artificial diffusion, for solving the steady-state incompressible Navier-Stokes equations is presented. The scheme uses a false transient approach with a combination of variable time step size and over-relaxation for the convection and diffusion terms. The driven flow in a square cavity is used as the model problem. The convergence is much better than the conventional Gauss-Seidel type iterative process. Results for Reynolds numbers up to 5000 are presented.

A many-fibre theory of airflow through a fibrous filter—II: Fluid inertia and fibre proximity

The problem of Stokes flow through a fibrous filter modelled as a two-dimensional array has previously been solved by direct application of the variational principle to a generalized stream function, expressed as a two-dimensional Fourier series in Cartesian coordinates. In this paper the effect of fluid inertia at low Reynolds number is described, by solution of the steady state Navier-Stokes equation in the form of a perturbation expansion, leading to a system of linear partial differential equations with solutions in the form of Fourier series. Pressure drops across model filters, calculated by this method, are compared with experimental results. Both theory and experiment illustrate that the lowest order correction to Darcy's Law is a cubic term in the flow velocity, and that this term is strongly influenced by the filter structure. The details of the flow pattern close to the fibre surface, in particular the presence or absence of standing eddies at finite Reynolds number, is shown to depend critically on the filter structure; eddies are shown to occur in Stokes flow when the fibre layers are sufficiently close together.

Penalty finite-element analysis of coupled fluid flow and heat transfer for in-line bundle of cylinders in cross flow

The two-dimensional steady state Navier-Stokes equations and the energy equation governing laminar incompressible flow are solved using a penalty finite-element model for the case of flow across an in-line bundle of cylinders. Two cases of in-line cylinder bundles, one five rows deep and the other an infinite bundle, are considered with pitch-diameter ratios of 1.25, 1.5 and 1.8 Reynolds numbers studied range from 100 to 600 and Prandtl number is taken as 0.7. Velocity field vectors, stream lines, vorticity, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distribution around the cylinder walls and drag coefficients are presented. The results obtained agree well with available experimental and numerical data.

Validation of a computer simulation of wind flow over a building model

In an attempt to test a computer simulation of wind flow around a block-shaped building against corresponding flows observed in a boundary-layer wind-tunnel, numerical solutions to the steady-state Navier-Stokes equations are compared with the three time-averaged components of velocity on a 200-point grid surrounding the block in the tunnel. A comparison of the pressure fields is also made. Over a large part of the flow the comparisons are encouraging, but the wake region presents unresolved problems.

Boundary integral equation formulations for steady Navier-Stokes equations using the Stokes fundamental solutions

Boundary integral equations for the steady-state flow of an incompressible viscous fluid in two and three dimensions are presented. The steady-state Navier-Stokes equations are transformed into the integral equations by the method of weighted residuals. The fundamental solutions of the Stokes approximate equations are used as the weight functions. The fundamental solotions are constructed by Hörmander's method. The boundary integral equations for unknown pressure are also derived by using the fundamental solution of the Laplacian. A numerical example of the driven cavity flow at Reynolds number 100 is given to show the validity of the formulation.

A one-parameter family of LAD methods for the steady-state Navier-Stokes equations

A one-parameter family of LAD methods for the steady-state Navier-Stokes equations

Penalty-finite-element analysis of 3-D Navier-Stokes equations

Abstract The steady-state Navier-Stokes equations in three dimensions are solved by a penalty-finite-element method for the problems of wall-driven cavity flow in a cubical box and natural convection in a cubical cavity subjected to differential side heating. The present solutions are compared with recent finite-difference solutions of the wall-driven cavity problem for Reynolds numbersRE = 100 and 400. The agreement is found to be very good.

Submerged laminar jet impingement on a plane

Submerged laminar jet impingement on a plane is studied using computation. Steady-state Navier-Stokes equations for the axisymmetric case are solved numerically. The extent of the infinite flow is approximated by applying the boundary conditions at a finite but sufficiently large distance. The tube-exit velocity profile is assumed to be either a fully developed parabolic profile or a flat profile. For the former case, two different nozzle heights from the target plane are considered. The presence of a toroid-shaped eddy at low values of Reynolds number, Re, leads to some interesting observations such as the manner in which the wall shear stress depends on Re. An increase in the height of the nozzle exit from the target plane decreases the wall shear stress, more so at lower values of Re. A change from the parabolic exit velocity profile to the flat profile leads to a decrease in wall shear stress due to decreased momentum flux. The study was motivated by experiments designed to measure the yield shear strength of the vascular endothelium wherein a small saline jet was used to erode the tissue by normal impingement.

Nature of viscous flows near sharp corners

Explicit solutions of two-dimensional, steady-state Navier-Stokes equations are derived in the neighborhood of sharp corners where a sliding wall meets a stationary wall and causes a mathematical singularity. These solutions are valid for small Reynolds numbers. A semi-analytic technique is used to derive these solutions. Some comparisons with numerical solutions are also carried out.

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.

Methodes d'elements finis mixtes pour les equations de stokes et de Navier-Stokes dans un polygone non convexe

We study a mixed finite element method for the steady-state Navier-Stokes equations in a polygon which is not necessarily convex. To take into account the signularities of the solution near the corners, we introduce weighted Sobolev spaces and prove the convergence of the method. The use of a non-uniformly regular family of triangulations allows us to get best error estimates.