Showing papers on "Navier–Stokes equations published in 1981"

The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2

Abstract: SUMMARY The spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier-Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of both mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two-dimensions and one element in three-dimensions. The automatic pressure filter associated with the penalty method is also explained. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented herein.

Implementation of finite element methods for Navier-Stokes equations

Abstract: Notations.- 1. Elliptic Equations of Order 2: Some Standard Finite Element Methods.- 1.1. A 1-Dimensional Model Problem: The Basic Notions.- 1.2. A 2-Dimensional Problem.- 1.3. The Finite Element Equations.- 1.4. Standard Examples of Finite Element Methods.- 1.4.1. Example 1: The P1-Triangle (Courant's Triangle).- 1.4.2. Example 2: The P2-Triangle.- 1.4.3. Example 3: The Q1-Quadrangle.- 1.4.4. Example 4: The Q2-Quadrangle.- 1.4.5. A Variational Crime: The P1 Nonconforming Element.- 1.5. Mixed Formulation and Mixed Finite Element Methods for Elliptic Equations.- 1.5.1. The One Dimensional Problem.- 1.5.2. A Two Dimensional Problem.- 2. Upwind Finite Element Schemes.- 2.1. Upwind Finite Differences.- 2.2. Modified Weighted Residual (MWR).- 2.3. Reduced Integration of the Advection Term.- 2.4. Computation of Directional Derivatives at the Nodes.- 2.5. Discontinuous Finite Elements and Mixed Interpolation.- 2.6. The Method of Characteristics in Finite Elements.- 2.7. Peturbation of the Advective Term: Bredif (1980).- 2.8. Some Numerical Tests and Further Comments.- 2.8.1. One Dimensional Stationary Advection Equation (56).- 2.8.2. Two Dimensional Stationary Advection Equation.- 2.8.3. Time Dependent Advection.- 3. Numerical Solution of Stokes Equations.- 3.1. Introduction.- 3.2. Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure.- 3.2.1. uh: P1 Nonconforming Triangle ( 1-4-5) ph: Piecewise Constant.- 3.2.2. uh: P2 Triangle ph: P0 (Piecewise Constant).- 3.2.3. uh: "P2+bubble" Triangle (or Modified P2) ph: Discontinuous P1.- 3.2.4. uh: Q2 Quadrangle ph: Q1 Discontinuous.- 3.2.5. Numerical Solution by Penalty Methods.- 3.2.6. Numerical Results and Further Comments.- 3.3. Velocity-Pressure Formulations: Continuous Approximation of the Pressure and Velocity.- 3.3.1. Introduction.- 3.3.2. Examples and Error Estimates.- 3.3.3. Decomposition of the Stokes Problem.- 3.4. Vorticity-Pressure-Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 3.5. Vorticity Stream-Function Formulation: Decompositions of the Biharmonic Problem.- 4. Navier-Stokes Equations: Accuracy Assessments and Numerical Results.- 4.1. Remarks on the Formulation.- 4.2. A review of the Different Methods.- 4.2.1 Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure.- 4.2.2. Velocity-Pressure Formulations: Continuous Approximations of the Pressure.- 4.2.3. Vorticity-Pressure-Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 4.2.4. Vorticity Stream-Function Formulation.- 4.3. Some Numerical Tests.- 4.3.1. The Square Wall Driven Cavity Flow.- 4.3.2. An Engineering Problem: Unsteady 2-D Flow Around and In an Air-Intake.- 5. Computational Problems and Bookkeeping.- 5.1. Mesh Generation.- 5.2. Solution of the Nonlinear Problems.- 5.2.1. Successive Approximations (or Linearization) with Under Relaxation.- 5.2.2. Newton-Raphson Algorithm.- 5.2.3. Conjugate Gradient Method (with Scaling) for Nonlinear Problems.- 5.2.4. A Splitting Technique for the Transient Problem.- 5.3. Iterative and Direct Solvers of Linear Equations.- 5.3.1. Successive Over Relaxation.- 5.3.2. Cholesky Factorizations.- 5.3.3. Out of Core Factorizations.- 5.3.4. Preconditioned Conjugate Gradient.- Appendix 2. Numerical Illustration.- Three Dimensional Case.- References.

Instability and transition in rotating disk flow

Abstract: The stability of three dimensional rotating disk flow and the effects of Coriolis forces and streamline curvature were investigated It was shown that this analysis gives better growth rates than Orr-Sommerfeld equation Results support the numerical prediction that the number of stationary vortices varies directly with the Reynolds number

Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier—Stokes equations with reverse flow

Abstract: An exact solution to the Navier–Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an O(R−1) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions.For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R [Lt ] 10[sdot ]25, two solutions exist which form a closed branch. Beyond 10[sdot ]25, no similarity solutions exist within the range 10[sdot ]25 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R.

Rates of convergence for viscous splitting of the Navier-Stokes equations

Abstract: Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity v as it becomes small for either twoor three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate C^lAt, Strang-type splitting converges at the rate CP(At)2, and also that solutions of the Navier-Stokes and Euler equations differ by Cp in this case. Here C depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional NavierStokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.

Conservation form of the Navier-Stokes equations in general nonsteady coordinates

Abstract: Introduction R interest in the generation of general bodyoriented curvilinear coordinate systems," for the purpose of solving the complete Navier-Stokes system of equations for subsonic and supersonic flows, has given rise to many forms of presentations of the equations both in conservative and nonconservative formulations. Based on the available solutions of the gasdynamic equations (e.g., Ref. 5) the conservation-law form of the equations seems definitely preferable, particularly when shocks are present. Although the above statement cannot be repeated in a definitive sense for the case of viscous subsonic and supersonic flow prediction through the Navier-Stokes equations, nevertheless, it is expected that the conservation-law form may eventually be more acceptable for numerical purposes. The purpose of this paper is to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems in a simple and direct fashion. Previous work on this subject has been done by McVittie, Viviand, and Vinokur. It will be shown in this Note that the equations in the conservation-law form can be obtained simply by a little manipulation of some standard vector and tensor formulas.

The infinitesimal group of the Navier-Stokes equations

Abstract: The Navier-Stokes equations for an incompressible viscous fluid admit time translation, time dependent change of the pressure origin, a scale change, rotation of axes, and time dependent spatial translation. No other transformations appear if dependence on derivatives is allowed.

Two-Dimensional Inlet Simulation Using a Diagonal Implicit Algorithm

Abstract: A modification of an implicit approximate-factorization finite-difference algorithm applied to the two-dimensional Euler and Navier-Stokes equations in general curvilinear coordinates is presented for supersonic freestream flow about and through inlets. The modification transforms the coupled system of equations Into an uncoupled diagonal form which requires less computation work. For steady-state applications the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. Solutions are given for inviscid and laminar flow about a two-dimensional wedge inlet configuration. Comparisons are made between computed results and exact theory.

Calculation of Unsteady Transonic Flow over an Airfoil

Abstract: An implicit finite-difference solver for either the Euler equations or the "thin-layer" Navier-Stokes equations was used to calculate a transonic flow over the NACA 64A010 airfoil pitching about its one-quarter chord. An unsteady automatic grid-generation procedure that will improve significantly the computational efficiency of various unsteady flow problems is described. The calculated results for both inviscid and viscous flows at Mach number 0.8 over the airfoil oscillating with reduced frequency referenced to one-half chord, 0.2, are compared with experimental data measured in the Ames 11 x 11 ft Transonic Wind Tunnel. Nonlinear, unsteady effects of the flow on the surface pressure variations, shock-wave excursions, and overall airloads are examined. Good agreements between the results of computations and experiments were obtained. In the shock-wave region, however, the results of the viscous-flow computations showed closer agreement with the experimental data.

VNAP2: A Computer Program for Computation of Two-dimensional, Time-dependent, Compressible, Turbulent Flow

Abstract: A computer program, VNAP2, for calculating turbulent (as well as laminar and inviscid), steady, and unsteady flow is presented. It solves the two dimensional, time dependent, compressible Navier-Stokes equations. The turbulence is modeled with either an algebraic mixing length model, a one equation model, or the Jones-Launder two equation model. The geometry may be a single or a dual flowing stream. The interior grid points are computed using the unsplit MacCormack scheme. Two options to speed up the calculations for high Reynolds number flows are included. The boundary grid points are computed using a reference plane characteristic scheme with the viscous terms treated as source functions. An explicit artificial viscosity is included for shock computations. The fluid is assumed to be a perfect gas. The flow boundaries may be arbitrary curved solid walls, inflow/outflow boundaries, or free jet envelopes. Typical problems that can be solved concern nozzles, inlets, jet powered afterbodies, airfoils, and free jet expansions. The accuracy and efficiency of the program are shown by calculations of several inviscid and turbulent flows. The program and its use are described completely, and six sample cases and a code listing are included.

Numerical analysis of the scramjet inlet flow field using two-dimensional Navier-Stokes equations

Abstract: A computer code was developed to solve the full two dimensional Navier-Stokes equations in a supersonic combustion ramjet (scramjet) inlet. In order to be able to consider a general inlet geometry with embedded bodies, a numerical coordinate transformation is used which generates a set of boundary-fitted curvilinear coordinates. The explicit finite difference algorithm of MacCormack is used to solve the governing equations. An algebraic, two-layer eddy-viscosity model is used for the turbulent flow. The code can analyze both inviscid and viscous flows with no strut, one strut, or multiple struts in the flow field. The application of the two dimensional analysis in the preliminary parametric design studies of a scramjet inlet is discussed. Detailed results are presented for one model problem and for several actual scramjet-inlet configurations.

Viscous Flow over Arbitrary Geometries at High Angle of Attack

Abstract: : This paper presents a numerical method for obtaining the supersonic, laminar viscous flow about arbitrary geometries without compression surfaces at high angle of attack In particular, results are presented for blunt biconic bodies with windward and leeward cuts The basic approach used is to solve the steady three-dimensional 'Parabolized Navier-Stokes Equations' (PNS), first derived for circular cones by Lubard and Helliwell These equations have been used to predict the flowfield for a variety of different problems, including flow over sharp and blunt cones at angle of attack up to 40 deg, flow over spinning cones at angle of attack, and flow over cones with mass transfer and temperature variations at the surface In addition to these results which were confined to circular cones, some limited results have been obtained for biconic geometries, non-circular cones and the NASA space shuttle

A seven-mode truncation of the plane incompressible Navier-Stokes equations

Abstract: A model obtained by a seven-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied This model, extending a previously studied five-mode one, exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis (ie, coexistence of attractors) A stochastic behavior is found for high values of the Reynolds number, when no stable fixed points, closed orbits, or tori are present

Improved Calculation of High Speed Inlet Flows: Part I. Numerical Algorithm

Abstract: An improved numerical algorithm has been developed for the calculation of flowfields in two-dimensional high speed inlets. The full mean compressible Navier-Stokes equations are employed, with turbulence represented by an algebraic turbulent eddy viscosity. A body-oriented coordinate transformation is used to facilitate treatment of arbitrary inlet contours. The explicit finite-difference algorithm of MacCormack is utilized. Several well-known techniques for improving computational efficiency are incorporated, including time-splitting of the finite-difference operators and splitting of the mesh into several regions in the cross-stream direction. A number of new computational techniques are introduced; namely, a procedure for automatic determination of the optimal mesh splitting, and a separate treatment of the viscous sublayer and transition wall region of the turbulent boundary layers. The accuracy and efficiency of the approach is demonstrated for the specific examples of the development of a turbulent boundary layer on a flat plate, and the interaction of a shock wave with a flat plate turbulent boundary layer. In all cases, the results compare very favorably with previous numerical calculations and experimental results.

Steady Laminar Flow through Twisted Pipes: Fluid Flow in Square Tubes

Abstract: The Navier-Stokes equation in a rotating frame of reference is solved numerically to obtain the flow field for a steady, fully developed laminar flow of a Newtonian fluid in a twisted tube having a square cross-section. The macroscopic force and energy balance equations and the viscous dissipation term are presented in terms of variables in a rotating reference frame. The computed values of friction factor are presented for dimensionless twist ratios, (i.e., length of tube over a rotation of π radians normalized with respect to half the width of tube) of 20, 10, 5 and 2.5 and for Reynolds numbers up to 2000. The qualitative nature of the axial velocity profile was observed to be unaffected by the swirling motion. The secondary motion was found to be most important near the wall.

Computation of Three-Dimensional Horseshoe Vortex Flow Using the Navier-Stokes Equations,

Abstract: Numerical solutions of the compressible Navier-Stokes equations are presented for a laminar horseshoe vortex flow created by the interaction of a boundary layer on a flat surface and an elliptical strut leading edge mounted normal to the flat surface The computational approach utilizes “zone embedding”, surface-oriented elliptic-cylindrical coordinates, interactive boundary conditions, and a consistently-split linearized block implicit (LBI) scheme developed by the authors Mesh resolution tests are performed, and the horseshoe vortex flow is discussed

Finite element solution of the Navier-Stokes equations in rotating flow

Abstract: The viscous incompressible rotating flow in a shrouded rotor geometry is solved by the finite element method, using the complete three-dimensional (axisymmetric) Navier-Stokes equations. Solutions to classical as well as to previously unsolved rotating flow problems are given and discussed.

A Numerical Solution for Gas-Particle Flows at High Reynolds Numbers

Abstract: Predicting the fluid mechanical characteristics of a gas-solid two-phase flow is critical for the successful design and operation of coal gasification systems, coal fired turbines, rocket nozzles, and other energy conversion systems. This work presents a general grid-free numerical solution which extends a numerical solution of the Navier-Stokes equations developed by Chorin to a solution suitable for unsteady or steady dilute gas-solid particle flows. The method is applicable to open or closed domains of arbitrary geometry. The capability of the method is illustrated by analyzing the flow of gas and particles about a cylinder. Good agreement is found between the numerical method and experiment.

MRS criterion for flow separation over moving walls

Abstract: Steady, incompressible, laminar boundary-layer flows over moving walls are numerically investigated with special regard to the MRS criterion for flow separation. To analyze separated flows without meeting any singularity at the separation point, a set of approximate equations (instead of the boundary-layer equations) is solved under the condition of prescribed external velocity distributions. Three different types of flows are considered, one of which is quite similar to that treated by Tsahalis. It is found that, at least for the flows treated in the present paper, the MRS criterion is not satisfied for the case of upstream-moving walls, although it is satisfied for the case of downstream-moving walls.