Viscous Flow Equations Research Articles

Simulations of post-filling process considering the cooling effect of the mold cooling systems

Simulation of the post-filling process has been developed and performed to study the compressible polymer melt flow during the packing phase and the pressure development inside the mold cavity for the entire post-filling process. A mixed finite-element/finite-difference numerical scheme is implemented to solve the non-isothermal, compressible viscous flow equations. The transient, non-isothermal mold cavity surface temperatures which depend on the mold cooling channel arrangement and coolant flow conditions are also incorporated during simulation using hybrid finite-element/shape-factor method. It has been found that the difference in the pressure profile variation during the post-filling stage is quite distinguished for cases assuming constant mold wall temperature and cases with the consideration of the cooling effect of the cooling system configuration.

Essential Ingredients for the Computation of Steady and Unsteady Blade Boundary Layers

Two methods are described for calculating pressure distributions and boundary layers on blades subjected to low Reynolds numbers and ramp-type motion. The first is based on an interactive scheme in which the inviscid flow is computed by a panel method and the boundary layer flow by an inverse method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier–Stokes equations with an embedded grid technique that permits accurate calculation of boundary layer flows. Studies for the Eppler-387 and NACA-0012 airfoils indicate that both methods can be used to calculate the behavior of unsteady blade boundary layers at low Reynolds numbers provided that the location of transition is computed with the en method and the transitional region is modeled properly.

Interactive airfoil calculations with higher-order viscous-flow equations

A critical examination of the use of higher-order viscous-flow equations in the computation of airfoil flowfields with a viscid-inviscid interaction scheme is presented. Comparisons are made between experimental data and interactive solutions obtained using the boundary-layer equations, the second-order boundary-layer equations, and the Navier-Stokes equations, with corresponding approximations to the viscid-inviscid matching conditions, for flows about symmetric and aft-loaded airfoils.

On creeping flow between a source and a sink in the presence of a sphere

Through analyzing a simple solution of the axisymmetric Stokes equations for slow viscous flow, it is possible to show how placing a finite body between a source and sink tends to expand the domain through which the exchange of mass takes place, whereas placing it on the same side tends to contract this domain.

General three-dimensional viscous primary/secondary flow analysis

Generalized primary/secondary flow equations are derived as an approximation to the Navier-Stokes equations for three-dimensional viscous flows with a dominant flow direction. The primary/secondary flow equations are well posed for solution by spatial marching and can be solved one to two orders of magnitude faster than the Navier-Stokes equations. A key element in the approximations, which is a distinguishing feature of the present approach, is that accuracy is related to curvature terms obtained from a local primary flow direction rather than the coordinate system used to describe geometry of the flowfield. Potential flow streamlines for the flow geometry under consideration are a suitable choice for the primary flow direction. A sequentially decoupled implicit algorithm has been developed that exploits the form of the primary/secondary flow equations to obtain decoupled subsets of equations through choices for dependent variables, the sequence of equations, and the linearization scheme. Each of the equation subsets is solved by efficient and appropriate implicit numerical procedures. Computed solutions for flow in 90-deg bends agree very well with experimental data and NavierStokes solutions. The combined efficiency and accuracy of the approximate equations and solution algorithm make this approach attractive for problems in which a suitable primary flow direction can be identified.

Viscous flux flow in the Bean model of type II superconductors

The magnetic response of a type II superconductor to an applied magnetic field is described by solving the Bean model and viscous flux flow equations simultaneously. Two analytical solutions are presented, which detail the flux distribution within the superconductor induced by a sawtooth-like applied magnetic field. The resultant hysteresis loops show features which depend strongly on the 'sharpness' of the sawtooth waveform. As expected the magnetic hysteresis losses increase with frequency, reflecting the viscous nature of the flux flow. In the zero frequency limit, as the velocity of flux flow approaches zero, the usual Bean result is obtained.

Convergence acceleration of the rational Runge-Kutta scheme for the Euler and Navier-Stokes equations

An efficient method-of-lines approach is presented for the Euler and Navier-Stokes equations. The governing equations are spatially discretized by a central finite-difference approximation. The rational Runge-Kutta scheme is used for the time integration. Attention is focused on improving the efficiency and accuracy of the solution. A remarkable improvement in the efficiency is achieved by adopting a combination of the present scheme with the residual averaging and multigrid (M.G.) techniques. The M.G. method and the high suitability of the present scheme to a vector computer partly reduce the computational load imposed on a numerical simulation with a finer grid. The steady-state convergence obtained with the scheme is comparable with those of diagonalized implicit approximate factorization schemes for inviscid and viscous flow equations. The reliability and accuracy of the scheme have also been improved by adopting the artificial dissipation terms scaled down to the minimum level required for stability. The facilities of the scheme are demonstrated in a series of numerical experiments for two- and three-dimensional transonic flows.

Numerical Analysis for Viscous Shear Flows Past a Circular Cylinder at the Intermediate Reynolds Numbers.

Numerical solutions of the Navier-Stokes equations for viscous shear flows past a circular cylinder are obtained for the range of Reynolds numbers, Re, 2≤Re≤40 and a shear parameter, ε, 0.01≤ε≤1.0. The outer boundary values and the initial ones of numerical calculations are determined by our previous analytical solutions(1) utilizing Oseen's approximation, and numerical calculations are carried to be single-valued throughout the field of flow in the process of repetition. The flow patterns, drag, lift, moment and pressure distributions on a circular cylinder are compared with the results calculated in the same methods under the fixed boundary conditions and ones of other papers. From these comparisons, it is seen that the values of some properties are fairly affected by the outer boundary conditions, and also the present boundary values are favorable in comparison with the previous ones.

SLOW MOTION OF A SOLID SPHERE IN THE PRESENCE OF A NATURALLY PERMEABLE SURFACE

Exact solutions of the equations of slow viscous flow are obtained for a solid sphere moving in the presence of a naturally permeable plane surface bounding a porous medium for the cases of (a) normal approach to the surface and (b) rotation about a diameter perpendicular to the surface

Flow separation and unsteadiness in a rotating sliced cylinder

Abstract The rotating sliced cylinder model has its origins in simulations of the large scale wind-driven ocean circulation. The model consists of a rotating circular cylinder with a sloping planar base and a lid which rotates at a rate slightly different from that of the background rotation. Numerical models for the inviscid flow, the boundary-layer flow and the viscous flow equations in this configuration are described and the flow patterns produced by them are compared. Laboratory experiments were also performed and the results are presented as validation of the numerical models. The unsteadiness of the flow in some parameter regimes which was noted by other authors (Beardsley, 1969, 1973a, b and Beardsley and Robbins, 1975) appears to be caused by instability of the separated boundary-layer flow. The separated flow has some unique features due to the effect of the driving force exerted on the flow by the rotating lid and these are also described.

Boundary conditions and efficient solution algorithms for the potential function formulation of the 3-D viscous flow equations : Houseman, G A Geophys JV100, N1, Jan 1990, P33–38

Boundary conditions and efficient solution algorithms for the potential function formulation of the 3-D viscous flow equations : Houseman, G A Geophys JV100, N1, Jan 1990, P33–38

The Jeffery paradox as the limit of a three-dimensional Stokes flow

The solution of the biharmonic equation for the slow viscous flow resulting from a line rotlet in front of a circular cylinder indicates the physically unrealistic behavior of a uniform stream at infinity—an example of the Jeffery paradox. It is shown here, when the three-dimensional motion resulting from a finite line rotlet parallel to the (infinite) circular cylinder is investigated, that the length h of the cylinder has to be so large that O[(ln h)−1] terms are negligible before this two-dimensional result is correct.

Boundary Conditions and Efficient Solution Algorithms For the Potential Function Formulation of the 3-D Viscous Flow Equations

SUMMARY For incompressible viscous creeping flows that occur in a number of geophysical situations, the velocity field may be expressed as the curl of a vector potential function, the use of which allows the momentum equation to be written as a biharmonic equation. The 3-D Cartesian formulation for a constant viscosity fluid is summarized here with special reference to two important types of boundary condition: the stress-free boundary (with zero normal velocity and zero tangential stress) and the rigid boundary (with all components of velocity zero). Fast algorithms for inversion of the biharmonic operator with all boundaries stress-free are well established. There also exists a fast method for the solution of the biharmonic equation with a parallel pair of rigid boundaries with the other boundaries stress-free. This method has not previously been applied, but it is a relatively straightforward extension of the Fourier transform based algorithm for the stress-free problem, using an analytical solution to enforce the required boundary conditions for each horizontal harmonic component. The method is easily vectorized and allows solutions to be obtained that compare very favourably in accuracy and solution time with those for the stress-free problem. The errors are of comparable magnitude given the differing harmonic content required by the boundary conditions, and the solution requires between 20 per ce-t (for a 3-D problem) and 30 per cent (for a 2-D problem) more processing time than does the solution of a comparable stress-free problem.

Viscous liquid flows that are independent of the reynolds number

All solutions of the Navier-Stokes equations for stationary plane viscous fluid flow that are independent of the Reynolds number are investigated. All solutions with variable vorticity are given explicitly. Constant vorticity solutions are known to include an arbitrary harmonic function. Examples of analytic solutions of the latter type which satisfy adhesion conditions on a quadratic parabola and an ellipse are given.

Numerical solution of Navier-Stokes equations for two-dimensional viscous compressible flows

A nodal-point, finite-volume space discretization of viscous fluxes in compressible Navier-Stokes equations is presented. To advance the solution in time, an explicit five-stage Runge-Kutta scheme has been used. To accelerate the rate of convergence to steady state, local time stepping, residual averaging, and enthalpy damping have been employed. The scheme has been evaluated by solving laminar flow over a semi-infinite flat plate and an NACA 0012 air foil using thin-layer approximation. It has been observed here that fourth-order artificial dissipation is sufficient for numerical stability. The results have been compared with available theoretical and numerical solutions.

II. Determination of constitutive coefficients and illustrative examples

This paper is a continuation of Part I under the same title and is concerned mainly with the determination of the constitutive response coefficients, as well as some simple illustrative examples. First, a system of simplified constitutive equations for incompressible viscous turbulent flow is obtained from the more general system of equations in Part I through a judicial choice of retaining only those terms which appear to represent major features of the turbulent flow. Even for this simplified system of equations, the identification of some of the constitutive coefficients presents a formidable task; and this is especially true in the case of those coefficients that are associated with the presence of the additional independent variables of the theory due to the manifestation of the alignment of eddies (on the microscopic scale), turbulent fluctuation and eddy density. Because of this difficulty, the present effort for identification of the various constitutive coefficients must be regarded partly as tentative, pending future availability of suitable relevant experimental data and/or pertinent numerical simulation results. Keeping this background in mind, most of the relevant coefficients in the constitutive equations are determined, or the nature of their functional forms are estimated, through consideration of‘cartoon-like’ models on the microscopic level and these results are then used in conjunction with the macroscopic equations of motion to examine a number of simple solutions. These include the possibility of a flow possessing a constant uniform velocity gradient and solutions pertaining to decay of flow anisotropy and plane turbulent channel flows. The predicted theoretical calculations are in general accord with experimental observations. In addition, for plane channel flow, plots of variation along the width of the channel for the turbulent temperature and the macroscopic velocity compare favourably with corresponding known experimental results.

Interactive and large-domain solutions of higher-order viscous-flow equations

Two approaches for the solution of the partially parabolic Reynolds equations are evaluated using the same numerical techniques and turbulence model. Comparisons are made between interactive viscous-inviscid solutions and non-interactive large-domain solutions for the flow over the tail and in the wake of axisymmetric and three-dimensional bodies to highlight the differences between the two strategies. Both approaches yield satisfactory results, though the interactive solutions appear to be computationally more efficient for the three-dimensional bodies.

Similarity Transformations of Compressible Viscous Flow in Conservative Force Fields. I. Two-Dimensional Problem

This paper treats the similarity transformation of the Navier-Stokes equations for a two-dimensional compressible viscous flow. The transformation of the basic equations with a certain conservative force to those without an external force is considered. It is assumed that the spatial variation of the viscosity and thermal conductivity may be neglected and the dissipation function in the energy equation is also neglected. A simple example of a flow is given for a centrifugal force field by applying the transformation obtained. Several other transformations for more generalized conservative force fields are obtained for the case where the flow of two-dimensions is reduced to that of one-dimension by them.

非圧縮性粘性流れと弾性シェルの相互作用に関する数値シミュレーション

The oscillating modes of thin elastic shells under unsteady fluid forces are numerically analyzed. The governing equations used in this paper are the Navier-Stokes equations, continuity equation and ones of motion of thin elastic shells. The spatial discretization of each equation is achieved with the finite element method. In the proposed numerical method, the finite element equations for incompressible viscous flows and shells are solved simultaneously at each time step of numerical simulation. Consequently, both the flow profiles around the shells and oscillating modes of ones can be obtained. This method is applied to numerical calculation of a coupled problem of an incompressible viscous flow and an elastic cylindrical shell in order to show the validity of our procedure.

On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics

The mathematical properties of the pressure-velocity and vorticity-velocity formulations of the equations of viscous flow are compared. It is shown that a vorticity-velocity formulation exists which has the interesting property that non-inertial effects only enter the problem through the implementation of initial and boundary conditions. This valuable characteristic, along with other advantages of the vorticity-velocity approach, are discussed in detail.