Showing papers on "Stream function published in 1987"

Review of the entry flow problem: Experimental and numerical

Abstract: This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.

Nonlinear stability and statistical mechanics of flow over topography

Abstract: The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable ; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the b-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.

Vortex Motion on a Sphere

Abstract: The theory of the vortex motion of two-dimensional incompressible inviscid flow on a sphere is presented. Vorticity and stream function, which are related by the Laplace-Beltrami operator, are initially outlined. Green's function of the equation is obtained in which the stream function is expressed as an integral form. The equations of motion for two vortex models on a sphere are derived. In particular, the equation for vortex patches with constant vorticity is given in terms of the contour integral appropriate for the contour dynamics.

Modeling ground water flow and pollution : with computer programs for sample cases

Abstract: 1. Introduction.- 1.1. Groundwater and Aquifers.- 1.2. Management of Groundwater.- 1.3. Groundwater Modeling.- 1.4. Continuum Approach to Porous Media.- 1.5. Horizontal Two-Dimensional Modeling of Aquifers.- 1.6. Objectives and Scope.- 2. Groundwater Motion.- 2.1. Darcy's Law and its Extensions.- 2.2. Aquifer Transmissivity.- 2.3. Dupuit Assumption.- 3. Modeling Three-Dimensional Flow.- 3.1. Effective Stress in Porous Media.- 3.2. Mass Storage.- 3.3. Fundamental Mass Balance Equation.- 3.4. Initial and Boundary Conditions.- 3.5. Complete Statement of Mathematical Flow Model.- 3.6. Modeling Soil Displacement.- 4. Modeling Two-Dimensional Flow in Aquifers.- 4.1. Aquifer Storativity.- 4.2. Fundamental Continuity Equations.- 4.3. Initial and Boundary Conditions.- 4.4. Complete Statement of Aquifer Flow Model.- 4.5. Regional Model for Land Subsidence.- 4.6. Streamlines and Stream Function.- 5. Modeling Flow in the Unsaturated Zone.- 5.1. Capillarity and Retention Curves.- 5.2. Motion Equations.- 5.3. Balance Equations.- 5.4. Initial and Boundary Conditions.- 5.5. Complete Statement of Unsaturated Flow Model.- 6. Modeling Groundwater Pollution.- 6.1. Hydrodynamic Dispersion.- 6.2. Advective, Dispersive, and Diffusive Fluxes.- 6.3. Balance Equation for a Pollutant.- 6.4. Initial and Boundary Conditions.- 6.5. Complete Statement of Pollution Model.- 6.6. Pollution Transport by Advection Only.- 6.7. Macrodispersion.- 7. Modeling Seawater Intrusion.- 7.1. The Interface in a Coastal Aquifer.- 7.2. Modeling Seawater Intrusion in a Vertical Plane.- 7.3. Modeling Regional Seawater Intrusion.- 8. Introduction to Numerical Methods.- 8.1. Analytical versus Numerical Solutions.- 8.2. Survey of Numerical Methods.- 8.3. Computer Programming.- 9. The Finite Difference Method.- 9.1. Steady Flow.- 9.2. Unsteady Flow.- 9.3. Accuracy and Stability.- 9.4. Generalizations.- 10. The Finite Element Method.- 10.1. Steady Flow.- 10.2. Steady Flow in a Confined Aquifer.- 10.3. Steady Flow with Infiltration and Leakage.- 10.4. Steady Flow through a Dam.- 10.5. Unsteady Flow in an Aquifer.- 10.6. Generalizations.- 11. Transport by Advection.- 11.1. Basic Equations.- 11.2. Semi-Analytic Solution.- 11.3. System of Wells in an Infinite Field.- 11.4. System of Wells in an Infinite Strip.- 11.5. Numerical Solution in Terms of the Piezometric Head.- 11.6. Numerical Solution in Terms of the Stream Function.- 11.7. Tracing Particles Along a Stream Line.- 12. Transport by Advection and Dispersion.- 12.1. Dispersion in One-Dimensional Flow.- 12.2. Numerical Dispersion.- 12.3. A Finite Element Model for Two-Dimensional Problems.- 12.4. Random Walk Model.- 13. Numerical Modeling of Seawater Intrusion.- 13.1. Model for Flow in a Vertical Plane.- 13.2. Basic Equations for a Regional Model of Seawater Intrusion.- 13.3. Finite Element Model for Regional Interface Problems.- Appendix. Solution of Linear Equations.- References.- Problems.- Index of Subjects.

Exact solutions of unsteady two-dimensional Navier-Stokes equations

Abstract: This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

An analytical study of the cone penetration test

Abstract: The quasi-static penetration of a cone penetrometer into clay can be formulated as a steady state problem by considering a steady flow of soil past a stationary cone. The soil velocities are estimated from the flow field of an inviscid fluid, and the incompressibility condition is achieved by adopting a stream function formulation. Emphasis is placed on obtaining an accurate velocity estimate and this is accomplished by a solution of the Navier-Stokes equations. The strain rates are evaluated from the flow field using a finite difference scheme. The clay is modelled as a homogeneous incompressible elastic-perfectly plastic material and the soil stresses are computed by integrating along streamlines from some initial stress state in the upstream region. These stresses do not in general obey the equilibrium equations, although one of the two equations can be satisfied by an appropriate choice of the mean stress. Several attempts have been made to use the remaining equilibrium equation to obtain an improved velocity estimate and three plausible iterative methods are detailed in this thesis. In a second study, a series of finite element calculations on the cone penetration problem is performed. In modelling the penetration process, the cone is introduced in a pre-formed hole and some initial stresses assumed in the soil, incremental displacements are then applied to the cone until a failure condition is reached. Although the equilibrium condition is satisfied very closely in the finite element calculations, it is extremely difficult to achieve a steady state solution. In a third series of computations, the stresses evaluated by the strain path method are used as the starting condition for the finite element analysis. This is believed to give the most realistic solution of the cone penetration problem because both the steady state and equilibrium conditions are approximately satisfied. Numerically derived cone factors are presented and these are found to depend on the rigidity index of the soil and the in situ stresses. The pore pressure distribution in the soil around the penetrometer is estimated using Henkel's empirical equation. The dissipation analysis is based on Terzaghi's uncoupled consolidation theory. The governing equation is formulated in the Alternating-Direction-Implicit finite difference scheme. This formulation is unconditionally stable and variable time steps are used to optimise the solution procedure. The dissipation curves are found to be significantly affected by the rigidity index of the soil and a dimensionless time factor is proposed to account for this effect.

A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations

Abstract: A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.

Hydraulic Control of Flows with Nonuniform Potential Vorticity

Abstract: The hydraulics of flow contained in a channel and having nonuniform potential vorticity is considered from a general standpoint. The channel cross section is rectangular and the potential vorticity is assumed to be prescribed in terms of the streamfunction. We show that the general computational problem can be expressed in two traditional forms, the first of which consists of an algebraic relation between the channel geometry and a single dependent flow variable and the second of which consists of a pair of quasi-linear differential equations relating the geometry to two dependent flow variables. From these forms we derive a general “branch condition” indicating a merger of different solutions having the same flow rate and energy and show that this condition implies that the flow is critical with respect to a certain long wave. It is shown that critical flow can occur only at the sill in a channel of constant width (with one exception) at a point of width extremum in a flat bottom channel. We als...

Numerical simulation of viscoelastic flow through a sudden contraction using a type dependent difference method

Abstract: Numerical simulation of the flow of upper convected Maxwell fluid through a planar 4:1 contraction has been performed using type dependent difference approximation of the vorticity equation. For creeping flow assumption, the numerical convergence has been achieved up to much higher values of the elasticity parameter than those obtained by conventional finite difference methods. For non-vanishing Reynolds number flow, it is shown that the corner vortices disappear, which is in good qualitative agreement with existing experimental results. In doing so, spatial distributions of stream function, vorticity and stresses are considered in relation to a change of type of vorticity.

On the finite element approximation of a cascade flow problem

Abstract: The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.

Finite element stream function-vorticity solutions of the incompressible Navier-Stokes equations

Abstract: SUMMARY The incompressible, two-dimensional Navier-Stokes equations are solved by the finite element method (FEM) using a novel stream function/vorticity formulation. The no-slip solid walls boundary condition is applied by taking advantage of the simple implementation of natural boundary conditions in the FEM, eliminating the need for an iterative evaluation of wall vorticity formulae. In addition, with the proper choice of elements, a stable scheme is constructed allowing convergence to be achieved for all Reynolds numbers, from creeping to inviscid flow, without the traditional need for upwinding and its associated false diffusion. Solutions are presented for a variety of geometries.

Creeping flow around and through a permeable sphere moving with constant velocity towards a solid wall

Abstract: The problem of creeping flow of a newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity is solved in terms of the streamfunction and the pressure. The permeability of the sphere is assumed to be continuous, uniform and isotropic, The flow in the sphere is modelled with Darcy's law, and a Beavers-Joseph-Saffman slip-flow boundary condition is assumed at the boundary. Sample streamlines and isobars are calculated. The hydrodynamic correction factor to Stoke's law, ƒ is calculated as a function of the dimensionless permeability, κ, the dimensionless slip factor, β, and the dimensionless gap length, δ. As expected, for typical κ and β values the values of ƒ are substantially smaller than those for an impermeable sphere. An important result is that as δ decreases the value of ƒincreases much more slowly than it does for an impermeable sphere; furthermore, ƒ is finite as δ→0. For moderate and large κ values the ƒ vs δ curve has a maximum.

Three-dimensional viscous flow solutions with a vorticity-stream function formulation

Abstract: : A three-dimensional streamlike function/vorticity transport procedure has been developed to analyze two- and three-dimensional inviscid and viscous flows. Both the formulation and the numerical techniques used to solve these equations contain many of the advantages of interacting boundary layer theory for strongly interacting viscous and inviscid flows. An algorithm which involves the solution of two uncoupled Poisson/vorticity transport equation sets is described. An implicit line relaxation scheme is used to solve a 2 x 2 block-tridiagonal system for the Poisson and vorticity transport equations in each of the x- and z-directions. Solutions for 2-D and 3-D inviscid and viscous flows are compared with other numerical solutions demonstrating the stability and accuracy of the current procedure. Favorable agreement with the recently obtained 3-D interacting boundary layer solutions of Edwards demonstrates the overall accuracy of this new approach for 3-D viscous flows including flow separation. This streamlike function/vorticity transport procedure has been found to yield smooth solutions without the need to add explicit artificial viscosity. Keywords: stream function; finite difference coefficients; Navier Stokes equations.

Nonlinear magnetohydrodynamic waves in a steady zonal circulation for a shallow fluid shell on the surface of a rotating sphere

Abstract: This paper considers two-dimensional nonlinear MHD waves of large horizontal spatial scales for a thin magnetofluid layer on the surface of a rotating sphere The 'shallow fluid' hydrodynamic equations are generalized to include the effects of magnetic fields, and it is shown that the resulting MHD equations can be reduced to a single scalar equation for a stream function involving several free functions For special choices of these free functions, two kinds of finite-amplitude MHD waves are obtained, propagating in the azimuthal direction relative to the uniformly rotating background atmosphere in the presence of a background zonal magnetic field and a steady differential zonal flow These two kinds of MHD waves are fundamentally due to the joint effects of the uniform rotation of the background atmosphere and background magnetic field; the first is an inertial wave of the Rossby (1939) and Haurwitz (1940) type, modified by the presence of the background zonal magnetic field, while the second is a magnetic Alfven-like wave which is modified by the uniform rotation of the background atmosphere

3D eddy currents in multiply connected thin sheet conductors

Abstract: A method for calculating 3D eddy current distributions in thin sheet conductors of arbitrary shape and of small thickness relative to the skin depth of the material at the excitation frequency is described. The regions surrounding the conducting sheets are modelled using magnetic scalar potentials and the sheet current is represented by a scalar stream function. The field equations are in differential form and are modelled numerically using finite elements. The resulting sparse symmetric set of equations is solved using the pre-conditioned bi-conjugate gradient technique. Discontinuities in sheet conductivity and thickness can be modelled. This is used to simulate the effect of holes in the sheets.

Two-Dimensional Numerical Model of Thermal Discharges in Coastal Regions

Abstract: This paper describes a two-dimensional numerical model of thermal discharge in coastal regions. The model uses potential flow theory to calculate the ambient velocity field of the coastal water. Using the calculated velocity field, the heat transport equation is solved to define the excess temperature field. The finite difference method for arbitrary boundary shapes is used to solve the stream function for the current velocity calculation and the two-dimensional heat transport equation for the excess temperature calculation. To illustrate the computational procedure, the model is applied to the case of a recently developed industrial complex in the Red Sea region. The power and desalination plants of the industrial complex require a large amount of seawater for cooling purposes. The used cooling water is discharged into the Red Sea. Hypothetical values for the various parameters are used in this case study. The model is run on a minicomputer, a VAX-11/730. The case study demonstrates the usefulness of thestream function to obtain the velocity field. It also demonstrates that the model can provide a first approximation of the temperature field. Therefore, the use of the model is justified where field data are not readily available.

Numerical simulation of the die swell problem of a Newtonian fluid by using the concept of stream function and a local analysis of the singularity at the corner

Abstract: The die-swell problem is reconsidered by using the concept of stream tubes for the incompressible axisymmetric case. Using some assumptions and analytical equations, it is possible to study the influence of the singularity at the corner where the free surface forms, in addition to the upstream and downstream boundary conditions. A minimization technique is used for the determination of the coefficients related to the analytical equations of the streamlines. This enables us to compute the flow field in the jet for a Newtonian fluid, the swelling ratio of which is found to be 12%.

Transformation of regular waves

Abstract: Results are presented from calculations for shoaling waves using a development of Dean's non-linear stream function which display better agreement with previous experimental data for parallel waves in shallow water than other currently available theories. The results are presented in a manner suitable for design applications. In addition to wave height evaluation, curves have been given to aid the prediction of other parameters of design interest, such as crest elevation and horizontal water particle velocity. It is demonstrated that the stream function series type of solution is particularly suitable for this application, because the integral wave properties required for shoaling predictions may be obtained directly from the stream function solution, without performing integrations in the body of one wave. It is also shown that the conservation of wave action flux and wave energy flux lead to identical results in terms of wave shoaling predictions.

The Numerical and Analytical Study of Bifurcation and Multicellular Flow Instability Due to Natural Convection Between Narrow Horizontal Isothermal Cylindrical annuli at High Rayleigh Numbers.

Abstract: : This effort deals with a numerical and analytical study of multicellular flow instability due to natural convection between narrow horizontal isothermal cylindrical annuli. Buoyancy-induced steady or unsteady flow fields between the annuli are determined using the Boussinesq approximated two-dimensional Navier-Stokes equations and the viscous-dissipation neglected thermal-energy equation. The vorticity stream function formulation of the Navier Stokes equations is adopted. Both thermal and hydrodynamic instabilities are explored. An asymptotic expansion theory is applied to the Navier-Stokes equations in the double-limit of Rayleigh number approaching infinity and gap width approaching zero. Thermal instability of air near the top portions of narrow annuli is considered for various size small gap widths. For these narrow gaps, the Rayleigh numbers corresponding to the onset of steady multicellular flow are predicted. Numerical solutions of the 2-D Navier Stokes equations also yield hysteresis behavior for the two-to-six and two-to-four cellular states, with respect to diameter ratios of 1.100 and 1.200. In contrast, an unsteady hydrodynamic multicellular instability is experienced near the vertical sections of narrow annuli when the Pr approaches 0 boundary layer equations are solved numerically. (Theses).

Introduction of the Stream Function Concept to the analysis of hydrodynamic dispersion in porous media

Abstract: The stream function concept is introduced to the analysis of hydrodynamic dispersion in porous media. The governing partial differential equation for the stream function for two-dimensional flow under steady state water and solute transport conditions in an isotropic porous medium is derived. For a uniform seepage velocity, the corresponding differential equation is obtained as a special case. The equation for axially symmetric solute transport case is also derived. It is shown that the isoconcentration curves and the hydrodynamic dispersion stream lines are not orthogonal to each other. For the uniform seepage velocity field case, general solutions are presented for concentration distribution and hydrodynamic dispersion stream function (HDSF) for strip sources having variable concentration distributions. Special solutions are presented for concentration, HDSF, and convective-dispersive flux components for a strip source having constant concentration. The concept of hydrodynamic dispersion bounding streamline separating the solute transport zone from the other zones is also introduced. The methodology of the HDSF concepts and the solutions for some idealized cases can be applied for contaminant plume analysis, computer model validation, and other hydrogeological studies.

The Turbomachine Blading Design Achieved by Solving the Inverse Field Problem

Abstract: This paper describes a design procedure for the determination of the geometry of the blading of the turbomachine with prescribed thickness and bound vorticity distribution. The boundary conditions are discussed in order to have a properly posed field problem. Optimised 2D cascade design example is shown. The quasi 3D “S2” - “S1” stream function formulation is developed. The design of guide vanes downstream of a lateral inlet casing is described. A new approach by introducing a potential like function to treat the 3D rotational flow is also formulated.Copyright © 1987 by ASME

Simulation of separated flow past a bluff body using Navier-Stokes equations

Abstract: An analysis is developed for simulating two-dimensional flow past a bluff body, using the incompressible unsteady Navier-Stokes equations in terms of vorticity and stream function. The conservation-law form of the equations is employed, in generalized orthogonal curvilinear coordinates, using the contravariant component of the vorticity. The fully implicit time-marching alternating-direction implicit-block Gaussian elimination (ADI-BGE) method employed is a direct method, with second-order spatial accuracy and, hence, avoids introduction of any artificial viscosity. This unsteady analysis has been carefully applied to simulate flow past a circular cylinder with and without symmetry, requiring the use of either the half or the full cylinder, respectively. For the latter configuration, at early time levels, the evolution of some flow characteristics in the near wake is compared extensively with existing experimental data, in order to partially validate the present analysis. For the half-cylinder configuration, the asymptotic flow structure predicted by the present analysis conforms, to some extent, with the existing numerical and analytical solutions. This analysis has the potential advantage of computing symmetric unstable modes as well as solution bifurcations which may result for flow up to Reynolds number Re = 104 or even Re = 105.

Numerical solution of stream function equations in transonic flows

Abstract: In order to develop the transonic stream function approach, in this paper one of the momentum equations is employed to form the principal equation of the stream function which does not contain vorticity and entropy terms, and the other one is used to calculate the density directly. Since the density is uniquely determined, the problem that the density is a double-valued function of mass flux in the stream function formulation disappears and the entropy increase across the shock is naturally included. The numerical results for the transonic cascade flow show that the shock obtained from the present method is slightly weaker and is placed farther downstream compared to the irrotational stream function calculation, and is closer to the experimental data. From a standpoint of computation the iterative procedure of this formulation is simple and the alternating use of two momentum equations makes the calculation more effective.

Numerical solution of a viscous incompressible flow problem through an orifice

Abstract: A steady flow problem of a viscous, incompressible fluid through an orifice is widely applicable to many physical phenomena and has been studied previously by many researchers. A problem of such type has been solved by applying LAD method given by Roache [1]. The resulting system of linear equations is solved by Hockney's method [2].

Inverse Design Calculations for Transonic Cascades

Abstract: The computation methods of the inverse problem and several mixed aerodynamic problems for transonic cascades are presented. The calculations are completed directly on the physical plane by the stream function equation. The corresponding boundary conditions of these problems are transfered into the boundary conditions expressed by stream function with the help of the relations of the basic equations. The methods presented have a clear physical concept and use similar manners of iteration for different problems. The calculations of some examples for the transonic inverse problem indicate this method is very effective. An example is given of a shockless supercritical cascade designed successfuly by the transonic inverse code.Copyright © 1985 by ASME