Showing papers on "Stream function published in 1994"

Hiemenz flow in hydromagnetics

Abstract: The laminar flow of an incompressible, viscous, electrically conducting fluid impinging normal to a plane in the presence of a transverse magnetic field is investigated. Using finite-differences and quasilinearization, an exact numerical solution is presented which takes into account the asymptotic boundary condition. It is demonstrated that iff denotes the dimensionless stream function, the value off″(0) increases monotonically withM, the Hartmann number, where a prime denotes the derivative normal to the plane.

An experimental study of dipolar vortex structures in a stratified fluid

Abstract: This paper describes laboratory experiments on dipolar vortex structures in a linearly stratified fluid. The dipoles are generated by a pulsed horizontal injection of a small volume of fluid, by which a localized three-dimensionally turbulent flow region is created. After the subsequent gravitational collapse the flow becomes approximately two-dimensional, and eventually a single vortex dipole emerges, as the result of the self-organizing properties of such flows. The flow evolution has been visualized both by dye and tracer particles, through which qualitative as well as quantitative information was obtained. By application of digital image analysis, the spatial distribution of the velocity ν, vorticity ω and stream function ψ were determined. It was found that dipoles in the turbulent-injection experiments are characterized by a nonlinear sinh-like relationship between ω and ψ, whereas in the case of laminar injection the (ω, ψ)-scatter plots of the dipoles reveal a linear relationship. Notwithstanding these differences, both types of dipoles show a dynamical structure that agrees very well with the Lamb–Chaplygin dipole, as was found by comparing the size, position of maximum vorticity, cross-sectional distributions of ν and ω, characteristics of the (ω, ψ)-relationship, and the translation speed of the experimental and the model dipole.

Quantitative experimental study of the free decay of quasi-two-dimensional turbulence

Abstract: A quantitative experimental study of freely decaying quasi-two-dimensional turbulence is presented. The flow is produced in a thin layer of electrolyte by using a steady, spatially periodic, electromagnetic forcing. The particle-image-velocimetry method is used to determine the instantaneous velocity field, the vorticity field, and the stream function. Global quantities, such as the energy, the enstrophy, and the kurtosis of the vorticity distribution are measured, and geometrical properties, such as the number of eddies, their size, and their mean separation, are determined. Unexpected characteristics are obtained, revealing the existence of a new form of quasi-two-dimensional turbulence, which we call liquid.

Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates

Abstract: The stream function y/ for axisymmetric Stokes flow satisfies the wellknown equation E4y/ = 0. In spheroidal coordinates the equation E2y/ = 0 admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation E4i// = 0. The nonseparability of Ea >// = 0 in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for ^ in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E2 is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator E . A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions. Received March 18, 1992. 1991 Mathematics Subject Classification. Primary 76D07; Secondary 33A50, 33A70, 35C10, 35D99, 35J30.

Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows

Abstract: It is well known that the equations of motion of a fluid particle in a two-dimensional flow can be written in the canonical form with the stream function playing the role of the Hamiltonian. Here we extend this canonical formalism to three-dimensional steady flows. We show how the methods of the theory of Hamiltonian systems can be successfully used to investigate the advection of a passive particle. As an example of the perturbed closed streamline flow we consider a rigid rotation with added small quadratic velocity field and explain the structure of streamlines by averaging the corresponding Hamiltonian. We also show how the Hamiltonian formulation can be used to find the invariants of the fluid particle motion which are then used for non-canonical averaging.

The full approximation accuracy for the stream function-vorticity-pressure method

Abstract: By means of identity techniques, in this paper, we develop the stream function-vorticity-pressure method and obtain the full approximation convergence and global superconvergence estimates for the Stokes equations.

Numerical analysis of oscillatory instability of buoyancy convection with the galerkin spectral method

Abstract: The Galerkin spectral method with basis Junctions previously introduced by Gelfgat 111 is applied for analysis of oscillatory instability of convective flows in laterally heated rectangular cavities. Convection of water and air in a square cavity, and convection of a tow-Prandtl-number fluid in a square cavity, and a cavity with a ratio length / height of 4 are considered. Patterns of the most unstable perturbations of the stream function and the temperature are presented, and mechanisms of oscillatory instability are discussed. Comparison with other numerical investigations shows that the Galerkin method with divergent-free basis functions, which satisfy all the boundary conditions, needs fewer modes than other methods using discretization of the flow region.

Finite element solution of the streamfunction‐vorticity equations for incompressible two‐dimensional flows

Abstract: The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework or the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution or two well-known benchmark problems: flow in a lid-driven cavity at Re≤10,000 and flow over a backward-facing step at Re=800

Newton's method for solving heat transfer, fluid flow and interface shapes in a floating molten zone

Abstract: Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier–Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.

Boundary layer receptivity to free‐stream vorticity

Abstract: The receptivity to free‐stream vorticity of the boundary layer over a flat plate with an elliptic leading edge is investigated numerically by solving the incompressible Navier–Stokes system in general curvilinear coordinates with the vorticity and streamfunction as dependent variables. A small‐amplitude vortical disturbance is introduced at the upstream boundary and the governing equations solved time accurately to evaluate the spatial and temporal growth of the perturbations leading to instability waves [Tollmien–Schlichting (TS) waves] in the boundary layer. The effect of disturbance amplitude, orientation, and the effect of the leading edge and of surface curvature are investigated for the case of spanwise vorticity. Simulations reveal, for the conditions considered, a linear variation in the TS response with forcing amplitude for perturbations of free‐stream velocity that are either symmetrical or asymmetrical with respect to the basic‐state stagnation streamline. The presence near the leading edge of a large, oscillating component of velocity normal to the airfoil axis for the case of asymmetrical forcing results, for the same strength of input disturbance, in an increase in the TS response aft of the juncture and in the appearance of a superharmonic component of the disturbance motion near the tip of the nose. This superharmonic decays rapidly in the streamwise direction. In all cases considered, the first clear appearance of the TS mode occurs aft of the surface pressure‐gradient maximum. Changes to the geometry that increase the maximum in steady surface pressure gradient are found to increase receptivity.

A scalar function formulation for optical flow

Abstract: In this work, we present results from a new formulation for determining image velocities from a time-sequence of X-ray projection images of flowing fluid. Starting with the conservation of mass principle, and physics of X-ray projection, we derive a motion constraint equation for projection imaging, a practical special case of which is shown to be the Horn and Schunck's optical flow constraint. We are interested in the study of non-rigid motion of blood which is an incompressible fluid, and as such have developed a formulation for optical flow which is applicable to such media. The formulation is particularly efficient, as the flow field is obtained from a 90 degrees rotation applied to the gradient of a scalar function. It is shown that if specific criteria are met, in addition to normal flow which is commonly recoverable, the tangential component of the flow field is also recoverable, bypassing the aperture problem. An algorithm is presented to illustrate this. Preliminary results from the optical flow formulation applied to synthetic images, as well as contrast-injected X-ray images of flowing fluid, in a cylindrical fluid phantom are presented.

Wave-induced abyssal recirculations

Abstract: The forcing of abyssal recirculation gyres by cross-isopycnal mixing and wave fluxes near the deep western boundary is investigated. A three-layer isopycnal primitive equation model is applied in a series of experiments to an idealized basin with bottom topography. In the absence of deep western boundary current instabilities, cross-isopycnal mixing forces a cyclonic recirculation gyre, modified by topography, which is consistent with the traditional Stommel-Arons model. Instabilities of the boundary current fundamentally alter the mean basin-scale deep flow from a cyclonic recirculation to an anticyclonic recirculation. Bottom topography plays a key role in destabilizing the mean flow. The forcing mechanism for the interior recirculation is the horizontal divergence of momentum and potential vorticity fluxes carried by topographic waves that are forced by the boundary current instabilities. The strength of the recirculation gyre is linearly proportional to the kinetic energy of the waves, which is controlled in the present model by bottom drag, and well predicted by a simple scale analysis. This is essentially an adiabatic process. The addition of cross-isopycnal mixing forces the large-scale interior recirculation toward the pole, partially into boundary currents. through linear vorticity dynamics. Vorticity budgets reveal three dynamical regimes for the eddy-driven flows, the western boundary current, the recirculation region, and the interior. Similarities and differences between the mean flow and recent observations in the Brazil Basin are discussed.

Cross-Frontal Mixing in a Meandering Jet

Abstract: A Gulf Stream-like meandering current is kinematically modeled using a streamfunction ψ = Uλ(1 − tanh[(y − yc)/λ cos(α)]. Under suitable parameter values a “front” exists along the velocity maximum across which there is no transport, but with mixing occurring on either side. An analogous barrier exists in the Gulf Stream but appears to fade with depth. The conditions for mixing across the model front are studied using the Chirikov overlap criterion, which indicates the front breaks down for meander amplitude above a critical threshold that is inversely dependent on the ratio of meander phase speed and current speed, c/U. It is suggested that the increase in cross-frontal mixing in the deeper levels of the Gulf Stream is the result of current meandering and the decrease of current velocity with depth. The mechanism for this is interaction of different meander modes traveling along the Gulf Stream. These ideas are shown to be consistent with field measurements of tracers in the ocean.

Model for heat and moisture transfer in arbitrarily shaped two-dimensional porous media

Abstract: A model was developed to predict heat and moisture transfer due to natural convection and diffusion in arbitrarily shaped two-dimensional porous media. Boundary conditions were diurnally varying ambient temperature on the outside of walls with moderate Biot number. Other important boundary conditions were developed for typical storage and transportation situations, A two-energy equation model was used to allow for the difference between the fluid and solid temperatures and its effect on mass transfer in the porous medium. The governing equations were solved with a finite-difference method in a generalized coordinate system using a stream function formulation. It was found that the energy and moisture transport equations were best solved using a modified Crank-Nicolson method that was developed to control the tendency for instability caused by the source terms in these equations. All of the boundary conditions that were developed worked satisfactorily The two-energy equation model predicted small differences between the fiuid and solid particle temperatures and natural convection only impacted the temperature solution significantly in the upper comers of the porous media.

Finite element method for incompressible viscous flows with adaptive hybrid grids

Abstract: A new adaptive finite element numerical method has been developed for the unsteady Navier-Stokes equations of incompressible flow in two dimensions. The momentum equations combined with a pressure correction equation are solved employing a nonstaggered grid. The solution is advanced in time with an explicit/implicit marching scheme. An adaptive algorithm has been implemented, which refines the grid locally to resolve detected flow features. A combination of quadrilateral as well as triangular cells provides a stable and accurate numerical treatment of grid interfaces that are located within regions of high gradients. Applications of the developed adaptive algorithm include both steady and unsteady flows, with low and high Reynolds numbers. Comparisons with analytical as well as experimental data evaluate accuracy and robustness of the method. I. Introduction I NCOMPRESSIBLE flows are frequently encountered in engineering applications. During the past two decades a significant number of numerical algorithms have been developed for solution of the incompressible Navier-Stokes equations.1 liie lack of pressure term in the continuity equation makes solution of the momentum equations with the divergence-free constraint more difficult. In the case of incompressible flows, the conservation of mass acts as a constraint condition that the velocity field must satisfy, whereas in compressible flows, the conservation of mass is given through a partial differential equation for the temporal variation of density. The infinite speed of sound in the incompressible case requires an implicit treatment of the pressure. Furthermore, spatial discretization for pressure and velocity may produce oscillatory solutions. One approach followed is to formulate the equations in terms of a stream function and a vorticity. Extension of this method to three dimensions is not possible. Different formulations have been used in three dimensions, such as the vorticity-velocity approach. Another approach is to use the compressible flow equations and solve them for low Mach numbers. The required time step for such computations is very small because the speed of sound approaches infinity at the incompressible limit. A method that uses compressible-like governing equations is the artificial compressibility approach.2"^ A time derivative of the pressure is added to the continuity equation, and the incompressible flowfield is treated as compressible during the transient stage. Time accuracy of the simulation is usually not preserved. Another class of algorithms uses a special Poisson equation for the pressure field.5"8 The usual computational procedure is to assume an initial pressure field, and then an iterative process is defined until the continuity equation is satisfied. A major issue of the corresponding pressure and velocity spatial discretization is oscillations in the pressure field. To reject these modes, staggered grids have been employed by several of these algorithms.9'10 On the other hand, employment of nonstaggered grids11"13 requires dissipation in the algorithms. Stability of both approaches with high-Reynolds-number flows is an important issue. A review of numerical methods for incompressible flows, as well as references

Mixing the Earth's mantle by thermal convection: A scale dependent phenomenon

Abstract: We have investigated the mixing properties of 2D Rayleigh Benard convection in infinite Prandtl number flows. The mixing properties were monitored by injecting tracer particles into previously calculated flow fields. A particle correlation function H(r) and the corresponding correlation dimension have been used to characterize the mixing efficiency of the flow as function of its vigor and it's structure. We demonstrate that the chosen method captures the mixing process in a detailed manner. Our study suggests that mixing properties of the flow depend on the spatial scale. At Rayleighnumbers 106 < Ra < 108 heterogeneities within one circulation cell are destroyed rapidly while two adjacent cells can remain unmixed substantially longer.

Unsteady second grade aligned MHD fluid flow

Abstract: Exact solutions are established for the equations of a class of an unsteady, plane, second grade, electrically conducting, MHD aligned fluid which undergoes isochoric motion This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream

On the solution of Stokes’ equations between confocal ellipses

Abstract: The analytical solution of Stokes’ equations between two concentric, confocal ellipses is derived here. This bounded flow, similar in certain respects to the journal bearing flow, was imagined in order to investigate two‐dimensional mixing and Lagrangian chaos in a bounded flow with two symmetry axis. The derived streamfunction is in the form of a Fourier cosine series and, when the eccentricity ratio of the inner ellipse is not very low, the solution converges very rapidly. When the ellipses turn in opposite directions, there are cases where two saddle points are connected by two different streamlines, a necessary and sufficient condition for structural instability according to Peixoto’s theorem. This flow geometry could be particularly effective for mixing of viscous fluids since the number of low period hyperbolic and elliptical points during time periodic boundary motion is greater than for the eccentric rotating cylinder system. The Poincare sections obtained with a discontinuous velocity protocol su...

Creeping flow about a slightly deformed sphere

Abstract: The problem of slow streaming flow of a viscous incompressible fluid past a spheroid which departs but little in shape from a sphere with mixed slip-stick boundary conditions, is investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterising the deformation. The case of an oblate spheroid is considered as a particular example and the force on this non-spherical body is evaluated. It is found that the parameter λ1, which arises in connection with the boundary condition, has significant effect upon the hydrodynamic force. In fact, it is shown that, the force is a quadratic function of this parameter up to the first order of deformation. Also, it is observed that the drag in the present case is less than that of the Stokes resistance for a slightly oblate spheroid. Some other special cases are also deduced from the present result. A brief discussion of the results to other body shapes is presented.

Robust velocity estimates, stream functions, and simulated Lagrangian drifters from sequential spacecraft data

Abstract: The maximum cross-correlation (MCC) method has been used to compute both oceanic and cloud velocity vectors from sequences of satellite data [e.g., advanced very high resolution radiometer (AVHRR), coastal zone color scanner (CZCS), geostationary observing earth satellite (GOES)]. Unfortunately, the two-dimensional cross-correlation functions used in the computation often contain saddlepoints, which can give rise to large magnitude and direction uncertainties in the derived velocity estimates. This paper develops a numerical iterative procedure that combines image analysis methods and dynamical constraints to minimize these difficulties. The resultant velocities are both physically realistic and numerically stable. Thus, it is also possible to compute stream functions and simulated Lagrangian drifters. The validity of these results are confirmed with independent oceanic observations. Finally, the advective-diffusive equation is solved for a few oceanic applications (e.g., prediction of sea-surface temperature, dispersal of anchovy eggs and larvae) using the derived velocities. >

Simple Adjoint Methods for Single-Doppler Wind Analysis with a Strong Constraint of Mass Conservation

Abstract: Three schemes are developed to incorporate a strong constraint of (incompressible) mass conservation into the basic scheme (scheme B) of the simple adjoint method of Qiu and Xu for retrieving the time-mean wind field from a sequence of single-Doppler scans. In the first scheme (S1), the two-dimensional wind field on a surface normal to the radar beam is partitioned into an irrotational component expressed by the velocity potential and a nondivergent component expressed by the streamfunction. The velocity potential can be obtained directly from single-Doppler observations, and the streamfunction is retrieved by the adjoint method. In this way, the retrieved wind field satisfies the mass conservation equation precisely. The second scheme (S2) is the same as scheme S1 except that a spectral expression is used to replace the grid representation of the streamfunction. The third scheme (S3) imposes a strong mass conservation constraint through a postadjustment after the wind field is retrieved by schem...

Relationship between tropical heating and global circulation: Interannual variability

Abstract: This study examines the relationship between tropical heating and global circulation in the troposphere through singular value decomposition (SVD) analysis and numerical simulations based on a global barotropic spectral model. SVD analysis was applied to the 200-mbar stream function field and the outgoing longwave radiation (OLR) to extract the most recurrent coupled mode. The stream function vector of the first mode indicates an out-of-phase relationship between the stream function in the northern and southern hemispheres. Tight gradients in the pattern along the equator suggest a large fluctuation of zonal-mean zonal wind associated with this seesaw. Its eddy component appears to modulate the stationary eddies. The major structure of the corresponding OLR vector is a dipole near the equator with one center at 120°E and another one at 160°W. The first mode exhibits a strong interannual variability and is strongly correlated with the occurrence of El Nino and La Nina. To understand the relationship between the stream function and OLR, a series of numerical experiments using a barotropic model was carried out, specifying Rossby wave source associated with differing idealized divergence patterns in the tropics. The steady state responses in all experiments exhibit a zonally symmetric structure resembling the stream function vector of the first mode. The pattern was mainly forced by the zonal-mean component of Rossby wave source that is contributed mostly by zonal-mean divergent winds under the constraint of zero global-mean divergence. The same zonal-mean forcing is also responsible for the forced eddy stream function perturbations that exhibit similar structures in all experiments.

Nonlinear Equilibration of Two-Dimensional Eady Waves: Simulations with Viscous Geostrophic Momentum Equations

Abstract: Equilibration of two-dimensional Eady waves is numerically investigated using the geostrophic momentum equations incorporating heat and momentum diffusion. Extended solutions are obtained beyond what would be the collapse of surface fronts in the inviscid theory, and are found to accurately reproduce the equilibration of baroclinic waves simulated with the primitive equations. Potential vorticity anomalies produced at the surface fronts are essential in the initial amplitude saturation and in the asymptotic behavior of the equilibrated flow. A supergeostrophic shear spun up nonlinearly in the zonal flow also plays an important role in causing the reversal of the tilt. The zonal mean potential temperature profile in the equilibrium state is similar to the prediction by the adjustment hypothesis of Gutowski when only horizontal diffusion is present. However, it is closer to an Eady's basic state with enhanced static stability when vertical diffusion is also present. The difference in the ageostroph...

Source-sink turbulence in a stratified fluid

Abstract: A new method of generating turbulence in a stratified fluid is presented. The flow is forced by a symmetric array of sources and sinks placed around the perimeter of a tank containing stratified fluid. The sources and sinks are located in a horizontal plane and the flow from the sources is directed horizontally, so that fluid is withdrawn from and re-injected at its neutral density level with some horizontal momentum. The sources and sinks are arranged so that no net impulse or angular momentum is imparted to the flow. Measurements of the mixing produced by the turbulence are made using a conductivity probe to record the vertical density profile. The flow field is measured by tracking small neutrally buoyant particles which are placed within the fluid. The tracking of the particles and analysis of the flow fields are done automatically using DigImage, a recently developed suite of particle tracking software. The characteristics of the flow are found to depend on the forcing parameter F = V/Nd , where V is the mean velocity of the flow through the source orifices, d is the diameter of the sources and sinks and N is the buoyancy frequency of the stratification. At large F three-dimensional turbulence is produced within a mixed layer centred on the level of the sources and sinks. A comparison of mixing rates measured in this and more conventional experiments is made, and it is concluded that in terms of the local turbulence parameters the entrainment rates are similar. At low F , no significant mixing occurs and the flow is approximately two-dimensional with very small vertical velocities. Under these circumstances a qualitative change in the characteristics of the flow occurs after the experiment has been running for some hours. It is observed that the scale of the motion increases until there is an accumulation of the energy at the largest scale that can be accommodated within the tank. The structure of this large-scale circulation is analysed and it is found that a form of vorticity expulsion from the interior of the circulation has occurred. These results are compared with numerical simulations of two-dimensional turbulence, and some measurements of turbulent decay are discussed.

Some exact pseudo-plane solutions of the first kind for the Navier-Stokes equations

Abstract: We study pseudo-plane flows of the first kind generated by a stream function ψ=f(x, z, t) +g(y, z, t) which are generalized Beltrami flows in every plane parallel to thexy-plane. This problem is solved completely resulting in several new families of exact solutions for the Navier-Stokes equations.