Showing papers on "Stream function published in 2003"
TL;DR: In this paper, the authors present an efficient method for solving 2D incompressible viscous flows around multiple moving objects using an underlying regular Cartesian grid to solve the system using a streamfunction-vorticity formulation and with discontinuities representing the embedded objects.
247 citations
TL;DR: In this paper, a self-similarity solution of Navier-Stokes equations for a laminar, incompressible, and time-dependent flow that develops within a channel possessing permeable, moving walls is presented.
Abstract: This article describes a self-similarity solution of the Navier–Stokes equations for a laminar, incompressible, and time-dependent flow that develops within a channel possessing permeable, moving walls. The case considered here pertains to a channel that exhibits either injection or suction across two opposing porous walls while undergoing uniform expansion or contraction. Instances of direct application include the modeling of pulsating diaphragms, sweat cooling or heating, isotope separation, filtration, paper manufacturing, irrigation, and the grain regression during solid propellant combustion. To start, the stream function and the vorticity equation are used in concert to yield a partial differential equation that lends itself to a similarity transformation. Following this similarity transformation, the original problem is reduced to solving a fourth-order differential equation in one similarity variable η that combines both space and time dimensions. Since two of the four auxiliary conditions are of the boundary value type, a numerical solution becomes dependent upon two initial guesses. In order to achieve convergence, the governing equation is first transformed into a function of three variables: The two guesses and η. At the outset, a suitable numerical algorithm is applied by solving the resulting set of twelve first-order ordinary differential equations with two unspecified start-up conditions. In seeking the two unknown initial guesses, the rapidly converging inverse Jacobian method is applied in an iterative fashion. Numerical results are later used to ascertain a deeper understanding of the flow character. The numerical scheme enables us to extend the solution range to physical settings not considered in previous studies. Moreover, the numerical approach broadens the scope to cover both suction and injection cases occurring with simultaneous wall motion.
140 citations
TL;DR: In this paper, the authors studied the two-dimensional flow of a Johnson-Segalman fluid in a planar channel having walls that are transversely displaced by an infinite, harmonic travelling wave of large wavelength.
Abstract: This paper is devoted to the
study of the two-dimensional flow of a Johnson-Segalman fluid in
a planar channel having walls that are transversely displaced by
an infinite, harmonic travelling wave of large wavelength. Both
analytical and numerical solutions are presented. The analysis
for the analytical solution is carried
out for small Weissenberg
numbers. (A Weissenberg number is the ratio of the relaxation
time of the fluid to a characteristic time associated with the
flow.) Analytical solutions have been obtained for the stream
function from which the relations of the velocity and
the longitudinal pressure gradient have been derived. The
expression of the pressure rise over a wavelength has
also been determined. Numerical computations are performed and
compared to the perturbation analysis. Several limiting
situations with their implications can be examined from the
presented analysis.
104 citations
TL;DR: A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation, which is especially suitable for moderate to large Reynolds number flows.
Abstract: A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.
81 citations
TL;DR: In this article, a fully predictive mathematical model was developed for membrane distillation considering heat and mass transfers and concentration polarisation, and the effect of parameters such as thickness of cavity, temperature and feed flow rate were investigated.
75 citations
TL;DR: In this paper, a fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation.
Abstract: A fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive under-relaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature
71 citations
TL;DR: In this paper, two different formulations of maximum entropy predictions for decaying, two-dimensional Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. But these predictions might be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly understood correspondences between the long-time computations of decaying states of NS flows and the results of the maximum entropy analyses.
Abstract: Numerical and analytical studies of decaying, two-dimensional Navier–Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Though these predictions might be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly understood correspondences between the long-time computations of decaying states of NS flows and the results of the maximum entropy analyses. Both formulations define an entropy using a somewhat ad hoc discretization of vorticity into “particles.” Point-particle statistical methods are used to define an entropy, before passing to a mean-field approximation. In one case, the particles are delta-function parallel “line” vortices (“points,” in two dimensions), and in the other, they are finite-area, mutually exclusive convected “patches” of vorticity which only in the limit of zero area become “points.” The former are assumed to obey Boltzmann statistics, and the latter, Lynden-Bell statistics. Clearly, there is no unique way to reach a continuous, differentiable vorticity distribution as a mean-field limit by either method. The simplest method of taking equal-strength points and equal-strength, equal-area patches is chosen here, no reason being apparent for attempting anything more complicated. In both cases, a nonlinear partial differential equation results for the stream function of the “most probable,” or maximum entropy, state, compatible with conserved total energy and positive and negative velocity fluxes. These amount to generalizations of the “sinh-Poisson” equation which has become familiar from the “point” formulation. They have many solutions and only one of them maximizes the entropy from which it was derived, globally. These predictions can differ for the point and patch discretizations. The intent here is to use time-dependent, spectral-method direct numerical simulation of the Navier–Stokes equation to see if initial conditions which should relax toward the different late-time states under the two formulations actually do so.
55 citations
TL;DR: In this paper, the authors derived the flow field derived from the potential function in a deep basin with flat bottom, and treated departures from this situation with a stream function, that is, a correction for the potential functions solution, and is solved through the vorticity equation.
Abstract: [1] The tidal flow field in a basin of small dimensions with respect to the tidal wavelength is calculated. Under these conditions, the tide becomes a standing wave oscillating synchronously (with a flat water surface) over the whole basin. The shallow water equations can thus be strongly simplified, expressing the discharge vector field in terms of a potential function and a stream function. The potential function can be independently solved with the continuity equation, and is responsible for the total water balance in the basin. Moreover, the flow field derived from the potential function is shown to represent the tidal motion in a deep basin with flat bottom. Departures from this situation are treated with a stream function, that is, a correction for the potential function solution, and is solved through the vorticity equation. The stream function accounts for the nonlinear inertial terms and the friction in the shallow water equations, as well as bottom topography. In basins where channels incise within shallow tidal flats, the solution demonstrates that friction redistributes momentum, increasing the flow in the channels and decreasing it on the flats. The model is tested in San Diego Bay, California, with satisfactory results.
50 citations
TL;DR: A novel variational approach to dense motion estimation of highly non-rigid structures in image sequences with regularizers with derivatives up to third order to obtain unbiased high-quality solutions.
Abstract: We present a novel variational approach to dense motion estimation of highly non-rigid structures in image sequences Our representation of the motion vector field is based on the extended Helmholtz Decomposition into its principal constituents: The laminar flow and two potential functions related to the solenoidal and irrotational flow, respectively The potential functions, which are of primary interest for flow pattern analysis in numerous application fields like remote sensing or fluid mechanics, are directly estimated from image sequences with a variational approach We use regularizers with derivatives up to third order to obtain unbiased high-quality solutions Computationally, the approach is made tractable by means of auxiliary variables The performance of the approach is demonstrated with ground-truth experiments and real-world data
49 citations
TL;DR: In this paper, the fourth-order compact finite difference schemes were employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers.
Abstract: Fourth-order compact finite difference schemes are employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers. The governing Navier-Stokes equation is linearized in streamfunction and vorticity formulation. The fourth-order compact approximation schemes are coupled with fourth-order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods.
47 citations
TL;DR: In this article, the authors studied the Stokes problem of incompressible fluid dynamics in two and three-dimensional spaces, for general bounded domains with smooth boundary, and developed an abstract mixed formulation that gives a precise variational frame and conducts to a well-posed STokes problem involving a new velocity-vorticity boundary condition.
TL;DR: A fast finite difference method is proposed to solve the incompressible Navier-Stokes equations defined on a general domain using the immersed interface method and the fast Poisson solver for general domains and the interpolation scheme for the boundary condition of the stream function.
Abstract: A fast finite difference method is proposed to solve the incompressible
Navier-Stokes equations defined on a general domain. The method is based
on the voricity stream-function formulation and a fast Poisson solver defined
on a general domain using the immersed interface method. The key to the new
method is the fast Poisson solver for general domains and the interpolation
scheme for the boundary condition of the stream function. Numerical examples
thats show second order accuracy of the computed solution are also provided.
TL;DR: A new numerical formulation for improving the accuracy of PC velocity fields and corresponding streamlines is introduced, based on a physical constraint from fluid dynamics, within a regularization framework.
Abstract: Magnetic resonance angiography (MRA) has become an important tool for the clinical evaluation of vascular disease. Flow measurement with phase-contrast (PC) magnetic resonance (MR) imaging provides a powerful method for evaluation of blood velocity information inside vessels. However, image artifacts from complex flow patterns including slow flow, recirculation zone, and pulsatile flow can adversely affect accuracy of results. In this paper, we introduce a new numerical formulation for improving the accuracy of PC velocity fields and corresponding streamlines, based on a physical constraint from fluid dynamics, within a regularization framework. The formulation which makes use of a stream function, automatically enforces continuity constraint of incompressible flow and reconstructs the flow streamlines from PC images. We applied the algorithm to complex MR imaging flow velocities obtained in a flow phantom of an axisymmetric abdominal aortic aneurysm. The algorithm significantly improved streamline results especially inside the recirculation zone, where artifacts are more pronounced. A velocity reconstruction method in primitive variable form is also presented and results are compared with the stream function method. In order to validate flow characteristics derived from PC MR images, we used the FLUENT computational fluid dynamics software package, to simulate flow patterns within the same geometry as our phantom. There was a good agreement between the numerical simulations and recovered PC streamline results. Processed streamlines, in both stream function and primitive variable methods, were more realistic and provided more precise flow patterns than unprocessed PC data. Additionally, the feasibility of the method was demonstrated in the aorta of a normal volunteer.
TL;DR: In this article, a numerical method is described for the resolution of the incompressible Navier-Stokes equations, based on a stream function/vorticity formulation in relative frame.
TL;DR: In this paper, coherent vortices and spatial distributions of energy density in numerically simulated, horizontally homogeneous, doubly periodic, quasi two-dimensional f-plane turbulence are examined.
Abstract: This paper examines coherent vortices and spatial distributions of energy density in asymptotic states of numerically simulated, horizontally homogeneous, doubly periodic, quasi two-dimensional f-plane turbulence. With geophysical applications in mind, the paper progresses from freely decaying two-dimensional flow to freely decaying equivalent barotropic flow, freely decaying two-layer quasi-geostrophic (QG) flow, and, finally, statistically steady two-layer QG turbulence forced by a baroclinically unstable mean flow and damped by bottom Ekman friction. It is demonstrated here that, with suitable elaborations, a two-vortex state having a sinh-like potential vorticity/streamfunction (q/ψ) scatter plot arises in all of these systems. This extends, at least qualitatively, previous work in inviscid and freely decaying two-dimensional flows to flows having stratification, forcing, and dissipation present simultaneously. Potential vorticity steps and ribbons of kinetic energy are shown to form in freely decaying equivalent barotropic flow and in the equivalent barotropic limit of baroclinically unstable flow, which occurs when Ekman damping is strong. Thus, contrary to expectations, strong friction can under some circumstances create rather than hinder the formation of sharp features. The ribbons are present, albeit less dramatically, in moderately damped baroclinically unstable turbulence, which is arguably a reasonable model for mid-ocean mesoscale eddies.
TL;DR: In this article, mesh-free variational methods are used for the solution of incompressible fluid dynamics problems using the R-function method (RFM), which constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non-conforming spatial grid.
Abstract: We show that meshfree variational methods may be used for the solution of incompressible fluid dynamics problems using the R-function method (RFM). The proposed approach constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non-conforming spatial grid. We give detailed implementation of the method for two common formulations of the incompressible fluid dynamics problem: first using scalar stream function formulation and then using vector formulation of the Navier-Stokes problem with artificial compressibility approach. Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back-facing step channel problem reveal strengths and weaknesses of the proposed meshfree method.
01 Oct 2003
TL;DR: In this article, a new formulation is proposed for the simulation of viscous flows around structures in waves, which consists in modifying the Reynolds-averaged Navier-Stokes equations: velocity, pressure or free-surface elevation fields are split into incident and diffracted fields to compute the diffracted flow only.
Abstract: A new formulation is proposed for the simulation of viscous flows around structures in waves. It consists in modifying the Reynolds-averaged Navier-Stokes equations: velocity, pressure or free-surface elevation fields are split into incident and diffracted fields to compute the diffracted flow only. The incident flow may be explicitly given by a stream function theory for non-linear regular waves, or by a spectral method for irregular waves. This method avoids classical problems (large CPU time, poor quality of generated wave) of numerical generation of waves in a viscous flow solver. The 2D flow around an immersed square in regular waves demonstrates the effectiveness of the method.
01 Jan 2003
TL;DR: In this article, a high-order immersed boundary method for the Navier-Stokes equations in stream function-vorticity formulation is presented, which employs an explicit Runge-Kutta (second or fourth order) time integration scheme, fourth-order compact finite-differences for computation of spatial derivatives, and a nine-point, fourthorder compact discretization of the Poisson equation for computations of the stream function.
Abstract: Immersed boundary methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a Cartesian grid. While good results, both qualitative and quantitative, have been obtained, many of the methods rely on low-order corrections to account for the immersed boundary. The objective of the present work is to present a high-order immersed boundary method for the 2-D, unsteady, incompressible Navier-Stokes equations in stream function–vorticity formulation. The method employs an explicit Runge-Kutta (second or fourth order) time integration scheme, fourth-order compact finite-differences for computation of spatial derivatives, and a nine-point, fourth-order compact discretization of the Poisson equation for computation of the stream function. Corrections to the finite-difference stencils are used to maintain high formal accuracy at the immersed boundary, as confirmed by analytical tests. To validate the method in its application to incompressible flows, several physically relevant test cases are computed, including uniform flow past a circular cylinder and Tollmien-Schlichting waves in a boundary layer.
TL;DR: The problem of an incompressible viscous fluid moving through a porous medium between two inclined wavy porous plates under the effects of a constant inclined magnetic field that makes an angle with the vertical axis and constant suction (or injection) is studied numerically by a method related to that of Takabatake-Ayukawa in 1982.
TL;DR: In this paper, a doubly periodic array of vortices is expressed in terms of the Jacobi elliptic functions, and a threshold value of the period parameter is defined such that a transition from globally smooth distributions of Vorticity to singular distributions occurs.
Abstract: Arrays of vortices are considered for two-dimensional, inviscid flows when the functional relationship between the stream function and the vorticity is a hyperbolic sine. An exact solution as a doubly periodic array of vortices is expressed in terms of the Jacobi elliptic functions. There is a threshold value of the period parameter such that a transition from globally smooth distributions of vorticity to singular distributions occurs.
TL;DR: In this article, an analytic solution for the stream function, ψ, expressed as an infinite series of Papkovich-Fadle eigenfunctions is used to reveal changes in flow structures as A is varied.
Abstract: Stokes flow in a rectangular cavity with two moving lids (with equal speed but in opposite directions) and aspect ratio A (height to width) is considered. An analytic solution for the streamfunction, ψ, expressed as an infinite series of Papkovich–Fadle eigenfunctions is used to reveal changes in flow structures as A is varied. Reducing A from A=0.9 produces a sequence of flow transformations at which a saddle stagnation point changes to a centre (or vice versa) with the generation of two additional stagnation points. To obtain the local flow topology as A→0, we expand the velocity field about the centre of the cavity and then use topological methods. Expansion coefficients depend on the cavity aspect ratio which is considered as a separation parameter. The normal-form transformations result in a much simplified system of differential equations for the streamlines encapsulating all features of the original system. Using the simplified system, streamline patterns and their bifurcations are obtained, as A→0.
TL;DR: In this article, the authors focus on mathematical modeling of the Bloch NMR flow equations into the harmonic differential equation to explain qualitatively, the effects of coriolis force on the motion of flowing fluid.
Abstract: Unlike most medical imaging modalities, magnetic resonance imaging (MRI) is based on effects that cross multiple biological levels: contrast depends on interactions between the local chemistry, water mobility, microscopic magnetic environment at the subcellular, cellular or vascular level, cellular integrity, etc. These interactions potentially allow for imaging functional changes in the same reference frame as the anatomic information. However, to tap this potential, we need methodologies that robustly incorporate the best models of the underlying physics interactions in order to extract the best possible interaction obtainable on flow velocity and rates. Due to the fundamental role the Bloch NMR equations play in the analysis of the properties of magnetic resonance imaging, this presentation will focus on mathematical modeling of the Bloch NMR flow equations into the harmonic differential equation. This simplification allows us to explain qualitatively, the effects of coriolis force on the motion of flowing fluid. The Transverse magnetization My, is introduced as a stream function. Our choice of conditions has led to a linear equation for My. We derived the stream function as a form of solution which contains the linearity property demanded by conditions at x=0. The resulting flow reveals some interesting wave-like properties which were examined directly. The existence of the waves is associated with the non-uniformity of the Coriolis parameter, and it is not difficult to see the general mechanism. The quantum mechanical models of Bloch NMR equations describe dynamical states of particles in flowing fluid. We introduce the basic background for understanding some of the applications of quantum mechanics to NMR and explain their significance and potentials. It also describes the behavior of the rF B1 field when the fluid particles flow under physiological and some modeled pathological conditions. The wave function is explored to determine the minimum energy, a function of the rF B1 field for the fluid particle to be located in the non-classical region. These models can be invaluable to understand the basic Physics of extracting the relevant flow parameters by which velocity quantification can be made in Blood vessels with changing cross-section.
TL;DR: An a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows to obtain well adapted meshes and is efficient to eliminate the oscillations around the boundary layer.
TL;DR: In this article, the creeping flow of incompressible viscous fluid past and through a porous approximate sphere has been studied in terms of Bessel and Gegenbauer functions of the first kind.
Abstract: This paper concerns the creeping flow of incompressible viscous fluid past and through a porous approximate sphere. The Brinkman model for the flow inside the porous particle and Stokes model for the flow outside the particle in their stream function formulations are used. The stream function and the pressure distribution, both for the flow inside and outside are obtained in terms of Bessel and Gegenbauer functions of the first kind. The drag force experienced by the particle is determined and its variation with respect to permeability parameter is studied numerically. The special cases of flow past a porous sphere and spheroid are obtained from the present analysis.
TL;DR: In this paper, the motion of a solid spherical particle at the instant it passes the center of a spheroidal container has been investigated using the no-slip condition at the surfaces.
TL;DR: In this article, the finite difference method of calculation of non-steady-state ion transfer in electrochemical systems under the conditions of natural convection is elaborated, based on the mathematical model involving the continuity equations for electrolyte species, the condition of electroneutrality, and the Navier-Stokes equations for a viscous incompressible liquid with corresponding initial and boundary conditions.
TL;DR: In this article, the steady incompressible magnetohydrodynamic (MHD) flow past a sphere with an applied magnetic field parallel to the main flow is studied for the Reynolds number, Re, up to 100 and interaction parameter, N up to 0.7.
Abstract: The steady incompressible magnetohydrodynamic (MHD) flow past a sphere with an applied magnetic field parallel to the main flow is studied for the Reynolds number, Re, up to 100 and interaction parameter, N, up to 0.7. The magnetic Reynolds number is assumed to be small. It is observed that drag coefficient increases as N increases. The Finite difference method is used to solve the vorticity-stream function form of the non-linear Navier-Stokes equations. The multigrid method is used to solve the finite difference equations. The fifth decimal place convergent solutions are obtained upto the finest grid of 1024×512. Graphs of the streamlines, vorticity lines, surface vorticity and drag coefficient are presented.
TL;DR: In this paper, a complex variable boundary element method (CVBEM) is developed for potential flow in the presence of thin objects, and singularities caused by such objects are expressed analytically through the conformal mapping technique, and the resultant singular solution is coupled with the CVBEM by the principle of superposition.
Journal Article•
TL;DR: Inverse solutions of the equations of motion of an incompressible second-grade fluid are obtained by assuming certain forms of the stream function as mentioned in this paper, and expressions for streamlines, velocity components and pressure distributions are given in each case and compared with the known results.
Abstract: Inverse solutions of the equations of motion of an incompressible second-grade fluid are obtained by assuming certain forms of the stream function. Expressions for streamlines, velocity components and pressure distributions are given in each case and compared with the known results.
TL;DR: In this paper, the influence of uniform and Poiseuille flows on the wall correction factor is numerically and asymptotically investigated using a finite differences method, where the Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear coordinates.
Abstract: To contribute to the existing knowledge of the hydrodynamic force exerted on a spherical particle placed in the axis of a cylinder, at small Reynolds numbers, the influence of the uniform and Poiseuille flows on the wall correction factor are numerically and asymptotically investigated. The Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear coordinates. These equations are solved using a finite differences method. The generation of the grid was carried out by the singularities method. The accuracy of the numerical code is tested through comparison with theoretical and experimental results. In both cases we numerically calculated the separate contributions of the pressure and viscosity forces. In concentrated regime these numerical calculations are in very good agreement with those obtained by asymptotic expansions. This analysis allowed us to show the prevalence of the pressure term over the viscosity one in the lubrication regime contrary to what happened for the dilute regime. All our numerical and asymptotical results compared with those of Bungay et al. (Int. J. Multiphase Flow 1 , 25–56 (1973)) seem to give a response to this problem argued for a long time.