Showing papers on "Stream function published in 1976"

Time-dependent laminar incompressible flow through a spherical cavity.

Abstract: A finite-difference numerical method for the solution of the unsteady flow of a viscous incompressible fluid through axisymmetric circular ducts of variable axial geometry is developed and applied to the flow in a spherical-cavity geometry approximating the human aortic valve. The presence and motion of the valve leaflets are considered only as long as they can be assumed to present negligible impedance to the flow. The numerical solution is based on the vorticity/streamfunction approach, and is carried out for the systolic acceleration phase of the heart beat. A hybrid-mesh design consisting of a fine cell structure in the region close to the solid walls and a coarser grid in the core region is used. An experimental flow-visualization study in an acrylic model of the spherical cavity shows good agreement with the numerical simulation. An early separation of flow occurs at the entrance to the cavity, and an annular eddy grows in the wake until it occupies most of the cavity. The use of the hybrid mesh also makes possible the simulation of fine secondary-flow features in the cavity under peak-flow conditions.

A computer solution for two‐dimensional fluid‐particle flows

Abstract: A computer solution for two-dimensional fluid-particle flows using the cellular approach is presented. The euqations describing the flow of the continuous (fluid) phase are formulated using stream function, vorticity and enthalpy as the dependent variables. The effect of the disperse (particle) phase on the continous phase is represented by vorticity and energy sources in each cell. To illustrate the capability of the solution scheme, the flow of gas laden with hot particles issuing into a sudden expansion is analysed.

A numerical study of viscous flow around an airfoil

Abstract: An integrodifferential method, previously formulated in terms of velocity and vorticity vectors, is reformulated in terms of stream function and vorticity for two-dimensional incompressible viscous flows The reformulated integrodifferential method is shown to retain the distinguishing feature of the previous formulation in permitting the confinement of the solution field to the viscous region of the flow and consequently offers great computational advantages The application of this procedure in a study of an incompressible flow around an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000 is discussed Numerical results are presented and compared with available finite-difference results

Rip current spacing

Abstract: Mass, momentum and energy conservation laws, including the radiation stress, are used to derive an equation of the eigenvalues of rip current spacing. A coastal region with linear bottom slope is divided into two parts: Offshore region and surfzone separated by the breaker line. Wave set-up, wave energy and mean current are assumed to be composed of basic state, which is a function of the distance from the coast to offshore only, and of superposed two-dimensional perturbations. In the case of normal incidence of waves, basic steady current system vanishes and perturbations are found to be of cellular shape. According to the boundary conditions at the coast, stream function of perturbed motion in the surfzone can be represented by the confluent hypergeometric function, while in the offshore zone it is approximated by the modified Bessel function. Interpolation of the stream functions in the surf and offshore regions enables us to obtain a characteristic relation which gives the eigenvalues of nondimensional alongshore spacing of rip current system as a function of a parameter determined by the bottom friction coefficient, width of the surfzone and breaker height.

Symmetric Finite-Amplitude Rotational Water Waves

Abstract: Two forms of a tw0-dimensional streamfunction solution for symmetric periodic water waves on a fluid with a vertical distribution of vorticity are presented. The magnitude of the vorticity varies linearly with the magnitude of the streamfunction, while remaining constant on a particular streamline. The analysis utilizes a numerical perturbation technique, which converges rapidly to a wave of given height and period in water of a specified depth with a given vorticity distribution. Computed results show the influence of the vorticity on the wavelength and crest elevation of the wave.

Near-bottom water motion under ocean waves

Abstract: A two-year ocean experiment involving wave-induced forces on a test pipe mounted on the sea floor [Grace and Nicinski (1976)] involved the measurement of various quantities other than the pipe forces per se. A pair of these involved surface wave characteristics and wave-induced water motion at the level of the pipe centerline but off to one end of the pipe. These wave-kinematics data have been combined, and the results of this work make up this paper in which the emphasis is on the deterministic approach to data interpretation. Presented are comparisons of the velocity and acceleration data with the predictions of Airy and stream function theories plus discussion of the dispersion of the field data. The primary intent of the paper is to suggest to designers of bottom-laid structures, such as pipes, how values of the peak velocity and maximum acceleration of the water motion associated with a non-breaking design wave of specified characteristics can be chosen.

Steady Flow in a Curved Tube of Triangular Cross Section

Abstract: The steady flow of a viscous fluid moving under a constant pressure gradient in a curved tube with a uniform triangular cross section is investigated. Numerical solutions of the equations of motion have been found for the range 100-12 000 of the Dean number D = Ga 3 √(2a/ L )/ μv , where G is the constant pressure gradient, a is a dimension of the triangle, L the radius of the circle in which the tube is coiled, μ the viscosity and v the coefficient of kinematic viscosity of the fluid. The results for low D have been checked by an independent numerical method in which the stream function is expanded in a series of powers of D following the method of Dean (1928). All the results have been checked for accuracy by varying the grid size used in the numerical computations. The trend of the results as D increases is examined for evidence of the development of a boundary-layer structure as D → ∞. Some indication is found of the formation of a boundary layer of thickness proportional to D –1/3 near the side walls of the tube with an associated inviscid core region in the centre of the tube. In particular, comparison is made with details of an asymptotic model as D → ∞ proposed by Smith (1976). A measure of agreement with the general characteristics of this model is obtained, although there are some discrepancies in the precise details. It is possible that the range of D considered in the present work is not great enough to form any definite conclusions regarding the precise nature of the flow as D → ∞. A feature of the present results which develops for D > 3000 is that the maximum axial velocity in the tube ceases to occur on the axis of symmetry of the cross section. This feature appears to be generally consistent with numerical results obtained by Cheng & Akiyama (1970) and Hocking (unpublished) for a tube of rectangular cross section. The sequence of corner vortices of the type identified by Moffatt (1964) is found to occur in the numerical solutions. A detailed study of the vortices has already been published (Collins & Dennis 1976).

Turbulent atmospheric flow over a backward-facing step

Abstract: The phenomenon of atmospheric shear layer separation over a man-made structure such as a building (modeled as a backward-facing step) has been analyzed theoretically by (1) solving the two-dimensional equations of motion in the two variables, stream function and vorticity, and by (2) employing an approximate integral technique. Boundary conditions for the undisturbed flow are that of the turbulent atmospheric shear flow over a rough terrain. In the first approach a two-equation model of turbulence was used. In the second approach an approximate technique was utilized in an attempt to describe the details of the flow in the recirculation zone behind the step. The results predict velocity profiles in sufficient detail that the presence of the corner eddy in the region of negative surface pressure gradient is evident. The magnitude of the reversed flow velocity in the recirculation eddy has been found to agree with that found from experiments. Also, a surface eddy viscosity distribution has been an outgrowth of the method which realistically follows the magnitude of the surface pressure gradient distribution as found experimentally.

Slow flow of a viscoelastic fluid through a contraction

Abstract: The creeping flow of a viscoelastic fluid through a tapered contraction is analysed by using a perturbation method. Finite element method has been applied to solve the vorticity transport and stream function equations.

Liquid jet impingement normal to a disk in zero gravity

Abstract: The free surface shapes of circular liquid jets impinging normal to sharp-edged disks in zero gravity are determined. Zero gravity drop tower experiments yielded three distinct flow patterns that were classified in terms of the relative effects of surface tension and inertial forces. An order of magnitude analysis was conducted that indicated regions where viscous forces were not significant in the computation of free surface shapes. The free surface analysis was simplified by transforming the governing potential flow equations and boundary conditions into the inverse plane, where the stream function and velocity potential became the coordinates. The resulting nonlinear equations were solved by standard finite difference methods, and comparisons were made with the experimental data for the inertia dominated regime.

Barotropic disturbances caused by a point storm in aβ-plane ocean

Abstract: Barotropic equation for vorticity balance in theβ-plane is solved for a volume transport stream function with a northward moving wind stress torque concentrated at a point (tweak). Asymptotic techniques are applied to the solution expressed by Fourier type integrals. In the front zone (northern half from the tweak) the stream function behaves asO(¦X¦−5) whereX is a scaled eastward coordinate. In the wake, the stream function behaves asO(¦X¦−1/2) within a wedge bounded by the westward axis and a line directed SSE approximately, but behaves asO(¦X¦−5) in the rest of the half plane. On these boundaries it behaves asO(¦X¦−5/2) andO(¦X¦−1/3), respectively. East-west asymmetry is a result of asymmetry in zonal propagation of the planetary waves, which also cause the wavy pattern of the streamline in the western half of the wake. Friction is most effective in decreasing the stream function in the south to SSW sector and least effective to the west. Effect of divergence changes the wave pattern in the western half of the wake but does not change magnitude of the stream function. It is speculated that abnormal changes in daily mean sea levels observed in 1965 to 1971 along the south coast of Japan may be caused by the wake effect of moving typhoons east off Japan.

Deep‐ocean dynamics for environmental acoustics models

Abstract: A hydrodynamic model for flows in the deep ocean is developed in order to determine the velocity field and sound‐speed distribution for use in acoustic transmission problems. A scaling of the governing equations is constructed that explicitly includes sound speed. A subsequent perturbation expansion yields a set of approximate equations for motions nearly in geostrophic and hydrostatic balance, such as large‐scale, quasisteady currents and Rossby waves. The quasigeostrophic potential vorticity equation or a simpler limiting case of this equation arises from the perturbation scheme to govern higher‐order dynamics of the stream function for these flows. The results of the analysis are used to obtain a significant simplification of the ray equations of geometrical acoustics for moving media. For the particular class of flows considered here, the model equations are applicable if the ocean depth is about 1 km or greater and if the spatial and temporal scales of variation of the motions are of the order of 100...

Relaxation Solution for the Transonic Flow Through a Cascade

Abstract: The finite difference relaxation method is developped to calculate the performances of the cascade up to transonic range with occurring of shocks. The full potential equation with exact boundary conditions is solved in a conformal coordinate system constitued by the stream function and the potential function of the incompressible flow through the same cascade. The grid points on the boundary of the computational region are established after resolving the incompressible flow by the singularities method whereas the internal grid points are determined by means of the finite difference technic. The transonic potential field is computed using Jameson’s rotated upwind difference scheme in the supersonic region, and central difference scheme in the subsonic region. Thus in the supersonic regions, the disturbances are propagated from upstream to downstream; and terms provided from the truncation error acting as the artificial viscosity are introduced into the governing equation, making the occurring of the shocks automatically during the iterative process. Examples applied to cascades of compressor and turbine are also given.

Effect of latitudinal temperature gradient on spherical Couette flow

Abstract: The motion of a viscous thermally stratified liquid in a spherical layer due to rotation of the boundaries of the layer at different angular velocities with nonuniform (latitudinal) heating of the outer boundary is investigated. The investigated problem can be regarded as a very simple qualitative model of some astrophysical and geophysical effects, e.g., flows in planetary atmospheres. The characteristic parameters of the problem are the Reynolds number Re=ri2Ωi/Υ, the ratios of the angular velocities of rotation Ω=Ωe/Ωi and of the radiia=re/ri of the outer and inner spheres, and the heating nonuniformity parameter α. The Boussinesq approximation is adopted in the investigation: It is assumed that the Reynolds numbers Re are fairly small and the solution is sought in the form of a series of whole positive powers of Re. The first two terms of the series are found in analytical form for arbitrary values of the other characteristic parameters. Possible types of meridional flows are established and the regions of the parameters in which a particular type of circulation takes place are determined. It is shown that a latitudinal temperature gradient on the outer boundary leads to the appearance of new (in comparison with the isothermal case) types of meridional flows. The asymptotic form of the stream function of meridional flow in very thin and very thick layers is obtained.

A relativistic spherical vortex.

Abstract: This investigation is concerned with stationary relativistic flows of an inviscid and incompressible fluid. In choosing a density-pressure relation to represent relativistic “incompressibility,” it is found that a fluid in which the velocity of sound equals the velocity of light is to be preferred for reasons of mathematical simplicity. In the case of axially symmetric flows, the velocity field can be derived from a stream function obeying a partial differential equation which is nonlinear. A transformation of variables is found which makes the relativistic differential equation linear. An exact solution is obtained for the case of a vortex confined to a stationary sphere. One can make all three of the components of velocity vanish on the surface of the sphere, as in the nonrelativistic Hicks spherical vortex. In the case of an isolated vortex on whose surface the pressure is made to vanish, it is found that the pressure at the center of the sphere becomes negative, as in the nonrelativistic case. A solution is also obtained for a relativistic vortex advancing in a fluid. The sphere is distorted into an oblate spheroid. The maximum possible velocity of advance of the vortex is (2/3) c.

Sparse Matrix Problems in a Finite Element Open Ocean Model

Abstract: Publisher Summary This chapter discusses the sparse matrix problems in a finite element open ocean model. The earliest efforts in numerical oceanography were devoted to large-scale models that typically encompassed an entire ocean basin. These models furthered the understanding of the general features of ocean circulation such as the Gulf Stream but left unresolved the dynamics of sub-basin or mesoscale phenomena. Two such features are the meandering of the Gulf Stream and mid-ocean eddies. Both of these phenomena are highly non-linear, time-dependent, and almost certainly affect the global circulation. In an attempt to resolve these mesoscale problems, several models have been constructed. However, these models have the restriction of either periodic boundary conditions or of a closed basin domain. The model removes these restrictions. The chapter presents equations of motion and boundary conditions for an open-sided ocean. The equations consist of an advection equation for the quasigeostrophic potential vorticity and a three-dimensional Poisson equation for the streamfunction.

The flow of a non-Newtonian liquid due to the symmetric rotation of a body of revolution, using a natural coordinate system

Abstract: This paper looks at the non-Newtonian flow between symmetric rotating bodies and it establishes the condition for a coordinate surface to coincide with a surface of constant angular velocity in the primary flow. Then using a natural coordinate system based on the primary flow the equation for the stream function, for the secondary flow, is given in a neat and compact form which is suitable for further analysis and also for numerical work.