Showing papers on "Stream function published in 1978"

Structure and entrainment in the plane of symmetry of a turbulent spot

Abstract: Laser-Doppler velocity measurements in water are reported for the flow in the plane of symmetry of a turbulent spot. The unsteady mean flow, defined as an ensemble average, is fitted to a conical growth law by using data at three streamwise stations to determine the virtual origin in x and t. The two-dimensional unsteady stream function is expressed as ψ=U^2_∞tg(ξ,η) in conical similarity co-ordinates ζ = x/U_∞t and η = y/U_∞t. In these co-ordinates, the equations for the unsteady particle displacements reduce to an autonomous system. This system is integrated graphically to obtain particle trajectories in invariant form. Strong entrainment is found to occur along the outer part of the rear interface and also in front of the spot near the wall. The outer part of the forward interface is passive. In terms of particle trajectories in conical co-ordinates, the main vortex in the spot appears as a stable focus with celerity 0·77U_∞. A second stable focus with celerity 0·64U_∞ also appears near the wall at the rear of the spot. Some results obtained by flow visualization with a dense, nearly opaque suspension of aluminium flakes are also reported. Photographs of the sublayer flow viewed through a glass wall show the expected longitudinal streaks. These are tentatively interpreted as longitudinal vortices caused by an instability of Taylor-Gortler type in the sublayer.

Dynamic stall of an oscillating airfoil

Abstract: Unsteady separated boundary layers and wakes were studied by investigating flow past an oscillating airfoil which in part models the retreating blade stall on the helicopters. The Navier-Stokes equations in terms of the vorticity and stream function for laminar flow were solved to determine the flow field around a modified NACA 0012 airfoil. After a fully developed flow was determined at zero incidence, the airfoil was oscillated in pitch through an angle of attack range from 0 deg to 20 deg. The computed streamlines during this pitch-up motion are in qualitative agreement with the trajectories of air bubbles observed in water tunnel experiments conducted with a NACA 0012 airfoil under the same conditions. During the pitch-down motion of the airfoil, the computed flow patterns cannot be compared with the experiments because the trajectories of air bubbles intersect.

On the stream function‐vorticity finite element solutions of Navier‐Stokes equations

Abstract: The stream function-vorticity method for solving Navier-Stokes equations with finite elements is often disregarded in view of the necessity of an iterative proceudre for satisfying the boundary conditions. The present work shows briefly that such iterations are unnecessary. Sample solutions for flows in a square cavity and in a channel with a step show good agreement with existing solutions.

Finite-element solution procedures for solving the incompressible, Navier-Stokes equations using equal order variable interpolation

Abstract: The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.

Collocation solution of creeping newtonian flow through periodically constricted tubes with piecewise continuous wall profile

Abstract: A collocation solution of creeping Newtonian flow through periodically constricted tubes is obtained. The profile of the wall of the type of tube considered is piecewise continuous, composed of symmetric parabolic segments. A transformation of the domain of interest into a rectangular one is obtained, which allows satisfaction of all boundary conditions. The collocation solution gives the stream function in terms of the new independent variables and can easily be converted to the original cylindrical coordinates. Axial and radial velocity components are obtained in analytical form, and the pressure drop is calculated from a volume integration of the viscous dissipation function as well as from line integration of the Navier-Stokes equation. The results are compared with the finite-difference solution by Payatakes et al. (1973b) and are found in good agreement. Differences between the two solutions are attributed mainly to discretization error in the finite-difference solution. The analytical expressions obtained from the collocation solution can be used together with porous media models of the constricted unit cell type for the modeling of processes taking place in granular porous media.

Use of variational methods to the eddy currents calculation in thin conducting plates

Abstract: In the paper eddy-currents are calculated in thin conducting plates of chosen shapes. It is assumed that the secondary magnetic field due to eddy-currents is negligible compared with the exciting flux. The stream function I is introduced to formulate field equations. The Ritz and Kantorovich varia - tional methods are applied for solving the field equation and calculating the power losses.

Recycling flow over bottom topography in a rotating annulus

Abstract: Steady flow of an incompressible homogeneous fluid over shallow topography in a rotating annulus is considered. The flow is driven by a differentially rotating lid. The recycling nature of the system means that prescribed upstream conditions are not available to close the problem. Consideration of the balance of transport across streamlines for the geostrophic flow leads to a general circulation condition; namely $\Gamma(\psi) = \frac{1}{2}\Gamma_T(\psi)$ , where ϕ is the stream function for the geostrophic flow, Λ is the circulation around a streamline and Λ T is the circulation around the same path calculated using the prescribed upper surface velocity. Using this condition, stream functions for the linear viscous and nonlinear quasi-inviscid flow can be found. Solutions for these two limits, and linearized perturbation solutions for the transition regime between them, are presented for flow over ridges in the annulus.

Laser-Velocimeter Surveys of Merging Vortices in a Wind Tunnel

Abstract: The merger of two vortices was studied with a laser velocimeter designed to measure the two cross-stream components of velocity. Measurements were made at several downstream distances in the vortex wake shed by two semispan wings mounted on the wind tunnel walls. The velocity data provided well difined contours of cross-flow velocity, stream function and vorticity. Downstream of the merger point the vorticity was shown to be independent of the downstream distance for small radii, and at larger radii was dependent on the distance from the wing rather than from the merger point. Upstream of the merger point a multicell vorticity pattern was shown.

A variational inequality method applied to free surface seepage from a triangular ditch

Abstract: Two-dimensional seepage from a single triangular channel into permeable soil underlain at a finite depth by a drain has been solved by reducing the problem to a variational inequality. The results obtained consist of the location of the free surface and thus the shape of the seepage region, the velocity potential and stream function at a series of mesh points, and the seepage flow rate. The successive overrelaxation method with projection was used to solve the numerical problem. The numerical results compared very favorably with the analytical solution to the same problem.

Non-Newtonian flow characteristics in a steady two-dimensional flow

Abstract: The two-dimensional steady flow of a non-Newtonian fluid (a dilute polymer solution) is examined. The flow domain is composed of a parallel-walled inflow region, a contraction region in which the walls are rectangular hyperbolae, and a parallel-walled outflow region. The problem is formulated in terms of the vorticity, stream function and appropriate rheological equation of state, i.e. an Oldroyd-type constitutive equation (with no shear-thinning) for the total shear and normal-stress components. Computational results from the numerical solution of the equations are presented. In particular, the molecular extension and pressure distribution along the centre-line are presented as well as contour plots of the different flow variables. The alignment of the molecules with the principal axes of strain rate is shown by a qualitative comparison of the streamwise normal-stress contours with contours of the eigenvalues of the strain-rate matrix.

Estimation of water particle velocity of breaking wave

Abstract: A possibility to estimate the water particle velocity of breaking wave on beaches by giving only the values of the beach slope i and the deep-water wave steepness H0/Lo is investigated. The data of the water particle velocity and profile of breaking wave on the beaches measured by Iwagaki, Sakai, Tsukioka and Sawai(1974) are used for this purpose. A method of using empirical curves relating the coefficients of Dean's stream function to i and H0/L0 does not give good results. The relations between the asymmetric breaking wave profile, i and Ho/L0 are examined. As well as the front face of the profile, the slope behind the crest above some level is found to become steep when H0/L0 becomes small. The breaking wave profiles are reproduced by using the empirical curves relating the parameters characterizing the breaking wave profile to i and Ho/L0. The velocity of Dean's stream function calculated from the reproduced profile explains the measured velocity better than that of Stokes waves. It is therefore possible to estimate the water particle velocity of breaking wave on beaches by giving only the values of i and Ho/Lo.

Longshore currents due to surf zone barrier

Abstract: We consider a straight coastline exposed to large regular waves, of typical wave length, 100 m amplitude 1.6 m, and period 12 sec. The radiation stress gradients in the extensive (up to 2 km wide) surf zone cause set up and long-shore currents. Despite these currents, the beach is known to be fairly stable. If now a cooling water intake basin is introduced on the coast, it is required to determine first whether the wave induced currents in the vicinity of the basin will affect the circulation of cooling water and second, whether sediment transport will occur, leading to a dredging requirement for the basin. An extensive programme of physical model testing and numerical studies is being undertaken, in order to answer the above questions, and this paper will survey the progress made to date. At the 15th Coastal Engineering Conference a paper on the application of a mathematical model to the prediction of dredging properties inside a cooling water intake basin was 9 presented by Fleming and Hunt, which described the first stage of this work . - In that paper a sediment transport model was combined interactively with numerical models of wave refraction, wave diffraction, long shore currents and circulation currents. The last of these numerical models was used to evaluate the current patterns due to the interruption of the continuity of the longshore currents, together with the cooling water flows in the vicinity of the basin. In this paper we describe the development of more sophisticated numerical models for the first three stages of the process. An understanding of the process of longshore current and set up creation, depends on the concept of radiation stress, introduced by Longuet-Higgins and Stewart,12'13' ' in a series of papers. A number of workers have since used the radiation stress to determine coastline phenomena, and we now describe a few of the relevant papers, without any attempt at a comprehensive survey. Bowen 5'6 considered a straight coastline with parallel contours, and determined near shore circulation patterns, using a stream function formulation of the shallow water equations, for normally incident waves, with a sinusoidal coastwise variation in wave amplitude. He used a finite difference method to solve for the stream function. Longuet-Higgins10'11 criticized Bowen's use of a constant mixing length (horizontal) viscosity, and introduced a viscosity which varied directly with the distance from the shore, in his one dimensional analytical model for obliquely incident waves. He was able to obtain analytically longshore velocity profiles, which he plotted for a range of viscosities.

Wind‐driven, steady flows in Lake Superior 1

Abstract: The wind-induced, steady state circulations in Lake Superior were investigated by use of , both mathematical and physical models. The mathematical model was developed from the equations of motion, assuming a homogeneous lake, negligible nonlinear accelerations, constant Coriolis force, hydrostatic pressure distribution, negligible horizontal turbulent shear stresses, and constant vertical eddy viscosity. With these assumptions, the equations of motion were solved analytically to give expressions for the velocities in terms of position in lake, wind stress, water surface slopes, depth, vertical eddy viscosity, and bottom condition (i.e. no-slip or slip). Water surface slopes were obtained from a numerical solution to the vertically integrated equations of motion, expressed in terms of a mass transport stream function and subject to appropriate wind stress and inflow-outflow conditions. Two- and three-dimensional velocity fields were obtained for uniform and curl wind stress distributions, for constant (average) and actual (variable) depth conditions, and for slip and no-slip on bottom. Laboratory experiments on the wind-generated circulation were conducted in a rotating, vertically distorted model of Lake Superior. Surface current patterns were obtained for steady, westerly winds that were constant in magnitude (uniform) over the lake and that had a constant curl (speeds decreased linearly from south to north) over the lake. The laboratory circulation patterns are in reasonable agreement with results from the numerical model.

Extension of Stokes series for flow in a circular boundary

Abstract: The expansion in powers of Reynolds number for steady, viscous, incompressible flow within a circular boundary, known to eighth order from the work of Kuwahara and Imai, is extended by computer to 22 terms. An attempt is made to analyze and improve the extended series.

Centerline formulation in the numerical computation of axisymmetric flows

Abstract: = parameter defined in Eq. (3) = Prandtl number for turbulent diffusion = constant in effective viscosity relation = dependent variable of interest = turbulence kinetic energy = index of radial point in finite-difference grid = index of axial point in finite-difference grid = production term in conservation equation = radial position = axial velocity = radial velocity = axial distance = increment in axial distance — turbulence energy dissipation = stream function — effective turbulent viscosity = density = turbulence vorticity a b CM F k m n P r u v x AJC e \l/ [L p co

Transonic nozzle flows of real gases with nonuniform gas properties

Abstract: The transonic flow of a vibrationally relaxing or a chemically reacting gas in a convergent‐divergent nozzle is considered. The procedure used to solve the flow equations is similar to that first used by Hall, and the solution is based on a series expansion in terms of the wall geometry in the transonic region. The nonuniformities of the flow properties across the flow field are taken into consideration. Taulbee and Boraas have already solved such a problem for a frozen flow by a sort of inverse method, in which the governing equations are written with the stream function as an independent variable so as to accomodate the variations in flow properties across the flow field. The problem is solved by a direct method, in which the flow equations are written with generalized coordinates of curvilinear coordinates comprising the streamlines and the system of lines normal to the streamlines, and analytical solutions are obtained for both frozen and equilibrium flows.

A new finite-difference representation for the vorticity at a wall with suction

Abstract: In two-dimensional convective transport, the vorticity at a wall is usually not well specified. There exist several representations to relate the vorticity at a wall to the stream function and interior vorticity values. These are shown to be inadequate when suction occurs at the wall. A new representation is proposed and shown to be significantly superior to the existing ones. The analysis can also be applied to heat transfer calculations at a wall with suction.

Laser-velocimeter surveys of merging vortices in a wind tunnel

Abstract: The merger of two vortices was studied with a laser velocimeter designed to measure the two cross-stream components of velocity. Measurements were made at several downstream distances in the vortex wake shed by two semispan wings mounted on the wind tunnel walls. The velocity data provided well difined contours of cross-flow velocity, stream function and vorticity. Downstream of the merger point the vorticity was shown to be independent of the downstream distance for small radii, and at larger radii was dependent on the distance from the wing rather than from the merger point. Upstream of the merger point a multicell vorticity pattern was shown.

Natural Convection in a Confined Region between Two Concentric Square Ducts

Abstract: Two-dimensional steady state laminar natural convection in a region between two infinitely long horizontal concentric square ducts is studied numerically. The basic differential equations for the conservation of mass, momentum and energy, using the Boussinesq approximation, are computed applying straightforward explicit finite difference scheme. The ratio of the inner (hot) and outer (cold) ducts is considered to be 1 to 5. Calculations have been made for Rayleigh number, Ra, up to 20000 and Prandtl number, Pr, in the range of 0.7 to 1000. Numerical values are presented for the flow field (stream function), temperature distributions, and both local and overall heat transfer rates.