Showing papers on "Stream function published in 1997"

The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution

Abstract: An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ[Lt ]1 (or equivalently when the Reynolds number R[Gt ]1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ[Lt ]1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ→0 the proposed solution becomes the inviscid one at each point in the flow field.For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided λs/a is less than about 10−3. This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs/a is about 10−3. For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when λs/a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs/a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

On the three-dimensional instabilities of plane flows subjected to Coriolis force

Abstract: Linear stability of two-dimensional flows in a frame rotating with angular velocity vector Ω=Ωez perpendicular to their plane is considered. Sufficient conditions for instability have been derived for simple inviscid flows, namely parallel shear flows (characterized by the “Pedley” or “Bradshaw-Richardson” number), circular vortices (by the “generalized Rayleigh” discriminant) and unbounded flows having a quadratic streamfunction (with elliptical, rectilinear or hyperbolic streamlines). These exact criteria are reviewed and contrasted using stability analysis for both three-dimensional disturbances and oversimplified “pressureless” versions of the linear theory. These suggest that one defines a general inviscid criterion for rotation and curvature, based on the sign of the second invariant of the “inertial tensor,” and stating that, in a Cartesian coordinate frame: a sufficient condition for instability is thatΦ(x,y)=−12S:S+14Wt⋅Wt<0 somewhere in the flow domain. It involves the “tilting vorticity” Wt=W+4...

An Introduction to Fluid Dynamics: Principles of Analysis and Design

Abstract: What is Fluid Dynamics? Statics, Dynamics, and Surface Tension. Forces On, and Within, a Flowing Medium. Conservation of Mass and Momentum in a Continuous Fluid. Dimensional Analysis and Dynamic Similarity. Nearly Parallel Flows. Unsteady Flows. The Stream Function. Turbulent Flow and the Laminar Boundary Layer. Flow through Porous Media. Macroscopic Balances. Appendix. Index.

Stokes flow in a slowly varying two-dimensional periodic pore

Abstract: This article presents a series solution to the velocity in a two-dimensional long sinusoidal channel. The approach is based on a stream function formulation of the Stokes problem and a series expansion in terms of the width to the length ratio, which is considered small. Results show how immobile zones may appear and even flow separation and nonturbulent eddies, even in the absence of prima facie dead-end pores. It is shown that the flow tends to concentrate in strips connecting pore throats.

Convection-enhanced diffusion for random flows

Abstract: We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of Isichenko and Kalda that the effective diffusivity behaves likeɛ 3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows.

The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution

Abstract: An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter k which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When k ’ 1 (or equivalently when the Reynolds number R ( 1), the viscous boundary condition at the surface of the cylinder may to first order in k be replaced by the inviscid one. A viscous solution is proposed for the case k ’ 1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit k U 0 the proposed solution becomes the inviscid one at each point in the flow field. For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of k above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if k is suciently small, except in a small region near where the characteristics touch the cylinder where viscous eects dominate. Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1%, with the (constant) inviscid values provided ks}a is less than about 10’$. This result is interpreted as indicating that those viscous eects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when ks}a is about 10’$. For larger values of s, viscous eects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1% when ks}a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of ks}a, their similarity solution applies. In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero. Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

Dynamics of vorticity defects in shear

Abstract: Matched asymptotic expansions are used to obtain a reduced description of the nonlinear and viscous evolution of small, localized vorticity defects embedded in a Couette flow. This vorticity defect approximation is similar to the Vlasov equation, and to other reduced descriptions used to study forced Rossby wave critical layers and their secondary instabilities. The linear stability theory of the vorticity defect approximation is developed in a concise and complete form. The dispersion relations for the normal modes of both inviscid and viscous defects are obtained explicitly. The Nyquist method is used to obtain necessary and sufficient conditions for instability, and to understand qualitatively how changes in the basic state alter the stability properties. The linear initial value problem is solved explicitly with Laplace transforms ; the resulting solutions exhibit the transient growth and eventual decay of the streamfunction associated with the continuous spectrum. The expansion scheme can be generalized to handle vorticity defects in non-Couette, but monotonic, velocity profiles.

Linear and angular momentum conservation in hydraulic jump

Abstract: The present paper deals with the integral conservation of linear momentum and angular momentum in the stationary hydraulic jump in a wide rectangular channel. The flow is considered to be divided into a mainstream, that conveys the total liquid discharge, and a roller, in which no average mass transport occurs. Referring to the infinitely large case, a purely two dimensional motion is considered. The interface between the two flow regions is a streamline, corresponding to a stream function value equal to the total discharge per unit width. The present approach consists in satisfying the mechanical balances of mass, momentum and angular momentum, while no (large scale) constitutive relation is assumed for the turbulent motion of the liquid. Regarding the stress tensor, hydrostatic normal pressure distribution is assumed, while nothing is assumed regarding shear stresses, except that viscous stresses are negligible with respect to turbulent stresses. A paradox is put in evidence, that in the classical hydra...

Numerical simulations of experiments on quasi-two-dimensional turbulence

Abstract: We report direct numerical simulations ~DNS! of the two-dimensional ~2D! Navier-Stokes equation, including a linear friction term which parallels a series of recent experiments on decaying quasi-2D turbulence in thin, stably stratified, fluid layers. If we start the DNS from the experimental situation when transient processes within the layers have died away, then quantitative comparison between simulation and experiment shows a remarkable agreement for the temporal evolution of the stream function and the vorticity field. The results confirm the two dimensionality of the experimental dynamics after a short initial period of relaxation and suggest the use of 2D simulations for extrapolating the observations to the case without bottom friction, inaccessible to experiments with stratified fluids. @S1063-651X~97!07804-5# PACS number~s!: 47.27.Eq, 47.27.Jv

Finite element solution of incompressible fluid–structure vibration problems

Abstract: In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases. © 1997 by John Wiley & Sons, Ltd.

Discrete Vortex Method for Simulating Unsteady Flow Around Pitching Aerofoils

Abstract: Amodi® ed discretevortex method to simulatetheseparatedowaround an aerofoil undergoing pitching motion is described. The vorticity generated in the thin layer around the body is discretized into vortices in accordance withthemultipanel surfacerepresentation. Byconvectionand diffusionthevorticesarereleased fromthebody and advanced in the wake as determined by the Biot± Savart law and random-walk model, respectively. Both unsteady static and pitching casesare presented, and comparison with the test data illustratesthat, without priorknowledge of the developing separation and reattachment points for the model, good agreement has been achieved. Nomenclature A = area of body (section) B = volume within the body c = aerofoil chord Fb = volume within the control zone Fw = volume outside the control zone K = number of subpanels for each panel k = unit vector k = reduced pitch rate k = X c/2V l = panel length m = index number of subpanel within the panel N = number of panels for the body P = static pressure Re = Reynolds number r, r = position vector and its magnitude S = surface of the body s, n = unit vector along and normal to the surface t = time U = ¯ ow velocity V = velocity Z = position in the form of complex number z = vortex position in the form of complex number a = angle of attack C = circulation c = circulation density 4 t = time step d = distance of nascent vortex off the body m = kinematic viscosity q = ¯ uid density r = vortex core radius W ,W = vector potential and stream function X = rotational velocity x = vorticity Subscripts

Temperature and flow fields for the flow of a second grade fluid past a wedge

Abstract: The effects of fluid viscoelasticity on the temperature and flow fields of the flow past a wedge were studied. The combined effects of the shape factor, suction/injection rates and viscoelasticity were analysed by the series expansion method, similarity transformation, fourthorder Runge-Kutta integration and the shooting method. The stream function was introduced into the momentum boundary layer equations of a second-grade fluid to eliminate the pressure terms. The energy equation is also analysed by the similarity transformation. The results indicated that the velocity profiles, temperature distributions, surface friction and heat transfer are greatly affected by the elastic parameter, the shape factor, the Prandtl number and the suction/injection rates.

The influence of rollers on longshore currents

Abstract: A computational model is developed for depth-averaged cross-shore and longshore currents which includes the effects of the surface roller generated by wave breaking. The creation and evolution of the roller itself is modeled explicitly (Dally and Brown, 1995), and convective acceleration terms are included in both the crossshore and longshore momentum equations. Lateral mixing is parameterized in terms of the local cross-shore current and a turbulent eddy viscosity, as proposed by Svendsen and Putrevu (1994); however, a new model for eddy viscosity is proposed which contains contributions from both the roller-induced and bed-induced turbulence. The laboratory measurements of quasi-uniform longshore currents reported by Visser (1991) are used to calibrate and verify the model. For driving the model, it is shown that using stream function wave theory produces significantly better results than linear wave theory. Also, comparisons of longshore current distributions with and without the roller terms included show that the roller plays an essential role in faithfully modeling the longshore current. The calibrated model also produces accurate results for the set-up/set-down using stream function theory, for the limited data available from Visser (1991).

A boundary element method for viscous gravity currents

Abstract: SUMMARY The viscous gravity spreading of a blob of fluid on a rigid, horizontal, no-slip surface is studied numerically by applying the boundary-element method to the Stokes equation in plane symmetry. The two-dimensional unsteady solution is obtained by solving the biharmonic equation for the streamfunction in a given domain to obtain the velocity field, which is then used to track the contour. The spreading is developed by letting adhere to the rigid boundary any fluid element set in contact with it. A detailed description of the two-dimensional flow near the head of a viscous gravity current shows a typical rolling motion which characterizes the advancing mechanism of the spreading. In particular, we obtain scaling laws for the shape and size of the current head in good agreement with previously reported experimental data. Attention is also paid to the validation of the numerical method. # 1997 by John Wiley & Sons, Ltd.

Singular points of intensity streamlines in two-dimensional sound fields

Abstract: Continuing earlier work on this subject, a more rigorous discussion is given of the singular points of streamlines and the critical points of the stream function. The results for vortex and saddle points obtained earlier in piecemeal fashion and by way of examples are obtained systematically and by generally utilizing the applicable theory of differential equations and calculus. New results are also obtained. For example, a saddle point can occur when the phase of pressure and velocity differ by π/2, and in certain parts of the sound field, specifically inside a closed streamline, the number of vortex points and saddle points are related. Finally, the streamlines and singular point are considered for a discrete source: the line source.

Stokes flow in a half-filled annulus between rotating coaxial cylinders

Abstract: A model is presented for viscous flow in a cylindrical cavity (a half-filled annulus lying between horizontal, infinitely long concentric cylinders of radii R-i,R-0 rotating with peripheral speeds U-i,U-0). Stokes' approximation is used to formulate a boundary value problem which is solved for the streamfunction, phi, as a function of radius ratio (R) over bar = R-i/R-0 and speed ratio S = U-i/U-0. Results show that for S > 0 (S 1, a sequence of 'flow bifurcations' leads to a flow structure consisting of a set of nested separatrices, and provides the means by which the two-dimensional cavity flow approaches quasi-unidirectional flow in the small gap limit. Control-space diagrams reveal that speed ratio has little effect on the flow structure when S 0 and aspect ratios are small (except near S = 1). For S > 0 and moderate to large aspect ratios the bifurcation characteristics of the two large eddies are quite different and depend on both (R) over bar and S.

Reservoir flow prediction by contravariant shallow water equations

Abstract: Jet-induced mixing is often used to prevent stagnation in shallow service reservoirs. This paper describes a nonorthogonal boundary-fitted model for simulating flows in reservoirs of arbitrary shape. The numerical model solves the curvilinear shallow water equations that are expressed in terms of the depth-averaged contravariant velocity components and free surface elevation. Results are presented for the case of jet-forced flow in a circular reservoir where the inlet and outlet stems are diametrically opposite. Excellent agreement is obtained with alternative analytical and numerical schemes, at inlet Reynolds numbers equal to 10 and 25. A further comparison is given between numerical simulations and experimental measurements of the steady-state velocities in a circular reservoir where the inlet and outlet stems are diametrically asymmetric. Although the present application concerns steady jet-forced circulation, the contravariant shallow water equations should be suitable for modeling wind-driven circulation or tidal flows.

Streamtube models of gravity currents in the ocean

Abstract: A streamtube model of an outflow is developed that accounts for changes in hydrostatic pressure owing to variations in the height of the plume. The resulting one-dimensional equations are similar in form to the St. Venant equations, but additionally specify the path of the outflow down the slope. For no entrainment, uniform steady solutions exist with the flow nearly geostrophic and gradually descending the slope. However, these solutions are unstable if the uniform Froude number is subcritical. Instead, the model predicts a flow straight down the slope with increasing spreading and decreasing fluid flow. Observational data for three outflows indicate that the flow is subcritical and flows predominantly along the slope. Consequently, a two-dimensional steady model is introduced that uses the stream function as the transverse coordinate. Subcritical flow is stable when the transverse pressure gradient (caused by changes in height) supports the fluid along the slope. A numerical simulation suggests that an outflow might be considered as a sheet of fluid in which the fluid velocity varies considerably from the downslope to upslope boundary. The bulk of the fluid flows along the slope but is drained by an Ekman-like layer at the base of the outflow. This picture and the stability calculations cast doubt on whether an outflow should be modelled as a steady tube of fluid with properties uniform across the tube.

Open boundary conditions for the streamfunction -vorticity formulation of unsteady laminar convection

Abstract: The streamfunction-vorticity formulation is used to analyze unsteady laminar-convection problems in two-dimensional incompressible flows. The Bubnov-Galerkin finite-element method and a sequential procedure are employed to discretize and solve the governing differential equations. Very accurate results are obtained by employing “advective derivative conditions” at the outflow for all the variables involved. The boundary conditions for the streamfUnction at internal walls are imposed during the assembly process, and the vorticity at inflow and wall boundaries is evaluated in the framework of the stream/unction equation. The accuracy of the approach is demonstrated by the solution of two well-known benchmark problems concerning forced convection over a circular cylinder in cross flow and mixed convection in a plane channel heated from below.

Uncoupled ω–ψ Formulation for Plane Flows in Multiply Connected Domains

Abstract: This work deals with the numerical solution of the unsteady Navier–Stokes equations in the vorticity and stream function representation for problems in multiply connected two-dimensional regions. A particular decomposition of the stream function space is proposed which leads to an uncoupled variational formulation of the equations linearized and discretized in time, thus extending to transient problems the celebrated method proposed by Glowinski and Pironneau for the biharmonic problem. Numerical results calculated by a mixed finite element implementation of the new uncoupled method are presented.

Principal stream surfaces

Abstract: The use of stream surfaces and streamlines is well established in vector visualization. However, the proper placement of starting points is critical for these constructs to clearly illustrate the flow topology. In this paper, we present the principal stream surface algorithm, which automatically generates stream surfaces that properly depict the topology of an irrotational flow. For each velocity point in the fluid field, we construct the normal to the principal stream surface through the point. The set of all such normal vectors is used to construct the principal stream function, which is a scalar field describing the direction of velocity in the fluid field. Volume rendering can then be used to visualize the principal stream function, which is directly related to the flow topology. Thus, topology in a fluid field can be easily modeled and rendered.

Stability Analysis of Navier-Stokes Flows in Annuli

Abstract: We study the linearized stability of planer flows of incompressible, viscous fluid in two-dimensional annular domains. A certain family of steady, explicit solutions which have spiral streamlines are considered. The Navier-Stokes equations are linearized at these solutions and we show analytically or numerically that these solutions are stable to perturbation of steady states, whatever the Reynolds number or the aspect ratio of the annuli may be. Hopf bifurcations from them are also examined numerically.