Stream Function

Objective streamfunction and velocity potential maps of three commonly observed flow states near Point Conception, California, are derived from moored current-meter and drifter observations. The states are defined using current-meter data. Drifter data are sorted by flow state and averaged within spatial bins to resolve their spatial structures. Spatial correlations of drifter data show the along-shelf and cross-shelf decorrelation distances are approximately 45 and 25 km, respectively. This information is used to make the objective streamfunction and velocity potential maps. The total velocity is represented as the sum of the rotational, nondivergent streamfunction component and the irrotational, divergent velocity potential component. The velocity derived from the streamfunction is much stronger than that derived from the velocity potential. The streamfunction is sufficiently well resolved to make meaningful vorticity maps. The velocity potential indicates a source term in the western Santa Barbara Channel consistent with wind stress curl-driven upwelling, but divergence maps derived from the velocity potential are very noisy, making comparisons with wind stress curl mapped on the same scales problematic.

Submesoscale eddies generated by baroclinic instability of upper ocean fronts lead to rapid restratification of the mixed layer on a time scale of days. This restratification can be opposed by a down‐front wind stress (acting in the direction of the geostrophic velocity) that drives a surface Ekman flow from the dense side to the light side of the front to arrest the slumping of isopycnals. A scaling diagnostic is suggested to determine whether the effect of eddies or wind dominates under different conditions. Using a numerical model, we investigate the juxtaposition of submesoscale eddies and down‐front winds acting on the mixed layer. By estimating the eddy‐induced overturning stream function in the mixed layer, we separate the along‐ and cross‐isopycnal fluxes of buoyancy associated with submesoscale mixed layer eddies and demonstrate the need for parameterization of the advective, along‐isopycnal flux. Though the cross‐front transport of buoyancy induced by the down‐front compon...

Using generalizations of the well known Rice formula and applying the general method related to m-dependent processes that we settled in earlier works, allow one to obtain representations into the Ito-Wiener Chaos and CLTs for curve-crossings number. This approach not only explains heuristic considerations of Longuet-Higgins on specular points and related problems in the context of sea modelling, but goes

Using generalizations of the well known Rice formula and applying the general method related to m-dependent processes that we settled in earlier works, allow one to obtain representations into the Itô-Wiener Chaos and CLTs for curve-crossings number. This approach not only explains heuristic considerations of Longuet-Higgins on specular points and related problems in the context of sea modelling, but goes far beyond when providing asymptotic results. These results on curve-crossings may also be applied in other fields. One example is the study of the estimator of the natural frequency of a harmonic oscillator.

We give an error estimate for the Energy and Helicity Preserving Scheme (EHPS) in second order finite difference setting on axisymmetric incompressible flows with swirling velocity. With careful and detailed truncation error analysis near the geometric singularity and far field decay estimate for the stream function, we have achieved optimal error bound using weighted energy estimate. A key ingredient in our a priori estimate is the permutation identity associated with the Jacobians, which is also a unique feature that distinguishes EHPS from standard finite difference schemes.

Buoyancy and Marangoni convection in melted zone generated by the melting and resolidification process of stainless steel under high power C0 2 laser irradiation was analyzed numerically. Governing equations with vorticity and stream function were solved for the flow field in the melted zone. Latent heat of fusion for melting and resolidification was treated with temperature recovery method. Development of the melted zone with time and the effect of the convection on heat transport process were discussed. Temperature in melted zone became more uniform and the depth of melted zone became thinner in depth and wider radially, as the convection was taken into account, in comparison with the cases in which it was neglected. Marangoni convection generated a high velocity convection with some small value of the temperature coefficient of surface tension. Comparison of the analytical results with the experimental results for several seconds irradiation under atmospheric condition showed that, the velocity of the surface of melted zone was almost in agreement with the results assuming the buoyancy convection, and the net evaporation coefficient was estimated to be much smaller in the experiment under atmospheric condition than that under vacuum condition.

Fluid flows occur due to internal or external forces such as wind, gravity, pressure gradients, side-wall motion, MHD, and free convection. This study examines how meanders impact heat transfer by studying the behavior of viscous fluid flow with streamwise vortices in a sinusoidal wavy meandering channel of non-uniform radius. The study simplifies the motion and energy equations governing the fluid flow using novel transformations and a regular perturbation method. By plotting graphs for different parameter values, such as Pr, Re, and Ec, it reveals that decreasing the wavelength leads to flow separation near the channel surface. However, the stream moves forward with a sudden meander disturbance, causing the flow to become rectilinear and independent of vertex-generating centrifugal forces. The study identifies a stream function using standard and established relations. The fluid flow patterns and temperature distribution behavior are shown in various plots, highlighting the signif...

The interaction between peristaltic transport of a dusty fluid (Saffman's model) in a 2-dimensional uniform channel and the elasticity of the channel's flexible walls was studied under long wavelength approx- imation. Perturbation solutions were obtained for the stream functions of both the fluid particles and solid particles, in terms of the wall slope parameter. The expressions for average velocity of the fluid particles and solid particles, and average fluid flow rate were derived. The effects of various elastic parameters and mass concentration of dust particles on the streamline pattern and average flow rate were studied. The phenomenon of trapping was observed and the area of the trapped bolus increased along with the tension parameter, but decreased with damping and mass concentration of dust particles.

Through calculating the scatter diagrams of the streamfunction ( P or T ) versus potential vorticity (PV) (q P or q T ), where P and T are the planetary-scale streamfunction and total streamfunction, respectively, and using a weakly nonlinear NAO model proposed in Part I of this paper, it is suggested that negative-and positive-phase NAO events may approximately correspond to free modes even though driven by synopticscale eddies. In a planetary-scale field, the q P ( P ) scatter diagram of an NAO event exhibits a linear multivalued functional relationship in a narrow region for the negative phase, but exhibits a linear singlevalued functional relationship during the positive phase. It was also found that there is no steepening of the slope of the main straight line in the q P ( P ) scatter diagrams for two phases of the NAO event. Instead, the slope of the straight line in the scatterplots is time independent throughout the life cycle of the NAO event. However, when synoptic-scale eddies are included in the streamfunction field, the q T ( T ) scatter diagram of the negative-phase NAO event shows a trend toward steepening during the intensification phase, and this tendency reverses during the decay phase. During the positive NAO phase the slope of the q t ( T ) scatter diagram shoals during the intensification phase and then steepens during the decay phase. Thus, it appears that the steepening and shoaling of the scatter diagrams of the streamfunction versus PV for the negativeand positive-phase NAO events are attributed to the effect of synoptic-scale eddies that force NAO events to form. Diagnostic studies using both composite and unfiltered fields of observed NAO events are presented to confirm these conclusions.

A simple theoretical model is proposed to clarify how synoptic-scale waves drive the life cycle of the North Atlantic Oscillation (NAO) with a period of nearly two weeks. This model is able to elucidate what determines the phase of the NAO and an analytical solution is presented to indicate a high similarity between the dynamical processes of the NAO and zonal index, which is not derived analytically in previous theoretical studies. It is suggested theoretically that the NAO is indeed a nonlinear initial-value problem, which is forced by both preexisting planetary-scale and synoptic-scale waves. The eddy forcing arising from the preexisting synoptic-scale waves is shown to be crucial for the growth and decay of the NAO, but the preexisting low-over-high (high-over-low) dipole planetary-scale wave must be required to match the preexisting positive-over-negative (negative-over-positive) dipole eddy forcing so as to excite a positive (negative) phase NAO event. The positive and negative feedbacks of the preexisting dipole eddy forcing depending upon the background westerly wind seem to dominate the life cycle of the NAO and its life period. An important finding in the theoretical model is that negative-phase NAO events could be excited repeatedly after the first event has decayed, but for the positive phase downstream isolated dipole blocks could be produced after the first event has decayed. This is supported by observed cases of the NAO events presented in this paper. In addition, a statistical study of the relationship between the phase of the NAO and blocking activity over Europe in terms of the seasonal mean NAO index shows that blocking events over Europe are more frequent and long-lived for strong positive-phase NAO years, indicating that the positivephase NAO favors the occurrence of European blocking events.

This paper presents a new domain embedding numerical scheme for the simulation of flows of a Newtonian fluid in multiply-connected domains. The governing equations are taken from the stream function-vorticity formulation. The problem domain is converted into a simply-connected domain that is then discretised using a Cartesian grid. Radial-basis-function networks, which are constructed through integration rather than the usual differentiation, are employed on grid lines to approximate the field variables. Each field variable is assumed to vary over interior holes according to appropriate polynomials that satisfy the boundary conditions. Point collocation is applied to discretise the governing equations. Several linear and nonlinear problems, including natural convection in the annulus between square and circular cylinders are simulated to verify the proposed technique.

High order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multiderivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p = 6, and define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

Horizontal convection in a rectangular enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically using a spectral-element discretization for velocity and temperature fields. A Boussinesq approximation is employed to model buoyancy. The emphasis of this study is on the scaling of mean Nusselt number and boundary layer quantities with aspect ratio and Rayleigh number. At low Rayleigh number, Nusselt number and boundary layer thickness are found to be independent of Rayleigh number but dependent on aspect ratio. At higher Rayleigh numbers, convective flow dominates, and Nusselt number, boundary layer thickness and peak boundary layer velocity become independent of aspect ratio. In this regime, the Rayleigh number scaling of these quantities agrees well with exponents predicted by theory, with respective values of 1/5, À1/5 and 2/5. Unsteady flow develops at a critical Rayleigh number independent of aspect ratio, and the development of unsteady flow is found to lead to an increase in the Nusselt number scaling exponent from 0.2 to approximately 0.3, which is closer to the theoretical upper bound than has yet been reported in the study of horizontal convection flows.

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H 1 -and L 2 -norms are proved and confirmed with numerical experiments.

Three types of previously used numerical methods are revisited for computing the streamfunction ψ and velocity potential χ from the horizontal velocity v in limited domains. The first type, called the SOR-based method, uses a classical successive over-relaxation (SOR) scheme to compute ψ (or χ) first with an arbitrary boundary condition (BC) and then χ (or ψ) with the BC derived from v. The second type, called the spectral method, uses spectral formulations to construct the inner part of (ψ, χ)-the inversion of (vorticity, divergence) with a homogeneous BC, and then the remaining harmonic part of (ψ, χ) with BCs from v. The third type, called the integral method, uses integral formulas to compute the internally induced (ψ, χ)-the inversion of domain-internal (vorticity, divergence) using the free-space Greenꞌs function without BCs and then the remaining harmonic ψ (or χ) with BCs from v minus the internally-induced part. Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains, their accuracy is compromised when applied to complex flows over mesoscale domains, as shown in this paper. To resolve this problem, two hybrid approaches, the integral-SOR method and the integral-spectral method, are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods, respectively. Upon testing these methods on real-case complex flows, the integral-SOR method is significantly more accurate than the integral-spectral method, noting that the latter is still generally more accurate than the three previously-used methods. The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.

In this paper the hydrodynamics and shell side mass transfer of laminar flow across in-line tube arrays is studied numerically. The hydrodynamics of a five-tube array were described by the two-dimensional Navier-Stokes equations, employing the stream function-vorticity method. After a converged solution for the hydrodynamics had been found, a transport equation for the concentration was solved. The numerical grid of the investigated geometries consists of a number of basic cells, which are generated with a domain decomposition method combined with orthogonal grid generation. The hydrodynamics and mass transfer were analysed for Reynolds = 2-200, Péclet = 10-300 and pitch-to-diameter ratios of 1.5, 1.85 and 2.0. It is shown that the strength of the recirculation occurring between the adjacent tubes increases with increasing Reynolds number, and with decreasing pitch-todiameter ratio. Cartesian coordinates A tube radius C concentration c dimensionless concentration D diffusivity f distortion function h η , h ξ scalar factor Pe Péclet number Re Reynolds number Sc Schmidt number Sh Sherwood number U ∞ bulk velocity Greek symbols ψ stream function ω vorticity ξ, η computational coordinates λ pitch-to-diameter ratio ν kinematic viscosity Subscripts avg.

A procedure used for expanding the height of a pressure surface over the Northern Hemisphere in a series of spherical-harmonic components is described. The corresponding spherical-harmonic representation of the stream function is obtained by utilizing the geostrophic balance condition. From this representation mean values for the spectral distribution of the kinetic energy at the 500 mb level for January 1957 is presented. The behaviour of the components with the largest horizontal scales is considered a t the 500 mb and the 1000 mb levels during the 90 days period from 1 December 1956 to 28 February 1957. I n general each of these components exhibits smaller or larger fluctuations, and it is attempted to investigate the character of the shorter periodic fluctuations by eliminating the constant and the long periodic parts of the stream field. For the components with wavenumber 1, 2, and 3, and with the most large-scale meridional variation, the 24 hours tendency fields show a more or less regular westward propagation with mean values for the velocity of propagation, corresponding to periods about 5 days. For the components with the same wavenumbers but with the second largest meridional scales we find for the daily deviations from the 15 days mean flow displacements also mainly towards the west and also with mean values for the velocity of propagation, decreasing with decreasing horizontal scale. The mean values of the velocity of propagation obtained in this way for the different components are nearly in accordance with the velocities determined by the Rossby effect, especially if this effect is reduced somewhat by a weak divergence effect. On the basis of the spectral form of the barotropic vorticity equation the time derivatives for the expansion coefficients as well as for the amplitude and phase angle of the stream function at the 500 mb level have been computed for each day in January 1957, for some components of the zonal flow and some of the components with wavenumber 1, 2, 5, and 6. From these time derivatives 48 hours tendencies have been evaluated and compared with the corresponding observed ones. In general it is found that the agreement is better for components with moderately large scales than for components with very large scales. To some extent this may be explained by the neglect of quasi-stationary effects, and as a very simple attempt, these are represented by a constant term. Finally the contributions from the barotropic interactions between different groups of components to the change of kinetic energy of individual components are considered.

The paper deals with surface tension driven flows, induced by imposed temperature differences, in axisymmetric liquid bridges. This configuration is connected to processes of crystal growth from melt by the floating zone method. The field equations are solved via a two-dimensional, unsteady, finite difference numerical code in terms of vorticity, stream function and temperature. The Poisson equation for the pressure field is solved, by using Briley's correction, and the free surface shape is computed under the assumption of small capillary number. The flow field structure and behaviour are discussed in terms of non-dimensional characteristic numbers and the aspect ratio A = R/L, with R and L, respectively, the radius and the length of the bridge. Stream lines, isotherms, velocity profiles, surface shapes are given and discussed. In particular, we report the maximum of the surface velocity, the maximum of the surface deformation and the bulk heat transfer coefficient (Nusselt number) as functions of the aspect ratio.

Energy-Conserving Semi-Lagrangian Discontinuous Galerkin Schemes for Vlasov-Poisson Systems 1 JAMES ROSSMANITH, Iowa State University -In many laboratory settings a common approach is to model the plasma via kinetic equations (i.e., some form of the Boltzmann equation). In this description the plasma is represented through a probability density function (PDF) that selfinteracts through electromagnetic forces. Kinetic models are valid over most of the spatial and temporal scales that are of physical relevance in many application problem; however, they are computationally expensive due to the high-dimensionality of phase space and the disparate length and time scales that they resolve. In this work we describe efforts to develop high-order accurate numerical methods for collisionless plasma. In this setting the canonical model is the Vlasov-Poisson system. After briefly reviewing key properties of the Vlasov-Poisson system, we describe a class of numerical methods for solving Vlasov-Poisson based on coupling high-order discontinuous Galerkin finite element methods with semi-Lagrangian time-stepping. We show how to modify such numerical schemes to maintain important physical properties such as charge, mass, and energy conservation, as well positivity of the probability density function. The resulting numerical method is tested on several numerical examples.

Energy-Conserving Semi-Lagrangian Discontinuous Galerkin Schemes for Vlasov-Poisson Systems 1 JAMES ROSSMANITH, Iowa State University -In many laboratory settings a common approach is to model the plasma via kinetic equations (i.e., some form of the Boltzmann equation). In this description the plasma is represented through a probability density function (PDF) that selfinteracts through electromagnetic forces. Kinetic models are valid over most of the spatial and temporal scales that are of physical relevance in many application problem; however, they are computationally expensive due to the high-dimensionality of phase space and the disparate length and time scales that they resolve. In this work we describe efforts to develop high-order accurate numerical methods for collisionless plasma. In this setting the canonical model is the Vlasov-Poisson system. After briefly reviewing key properties of the Vlasov-Poisson system, we describe a class of numerical methods for solving Vlasov-Poisson based on coupling high-order discontinuous Galerkin finite element methods with semi-Lagrangian time-stepping. We show how to modify such numerical schemes to maintain important physical properties such as charge, mass, and energy conservation, as well positivity of the probability density function. The resulting numerical method is tested on several numerical examples.

• Rectangular Jet's (RJ) are inherently unstable. • Secondary flows cause minor axis shear layer growth to exceed that of the major axis. • This instability (Kelvin Helmholtz) causes unsteadiness and axis-switching. • Makes RJ's more easily manipulated for flow control.

This study analyzes the convection heat transfer to a micropolar fluid in an enclosure with varied heating lengths and locations. The left wall is heated to higher temperatures, either fully or partially, while the right wall remains cooler. The heater is placed at three different positions along the left wall, and insulation is applied to the non-active areas. Using the finite volume method, the research investigates the effects of heater length, position, and material parameters, such as the Prandtl number and Rayleigh number. Results shown in graphs of Nusselt numbers, isotherms, and streamlines indicate that the mean Nusselt number decreases with increasing material parameters. Additionally, a shorter heating section enhances average heat transfer, particularly when the heater is located at the wall's lower part.

The present study investigates the spectral analysis for natural convection in a tilted rectangular cavity, lled with high Prandtl oil ”Pr =880” by the code CFD. A constant vertical temperature gradient has been performed by subjecting the horizontal walls to constant temperatures Th and Tc; respectively. Other walls are adiabatic except the left small sidewall is differentially heating with temperature TA creating the horizontal temperature gradient. The results are presented for different values of lateral heating and inclination angle. The spectral analysis is used to identify and show effects on the original oscillation of the natural convection by the various investigated parameters (TA and θ).

V-Uniform Ergodicity of Threshold Autoregressive Nonlinear Time Series. (December 2003) Thomas R. Boucher, B.S., University of Massachusetts-Lowell; M.S., University of Massachusetts-Lowell Chair of Advisory Committee: Dr. Daren B.H. Cline We investigate conditions for the ergodicity of threshold autoregressive time series by embedding the time series in a general state Markov chain and apply a Foster-Lyapunov drift condition to demonstrate ergodicity of the Markov chain. We are particularly interested in demonstratingV-uniform ergodicity where the test function V(·) is a function of a norm on the state-space. In this dissertation we provide conditions under which the general state space chain may be approximated by a simpler system, whether deterministic or stochastic, and provide conditions on the simpler system which imply V-uniform ergodicity of the general state space Markov chain and thus the threshold autoregressive time series embedded in it. We also examine conditions under...

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr-Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien-Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it. The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier-Stokes equations using a family of profiles based on the tanh function.

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by a fluid fictitious envelope. The fourth order partial differential equation governing the flow was completed with Happel-type boundary conditions which dictate no fluid slip on the inverted spheroid and a shear stress free non-permeable fictitious boundary. Through innovative means, such as the Kelvin inversion method and the R-semiseparation technique, a stream function was obtained as series expansion of Gegenbauer functions of the first and the second kinds of even order. Based on this, analytical expressions of meaningful hydrodynamic quantities, such as the velocity and the pressure field, were calculated and depicted in informative graphs. Using the first term of the stream function, the drag force exer...

The spreading and diffusion of two-dimensional vortices subject to weak, external, random strain fields is examined. The response to such a field of given angular frequency depends on the profile of the vortex and can be calculated numerically. An effective diffusivity can be determined as a function of radius and may be used to evolve the profile over a long time scale, using a diffusion equation that is both nonlinear and nonlocal. This equation, containing an additional smoothing parameter, is simulated starting with a Gaussian vortex. Fine scale steps in the vorticity profile develop at the periphery of the vortex and these form a vorticity staircase. The effective diffusivity is high in the steps where the vorticity gradient is low: between the steps are barriers characterised by low effective diffusivity and high vorticity gradient. The steps then merge before the vorticity is finally swept out and this leaves a vortex with a compact core and a sharp edge. There is also an increase in the effective diffusion within an encircling surf zone. In order to understand the properties of the evolution of the Gaussian vortex, an asymptotic model first proposed by Balmforth, Llewellyn Smith and Young (J. Fluid Mech. 426, 95-133, 2001) is studied. The model is based on a vorticity distribution that consists of a compact vortex core surrounded by a skirt of relatively weak vorticity. Again simulations show the formation of fine scale vorticity steps within the skirt, followed by merger. The diffusion equation we develop has a tendency to generate vorticity steps on arbitrarily fine scales; these are limited in our numerical simulations by smoothing the effective diffusivity over small spatial scales.

This paper considers the evolution of smooth, two-dimensional vortices subject to a rotating external strain field, which generates regions of recirculating, cat's eye stream line topology within a vortex. When the external strain field is smoothly switched off, the cat's eyes may persist, or they may disappear as the vortex relaxes back to axisymmetry. A numerical study obtains criteria for the persistence of cat's eyes as a function of the strength and timescale of the imposed strain field, for a Gaussian vortex profile. In the limit of a weak external strain field and high Reynolds number, the disturbance decays exponentially, with a rate that is linked to a Landau pole of the linear inviscid problem. For stronger strain fields, but not strong enough to give persistent cat's eyes, the exponential decay of the disturbance varies: as time increases the decay slows down, because of the nonlinear feedback on the mean profile of the vortex. This is confirmed by determining the decay rate given by the Landau pole for these modified profiles. For strain fields strong enough to generate persistent cat's eyes, their location and rotation rate are determined for a range of angular velocities of the external strain field, and are again linked to Landau poles of the mean profiles, modified through nonlinear effects.

Research in any field is a daunting task requiring perseverance and constant struggle against all odds and setbacks. I am thankful to almighty Allah for giving me courage and competence to complete my dissertation within the stipulated timeframe. I attribute my inquisitiveness to the last Messenger of Allah (SAW) who happened to be the beacon of light for all humanity. His preaching and actions are a great source of inspiration for all of us to pursue knowledge for human development.

Observations show the presence of localized regions in the atmosphere with diminished potential vorticity gradients, an example being the tropical upper troposphere where convective heating plays an important role. The present work investigates the effect of forcing on the evolution of Rossby waves in a zero potential vorticity gradient environment. As a preliminary investigation, the barotropic case is studied, where the analog of potential vorticity is absolute vorticity. A forcing term is employed to model thermal forcing in the real atmosphere. The analytic solution of the linearized problem shows that the streamfunction grows in time and eventually develops a nonlinear critical layer. The presence of forcing within the critical layer is found to drive the dynamics. The numerical solution of the nonlinear problem within the critical layer shows that the nonlinearity and the forcing act together to halt the growth as coherent vortices develop in a nonlinear oscillatory regime. At long times, the critical layer solution settles down to a quasi steady state consisting of relatively large amplitude stationary vortices with a set of small amplitude steadily propagating vortices superimposed. These results are contrasted with the results of previous unforced problems.

The role of baroclinicity in the dynamics of abyssal equator-crossing flows is examined by studying two-layer models of the flow valid in the equatorial region. Three new analytical models are derived from two-layer shallow-water theory. One of these models (Equatorial Model I, or EMI) reduces to the Swaters and Flierl coupled model in the midlatitude limit. In the equatorial limit, the lower-layer dynamics of EMI are that of the complete shallow-water equations, and the upper-layer dynamics are built upon quasigeostrophic potential vorticity conservation with a balance equation to relate the streamfunction and pressure. Simple numerical simulations are performed using this model to investigate its behavior in certain idealized situations, including equatorcrossing lenses and currents. In the midlatitudes, the dynamics of EMI are characterized by strong baroclinic interactions between the layers, while near the equator all three models exhibit a partial decoupling of the layers. This motivates the use of a one-layer reduced-gravity model to simulate abyssal dynamics in the immediate vicinity of the equator. Such simulations are reported elsewhere. A uniformly valid metamodel is derived that contains all of the necessary terms so that it may reduce, in the appropriate parameter limit, to any of the three models derived here.

We consider autonomous stochastic perturbations Ẋε (t) = ∇H(X ε (t)) + εb(X ε (t)) of Hamiltonian systems of one degree of freedom whose Hamiltonian H is quadratic in a neighborhood of the only saddle point of H. Assume that b = b 1 + ξb 2 for some random fields b i , i = 1, 2, and ξ is a random variable, and that divb i < 0 and ξ > 0 with probability 1. Also assume that H has only one saddle point and two minima. To consider the effects of the perturbations, we Math Department during the years, Millie Stengel, Bill Schildknecht,

The performance of a geometric multigrid method is analyzed for two-dimensional Laplace, Navier, Burgers and two formulations of Navier-Stokes (streamfunction-vorticity and streamfunction-velocity) equations. These equations are discretized with the Finite Difference Method on uniform grids with numerical approximations of first-and secondorders of accuracy. The systems of equations are solved with a Modified Strongly Implicit (MSI) and a Successive Over Relaxation (SOR) solver associated with the multigrid method with a V-cycle and a Correction Scheme (CS) and a Full Approximation Scheme (FAS). The effect of the number of inner iterations of the solver, the number of grid levels problems with grid sizes of 1025 Â 1025 points, the influence of differential equations numbers and Reynolds number up to 1000 on Central Processing Unit (CPU) time are investigated. The results show that (1) a solution of two coupled equations (Navier or Burgers) is obtained with the same efficiency multigrid textbook that occurs in the solution of only one equation (Laplace), (2) the efficiency of the multigrid method in the solution of two coupled equations (Navier-Stokes streamfunction-vorticity formulation) or only one equation (the Navier-Stokes streamfunction-velocity formulation) decreases with increasing Reynolds numbers, and (3) the poor performance of the multigrid method for solving the Navier-Stokes seems to be related to the physics of the problem and not to the type of formulation or coupling between the equations.

This communication investigates the unsteady flow of viscous incompressible fluid through porous medium induced by periodically heated half filled concentric cylindrical annulus placed horizontally. The boundaries of the annulus are rotating periodically with different angular velocities in the same or opposite directions about their common axis. The governing equations are expressed in terms of stream function and vorticity functions. The analytical solutions have been obtained by expanding the variables in a power series of the annulus aspect ratio. The expressions for streamlines, temperature distribution and rate of heat transfer are obtained and the effects of various parameters upon them have been examined.

We consider a parametric Robin problem driven by the p-Laplacian with an indefinite potential and with a superlinear reaction term which does not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions. We prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions as the parameter varies. We also show the existence of a minimal positive solution ũλ and establish the monotonicity and continuity of the map λ → ũλ .

The development of a new algorithm to solve the Navier-Stokes equations by an implicit formulation for the ÿnite di erence method is presented, that can be used to solve two-dimensional incompressible ows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to ÿve and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady uid mechanics problems: cavity-driven ow with Re = 100; 400 and 1000 and ow in a sudden expansion with expansion ratio H=h = 2 and Re = 50; 100 and 200. The results obtained for the stream function are compared with values obtained by di erent available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright ? 2002 John Wiley & Sons, Ltd.

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mec...

We present the mixed convective peristaltic motion of a magnetohydrodynamic (MHD) Jeffrey nanofluid in an asymmetric channel with Newtonian heating. In the peristaltic literature, Newtonian heating is used for the first time in the present article. The peristaltic flow of a nanofluid with Newtonian heating is not explored so far. So in the present problem, first we model the mixed convective peristaltic motion of a MHD Jeffrey nanofluid in an asymmetric channel with Newtonian heating. According to the realistic approch, the problem formulation is made under long wavelength and low Reynolds number approximation. We get the four coupled equations. Homotopy perturbation method (HPM) solutions are calculated for nanoparticle fraction and heat transfer phenomena, while exact solutions are evaluated for stream function and pressure gradient. The possessions of different parameters on the flow quantities of observation are analyzed graphically and physically. In the end, the streamlines are plotted and discussed.

We present the mixed convective peristaltic motion of a magnetohydrodynamic (MHD) Jeffrey nanofluid in an asymmetric channel with Newtonian heating. In the peristaltic literature, Newtonian heating is used for the first time in the present article. The peristaltic flow of a nanofluid with Newtonian heating is not explored so far. So in the present problem, first we model the mixed convective peristaltic motion of a MHD Jeffrey nanofluid in an asymmetric channel with Newtonian heating. According to the realistic approch, the problem formulation is made under long wavelength and low Reynolds number approximation.We get the four coupled equations. Homotopy perturbation method (HPM) solutions are calculated for nanoparticle fraction and heat transfer phenomena, while exact solutions are evaluated for stream function and pressure gradient. The possessions of different parameters on the flow quantities of observation are analyzed graphically and physically. In the end, the streamlines are...

Research in any field is a daunting task requiring perseverance and constant struggle against all odds and setbacks. I am thankful to almighty Allah for giving me courage and competence to complete my dissertation within the stipulated timeframe. I attribute my inquisitiveness to the last Messenger of Allah (SAW) who happened to be the beacon of light for all humanity. His preaching and actions are a great source of inspiration for all of us to pursue knowledge for human development.

Using a bi-cylindrical coordinates system and a vorticity-stream function formulation, the authors study numerically the two dimensional unsteady natural convection of a Newtonian fluid bounded by portions of cylinders. A spatial discretization based on the finite difference method is used to approximate the dimensionless equations. While a purely implicit scheme is adopted for the time discretization. The algebraic systems of equations resulting from the discretization are solved by a Successive under Relaxation (SUR) method. The authors analyzed the effects of the Rayleigh number on the dynamic of the system. The analysis of the thermal and flow field showed that for Rayleigh number greater than 5.10 6 the transfers inside the enclosure are dominated by convection.

This research is engaged to explore biological peristaltic transport under the action of an externally applied magnetic field passing through an asymmetric channel which is saturated with porous media. The set of governing partial differential equations for the present peristaltic flow are solved in the absence of a low Reynolds number and long wavelength assumptions. The governing equations are to be solved completely, so that inertial effects can be studied. The numerical simulations and results are obtained by the help of a finite element method based on quadratic six-noded triangular elements equipped with a Galerkin residual procedure. The inertial effects and effects of other pertinent parameters are discussed by plotting graphs based on a finite element (FEM) solution. Trapped bolus is discussed using the graphs of streamlines. The obtained results are also compared with the results given in the literature which are highly convergent. It is concluded that velocity and the num...

In this work, the movement of digestive juice in small intestine, food bolus through esophagus and the blood in arteries are addressed. Based on the characteristics of blood and digestive juices and the elements that affect them, a viscosity function adapted to an asymmetric channel is chosen to simulate some of the biological phenomena. A bivariate viscosity function reflects the natural phenomena where it is affected starting from the bottom and top walls as it occurs in the intestines and arteries. A peristaltic transport of Newtonian fluid is considered and the influence of the bivariate function in an asymmetric channel is studied. We were able to compute explicitly the pressure rise and the pressure gradient. Reflux, trapping, pumping and co pumping phenomena are studied. A graphical analysis of the effect of the viscosity variation is presented. Similar to the axisymmetric case, this work also illustrates that, the reflux limit and the free pumping do not depend on the viscos...

A numerical solution is presented for the motion of a neutrally buoyant circular cylinder in Poiseuille and Couette flows between two plane parallel boundaries. The force and torque on a stationary particle are calculated for a wide range of particle sizes and positions across the channel. The resistance matrix calculated in Ref. (henceforth referred to as Part 1) is utilized to find the translational and angular velocity for a drag-and torque-free particle. The results are compared with analytical perturbation solutions for a small cylindrical particle situated on the channel centerline, and for the motion of a spherical particle in a circular tube or between plane parallel boundaries. It is found the behavior of flow around a cylindrical particle in a channel is qualitatively similar to the behavior of flow around a spherical particle in a tube, while the flow around a spherical particle in a channel frequently exhibits different trends from the above two cases.

It is well known that an operator-valued function Θ from the Schur class S(M, N), where M and N are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function Θ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called "Schur parameters". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero-Moral-Velázquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

It is well known that an operator-valued function Θ from the Schur class S(M, N), where M and N are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function Θ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called "Schur parameters". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero-Moral-Velázquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

Анотація. З метою підвищення достовірності моделювання робочого процесу наддолотного струминного насоса за наявності відносного радіального зміщення елементів ежекційної системи розроблено метод моделювання циркуляційного інжектованого потоку за допомогою гідродинамічної вихрової функції, центр якої зміщений відносно осі камери змішування струминного насоса. В процесі моделювання обертового асиметричного руху інжектованого потоку використано поняття циркуляції вектора поступальної швидкості руху рідини по замкненому контуру. За допомогою складових комплексного потенціалу гідродинамічної циркуляційної функції отримана графічна інтерпретація еквіпотенціальних ліній та ліній течії просторового вихору із одностороннім та двостороннім зміщенням центру координат. Використовуючи аналітичні співвідношення для потенціалу швидкості і функції течії визначені компоненти циркуляційної швидкості та доведена аналітичність функції комплексного потенціалу із зміщеним центром вихору. Виконання умов Коши-Римана дозволило представити результуючу швидкість циркуляційної течії із зміщеним вздовж вертикальної осі центром координат у вигляді часткової похідної комплексного потенціалу. На відміну від комплексного потенціалу для симетричного вихору співвідношення для визначення циркуляційної швидкості у разі неспіввісності робочої насадки та камери змішування струминного насоса містить додаткову складову у вигляді величини зміщення центру вихору. В процесі аналізу результатів використання запропонованої математичної моделі встановлено, що результуюча швидкість циркуляційної течії та величина відносного зміщення центру координат просторового вихору зв'язані прямопропорційною залежністю. Проведеними дослідженнями доведено, що співвідношення швидкостей симетричної та асиметричної циркуляційної течії є функцією величини зміщення центру координат вихрового потоку та змінюється від 1 до нуля. Величина відносної швидкості циркуляційної течії обернено пропорційно залежить від зміщення центру координат та відстані до камери змішування струминного насоса. Розроблена математична модель може використовуватись для прогнозування впливу обертання ежекційної системи на її напірну характеристику у випадку радіального відносного зміщення робочої насадки та камери змішування струминного насоса. Ключові слова: ежекційна система, обертання струминного насоса, потенціальні потоки, комплексний потенціал, еквіпотенціальні поверхні, лінія течії, асиметрична вихрова функція.