Showing papers on "Stream function published in 2009"

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

Abstract: The Sawyer–Eliassen Equation (SEQ) is here rederived in height coordinates such that the sea surface is also a coordinate surface. Compared with the conventional derivation in mass field coordinates, this formulation adds some complexity, but arguably less than is inherent in terrain-following coordinates or interpolation to the lower physical boundary. Spatial variations of static stability change the vertical structure of the mass flow streamfunction. This effect leads to significant changes in both secondary-circulation structure and intensification of the primary circulation. The SEQ is solved on a piecewise continuous, balanced mean vortex where the shapes of the wind profiles inside and outside the eye and the tilt of the specified heat source can be adjusted independently. A series of sensitivity studies shows that the efficiency with which imposed heating intensifies the vortex is most sensitive to intensity itself as measured by maximum wind and to vortex size as measured by radius of ma...

Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum

Abstract: The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior. The system exhibits three f...

Diabatically Induced Secondary Flows in Tropical Cyclones. Part II: Periodic Forcing

Abstract: The linearized equation for the time-varying, axially symmetric circumferential component of the vorticity in a hurricane-like vortex closely resembles the classical Sawyer–Eliassen equation for the quasi-steady, diabatically induced secondary-flow streamfunction. The salient difference lies in the coefficients of the second partial derivatives with respect to radius and height. In the Sawyer–Eliassen equation, they are the squares of the buoyancy and isobaric local inertia frequencies; in the circumferential vorticity equation they are the differences between these quantities and the square of the frequency with which the imposed forcing varies. The coefficient of the mixed partial derivative with respect to radius and height is the same in both equations. Thus, for low frequencies the response to periodic forcing is a slowly varying analog to steady Sawyer–Eliassen solutions. For high frequencies, the solutions are radially propagating inertia-buoyancy waves. Since the local inertia frequency, ...

Large Eddy Simulation of Separated Flow over a Two-dimensional Hump with and without Control by Means of a Synthetic Slot-jet

Abstract: Large Eddy Simulations for two flows separating from a two-dimensional hump in a duct are reported and discussed. The flows differ through the presence or absence of a synthetic slot-jet injected in a sinusoidal manner, i.e. at zero net mass-flow rate, close to the location of separation and intended to reduce (“control”) the extent of the separated region. Results reported include instantaneous visualisations, pre-multiplied spectra, wall-pressure distributions, streamfunction fields and profiles of velocity and second moments. For both flows, agreement between the simulations and the experimental results is generally good, especially in respect of the overall control effectiveness of the synthetic jet, despite the use of an approximate wall treatment bridging the viscous sublayer. Proper Orthogonal Decomposition of the velocity field is used to study structural features, and this shows that the most energetic mode in the base flow is representative of large streamwise vortices in the separated region, while in the controlled flow, most of the energetically dominant modes are associated with large spanwise vortices.

A streamfunction-velocity approach for 2D transient incompressible viscous flows

Abstract: We recently proposed (J. Comput. Phys. 2005; 207(1):52-68) a new paradigm for solving the steady-state two-dimensional (2D) Navier-Stokes (N-S) equations using a streamfunction-velocity (Ψ-υ) formulation. This formulation was shown to avoid the difficulties associated with the traditional formulations (primitive variables and streamfunction-vorticity formulations). The new formulation was found to be second-order accurate and was found to yield accurate solutions of a number of fluid flow problems. In this paper, we extend the ideas and propose a second-order implicit, unconditionally stable Ψ-υ formulation for the unsteady incompressible N-S equations. The method is used to solve several 2D time-dependent fluid flow problems, including the flow decayed by viscosity problem with analytical solution, the lid-driven square cavity problem, the backward-facing step problem and the flow past a square prism problem. For the problems with known exact solutions, our coarse grid transient solutions are extremely close to the analytical ones even for high Reynolds numbers (Re). For the driven cavity problem, our time-marching steady-state solutions up to Re=7500 provide excellent matches with established numerical results, and for Re = 10000, our study concludes that the asymptotic stable solution is periodic as has been found by other authors in recent studies. For the backward step problem, our numerical results are in excellent agreement with established numerical and experimental results. Finally, for the flow past a square prism, we have very successfully simulated the von Karman vortex street for Re = 200.

Analytical solution for inviscid flow inside an evaporating sessile drop.

Abstract: Inviscid flow within an evaporating sessile drop is analyzed. The field equation E;{2}psi=0 is solved for the stream function. The exact analytical solution is obtained for arbitrary contact angle and distribution of evaporative flux along the free boundary. Specific results and computations are presented for evaporation corresponding to both uniform flux and purely diffusive gas phase transport into an infinite ambient. Wetting and nonwetting contact angles are considered, with flow patterns in each case being illustrated. The limiting behaviors of small contact angle and droplets of hemispherical shape are treated. All of the above categories are considered for the cases of droplets whose contact lines are either pinned or free to move during evaporation.

Slip Effects on the Peristaltic Flow of a Third Grade Fluid in a Circular Cylindrical Tube

Abstract: Peristaltic flow of a third grade fluid in a circular cylindrical tube is undertaken when the no-slip condition at the tube wall is no longer valid. The governing nonlinear equation together with nonlinear boundary conditions is solved analytically by means of the perturbation method for small values of the non-Newtonian parameter, the Debroah number. A numerical solution is also obtained for which no restriction is imposed on the non-Newtonian parameter involved in the governing equation and the boundary conditions. A comparison of the series solution and the numerical solution is presented. Furthermore, the effects of slip and non-Newtonian parameters on the axial velocity and stream function are discussed in detail. The salient features of pumping and trapping are discussed with particular focus on the effects of slip and non-Newtonian parameters. It is observed that an increase in the slip parameter decreases the peristaltic pumping rate for a given pressure rise. On the contrary, the peristaltic pumping rate increases with an increase in the slip parameter for a given pressure drop (copumping). The size of the trapped bolus decreases and finally vanishes for large values of the slip parameter.

Linear stability of Hunt's flow

Abstract: We analyse numerically the linear stability of the fully developed flow of a liquid metal in a rectangular duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Re_c 2018, based on the maximal velocity, is attained at Ha ~ 10. Further increase of the magnetic field stabilises this mode with Re_c growing approximately as Ha. For Ha>40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ~ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum Re_c ~ 1130 at Ha ~ 70 and increases as Re_c ~ Ha^1/2 for Ha >> 1. The asymptotics of the critical wavenumber is k_c ~ 0.525Ha^1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

Generalized terrain-based flow analysis of digital elevation models

Abstract: Digital Elevation Models (DEMs) of topography are widely used in Geographic Information Systems (GIS) to derive information for the modeling of hydrologic processes. We present a general method for flow analysis that builds upon the representation of downslope flow using the proportioning of flow among one or multiple downslope grid cells, such as with the D8 or D-infinity models for the representation of downslope flow used in hydrologic terrain analysis, to enable the calculation of a new set of flow-based derivative surfaces. This general method which we refer to as flow algebra, extends flow accumulation approaches commonly available in GIS through the integration of multiple inputs and a broad class of algebraic rules into the calculation of flow related quantities. It is based on first establishing a flow field through DEM grid cells, or more generally model elements of any spatial discretization of the domain, by removing spurious pits and then proportioning the flow from a grid cell among downslope neighbors based on elevation. The flow field is required to be non-circulating, and as such these methods are applicable for flow problems related to the gradient of any potential field. Once a flow field is defined we show how it may be used to evaluate any mathematical function that incorporates dependence on values of the quantity being evaluated at upslope (or downslope) grid cells as well as other input quantities. The definitions of flow algebra functions are thus recursive and allow the evaluation of concepts that extend the concepts captured by recursive calculations of weighted contributing area available in GIS, to a broad class of flow related hydrologic terrain analysis functionality. Flow algebra encompasses single and multi-directional flow fields, various topographic representations, weighted accumulation algorithms, and enables untapped potential for a host of application-specific functions. While flow algebra expressions are recursively defined and most easily implemented using recursive functions, we describe here both recursive and non-recursive approaches to their evaluation. We have used the non-recursive approach to develop implementations for parallel processing architectures thereby enabling the rapid processing of large datasets. We illustrate the potential of flow algebra by presenting examples of new functions enabled by this perspective that are useful for hydrologic and environmental modeling. A new weighted flow distance to stream function averages the "weight" moving through multiple flow paths from each point in the domain to the stream or other downslope reference point. This is an example of a general function which can specifically be used to estimate potential for sediment and nutrient filtering by streamside vegetation, by specifying vegetation cover as the weight. This function can also be used to provide a definition for connectivity in the context of multiple flow paths and identify areas that are connected to streams. Other new functions that examine elevations on upslope and downslope flow paths have been developed determine the average rise to ridge and drop to stream which together determine hillslope position useful for soil depth modeling. A new avalanche runout function determines the zone with downslope gradient greater than a critical angle (alpha) and is useful for mapping avalanche or potential landslide hazards. Software implementing the functions illustrated here is available as part of the TauDEM package: http://www.engineering.usu.edu/dtarb/taudem.

An Effective Integrated-RBFN Cartesian-Grid Discretization for the Stream Function–Vorticity–Temperature Formulation in Nonrectangular Domains

Abstract: This article presents a new numerical collocation procedure, based on Cartesian grids and one-dimensional integrated radial-basis-function networks (1D-IRBFNs), for the simulation of natural convection defined in two-dimensional, multiply connected domains and governed by the stream function–vorticity–temperature formulation. Special emphasis is placed on the handling of vorticity values at boundary points that do not coincide with grid nodes. A suitable formula for computing vorticity boundary conditions, which is based on the approximations with respect to one coordinate direction only, is proposed. Normal derivative boundary conditions for the stream function are forced to be satisfied identically. Several test problems, including natural convection in the annulus between square and circular cylinders, are considered to investigate the accuracy of the proposed technique.

Hamiltonian chaos in a model alveolus.

Abstract: In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincare sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincare sections. Elements of Kolmogorov-Arnol'd-Moser theory, the Poincare-Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Shallow Water Quasi‐Geostrophic Theory on the Sphere

Abstract: [1] Quasi-geostrophic theory forms the basis for much of our understanding of mid-latitude atmospheric dynamics. The theory is typically presented in either its f-plane form or its β-plane form. However, for many applications, including diagnostic use in global climate modeling, a fully spherical version would be most useful. Such a global theory does in fact exist and has for many years, but few in the scientific community seem to have ever been aware of it. In the context of shallow water dynamics, it is shown that the spherical version of quasigeostrophic theory is easily derived (re-derived) based on a partitioning of the flow between nondivergent and irrotational components, as opposed to a partitioning between geostrophic and ageostrophic components. In this way, the invertibility principle is expressed as a relation between the streamfunction and the potential vorticity, rather than between the geopotential and the potential vorticity. This global theory is then extended by showing that the invertibility principle can be solved analytically using spheroidal harmonic transforms, an advancement that greatly improves the usefulness of this “forgotten” theory. When the governing equation for the time evolution of the potential vorticity is linearized about a state of rest, a simple Rossby-Haurwitz wave dispersion relation is derived and examined. These waves have a horizontal structure described by spheroidal harmonics, and the Rossby-Haurwitz wave frequencies are given in terms of the eigenvalues of the spheroidal harmonic operator. Except for sectoral harmonics with low zonal wavenumber, the quasi-geostrophic Rossby-Haurwitz frequencies agree very well with those calculated from the primitive equations. One of the many possible applications of spherical quasi-geostrophic theory is to the study of quasi-geostrophic turbulence on the sphere. In this context, the theory is used to derive an anisotropic Rhines barrier in three-dimensional wavenumber space.

Analysis of induced-charge electro-osmotic flow in a microchannel embedded with polarizable dielectric blocks

Abstract: Within the frame work of classic electromagnetic theory, a general electrical boundary condition describing the induced-charge electrokinetic phenomena at the liquid-dielectric interface is proposed in the present study. Two well-known limiting cases, i.e., perfectly insulating and perfectly polarizable wall boundary conditions, can be recovered from the present electrical boundary condition. By utilizing the proposed boundary condition, the induced-charge electro-osmosis (ICEO) flow in an infinitely long microchannel patterned with two symmetric polarizable dielectric blocks is investigated analytically. Fourier transform method is invoked to solve a biharmonic equation, which governs the (ICEO) flow field described by the stream function. Dimensionless parameters are introduced, and their effects on flow characteristics are analyzed. It is found that an increase in polarizability of the dielectric block enhances the slip velocity on its surface and thus induces a pair of counter-rotating vortices. Also, increasing the natural zeta potential on the upstream and downstream of the insulating microchannel walls leads to extinction of the vortex near the upstream insulating microchannel and suppression of the vortex near the downstream insulating microchannel.

Emergent order in rheoscopic swirls

Abstract: We discuss the reflection of light by a rheoscopic fluid (a suspension of microscopic rod-like crystals) in a steady two-dimensional flow. This is determined by an order parameter which is a non-oriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a one-dimensional Schrodinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the long-time limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.

A family of helically symmetric vortex equilibria

Abstract: We present a family of steadily rotating equilibrium states consisting of helically symmetric vortices in an incompressible inviscid irrotational unbounded fluid. These vortices are described by contours bounding regions of uniform axial vorticity. Helical symmetry implies material conservation of axial vorticity (in the absolute frame of reference) when the flow field parallel to vortex lines is proportional to (1+ϵ 2 r 2 ) −1/2 , where ϵ is the pitch and r is the distance from the axis. This material conservation property enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterized by their mean radius and centroid position. In the case of a single vortex, parameter space cannot be fully filled by our numerical approach. We conjecture multiply connected contours will characterize equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. Stability properties are investigated numerically using a helical CASL algorithm.

The Kelvin transformation in potential theory and Stokes flow

Abstract: Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

Abstract: The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.

MHD peristaltic flow of a third order fluid in an asymmetric channel

Abstract: This study is concerned with peristaltic flow of a magnetohydrodynamic (MHD) fluid in an asymmetric channel. Asymmetry in the flow is induced by waves on the channel walls having different amplitudes and phase. A systematic approach based on an expansion of Deborah number is used for the solution series. Analytic expressions have been developed for the stream function, axial velocity and axial pressure gradient. The pressure rise over a wavelength has been addressed through numerical integration. Particular attention has been given to the effects of Hartman number and Deborah number on the pressure rise over a wavelength and the trapping phenomenon. Several limiting solutions of interest are obtained as the special cases of the presented analysis by taking the appropriate parameter(s) to be zero.

Numerical study of an inviscid incompressible flow through a channel of finite length

Abstract: A two-dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo-symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.

Stable fourth-order stream-function methods for incompressible flows with boundaries *

Abstract: Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.

Averaging of Hamiltonian flows with an ergodic component

Abstract: We consider a process on $\mathbb{T}^2$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine, then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.

Euler-like modelling of dense granular flows: application to a rotating drum

Abstract: General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method even though granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.