Vector Streamfunction Research Articles

Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.

Integrated climatology and trends in the subtropical Hadley cell, sunshine duration and cloud cover over South Africa

This study uses two methods to diagnose the local Hadley circulation; first the zonally averaged mass stream function, and second the stream function vector method. The two methods have been applied to the ERA‐Interim reanalysis data for the period 1979–2015, to calculate both the climatology and trends of the Hadley cell. Both diagnostics advocate downwards mass flux being dominant over the subtropics, particularly over South Africa, yet the strength of Hadley is seasonal. Contrasts have been found between linear trends of the two diagnostics. Zonally symmetric diagnostics indicate strengthening of the Hadley cell, particularly in the subtropics of the Southern Hemisphere in winter and weakening in summer. The zonally asymmetric results indicate maximum strengthening of the Hadley over South Africa to be in spring and weakening in summer. Furthermore, maximum decrease in cloud cover and increase in sunshine duration over South Africa is in spring, implying more opportunities for solar energy generation.

Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

Solution of Electromagnetic and Velocity Fields for an Electrohydrodynamic Fluid Dynamical System

We studied the temporal evolution of the electromagnetic and velocity fields in an incompressible conducting fluid by means of computer simulations from the Navier Stokes and Maxwell’s equations. We then derived the set of coupled partial differential equations for the stream function vector field and the electromagnetic field. These equations are first order difference equations in time and fetch simplicity in discretization. The spatial partial derivatives get converted into partial difference equations. The fluid system of equations is thus approximated by a nonlinear state variable system. This system makes use of the Kronecker Tensor product. The final system has taken account of anisotropic permittivity. The conductivity and magnetic permeability of the fluid are assumed to be homogeneous and isotropic. Present work in this paper describes characterization of magneto hydrodynamic anisotropic medium due to permittivity. Also an efficient and modified novel numerical solution using Tensor product has been proposed. This numerical technique seems to be potentially much faster and provide compatibility in matrices operation. Application of our characterization technique shall be very useful in tuning of permittivity in Liquid crystal polymer, Plasma and Dielectric lens antennas for obtaining wide bandwidth, resonance frequency reconfigure ability and better beam control.

Three dimensional application of the complementary mild-slope equation

The complementary mild-slope equation (CMSE) is a depth-integrated equation, which models refraction and diffraction of linear time-harmonic water waves. For 2D problems, it was shown to give better agreements with exact linear theory compared to other mild-slope (MS) type equations. However, no reference was given to 3D problems. In contrast to other MS-type models, the CMSE is derived in terms of a stream function vector rather than in terms of a velocity potential. For the 3D case, this complicates the governing equation and creates difficulties in formulating an adequate number of boundary conditions. In this paper, the CMSE is re-derived using Hamilton's principle from the Irrotational Green–Naghdi equations with a correction for the 3D case. A parabolic version of it is presented as well. The additional boundary conditions needed for 3D problems are constructed using the irrotationality condition. The CMSE is compared with an analytical solution and wave tank experiments for 3D problems. The results show very good agreement.

Linear stability of Hunt's flow

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square 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 Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec 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, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

Method of Fundamental Solutions for Stokes Problems by the Pressure-Stream Function Formulation

ABSTRACTThe main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.

Method of fundamental solutions for multidimensional Stokes equations by the dual-potential formulation

A simple and accurate boundary-type meshless method of fundamental solutions (MFS) is applied to solve both 2D and 3D Stokes flows based on the dual-potential formulation of velocity potential and stream function vector. Using the dual-potential concept, the solutions of both 2D and 3D Stokes flows are obtained by combining the much simpler fundamental solutions of Laplace (potential) and bi-harmonic equations without using the complicated singular fundamental solutions such as Stokeslets and their derivatives as well as source doublet hypersingularity. The developed algorithm is used to test five numerical experiments for 2D flows: (1) circular cavity, (2) wave-shaped bottom cavity and (3) circular cavity with eccentric rotating cylinder; and for 3D flows: (4) a uniform flow passing a sphere and (5) a uniform flow passing a pair of spheres. Good results are obtained as comparing with solutions of analytical and numerical methods such as FEM, BEM and other meshfree schemes.

Conservation of Turbulence in Rapidly Rotating Flow

The behavior of an incompressible homogeneous turbulence in a rapidly rotating system is investigated. Linearized equations based on Rapid Distortion Theory (RDT) are used to analyze the rotating turbulence. In the rotating system without mean velocity-gradients, it is shown that the trace, which is defined by the phase space of the Fourier spectrum of the velocity fluctuation and the time-derivative of the Fourier spectrum of the velocity fluctuation, has an elliptic trajectory-the system has periodicity. Such a rotating system involves the conservation of the sum of the Reynolds stress tensor and the tensor composed of the vector streamfunction gradient correlation. The latter tensor describes the large-scale structure of vorticity field [for example, Reynolds et al., Phys. Fluids, 14 (2002), 2485-2492]. The tensors were calculated numerically with various initial anisotropic conditions using the solutions derived from the linearized equations. The results were found to substantiate the conservation relation under various conditions.

The Role of Pressure Fluctuations in Rapidly Rotating Flow

The behavior of the pressure fluctuations in an incompressible homogeneous turbulence subjected to rapid rotation is investigated. Linearized equations based on Rapid Distortion Theory are used to analyze the turbulent motion in this study. We found a new expression, by which the pressure fluctuation is the dot product of the mean vorticity vector and the vector stream function of turbulence. It decomposes the pressure gradient of the linearized equations into a non-local vorticity-affected part and a local effect part. The former is orientated parallel to the mean vorticity direction, while the latter is similar to the Coriolis force and perpendicular to the mean vorticity direction. For the rapidly rotating frame with no mean velocity gradient, the Coriolis force can be canceled by the analogous one derived from the pressure gradient using the present new expression. As a result, the motion of turbulence is governed only by the non-local effects along the axis of rotation.

Numerical solution of three-dimensional stream function vector components of vorticity transport equations

The advantage of solving for the new three-dimensional stream function vector components is explored for the three-dimensional flow. To this end, a single scalar stream function for two dimensions is expanded into three stream function vector components for three dimensions. As a result, the fourth-order three component partial differential equations of vorticity transport in terms of three-dimensional stream function vector components are formulated. The generalized Galerkin finite element solutions of the governing equations for incompressible flow is then carried out, and the results are shown to be very accurate for the lid-driven three-dimensional cavity problems examined herein. This numerical accuracy is attributed to the correct definition of three-dimensional stream function vector components, appropriate finite element interpolation functions, associated physically simple boundary conditions, and the rigorous Newton—Raphson scheme of solution process, as well as the governing equations being free from pressure oscillations. To the best of our knowledge, the present study marks the first attempt to solve the three-dimensional fourth-order partial differential equations of vorticity transport for the three-dimensional stream function vector components. This will make it possible to obtain the three-dimensional version of the Orr—Sommerfeld type eigenvalue solutions for hydrodynamic instability in the future.

An alternative derivation of Howe’s (1975) formula for the scattering of hydrodynamic flow into sound by solid bodies

An alternative way to derive the Green’s function in the presence of a solid body is presented and applied to the case of scattering of hydrodynamic flow into sound. This method utilizes an imaging system of a point source outside a solid sphere. The ‘‘incident potential’’ and the ‘‘scattered potential’’ are obtained in terms of velocity potentials from the original source and the image sources. The total potential and the Green’s function can then be easily derived. The resultant Green’s function G=δ(t−τ−|x−Y|/c)/(4π|x−Y|) is exactly the same as that derived by Howe [J. Fluid Mech. 67, 597–610 (1975)] while the physical meaning is much more clear and less mathematics is involved. Here, x is the vector from the origin to the observation point, and Y is the incompressible flow vector potential in the presence of the solid body. For the case of scattering of hydrodynamic flow into sound, the above Green’s function is substituted in Lighthill’s solution p(x,t)=ρ0∫(∂2uiuj/∂yi∂yj)GdVd τ. The far-field scattered acoustic pressure is then obtained in terms of flow vorticity ζ and the vector stream function Ψx for a unit potential flow around the solid body in the x direction: p(x,t)=(ρ0/4πxc)∂2/∂t2∫Ψx⋅ζ(t−|x| /c)dV, as given by Dowling.

NUMERICAL SOLUTION OF VORTICITY TRANSPORT EQUATIONS IN TERMS OF THREEDIMENSIONAL STREAM FUNCTION VECtOR COMPONENTS

The fourth order three component partial differential equations of vorticity transport in terms of three-dimensional stream function vector components are formulated. The generalized Galerkin finite element solutions of the governing equations for incompressible flow is then carried out, and the results are shown to be very accurate for the lid-driven three-dimensional cavity problems examined herein. This numerical accuracy is attributed to the correct definition of three-dimensional stream function vector components, appropriate finite element interpolation functions, associated physically simple boundary conditions, and the rigorous Newton-Raphson scheme of solution process, as well as the governing equations being free from pressure oscillations. To the best of our knowledge, the present study marks the first attempt to solve the three-dimensional fourth order partial differential equations of voilicity transport for the three-dimensional stream function vector components.

A Technique for Diagnosing Three-Dimensional Ageostrophic Circulations in Baroclinic Disturbances on Limited-Area Domains

Abstract The methodology developed by Keyser et al. for representing and diagnosing three-dimensional vertical circulations in baroclinic disturbances using a two-dimensional vector streamfunction, referred to as the psi vector, is restricted to f-plane channel-model geometry. The vertical circulation described by the psi vector consists of the irrotational (or divergent) part of the ageostrophic wind and the vertical velocity. A key property of the psi vector is that its projections onto arbitrarily oriented orthogonal vertical planes yield independent vertical circulations, allowing separation of a three-dimensional vertical circulation into two two-dimensional components, and thus objective assessment of the extent to which a three-dimensional vertical circulation is oriented in a preferred direction. Here the methodology for determining the psi vector is modified to be suitable for real-data applications. The modifications consists of reformulating the diagnostic equations to apply to conformal map pr...

Relationship between tropical heating and global circulation: Interannual variability

This study examines the relationship between tropical heating and global circulation in the troposphere through singular value decomposition (SVD) analysis and numerical simulations based on a global barotropic spectral model. SVD analysis was applied to the 200‐mbar stream function field and the outgoing longwave radiation (OLR) to extract the most recurrent coupled mode. The stream function vector of the first mode indicates an out‐of‐phase relationship between the stream function in the northern and southern hemispheres. Tight gradients in the pattern along the equator suggest a large fluctuation of zonal‐mean zonal wind associated with this seesaw. Its eddy component appears to modulate the stationary eddies. The major structure of the corresponding OLR vector is a dipole near the equator with one center at 120°E and another one at 160°W. The first mode exhibits a strong interannual variability and is strongly correlated with the occurrence of El Niño and La Niña. To understand the relationship between the stream function and OLR, a series of numerical experiments using a barotropic model was carried out, specifying Rossby wave source associated with differing idealized divergence patterns in the tropics. The steady state responses in all experiments exhibit a zonally symmetric structure resembling the stream function vector of the first mode. The pattern was mainly forced by the zonal‐mean component of Rossby wave source that is contributed mostly by zonal‐mean divergent winds under the constraint of zero global‐mean divergence. The same zonal‐mean forcing is also responsible for the forced eddy stream function perturbations that exhibit similar structures in all experiments.

Vortex drift. II: The flow potential surrounding a drifting vortical region

The potential flowfield surrounding a vortical flow domain is investigated. The vortical region is restricted to a three-dimensional finite domain in an unbounded incompressible viscous fluid that is at rest at infinity. The vector streamfunction of the flow is obtained by the integrals over the given vorticity field, expressing the Biot–Savart law. In the approach proposed here, the asymptotic vector streamfunction in the far field is expressed by integrals that represent the moments of the vorticity vector field. The asymptotic far-field potential is then determined by an integration that is analogous to the procedure well known in two dimensions, where the conjugate potential is found in the case that the streamfunction is known. In three dimensions, no general formula was found but an explicit expression for the asymptotic potential was constructed for the far-field dipole and quadrupole. The integrability conditions that permit the determination of the potential from the vector streamfunction are interpreted as the consequence of the requirement that the underlying vorticity field has to be source free. The asymptotic potentials are discussed first using only the inherent symmetries of the general dipole and the quadrupole fields. The results are then specialized to the case of axial symmetry. It will be shown that (i) the dipole strength is invariant and that (ii) the strength of the flow quadrupole remains constant while it moves together with the dipole with the drift speed already postulated in Part I. The role of the pressure in the flow generating process is also briefly noted.

Three-dimensional baroclinic instability and summertime frontogenesis in the Australian region

The structure of a mature three-dimensional baroclinic wave and its attendant cold front is examined using an idealized numerical model with a simple parametrization of dry buoyant convection resulting from diurnal heating. Relative-flow isentropic analyses of the numerical solution and of an observed Australian summertime frontal system are compared. It is shown that the model well represents the typical synoptic environment accompanying frontogenesis in the Australian region. A diagnosis of the three-dimensional vertical circulation is obtained using a technique involving a vector streamfunction. The circulation in a cross-sectional plane parallel with the low-level temperature gradient is found to be highly two-dimensional, with geostrophic confluence constituting the principal frontogenetical forcing at low levels, while at upper levels the frontogentical forcing is dominated by geostrophic horizontal shear. In a second cross-section, taken zonally, the frontogenetic forcing is found to consist almost solely of geostrophic horizontal shear throughout the entire troposphere. However, in this case the circulation in the plane of cross-section is found to be only qualitatively two-dimensional, as a significant proportion of the low-level postfrontal subsidence is associated with circulation normal to the cross-section. It is shown that the frontal structure analysed is sensitive to the location of the chosen cross-sectional plane, and that this in turn provides an explanation for the observed relative shallowness of Australian summertime fronts. The effects of diurnal heating are important but confined to the surface-based mixed layer. With the addition of diurnal heating, the prefrontal updraught amplifies almost threefold and the frontal temperature contrast over the heated region approximately doubles. The diabatic forcing term in the well-known Sawyer-Eliassen equation is shown to be negligible. This suggests that the increased low-level convergence is a reflection of the model's enhanced response to frontogenetic forcing in the presence of low static stability in the surface-based well-mixed layer. Finally, the numerical model solution is compared with a simpler two-dimensional model specifically developed to study Australian summertime fronts.

Three-dimensional baroclinic instability and summertime frontogenesis in the Australian region

The structure of a mature three-dimensional baroclinic wave and its attendant cold front is examined using an idealized numerical model with a simple parametrization of dry buoyant convection resulting from diurnal heating. Relative-flow isentropic analyses of the numerical solution and of an observed Australian summertime frontal system are compared. It is shown that the model well represents the typical synoptic environment accompanying frontogenesis in the Australian region. A diagnosis of the three-dimensional vertical circulation is obtained using a technique involving a vector streamfunction. The circulation in a cross-sectional plane parallel with the low-level temperature gradient is found to be highly two-dimensional, with geostrophic confluence constituting the principal frontogenetical forcing at low levels, while at upper levels the frontogentical forcing is dominated by geostrophic horizontal shear. In a second cross-section, taken zonally, the frontogenetic forcing is found to consist almost solely of geostrophic horizontal shear throughout the entire troposphere. However, in this case the circulation in the plane of cross-section is found to be only qualitatively two-dimensional, as a significant proportion of the low-level postfrontal subsidence is associated with circulation normal to the cross-section. It is shown that the frontal structure analysed is sensitive to the location of the chosen cross-sectional plane, and that this in turn provides an explanation for the observed relative shallowness of Australian summertime fronts. The effects of diurnal heating are important but confined to the surface-based mixed layer. With the addition of diurnal heating, the prefrontal updraught amplifies almost threefold and the frontal temperature contrast over the heated region approximately doubles. The diabatic forcing term in the well-known Sawyer-Eliassen equation is shown to be negligible. This suggests that the increased low-level convergence is a reflection of the model's enhanced response to frontogenetic forcing in the presence of low static stability in the surface-based well-mixed layer. Finally, the numerical model solution is compared with a simpler two-dimensional model specifically developed to study Australian summertime fronts.

A Technique for Representing Three-Dimensional Vertical Circulations in Baroclinic Disturbances

Abstract The problem of representing vertical circulations in frontal zones is reexamined with the objective of devising a methodology sufficiently general to apply in situations where these circulations are no longer confined to the cross-front (transverse) vertical plane and therefore must be viewed as fully dimensional. The proposed methodology, which builds upon the earlier work of Hoskins and Draghici and of Eliassen, consists of adopting a vector streamfunction that describes the vertical velocity and the horizontal irrotational flow. This generalized streamfunction, referred to as the psi vector, may be determined uniquely from the vertical velocity field over a limited region provided that suitable lateral boundary conditions on the velocity potential for the irrotational part of the horizontal velocity can be specified. A key property of the psi vector is that its projections onto arbitrary orthogonal vertical planes yield two independent vertical circulations, providing an objective means for se...