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Journal articles on the topic 'Stream function-vorticity'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

Sousa, E., and I. J. Sobey. "Effect of boundary vorticity discretization on explicit stream-function vorticity calculations." International Journal for Numerical Methods in Fluids 49, no. 4 (2005): 371–93. http://dx.doi.org/10.1002/fld.1001.

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2

Papa, Touty Traore, Dieng Moussa, Faye Seydou, Ndiaye Mor, and Diagne Issa. "Numerical Study of Navier-Stokes Equations in Vorticity/Stream Function Formulation on the Torus in Two-Dimensional Flow under MATLAB." Journal of Scientific and Engineering Research 10, no. 5 (2023): 236–42. https://doi.org/10.5281/zenodo.10458711.

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<strong>Abstract </strong>The Navier-Stokes equations can be rearranged in 2D incompressible flows in terms of stream function and vorticity. In many areas of applications, the stream function and vorticity form of the Navier Stokes equations shows important effect into the mechanisms leading the flow than the first variables formulation. It’s also useful for numerical study it avoids some problems resulting from the discretization. In this work a numerical approach with the stream function and vorticity is used with a discretization of the governing equations and the boundary conditions in the domain included in a square of unit length where the upper boundary (the lid) at y = 1, moves with a constant velocity U = 1. The Reynolds number based on the size of the domain, the velocity of the moving wall, density ρ = 1 and viscosity µ = 0.001 is Re = 500. A code is developed under MATLAB software. The evolution of the vorticity field is determined in term of time. To do this, a differential finite method is used.

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3

Sommeria, J., C. Staquet, and R. Robert. "Final equilibrium state of a two-dimensional shear layer." Journal of Fluid Mechanics 233 (December 1991): 661–89. http://dx.doi.org/10.1017/s0022112091000642.

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We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.

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4

Waugh, D. W., and D. G. Dritschel. "The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models." Journal of Fluid Mechanics 231 (October 1991): 575–98. http://dx.doi.org/10.1017/s002211209100352x.

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The linear stability of filaments or strips of ‘potential’ vorticity in a background shear flow is investigated for a class of two-dimensional, inviscid, non-divergent models having a linear inversion relation between stream function and potential vorticity. In general, the potential vorticity is not simply the Laplacian of the stream function – the case which has received the greatest attention historically. More general inversion relationships between stream function and potential vorticity are geophysically motivated and give an impression of how certain classic results, such as the stability of strips of vorticity, hold under more general circumstances.In all models, a strip of potential vorticity is unstable in the absence of a background shear flow. Imposing a shear flow that reverses the total shear across the strip, however, brings about stability, independent of the Green-function inversion operator that links the stream function to the potential vorticity. But, if the Green-function inversion operator has a sufficiently short interaction range, the strip can also be stabilized by shear having the same sense as the shear of the strip. Such stabilization by ‘co-operative’ shear does not occur when the inversion operator is the inverse Laplacian. Nonlinear calculations presented show that there is only slight disruption to the strip for substantially less adverse shear than necessary for linear stability, while for co-operative shear, there is major disruption to the strip. It is significant that the potential vorticity of the imposed flow necessary to create shear of a given value increases dramatically as the interaction range of the inversion operator decreases, making shear stabilization increasingly less likely. This implies an increased propensity for filaments to ‘roll-up’ into small vortices as the interaction range decreases, a finding consistent with many numerical calculations performed using the quasi-geostrophic model.

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5

Dubois, François, Michel Salaün, and Stéphanie Salmon. "Vorticity–velocity-pressure and stream function-vorticity formulations for the Stokes problem." Journal de Mathématiques Pures et Appliquées 82, no. 11 (2003): 1395–451. http://dx.doi.org/10.1016/j.matpur.2003.09.002.

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6

Dubois, François, Michel Salaün, and Stéphanie Salmon. "Discrete harmonics for stream function-vorticity Stokes problem." Numerische Mathematik 92, no. 4 (2002): 711–42. http://dx.doi.org/10.1007/s002110100369.

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7

Sousa, E., and I. J. Sobey. "Numerical stability of unsteady stream-function vorticity calculations." Communications in Numerical Methods in Engineering 19, no. 6 (2003): 407–19. http://dx.doi.org/10.1002/cnm.600.

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8

Pokhrel, Puskar R., and Shiva P. Pudasaini. "Stream function - vorticity formulation of mixture mass flow." International Journal of Non-Linear Mechanics 121 (May 2020): 103317. http://dx.doi.org/10.1016/j.ijnonlinmec.2019.103317.

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9

GUERMOND, J. L., та L. QUARTAPELLE. "WEAK APPROXIMATION OF THE ψ–ω EQUATIONS WITH EXPLICIT VISCOUS DIFFUSION". Mathematical Models and Methods in Applied Sciences 10, № 01 (2000): 85–98. http://dx.doi.org/10.1142/s0218202500000070.

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This paper describes a variational formulation for solving the 2-D time-dependent incompressible Navier–Stokes equations expressed in the stream function and vorticity. The difference between the proposed approach and the standard one is that the vorticity equation is interpreted as an evolution equation for the stream function while the Poisson equation is used as an expression for evaluating the distribution of vorticity in the domain and on the boundary. A time discretization is adopted with the viscous diffusion made explicit, which leads to split the incompressibility from the viscosity. In some sense, the present method generalizes to the variational framework a well-known idea which is used in finite differences approximations and that is based on a Taylor series expansion of the stream function near the boundary. Some conditional stability results and error estimates are given.

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10

Mackinnon, R. J., G. F. Carey, and P. Murray. "A procedure for calcualting vorticity boundary conditions in the stream-function-vorticity method." Communications in Applied Numerical Methods 6, no. 1 (1990): 47–48. http://dx.doi.org/10.1002/cnm.1630060107.

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11

Borysyuk, A. O. "A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach." Reports of the National Academy of Sciences of Ukraine, no. 4 (August 27, 2022): 55–65. http://dx.doi.org/10.15407/dopovidi2022.04.055.

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A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

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12

Chudanov, V. V., A. G. Popkov, A. G. Churbanov, P. N. Vabishchevich, and M. M. Makarov. "Operator-splitting schemes for the stream function-vorticity formulation." Computers & Fluids 24, no. 7 (1995): 771–86. http://dx.doi.org/10.1016/0045-7930(95)00015-5.

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13

Karaa, Samir. "High-order ADI method for stream-function vorticity equations." PAMM 7, no. 1 (2007): 1025601–2. http://dx.doi.org/10.1002/pamm.200700414.

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14

He, Dongdong, and Kejia Pan. "A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries." East Asian Journal on Applied Mathematics 7, no. 4 (2017): 714–27. http://dx.doi.org/10.4208/eajam.200816.300517a.

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AbstractIncompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

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15

Khan, Waqar, and Faisal Yousafzai. "On the Exact Solutions of Couple Stress Fluids." Advanced Trends in Mathematics 1 (December 2014): 27–32. http://dx.doi.org/10.18052/www.scipress.com/atmath.1.27.

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Exact solutions of the momentum equations of couple stress fluid are investigated. Making use of stream function, the two-dimensional flow equations are transformed into non-linear compatibility equation, and then it is linearized by vorticity function. Stream functions and velocity distributions are discussed for various flow situations.

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16

Rylov, Yuri A. "Hydrodynamic equations for incompressible inviscid fluid in terms of generalized stream function." International Journal of Mathematics and Mathematical Sciences 2004, no. 11 (2004): 541–70. http://dx.doi.org/10.1155/s0161171204308021.

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Hydrodynamic equations for ideal incompressible fluid are written in terms of generalized stream function. Two-dimensional version of these equations is transformed to the form of one dynamic equation for the stream function. This equation contains arbitrary function which is determined by inflow conditions given on the boundary. To determine unique solution, velocity and vorticity (but not only velocity itself) must be given on the boundary. This unexpected circumstance may be interpreted in the sense that the fluid has more degrees of freedom than it was believed. Besides, the vorticity is a less observable quantity as compared with the velocity. It is shown that the Clebsch potentials are used essentially at the description of vortical flow.

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17

Barragy, E., and G. F. Carey. "Stream function-vorticity driven cavity solution using p finite elements." Computers & Fluids 26, no. 5 (1997): 453–68. http://dx.doi.org/10.1016/s0045-7930(97)00004-2.

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18

Auteri, F., and L. Quartapelle. "Galerkin Spectral Method for the Vorticity and Stream Function Equations." Journal of Computational Physics 149, no. 2 (1999): 306–32. http://dx.doi.org/10.1006/jcph.1998.6155.

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19

Fan, En Gui, and Man Wai Yuen. "Similarity reductions and exact solutions for two-dimensional Euler–Boussinesq equations." Modern Physics Letters B 33, no. 27 (2019): 1950328. http://dx.doi.org/10.1142/s0217984919503287.

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In this paper, by introducing a stream function and new coordinates, we transform classical Euler–Boussinesq equations into a vorticity form. We further construct traveling wave solutions and similarity reduction for the vorticity form of Euler–Boussinesq equations. In fact, our similarity reduction provides a kind of linearization transformation of Euler–Boussinesq equations.

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20

G, Krishna Veni, Satyanarayana Chirala, and Chenna Krishnareddy M. "Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations." Indian Journal of Science and Technology 16, no. 27 (2023): 2090–101. https://doi.org/10.17485/IJST/v16i27.336.

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Abstract <strong>Objective:</strong> The present method aims to solve and investigate the efficiency, accuracy, and stability of the 2D unsteady Navier-Stokes equation in stream function vorticity formulation and Taylor’s vortex problem. <strong>Method:</strong> RBF partition of unity method (RBF-PUM) was implemented to solve the twodimensional Navier- Stokes equations in stream function vorticity formulation and Taylor’s vortex problem. <strong>Findings:</strong> RBF-PUM results show good agreement with the exact solutions. The numerical approach is found to be efficient and accurate while maintaining stability even for a Reynolds number as high as 1000. The global RBF method’s high computational cost can be overcome by using the RBF-PUM. <strong>Novelty and applications:</strong> The RBF-PU methodology is extended to solve the two-dimensional Navier- Stokes equations in stream function vorticity formulation and Taylor’s vortex problem, which were not discussed earlier in the literature. The adaptive spatial refinement within the partitions may be performed independently using the RBF-PUM. Then it may be extended to the more complex problems in CFD. <strong>Keywords:</strong> Mesh Free Methods; RBF- PUM; Navier- Stokes Equations; Taylor’s Vortex Problem; CFD

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21

You, Xiaoguang, Aibin Zang та Yin Li. "Global well-posedness of 2D Euler-α equation in exterior domain". Nonlinearity 35, № 11 (2022): 5852–79. http://dx.doi.org/10.1088/1361-6544/ac9508.

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Abstract After casting Euler-α equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-α equations in 2D exterior domain provided that the initial data is regular enough.

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22

Brown, G. L., and J. M. Lopez. "Axisymmetric vortex breakdown Part 2. Physical mechanisms." Journal of Fluid Mechanics 221 (December 1990): 553–76. http://dx.doi.org/10.1017/s0022112090003676.

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The physical mechanisms for vortex breakdown which, it is proposed here, rely on the production of a negative azimuthal component of vorticity, are elucidated with the aid of a simple, steady, inviscid, axisymmetric equation of motion. Most studies of vortex breakdown use as a starting point an equation for the azimuthal vorticity (Squire 1960), but a departure in the present study is that it is explored directly and not through perturbations of an initial stream function. The inviscid equation of motion that is derived leads to a criterion for vortex breakdown based on the generation of negative azimuthal vorticity on some stream surfaces. Inviscid predictions are tested against results from numerical calculations of the Navier-Stokes equations for which breakdown occurs.

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23

CHAVANIS, P. H., and J. SOMMERIA. "Classification of robust isolated vortices in two-dimensional hydrodynamics." Journal of Fluid Mechanics 356 (February 10, 1998): 259–96. http://dx.doi.org/10.1017/s0022112097007933.

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We determine solutions of the Euler equation representing isolated vortices (monopoles, dipoles) in an infinite domain, for arbitrary values of energy, circulation, angular momentum and impulse. A linear relationship between vorticity and stream function is assumed inside the vortex (while the flow is irrotational outside). The emergence of these solutions in a turbulent flow is justified by the statistical mechanics of continuous vorticity fields. The additional restriction of mixing to a ‘maximum-entropy bubble’, due to kinetic constraints, is assumed. The linear relationship between vorticity and stream function is obtained from the statistical theory in the limit of strong mixing (when constraints are weak). In this limit, maximizing entropy becomes equivalent to a kind of enstrophy minimization. New stability criteria are investigated and imply in particular that, in most cases, the vorticity must be continuous (or slightly discontinuous) at the vortex boundary. Then, the vortex radius is automatically determined by the integral constraints and we can obtain a classification of isolated vortices such as monopoles and dipoles (rotating or translating) in terms of a single control parameter. This article generalizes the classification obtained in a bounded domain by Chavanis & Sommeria (1996).

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24

Alassar, R. S., and H. M. Badr. "OSCILLATING VISCOUS FLOW OVER PROLATE SPHEROIDS." Transactions of the Canadian Society for Mechanical Engineering 23, no. 1A (1999): 83–93. http://dx.doi.org/10.1139/tcsme-1999-0006.

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The axisymmetric viscous oscillating flow over a prolate spheroid is considered. The oscillations are harmonic and the free stream is always parallel to the spheroid major axis. The flow is governed by the Strouhal and the Reynolds numbers as well as the spheroid axis ratio. In the present paper, we only investigate the effect of Reynolds number while keeping the Strouhal number and the axis ratio unchanged. The results are presented in terms of the periodic variation of the drag coefficient, pressure, surface vorticity, separation angle, the wake length, and the streamline and vorticity patterns for Reynolds numbers ranging from 5 to 100. Upon averaging the stream function and vorticity over one complete oscillation, the double boundary-layer structure observed in the case of a sphere is confirmed for the range of parameters considered.

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25

Mestel, A. J. "An iterative method for high-Reynolds-number flows with closed streamlines." Journal of Fluid Mechanics 200 (March 1989): 1–18. http://dx.doi.org/10.1017/s0022112089000534.

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In steady, two-dimensional, inviscid flows it is well-known that, in the absence of rotational forcing, the vorticity is constant along streamlines. In a bounded domain the streamlines are necessarily closed. In some circumstances, investigated in this paper, this behaviour is exhbited also by forced viscous flows, when the variation of vorticity across the streamlines is determined by a balance between viscous diffusion and the forcing. Similar results hold in axisymmetry. For such flows, an iterative process for finding the vorticity as a function of the stream function is described. The method applies whenever the viscous boundary condition can be expressed in terms of the vorticity or tangential stress rather then the tangential velocity. When it is applicable, the iterative method is faster than direct solution of the Navier-Stokes equations at high Reynolds numbers. As an example, the method is used to calculate the flow in a model of the electromagnetic stirring process. In this model, a conducting fluid in an elliptical region is driven by a rotating magnetic field and resisted by a surface stress. The functional dependence of the vorticity on the stream function is found for various values of the magnetic skin depth, surface stress and eccentricity of the ellipse. The form of the flow is discussed with particular reference to whether it consists of a single circulatory region or separates into two or more such regions.

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26

Davis, R. L., J. E. Carter, and M. Hafez. "Three-dimensional viscous flow solutions with a vorticity-stream function formulation." AIAA Journal 27, no. 7 (1989): 892–900. http://dx.doi.org/10.2514/3.10197.

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27

Leriche, E., and G. Labrosse. "Stokes eigenmodes in square domain and the stream function–vorticity correlation." Journal of Computational Physics 200, no. 2 (2004): 489–511. http://dx.doi.org/10.1016/j.jcp.2004.03.017.

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28

Zhou, Aihui, and Jichun Li. "The full approximation accuracy for the stream function-vorticity-pressure method." Numerische Mathematik 68, no. 3 (1994): 427–35. http://dx.doi.org/10.1007/s002110050070.

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29

Barragy, E., and G. F. Carey. "Stream function vorticity solution using high-p element-by-element techniques." Communications in Numerical Methods in Engineering 9, no. 5 (1993): 387–95. http://dx.doi.org/10.1002/cnm.1640090504.

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30

Spotz, W. F., and G. F. Carey. "High-order compact scheme for the steady stream-function vorticity equations." International Journal for Numerical Methods in Engineering 38, no. 20 (1995): 3497–512. http://dx.doi.org/10.1002/nme.1620382008.

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31

Arul Prakash, M., K. Mayilsamy, and P. Rajesh Kanna. "Numerical Simulation of Two Dimensional Laminar Wall Jet Flow over Solid Obstacle." Applied Mechanics and Materials 592-594 (July 2014): 1935–39. http://dx.doi.org/10.4028/www.scientific.net/amm.592-594.1935.

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A Computational Fluid Dynamics code was developed to study the flow characteristics of two dimensional laminar incompressible flow. Stream function-vorticity formulation was used for solving two dimensional continuity and momentum equations. The unsteady vorticity transport equation is solved by alternate direction implicit scheme. The stream function equation is solved by the successive over relaxation method. A computational code in c-language was developed to solve the tridiagonal system of algebraic equations. Two dimensional flow through a channel with rectangular block at the bottom wall was considered for the validation. The streamline patterns obtained for different Reynolds number shows good agreement with published results. The code was modified to simulate an incompressible laminar wall jet flow around a solid obstacle. Simulations were carried out for different Reynolds numbers. Contour plots of Stream line, u-velocity and v-velocity were obtained. The variations of flow patterns and the development of vortices were studied and reported.

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32

Berger, B. S. "Asymptotic Approximation and Perturbation Stream Functions in Viscous Flow Calculations." Journal of Applied Mechanics 52, no. 1 (1985): 190–92. http://dx.doi.org/10.1115/1.3168993.

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An asymptotic approximation technique is shown to substantially decrease the number of interations necessary for the accurate computation of steady solutions for viscous incompressible flows. A numerical study of flows exterior to a circular cylinder indicates that the technique reduces the required number of iterations by a factor of 10. The flux formulation of the vorticity stream function solution for the flow of a viscous incompressible fluid external to a circular cylinder requires the solution of the Poisson equation for the stream function. The use of the FACR (1), fast direct Poisson equation solver leads to serious errors in the computation of the stream function near the cylinder. However it is shown that these errors are eliminated through the introduction of a perturbation stream function.

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33

Ralph, M. E., and T. J. Pedley. "Flow in a channel with a moving indentation." Journal of Fluid Mechanics 190 (May 1988): 87–112. http://dx.doi.org/10.1017/s0022112088001223.

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The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

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34

Jones, E. H., and R. A. Bajura. "A Numerical Analysis of Pulsating Laminar Flow Through a Pipe Orifice." Journal of Fluids Engineering 113, no. 2 (1991): 199–205. http://dx.doi.org/10.1115/1.2909480.

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A study of laminar pulsating flow through a 45 degree bevel pipe orifice was performed using finite-difference approximations to the governing stream function and vorticity transport equations. The distance from (−∞) to (+∞) was transformed into the region from (−1) to (+1) for the streamwise coordinate. Solutions were obtained for orifice bore/pipe diameter ratios of 0.2 and 0.5 for bore Reynolds numbers in the range from 0.8 to 64 and Strouhal numbers from 10−5 to 102. Stream and vorticity function plots were generated for all cases and the time-dependent discharge coefficient waws computed and averaged over a cycle of pulsation. The numerical solutions agree closely with available experimental data for steady flow discharge coefficients. The results show that the effects of pulsation on the discharge coefficient can be correlated by using the product of the orifice bore Reynolds and Strouhal numbers.

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35

Swaters, Gordon E. "A perturbation theory for the solitary-drift-vortex solutions of the Hasegawa-Mima equation." Journal of Plasma Physics 41, no. 3 (1989): 523–39. http://dx.doi.org/10.1017/s0022377800014069.

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A multiple-scales adiabatic perturbation theory is presented describing the adiabatic dissipation of the solitary vortex-pair solutions of the Hasegawa-Mima equation. The vortex parameter transport equations are derived as solvability conditions for the asymptotic expansion and are identical with the transport equations previously derived by Aburdzhaniya et al. (1987) using an energy- and enstrophy-conservation balance procedure. The theoretical results are compared with high-resolution numerical simulations. Global properties such as the decay in the enstrophy and energy are accurately reproduced. Local properties such as the position of the centre of the vortex pair, decay of the extrema in the vorticity and stream-function fields, and the dilation of the vortex dipole are also in good agreement. In addition, time series of vorticity–stream-function scatter diagrams for the numerical simulations are presented to verify the adiabatic ansatz.

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36

Paolino, M. A., R. B. Kinney, and E. A. Cerutti. "Numerical Analysis of the Unsteady Flow and Heat Transfer to a Cylinder in Crossflow." Journal of Heat Transfer 108, no. 4 (1986): 742–48. http://dx.doi.org/10.1115/1.3247007.

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The unsteady, two-dimensional viscous flow of an incompressible, constant-property fluid flowing over a cylinder is numerically analyzed by integrating the vorticity transport equation and the energy equation. Departing from the usual stream function approach, the velocity distribution is obtained from the vorticity distribution by integrating the velocity induction law. Calculations start with the impulsive motion of the free stream and a step change in the surface temperature of the cylinder. The solution is advanced in time until steady-state conditions are achieved. Results are obtained for a Prandtl number of 0.7 and Reynolds numbers of 3000 and 70,800. Local and average Nusselt numbers and force coefficients are presented and compared to available experimental data.

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37

Xu, J. Z., W. Y. Ni, and J. Y. Du. "Numerical Solution of Stream Function Equations in Transonic Flows." Journal of Turbomachinery 109, no. 4 (1987): 508–12. http://dx.doi.org/10.1115/1.3262140.

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In order to develop the transonic stream function approach, in this paper one of the momentum equations is employed to form the principal equation of the stream function which does not contain vorticity and entropy terms, and the other one is used to calculate the density directly. Since the density is uniquely determined, the problem that the density is a double-valued function of mass flux in the stream function formulation disappears and the entropy increase across the shock is naturally included. The numerical results for the transonic cascade flow show that the shock obtained from the present method is slightly weaker and is placed farther downstream compared to the irrotational stream function calculation, and is closer to the experimental data. From a standpoint of computation the iterative procedure of this formulation is simple and the alternating use of two momentum equations makes the calculation more effective.

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38

BASSOM, ANDREW P., and ANDREW D. GILBERT. "The spiral wind-up of vorticity in an inviscid planar vortex." Journal of Fluid Mechanics 371 (September 25, 1998): 109–40. http://dx.doi.org/10.1017/s0022112098001955.

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The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as ‘axisymmetrization’, is studied asymptotically and numerically. The vortex is perturbed at t=0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t−λ with λ=1+(n2+8)1/2, where n is the azimuthal wavenumber of the perturbation and n[ges ]1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

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39

Guermond, J. L., та L. Quartapelle. "Uncoupled ω–ψ Formulation for Plane Flows in Multiply Connected Domains". Mathematical Models and Methods in Applied Sciences 07, № 06 (1997): 731–67. http://dx.doi.org/10.1142/s0218202597000396.

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

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40

Shin, Myung Seob. "Grid Refinement Model in Lattice Boltzmann Method for Stream Function-Vorticity Formulations." Transactions of the Korean Society of Mechanical Engineers B 39, no. 5 (2015): 415–23. http://dx.doi.org/10.3795/ksme-b.2015.39.5.415.

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41

Bagai, Shobha, Manoj Kumar, and Arvind Patel. "The four-sided lid driven square cavity using stream function-vorticity formulation." Journal of Applied Mathematics and Computational Mechanics 19, no. 2 (2020): 17–30. http://dx.doi.org/10.17512/jamcm.2020.2.02.

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42

Abdellatif, Nehla. "A mixed stream-function and vorticity formulation for axisymmetric Navier–Stokes equations." Journal of Computational and Applied Mathematics 117, no. 1 (2000): 61–83. http://dx.doi.org/10.1016/s0377-0427(99)00333-7.

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43

Tezduyar, T. E., and J. Liou. "Computation of spatially periodic flows based on the vorticity-stream function formulation." Computer Methods in Applied Mechanics and Engineering 83, no. 2 (1990): 121–42. http://dx.doi.org/10.1016/0045-7825(90)90147-e.

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44

Zuykov, A. L. "Azimuthal vorticity and stream function in the creeping flow in a pipe." Vestnik MGSU, no. 4 (April 2014): 150–59. http://dx.doi.org/10.22227/1997-0935.2014.4.150-159.

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45

El-Beltagy, Mohamed A., and Mohamed I. Wafa. "Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation." Journal of Applied Mathematics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/903618.

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A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver. The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte-Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.

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46

Peeters, M. F., W. G. Habashi, and E. G. Dueck. "Finite element stream function-vorticity solutions of the incompressible Navier-Stokes equations." International Journal for Numerical Methods in Fluids 7, no. 1 (1987): 17–27. http://dx.doi.org/10.1002/fld.1650070103.

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47

Tzirtzilakis, E. E. "A simple numerical methodology for BFD problems using stream function vorticity formulation." Communications in Numerical Methods in Engineering 24, no. 8 (2007): 683–700. http://dx.doi.org/10.1002/cnm.981.

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48

Ren, W. W., J. Wu, C. Shu, and W. M. Yang. "A stream function-vorticity formulation-based immersed boundary method and its applications." International Journal for Numerical Methods in Fluids 70, no. 5 (2011): 627–45. http://dx.doi.org/10.1002/fld.2705.

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49

Ambethkar, V. "Numerical solutions of magneto-hydrodynamic flow past a sphere at high Reynolds numbers." Canadian Journal of Physics 86, no. 12 (2008): 1443–47. http://dx.doi.org/10.1139/p08-099.

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Numerical solutions of a steady, incompressible magneto-hydrodynamic flow past a sphere at high Reynolds numbers are presented by using finite differences in spherical polar coordinates with an applied magnetic field parallel to the main flow. Nonlinear coupled governing equations are solved numerically using finite-difference techniques. The results are presented up to Reynolds’ number R ≤ 10 000 and interaction parameter N = 25. Stability and convergence of the finite-difference technique has been discussed. Contours of stream lines and the vorticity are represented graphically up to high Reynolds number R ≤ 10 000 and N = 25. The values of the minimum stream function and vorticity at the primary vortex for different Reynolds numbers are tabulated and discussed.PACS Nos.: 47.11.Bc, 47.27.Jv, 47.63.mc, 52.65.Kj

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50

Burton, G. R. "Steady symmetric vortex pairs and rearrangements." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, no. 3-4 (1988): 269–90. http://dx.doi.org/10.1017/s0308210500014669.

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SynopsisWe prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equationin a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.

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