An unsteady two-cell vortex solution of the Navier—Stokes equations

1970 ◽  
Vol 41 (3) ◽  
pp. 673-687 ◽  
Author(s):  
P. G. Bellamy-Knights
Keyword(s):  
Radial Flow ◽  
Stokes Equations ◽  
Circumferential Velocity ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Radial Flux ◽  
Vortex Solution ◽  
Flow Systems ◽  
Vortex Solutions

The steady two-cell viscous vortex solution of Sullivan (1959) is extended to yield unsteady two-cell viscous vortex solutions which behave asymptotically as certain analogous unsteady one-cell solutions of Rott (1958). The radial flux is a parameter of the solution, and the effect of the radial flow on the circumferential velocity, is analyzed. The work suggests an explanation for the eventual dissipation of meteorological flow systems such as tornadoes.

Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

2017 ◽  
Vol 818 ◽  
pp. 241-259 ◽  
Author(s):  
F. A. T. Boshier ◽  
A. J. Mestel
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Vortex Solution ◽  
Vortex Solutions ◽  
Complex Solutions ◽  
Unknown Solution

Steady incompressible flow down a slowly curving circular pipe is considered, analytically and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Padé approximants, the Dean series solution is analytically continued outside its circle of convergence, where it predicts a complex solution branch for real positive Dean number, $K$. This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all $K>0$, which are related to an unforced complex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex $K$-plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at $K=0$ and non-uniqueness for $K>0$, culminating in a real bifurcation.

scholarly journals Numerical and Analytical Study of Fluid Dynamic Forces in Seals and Bearings

1988 ◽  
Vol 110 (3) ◽  
pp. 315-325 ◽  
Author(s):  
L. T. Tam ◽  
A. J. Przekwas ◽  
A. Muszynska ◽  
R. C. Hendricks ◽  
M. J. Braun ◽  
...  
Keyword(s):  
Flow Field ◽  
Stokes Equations ◽  
Fluid Dynamic ◽  
Three Dimensional ◽  
Circumferential Velocity ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Lumped Model ◽  
Destabilizing Factors ◽  
Recirculation Zones

A numerical model based on a transformed, conservative form of the three-dimensional Navier-Stokes equations and an analytical model based on “lumped” fluid parameters are presented and compared with studies of modeled rotor/bearing/seal systems. The rotor destabilizing factors are related to the rotative character of the flow field. It is shown that these destabilizing factors can be reduced through a descrease in the fluid average circumferential velocity. However, the rotative character of the flow field is a complex three-dimensional system with bifurcated secondary flow patterns that significantly alter the fluid circumferential velocity. By transforming the Navier-Stokes equations to those for a rotating observer and using the numerical code PHOENICS-84 with a nonorthogonal body fitted grid, several numerical experiments were carried out to demonstrate the character of this complex flow field. In general, fluid injection and/or preswirl of the flow field opposing the shaft rotation significantly intensified these secondary recirculation zones and thus reduced the average circumferential velocity, while injection or preswirl in the direction of rotation significantly weakened these zones. A decrease in average circumferential velocity was related to an increase in the strength of the recirculation zones and thereby promoted stability. The influence of the axial flow was analyzed. The lumped model of fluid dynamic force based on the average circumferential velocity ratio (as opposed to the bearing/seal coefficient model) well described the obtained results for relatively large but limited ranges of parameters. This lumped model is extremely useful in rotor/bearing/seal system dynamic analysis and should be widely recommended. Fluid dynamic forces and leakage rates were calculated and compared with seal data where the working fluid was bromotrifluoromethane (CBrF3). The radial and tangential force predictions were in reasonable agreement with selected experimental data. Nonsynchronous perturbation provided meaningful information for system lumped parameter identification from numerical experiment data.

Conical turbulent swirling vortex with variable eddy viscosity

1986 ◽  
Vol 403 (1825) ◽  
pp. 235-268 ◽  
Keyword(s):  
Eddy Viscosity ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Turbulent Vortex ◽  
The Core ◽  
Vortex Solutions ◽  
Variable Eddy Viscosity

In the one hundred years since Rankine suggested his well known two-dimensional vortex model with finite core, no one has ever found any exact vortex solutions of the Navier-Stokes equations that can satisfy a complete set of physical boundary conditions. In this paper a variable viscosity is introduced and the existence of conical turbulent vortex solutions of the Navier-Stokes equations is examined. It is found that for a class of deliberately chosen eddy viscosity function a steady turbulent vortex can, for the first time, satisfy both the regularity condition at the core and the adherence condition at the surface, except for a singularity at the origin inherent in all conical similarity solutions. In its asymptotic form, if the eddy viscosity only varies in a boundary layer near the surface or the core, outside the layer the solution given would approach one of the laminar solutions of Yih et al . ( Physics Fluids 25, 2147 (1982)) or that of Serrin ( Phil. Trans. R. Soc. Lond. A 271, 325 (1972)) respectively. These results reveal some remarkable relations between the behaviour, and even the existence, of a vortex and turbulence.

scholarly journals On baer-nunziato multiphase flow models

2019 ◽  
Vol 66 ◽  
pp. 61-83
Author(s):  
M. Hillairet
Keyword(s):  
Multiphase Flow ◽  
Stokes Equations ◽  
Analytical Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow Models ◽  
Flow Systems

In these notes, we present an analytical tool for the derivation of Baer-Nunziato multiphase flow systems with one velocity. We explain the method in the case of the isentropic Navier Stokes equations. We then apply this method to models with temperature and show the main computations which are necessary to the derivation.

Quantities which define conically self-similar free-vortex solutions to the Navier–Stokes equations uniquely

2001 ◽  
Vol 438 ◽  
pp. 159-181 ◽  
Author(s):  
CARL FREDRIK STEIN
Keyword(s):  
Asymptotic Analysis ◽  
Tangential Stress ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Opening Angle ◽  
Bounding Surface ◽  
Free Vortex ◽  
Navier Stokes Equations ◽  
Vortex Solutions ◽  
Self Similar

It is proved that if, in addition to the opening angle of the bounding conical streamsurface and the circulation thereon, one of the radial velocity, the radial tangential stress or the pressure on the bounding streamsurface is given, then a conically self-similar free-vortex solution is uniquely determined in the entire conical domain. In addition, it is shown that for ows inside a cone the same conclusion holds for the Yih et al. (1982) parameter T, but for exterior flows it is shown numerically that non-uniqueness may occur. For given values of the opening angle of the bounding conical streamsurface and the circulation thereon the asymptotic analysis of Shtern & Hussain (1996) is applied to obtain asymptotic formulae which interrelate the opening angle of the cone along which the jet fans out and the radial tangential stress on the bounding surface. A striking property of these formulae is that the opening angle of the cone along which the jet fans out is independent of the value of the viscosity as long as it is small enough for the first-order asymptotic expressions to apply. However, these formulae are shown to be inaccurate for moderate values of the ratio of the circulation at the bounding surface and the viscosity. To amend this shortcoming, an alternative, more accurate, asymptotic analysis is developed to derive second-order correction terms, which considerably improve the accuracy.

A Theoretical and Experimental Study of Confined Vortex Flow

1969 ◽  
Vol 36 (4) ◽  
pp. 687-692 ◽  
Author(s):  
G. J. Farris ◽  
G. J. Kidd ◽  
D. W. Lick ◽  
R. E. Textor
Keyword(s):  
Flow Field ◽  
Radial Flow ◽  
Stokes Equations ◽  
Numerical Technique ◽  
Tangential Velocity ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
General Validity ◽  
Wide Range

The interaction of a vortex with a stationary surface was studied both theoretically and experimentally. The flow field examined was that produced by radially inward flow through a pair of concentric rotating porous cylinders that were perpendicular to, and in contact with, a stationary flat plane. The complete Navier-Stokes equations were solved over a range of tangential Reynolds numbers from 0–300 and a range of radial Reynolds numbers from 0 to −13, the minus sign indicating radially inward flow. In order to facilitate the solution, the original equations were recast in terms of a dimensionless stream function, vorticity, and third variable related to the tangential velocity. The general validity of the numerical technique was demonstrated by the agreement between the theoretical and experimental results. Examination of the numerical results over a wide range of parameters showed that the entire flow field is very sensitive to the amount of radial flow, especially at the transition from zero radial flow to some finite value.

Fluid-Inertia Effects in Radial Flow Between Oscillating, Rotating Parallel Disks

1981 ◽  
Vol 103 (1) ◽  
pp. 144-149
Author(s):  
D. K. Warinner ◽  
J. T. Pearson
Keyword(s):  
Turbulent Flows ◽  
Radial Flow ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Fluid Inertia ◽  
Rotating Disks ◽  
Spiral Flow ◽  
Navier Stokes Equations ◽  
Inertia Effects

An order-of-magnitude analysis is applied to the Navier-Stokes equations and the continuity equation for isothermal, radial fluid flow between oscillating and rotating disks. This analysis investigates the four basic cases of 1) steady, radial flow, 2) unsteady, radial flow, 3) steady, spiral flow, and 4) unsteady, spiral flow. It is shown that certain values of particular dimensionless parameters for general cases will reduce the Navier-Stokes equations to simplified forms and thus render them amenable to closed-form solutions for, say, the pressure distribution between oscillating, rotating disks. The analysis holds for laminar and turbulent flows and compressible and incompressible flows. The conditions that must be satisfied for one to reasonably neglect 1) rotation, 2) unsteady terms, and 3) convective terms are set forth. One result shown is that only rarely could one reasonably neglect the radial convective acceleration while retaining the radial local acceleration.

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