Simulations of the formation of an axisymmetric vortex ring

1997 ◽  
Vol 339 ◽  
pp. 199-211 ◽  
Author(s):  
R. S. HEEG ◽  
N. RILEY
Keyword(s):  
Vortex Ring ◽  
Reasonable Agreement ◽  
Stokes Equations ◽  
Circular Tube ◽  
Inviscid Flow ◽  
High Rate ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Axisymmetric Vortex ◽  
High Reynolds

In this paper we present the results from numerical calculations, based upon the Navier–Stokes equations at relatively high Reynolds number, of the formation of a vortex ring when fluid is ejected from a circular tube. Our results are compared with the experiments of Didden (1979), and the inviscid flow calculations of Nitsche & Krasny (1994). Reasonable agreement is achieved except for the rate of shedding of circulation during the initial stages of ring formation. The theoretically predicted rate of shedding is substantially higher than that predicted by Didden. By contrast the inviscid theory predicts an anomalously high rate of initial shedding. We offer explanations for both of these apparent discrepancies.

Viscous oscillatory flow about a circular cylinder at small to moderate Strouhal number

1995 ◽  
Vol 303 ◽  
pp. 215-232 ◽  
Author(s):  
H. M. Badr ◽  
S. C. R. Dennis ◽  
S. Kocabiyik ◽  
P. Nguyen
Keyword(s):  
Circular Cylinder ◽  
Strouhal Number ◽  
Transient Flow ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
Method Of Solution ◽  
High Reynolds

The transient flow field caused by an infinitely long circular cylinder placed in an unbounded viscous fluid oscillating in a direction normal to the cylinder axis, which is at rest, is considered. The flow is assumed to be started suddenly from rest and to remain symmetrical about the direction of motion. The method of solution is based on an accurate procedure for integrating the unsteady Navier–Stokes equations numerically. The numerical method has been carried out for large values of time for both moderate and high Reynolds numbers. The effects of the Reynolds number and of the Strouhal number on the laminar symmetric wake evolution are studied and compared with previous numerical and experimental results. The time variation of the drag coefficients is also presented and compared with an inviscid flow solution for the same problem. The comparison between viscous and inviscid flow results shows a better agreement for higher values of Reynolds and a Strouhal numbers. The mean flow for large times is calculated and is found to be in good agreement with previous predictions based on boundary-layer theory.

scholarly journals Numerical stability analysis of a vortex ring with swirl

2019 ◽  
Vol 878 ◽  
pp. 5-36 ◽  
Author(s):  
Yuji Hattori ◽  
Francisco J. Blanco-Rodríguez ◽  
Stéphane Le Dizès
Keyword(s):  
Numerical Simulation ◽  
Growth Rate ◽  
Direct Numerical Simulation ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Base Flow ◽  
Critical Layer ◽  
Navier Stokes ◽  
Instability Mode ◽  
Navier Stokes Equations

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Some numerical experiments on developing laminar flow in circular-sectioned bends

1985 ◽  
Vol 154 ◽  
pp. 357-375 ◽  
Author(s):  
J. A. C. Humphrey ◽  
H. Iacovides ◽  
B. E. Launder
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Streamwise Velocity ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Numerical Predictions

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

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