scholarly journals The role of non-uniqueness in the development of vortex breakdown in tubes

1992 ◽  
Vol 242 ◽  
pp. 491-527 ◽  
Author(s):  
Philip S. Beran ◽  
Fred E. C. Culick
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Vortex Breakdown ◽  
Rotational Symmetry ◽  
Inviscid Flow ◽  
Critical Value ◽  
Area Ratio ◽  
Small Range ◽  
Circular Pipes ◽  
Constant Radius

Numerical solutions of viscous, swirling flows through circular pipes of constant radius and circular pipes with throats have been obtained. Solutions were computed for several values of vortex circulation, Reynolds number and throat/inlet area ratio, under the assumptions of steady flow, rotational symmetry and frictionless flow at the pipe wall. When the Reynolds number is sufficiently large, vortex breakdown occurs abruptly with increased circulation as a result of the existence of non-unique solutions. Solution paths for Reynolds numbers exceeding approximately 1000 are characterized by an ensemble of three inviscid flow types: columnar (for pipes of constant radius), soliton and wavetrain. Flows that are quasi-cylindrical and which do not exhibit vortex breakdown exist below a critical circulation, dependent on the Reynolds number and the throat/inlet area ratio. Wavetrain solutions are observed over a small range of circulation below the critical circulation, while above the critical value, wave solutions with large regions of reversed flow are found that are primarily solitary in nature. The quasi-cylindrical (QC) equations first fail near the critical value, in support of Hall's theory of vortex breakdown (1967). However, the QC equations are not found to be effective in predicting the spatial position of the breakdown structure.

Flow separation on a spheroid at incidence

1979 ◽  
Vol 92 (4) ◽  
pp. 643-657 ◽  
Author(s):  
Taeyoung Han ◽  
V. C. Patel
Keyword(s):  
Reynolds Number ◽  
Wind Tunnel ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Surface Flow ◽  
Laminar Separation ◽  
Low Reynolds Numbers ◽  
Turbulent Separation

Surface streamline patterns on a spheroid have been examined at several angles of attack. Most of the tests were performed at low Reynolds numbers in a hydraulic flume using coloured dye to make the surface flow visible. A limited number of experiments was also carried out in a wind tunnel, using wool tufts, to study the influence of Reynolds number and turbulent separation. The study has verified some of the important qualitative features of three-dimensional separation criteria proposed earlier by Maskell, Wang and others. The observed locations of laminar separation lines on a spheroid at various incidences have been compared with the numerical solutions of Wang and show qualitative agreement. The quantitative differences are attributed largely to the significant viscous-inviscid flow interaction which is present, especially at large incidences.

scholarly journals Rotation of a low-Reynolds-number watermill: theory and simulations

2018 ◽  
Vol 849 ◽  
pp. 57-75 ◽  
Author(s):  
Lailai Zhu ◽  
Howard A. Stone
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Rotational Symmetry ◽  
Low Reynolds Number ◽  
Rotational Velocity ◽  
Microfluidic Channel ◽  
Hydrodynamic Interactions ◽  
Small Scale ◽  
Low Reynolds

Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven ‘watermill’ at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill’s instantaneous rotational velocity as a function of its rod number $N$, position and orientation. When $N\geqslant 4$, the RF theory predicts that the watermill’s rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinite order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on $N$ and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.

Vortex breakdown mechanics of a laminar confined swirling flow with large expansion ratio using a non-uniform finite difference approximation

2021 ◽  
pp. 095440622110071
Author(s):  
F-J Granados-Ortiz ◽  
L Rodríguez-Tembleque ◽  
J Ortega-Casanova
Keyword(s):  
Reynolds Number ◽  
Finite Difference ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Critical Value ◽  
Expansion Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Large Expansion ◽  
Swirl Parameter

Abrupt expansions are a very frequent geometry in mechanical engineering systems, i.e. in combustion chambers, valves, heat exchangers or impinging cooling devices. However, despite the large number of devices that use this geometry, the expanded flow behaviour still needs further research to understand and predict the full system performance. This paper presents the application of the non-uniform finite difference approximation method developed in Sanmiguel et al. for the numerical characterisation of a confined swirling laminar jet discharging with a large expansion ratio. This investigation can be considered an extension of previous work by Revuelta, but now a swirling flow is generated by a rotating pipe upstream the expansion. The structures found when a fully-developed rotating Hagen-Poiseuille flow discharges into a much larger pipe section are summarised in a bifurcation diagram, whose coordinates are the Reynolds number of the jet ( Rej) and the swirl parameter ( L), for which the time-dependent, axisymmetric and incompressible Navier-Stokes equations are integrated numerically. For values of the jet Reynolds number below 200, there is a critical value of the swirl parameter above which stable vortex breakdown appears. For values of the Reynolds number above 200, three different behaviours are observed, and each performance appears for a critical value of the swirl parameter. When increasing the swirl parameter from zero, the flow becomes axisymmetrically unstable, showing an oscillatory behaviour. If further increasing the swirl intensity, the oscillatory flow coexists with a vortex breakdown bubble and, finally, a steady vortex breakdown is reached. The expansion ratio ε considered in all the simulations is 1[Formula: see text]. In previous literature, the exactness of the limiting critical Rej and L values that define these behaviours has been found to be influenced by the variability in the inlet profile conditions, which affects the expanded flow. This enhances the importance in the present investigation to accurately simulate the discharge pipe inlet profiles.

Nonlinear development of flow in channels with non-parallel walls

1996 ◽  
Vol 326 ◽  
pp. 265-284 ◽  
Author(s):  
O. R. Tutty
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Critical Value ◽  
Linear Increase ◽  
Navier Stokes ◽  
Local Geometry ◽  
Nonlinear Development ◽  
Unidirectional Flow ◽  
Critical Boundary

In Jeffery–Hamel flow, the motion of a viscous incompressible fluid between rigid plane walls, unidirectional flow is impossible if the angle between the walls exceeds a critical value of 2α2 which depends on the Reynolds number. In this paper the nonlinear development of the flow near this critical value is studied through numerical solutions of the two-dimensional Navier–-Stokes equations for flow in divergent channels with piecewise straight walls. It is found that if the angle between the walls exceeds 2α2 then Jeffery–-Hamel flow does not occur, and the solution takes the form of a large-amplitude wave with eddies attached alternately to the upper and lower walls. When viewed in the appropriate coordinate system, far downstream the wave has constant wavelength and strength, although, physically, there is a linear increase in wavelength with distance downstream, i.e. the wavelength is proportional to the channel width. If the angle between the walls is less than 2α2, then the existence (or otherwise) of the wave depends on the conditions near the inlet, in particular the local geometry of the channel. Jeffery–-Hamel flow is obtained downstream of the inlet for angles well below 2α2, but close to but below the critical value, solutions have been obtained with the wave extending (infinitely) far downstream. The wavelengths obtained numerically were compared with those from linear theory with spatially developing steady modes. No agreement was found: the wavelengths from the steady Navier–-Stokes solutions are significantly larger than that predicted by the theory. However, in other important aspects the results of this study are consistent with those from previous studies of the development/existence of Jeffery–-Hamel flow, in particular as regards the importance of the upstream conditions and the subcritical nature of the spatial development of the flow near the critical boundary in the Reynolds number–wall angle parameter space.

Generation (or Degeneration) of a Separation Bubble in a Liquid-Filled Cylinder Through Spin-Up (or Spin-Down) Process

1999 ◽  
Vol 66 (4) ◽  
pp. 1023-1026 ◽  
Author(s):  
S. Bhattacharyya ◽  
A. Pal
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Angular Velocity ◽  
Vortex Breakdown ◽  
Viscous Incompressible Fluid ◽  
Separation Bubble ◽  
Critical Value ◽  
Steady State Flow ◽  
Uniform Rotation ◽  
Small Rotation

The present study deals with the spin-up and spin-down of viscous incompressible fluid confined in a cylindrical enclosure. The spin-up (or spin-down) is developed by increasing (or decreasing) abruptly the angular velocity of the upper endwall where initial steady flow is either due to uniform rotation of the lower endwall or due to rotation of both end walls in the same direction with small rotation ratio of the top-bottom endwalls. We have seen numerically that through spin-up, an on-axis separatiorbubble can be generated (vortex breakdown) in a flow when the Reynolds number is much lower than the critical value of the Reynolds number (Re)c for the onset of vortex breakdown. At the ultimate steady state the separation bubble appears and a waviness in the stream surfaces in the vicinity of the lower endwall is found. The spin-down process acts in the opposite way. We found that for the flow with Reynolds number above the critical value (Re)c, the vortex breakdown can be prevented through spin-down and the ultimate steady-state flow is also free from it.

Performance Evaluation of Divided Intake Ducts: Effect of Area Ratio and Inlet Reynolds Number

2003 ◽  
Vol 30 (5) ◽  
pp. 525-541 ◽  
Author(s):  
Sanjeev Bharani ◽  
S.N. Singh ◽  
V. Seshadri ◽  
R. Chandramouli
Keyword(s):  
Reynolds Number ◽  
Performance Evaluation ◽  
Area Ratio

Rotating elliptic cylinders in a viscous fluid at rest or in a parallel stream

1977 ◽  
Vol 79 (1) ◽  
pp. 127-156 ◽  
Author(s):  
Hans J. Lugt ◽  
Samuel Ohring
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Elliptic Cylinder ◽  
The Body ◽  
Oscillatory Behaviour ◽  
Incompressible Fluid Flow ◽  
Transient Period ◽  
Parallel Stream ◽  
Rotating Bodies

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

On the Breakdown at High Incidences of the Leading Edge Vortices on Delta Wings

1960 ◽  
Vol 64 (596) ◽  
pp. 491-493 ◽  
Author(s):  
B. J. Elle
Keyword(s):  
High Speed ◽  
Recent Article ◽  
Vortex Breakdown ◽  
Leading Edge ◽  
Critical Value ◽  
Delta Wings ◽  
Edge Separation ◽  
Leading Edge Vortices

In a recent article, H. Werlé, has described how the free spiral vortices on delta wings with leading edge separation suddenly expand if the incidence is increased beyond a critical value. His description conforms to a great extent with the results, arrived at during an English investigation of the same phenomenon (called the vortex breakdown), but the interpretations of the observations, suggested by the two sources, are different. Against this background it is felt that some further comments and some pertinent high speed observations, may be of interest.

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