Transient torque in stirred tanks

The transient dynamics of stirred tanks whose impeller speed undergoes smooth or step changes is investigated. First, a low-order model is developed, linking the impeller torque with the ‘extent’ of the solid-body rotation in the tank, derived from an angular momentum balance in a control volume around the impeller. Utilisation of this model enables the prediction of the torque ‘spike’ appearing after an impulsive change of the shaft speed, and of the torque evolution during a quasi-steady transition. For the case of a small impulsive change in the shaft speed, a characteristic spin-up time is also proposed. Torque measurements performed in an unbaffled stirred tank show considerable agreement with the theoretical predictions.

Download Full-text

Taylor-vortex instability and annulus-length effects

Journal of Fluid Mechanics ◽

10.1017/s0022112076000098 ◽

1976 ◽

Vol 75 (1) ◽

pp. 1-15 ◽

Cited By ~ 132

Author(s):

J. A. Cole

Keyword(s):

Critical Speed ◽

Taylor Vortex ◽

Vortex Instability ◽

Critical Speeds ◽

Taylor Vortices ◽

Torque Measurements ◽

Concentric Cylinders ◽

Theoretical Predictions ◽

Range Of Values ◽

Visual Observations

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

Download Full-text

Oval Flow Mode Between Two Corotating Disks With Stationary Shroud

Journal of Fluids Engineering ◽

10.1115/1.4028729 ◽

2014 ◽

Vol 137 (3) ◽

Cited By ~ 1

Author(s):

Ching Min Hsu ◽

Jia-Kun Chen ◽

Min Kai Hsieh ◽

Rong Fung Huang

Keyword(s):

Flow Structure ◽

Solid Body ◽

Circumferential Velocity ◽

Body Rotation ◽

Flow Core ◽

Core Flow ◽

Node Point ◽

Solid Body Rotation ◽

Characteristic Flow ◽

Buffer Region

The characteristic flow behavior, time-averaged velocity distributions, phase-resolved ensemble-averaged velocity profiles, and turbulence properties of the flow in the interdisk midplane between shrouded two corotating disks at the interdisk spacing to disk radius aspect ratio 0.2 and rotation Reynolds number 3.01 × 105 were experimentally studied by flow visualization method and particle image velocimetry (PIV). An oval core flow structure rotating at a frequency 60% of the disks rotating frequency was observed. Based on the analysis of relative velocities, the flow in the region outside the oval core flow structure consisted of two large vortex rings, which move circumferentially with the rotation motion of the oval flow core. Four characteristic flow regions—solid-body-rotation-like region, buffer region, vortex region, and shroud-influenced region—were identified in the flow field. The solid-body-rotation-like region, which was featured by its linear distribution of circumferential velocity and negligibly small radial velocity, was located within the inscribing radius of the oval flow core. The vortex region was located outside the circumscribing radius of the oval flow core. The buffer region existed between the solid-body-rotation-like region and the vortex region. In the buffer region, there existed a “node” point that the propagating circumferential velocity waves diminished. The circumferential random fluctuation intensity presented minimum values at the node point and high values in the solid-body-rotation-like region and shroud-influenced region due to the shear effect induced by the wall.

Download Full-text

Liquid Films Falling Down a Vertical Fiber: Modeling, Simulations and Experiments

Fluids ◽

10.3390/fluids6080281 ◽

2021 ◽

Vol 6 (8) ◽

pp. 281

Author(s):

Yadong Ruan ◽

Ali Nadim ◽

Lekha Duvvoori ◽

Marina Chugunova

Keyword(s):

Velocity Profile ◽

Liquid Film ◽

Control Volume ◽

Plug Flow ◽

Traveling Wave Solutions ◽

Liquid Films ◽

Physical Experiments ◽

Theoretical Predictions ◽

Viscous Liquid Film ◽

Linear Drag

We provide a new framework for analyzing the flow of an axisymmetric liquid film flowing down a vertical fiber, applicable to fiber coating flows and those in similar geometries in heat exchangers, water treatment, and desalination processes. The problem considered is that of a viscous liquid film falling under the influence of gravity and surface tension on a solid cylindrical fiber. Our approach is different from existing ones in that we derive our mathematical model by using a control-volume approach to express the conservation of mass and axial momentum in simple and intuitively appealing forms, resulting in a pair of equations that are reminiscent of the Saint-Venant shallow-water equations. Two versions of the model are obtained, one assuming a plug-flow velocity profile with a linear drag force expression, and the other using the fully-developed laminar velocity profile for a locally uniform film to approximate the drag. These can, respectively, model high- and low-Reynolds number regimes of flow. Linear stability analyses and fully nonlinear numerical simulations are presented that show the emergence of traveling wave solutions representing chains of identical droplets falling down the fiber. Physical experiments with safflower oil on a fishing line are also undertaken and match the theoretical predictions from the laminar flow model well when machine learning methods are used to estimate the parameters.

Download Full-text

Azimuthal velocity profiles in Rayleigh-stable Taylor–Couette flow and implied axial angular momentum transport

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.275 ◽

2015 ◽

Vol 774 ◽

pp. 342-362 ◽

Cited By ~ 8

Author(s):

Freja Nordsiek ◽

Sander G. Huisman ◽

Roeland C. A. van der Veen ◽

Chao Sun ◽

Detlef Lohse ◽

...

Keyword(s):

Angular Momentum ◽

Couette Flow ◽

Outer Cylinder ◽

Azimuthal Velocity ◽

Velocity Profiles ◽

Body Rotation ◽

Solid Body Rotation ◽

Angular Momentum Transport ◽

Taylor Couette ◽

Taylor Couette Flow

We present azimuthal velocity profiles measured in a Taylor–Couette apparatus, which has been used as a model of stellar and planetary accretion disks. The apparatus has a cylinder radius ratio of ${\it\eta}=0.716$, an aspect ratio of ${\it\Gamma}=11.74$, and the plates closing the cylinders in the axial direction are attached to the outer cylinder. We investigate angular momentum transport and Ekman pumping in the Rayleigh-stable regime. This regime is linearly stable and is characterized by radially increasing specific angular momentum. We present several Rayleigh-stable profiles for shear Reynolds numbers $\mathit{Re}_{S}\sim O(10^{5})$, for both ${\it\Omega}_{i}>{\it\Omega}_{o}>0$ (quasi-Keplerian regime) and ${\it\Omega}_{o}>{\it\Omega}_{i}>0$ (sub-rotating regime), where ${\it\Omega}_{i,o}$ is the inner/outer cylinder rotation rate. None of the velocity profiles match the non-vortical laminar Taylor–Couette profile. The deviation from that profile increases as solid-body rotation is approached at fixed $\mathit{Re}_{S}$. Flow super-rotation, an angular velocity greater than those of both cylinders, is observed in the sub-rotating regime. The velocity profiles give lower bounds for the torques required to rotate the inner cylinder that are larger than the torques for the case of laminar Taylor–Couette flow. The quasi-Keplerian profiles are composed of a well-mixed inner region, having approximately constant angular momentum, connected to an outer region in solid-body rotation with the outer cylinder and attached axial boundaries. These regions suggest that the angular momentum is transported axially to the axial boundaries. Therefore, Taylor–Couette flow with closing plates attached to the outer cylinder is an imperfect model for accretion disk flows, especially with regard to their stability.

Download Full-text

Moment on a Liquid-Filled Spinning and Nutating Projectile: Solid Body Rotation

10.21236/ada125332 ◽

1983 ◽

Cited By ~ 6

Author(s):

Nathan Gerber ◽

Raymond Sedney

Keyword(s):

Solid Body ◽

Body Rotation ◽

Solid Body Rotation

Download Full-text

A Unified Theory for the Pressure Change of Sudden Expansions and Contractions Based on the Momentum Balance

10.1115/fedsm2021-65703 ◽

2021 ◽

Author(s):

Sebastian Müller ◽

Andreas Malcherek

Keyword(s):

Control Volume ◽

Pressure Change ◽

Sudden Expansion ◽

Momentum Balance ◽

Unified Approach ◽

Pressure Distributions ◽

Analytical Functions ◽

Specific Geometry ◽

Control Volumes ◽

Pressure Changes

Abstract In this paper a unified approach based on the momentum balance is presented, capable of predicting the pressure change of sudden contractions and sudden expansions. The use of empirically determined correction coefficients is not necessary. Therefore, the momentum balance is derived similarly for both applications but with different control volumes. The control volume takes into account the specific geometry of the hydraulic structure. With a properly chosen control volume, the unified approach requires coefficients that account for the velocity as well as pressure distributions on the boundaries of the control volume. These coefficients can be obtained by parameterizing the results of numerical simulations by simple analytical functions. The numerical model itself is validated by checking the simulated pressure change against calculated or measured pressure changes. It is found that the formulation of the momentum balance for the sudden expansion is more complex compared with the sudden contraction. The prediction of the pressure change of flows through sudden expansions can be improved by applying the momentum balance non-idealized. Most of the correction coefficients originate from an inappropriate application of Bernoulli’s energy conservation principle. Consequently, this leads to a gap between theory and experimental results. The proposed unified approach solely contains physical coefficients that are used to substitute integrals by averaged expressions.

Download Full-text

On a possible corrugation of the galactic plane

Symposium - International Astronomical Union ◽

10.1017/s0074180900000437 ◽

1970 ◽

Vol 38 ◽

pp. 147-150 ◽

Cited By ~ 1

Author(s):

C. M. Varsavsky ◽

R. J. Quiroga

Keyword(s):

Brightness Temperature ◽

Solid Body ◽

Rotation Curve ◽

Galactic Plane ◽

Maximum Temperature ◽

Body Rotation ◽

Maximum Brightness ◽

Solid Body Rotation ◽

The Galaxy ◽

Sinusoidal Pattern

We have studied the rotation curve of the Galaxy at different heights below and above the equator. In the course of this work we noticed that the maximum brightness temperature of hydrogen oscillates around the galactic plane following a fairly sinusoidal pattern. It is further noticed that the maximum temperature of hydrogen occurs right on the plane in the regions where the rotation curve has a form indicating solid body rotation. A rotation curve based on points of maximum hydrogen temperature does not differ appreciably from a rotation curve measured on the galactic plane.

Download Full-text

Binary Orbit Perturbations and the Rotational Angular Momentum of Stellar Interiors

International Astronomical Union Colloquium ◽

10.1017/s0252921100027184 ◽

1970 ◽

Vol 4 ◽

pp. 187-192

Author(s):

Stanley Sobieski

Keyword(s):

Angular Momentum ◽

Angular Velocity ◽

Significant Variation ◽

Eclipsing Binaries ◽

Model Parameters ◽

Body Rotation ◽

Solid Body Rotation ◽

Stellar Interiors ◽

Binary Orbit ◽

Photometric Variation

AbstractCalculations show that a significant variation in the minima of eclipsing binaries should arise for systems where axial precession exists. Several different angular velocity distributions are assumed in order to estimate the expected photometric variation as a function of the model parameters. It is found that the solid body rotation approximation is a reasonable representation unless interiors rotate more rapidly than present models predict.

Download Full-text

Measurements of Differential Rotation in Cool Stars

Symposium - International Astronomical Union ◽

10.1017/s0074180900195439 ◽

2004 ◽

Vol 215 ◽

pp. 138-143 ◽

Cited By ~ 2

Author(s):

A. Reiners ◽

J.H.M.M. Schmitt

Keyword(s):

Differential Rotation ◽

Rotation Parameter ◽

High Signal ◽

Body Rotation ◽

Signal To Noise ◽

Transform Method ◽

Solid Body Rotation ◽

Cool Stars ◽

Show Evidence ◽

The Fourier Transform

We have obtained high resolution (R ≈ 220000) - high signal-to-noise (S/N > 500) spectra of 142 field dwarfs of spectral types F–K and v sin i ≤ 45 km s–1. Using the Fourier Transform Method (FTM) we precisely determined rotational velocities (Δ v sin i < 1.0 km s–1). For stars with v sin i ≥ 12.0 km s–1 this method allows the detection of deviations from solid body rotation. In the case of symmetric profiles the differential rotation parameter α = (ωequator – ωpole) / ωequator can be determined. This was possible for 32 of our sample stars; ten stars show evidence for solar-like differential rotation with α > 0.0. Thus it becomes possible to search for connections between differential rotation, rotational velocities and other stellar parameters. Signatures of differential rotation could be found on stars rotating as fast as v sin i = 42 km s–1. Particularly the Li-depleted stars turned out to show strong signatures of differential rotation. Our measurements support the idea, that Li-depletion in fast rotators (v sin i > 15 km s–1) is closely connected to differential rotation.

Download Full-text