Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices

2007 ◽  
Vol 585 ◽  
pp. 153-180 ◽  
Author(s):  
STÉPHANE LE DIZÈS ◽  
DAVID FABRE
Keyword(s):  
Reynolds Number ◽  
Spatial Structure ◽  
Asymptotic Analysis ◽  
Layer Structure ◽  
Axial Flow ◽  
Multiple Scale ◽  
Large Reynolds Number ◽  
Leading Order ◽  
Vortex Model ◽  
Theoretical Predictions

This paper presents a large-Reynolds-number asymptotic analysis of viscous centre modes on an arbitrary axisymmetrical vortex with an axial jet. For any azimuthal wavenumber m and axial wavenumber k, the frequency of these modes is given at leading order by ω0 = mΩ0 + kW0 where Ω0 and W0 are the angular and axial velocities of the vortex at its centre. These modes possess a multi-layer structure localized in an O(Re−1/6) neighbourhood of the vortex. By a multiple-scale matching analysis, we demonstrate the existence of three different families of viscous centre modes whose frequency expands as ω(n) ∼ ω0 + Re−1/3ω1 + Re−1/2ω(n)2. One of these families is shown to have unstable eigenmodes when H0 = 2Ω0k(2kΩ0 − mW2) < 0 where W2 is the second radial derivative of the axial flow in the centre. The growth rate of these modes is given at leading order by σ ∼ (3/2)(H0/4)1/3Re−1/3. Our results prove that any vortex with a jet (or jet with swirl) such that Ω0W2 ≠ 0 is unstable if the Reynolds number is sufficiently large. The spatial structure of the viscous centre modes is obtained and simple approximations which capture the main feature of the eigenmodes are also provided.The theoretical predictions are compared with numerical results for the q-vortex model (or Batchelor vortex) for Re ≥ 105. For all modes, a good agreement is demonstrated for both the frequency and the spatial structure.

The linear and nonlinear stability of thread-annular flow

2005 ◽  
Vol 363 (1830) ◽  
pp. 1223-1233 ◽  
Author(s):  
Andrew G Walton
Keyword(s):  
Reynolds Number ◽  
Nonlinear Stability ◽  
Layer Structure ◽  
Large Reynolds Number ◽  
Concentric Cylinders ◽  
Axial Pressure Gradient ◽  
Nonlinear Equilibrium ◽  
Critical Reynolds Numbers ◽  
Multiple Regions ◽  
High Reynolds

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The linear stability of the flow to both axisymmetric and asymmetric perturbations is analysed asymptotically at large Reynolds number, and computationally at finite Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and thread velocity is determined. A discrepancy in critical Reynolds numbers and cut-off velocity is found to exist between experimental results and the predictions of the linear theory. In order to account for this discrepancy, the high Reynolds number, nonlinear stability properties of the flow are analysed and a nonlinear, equilibrium critical layer structure is found, which leads to an enhanced correction to the basic flow. The predictions of the nonlinear theory are found to be in good agreement with the experimental data.

Inviscid instability of a unidirectional flow sheared in two transverse directions

2019 ◽  
Vol 874 ◽  
pp. 979-994
Author(s):  
Kengo Deguchi
Keyword(s):  
Reynolds Number ◽  
Asymptotic Analysis ◽  
Transverse Direction ◽  
A Priori ◽  
Necessary Condition ◽  
Large Reynolds Number ◽  
Simple Shear Flow ◽  
Unidirectional Flow ◽  
Neutral Mode ◽  
Inviscid Instability

Linear inviscid stability of general unidirectional flows sheared in one transverse direction has long been investigated by numerous researchers using the Rayleigh equation. However, unlike the simple shear flow considered in this equation, most physically relevant unidirectional flows vary in two transverse directions. Here the inviscid instability of such flows is studied by the large-Reynolds-number limit asymptotic analysis. We derive an a priori necessary condition for the existence of a limiting neutral mode, and develop a new numerical method to accurately capture singular neutral modes.

scholarly journals Elliptic instability in a strained Batchelor vortex

2007 ◽  
Vol 577 ◽  
pp. 341-361 ◽  
Author(s):  
LAURENT LACAZE ◽  
KRIS RYAN ◽  
STÉPHANE LE DIZÈS
Keyword(s):  
Reynolds Number ◽  
Strain Field ◽  
Flow Parameter ◽  
Numerical Study ◽  
Normal Modes ◽  
Axial Flow ◽  
Base Flow ◽  
Large Reynolds Number ◽  
Resonant Coupling ◽  
Flow Parameters

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.

scholarly journals Dual subtractions

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Renato Maria Prisco ◽  
Francesco Tramontano
Keyword(s):  
Field Theory ◽  
Quantum Field ◽  
Matrix Elements ◽  
Leading Order ◽  
Initial State ◽  
Final State ◽  
Dual Representation ◽  
Subtraction Scheme ◽  
Theoretical Predictions ◽  
Perturbative Quantum

Abstract We propose a novel local subtraction scheme for the computation of Next-to-Leading Order contributions to theoretical predictions for scattering processes in perturbative Quantum Field Theory. With respect to well known schemes proposed since many years that build upon the analysis of the real radiation matrix elements, our construction starts from the loop diagrams and exploits their dual representation. Our scheme implements exact phase space factorization, handles final state as well as initial state singularities and is suitable for both massless and massive particles.

scholarly journals Constraints on neutrino non-standard interactions from LHC data with large missing transverse momentum

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
DianYu Liu ◽  
ChuanLe Sun ◽  
Jun Gao
Keyword(s):  
Transverse Momentum ◽  
Neutral Current ◽  
Data Sets ◽  
Leading Order ◽  
Missing Transverse Momentum ◽  
Upper Limit ◽  
Theoretical Predictions ◽  
Parton Showering ◽  
Single Jet

Abstract The possible non-standard interactions (NSIs) of neutrinos with matter plays important role in the global determination of neutrino properties. In our study we select various data sets from LHC measurements at 13 TeV with integrated luminosities of 35 ∼ 139 fb−1, including production of a single jet, photon, W/Z boson, or charged lepton accompanied with large missing transverse momentum. We derive constraints on neutral-current NSIs with quarks imposed by different data sets in a framework of either effective operators or simplified Z′ models. We use theoretical predictions of productions induced by NSIs at next-to-leading order in QCD matched with parton showering which stabilize the theory predictions and result in more robust constraints. In a simplified Z′ model we obtain a 95% CLs upper limit on the conventional NSI strength ϵ of 0.042 and 0.0028 for a Z′ mass of 0.2 and 2 TeV respectively. We also discuss possible improvements from future runs of LHC with higher luminosities.

Initial motion of a viscous vortex ring

1994 ◽  
Vol 446 (1928) ◽  
pp. 589-599 ◽  
Keyword(s):  
Reynolds Number ◽  
Asymptotic Analysis ◽  
Vortex Ring ◽  
Kinematic Viscosity ◽  
Reynolds Numbers ◽  
Initial Radius ◽  
Dimensionless Form ◽  
Ring Radius ◽  
Initial Motion ◽  
Matched Asymptotic Analysis

The behaviour of a viscous vortex ring is examined by a matched asymptotic analysis up to three orders. This study aims at investigating how much the location of maximum vorticity deviates from the centroid of the vortex ring, defined by P. G. Saffman (1970). All the results are presented in dimensionless form, as indicated in the following context. Let Γ be the initial circulation of the vortex ring, and R denote the ring radius normalized by its initial radius R i . For the asymptotic analysis, a small parameter ∊ = ( t / Re ) ½ is introduced, where t denotes time normalized by R 2 i / Γ , and Re = Γ/v is the Reynolds number defined with Γ and the kinematic viscosity v . Our analysis shows that the trajectory of maximum vorticity moves with the velocity (normalized by Γ/R i ) U m = – 1/4π R {ln 4 R /∊ + H m } + O (∊ ln ∊), where H m = H m ( Re, t ) depends on the Reynolds number Re and may change slightly with time t for the initial motion. For the centroid of the vortex ring, we obtain the velocity U c by merely replacing H m by H c , which is a constant –0.558 for all values of the Reynolds number. Only in the limit of Re → ∞, the values of H m and H c are found to coincide with each other, while the deviation of H m from the constant H c is getting significant with decreasing the Reynolds number. Also of interest is that the radial motion is shown to exist for the trajectory of maximum vorticity at finite Reynolds numbers. Furthermore, the present analysis clarifies the earlier discrepancy between Saffman’s result and that obtained by C. Tung and L. Ting (1967).

An example of steady laminar flow at large Reynolds number

1960 ◽  
Vol 9 (4) ◽  
pp. 593-602 ◽  
Author(s):  
Iam Proudman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Tangential Velocity ◽  
Fluid Flows ◽  
The Other ◽  
Large Reynolds Number ◽  
Rotational Flows ◽  
Flow Domain ◽  
Source Of Information

The purpose of this note is to describe a particular class of steady fluid flows, for which the techniques of classical hydrodynamics and boundary-layer theory determine uniquely the asymptotic flow for large Reynolds number for each of a continuously varied set of boundary conditions. The flows involve viscous layers in the interior of the flow domain, as well as boundary layers, and the investigation is unusual in that the position and structure of all the viscous layers are determined uniquely. The note is intended to be an illustration of the principles that lead to this determination, not a source of information of practical value.The flows take place in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. When fluid is extracted through both walls there are boundary layers on both walls and the flow outside these layers is irrotational. When fluid is extracted through one wall and injected through the other, there is a boundary layer only on the former wall and the inviscid rotational flow outside this layer satisfies the no-slip condition on the other wall. When fluid is injected through both walls there are no boundary layers, but there is a viscous layer in the interior of the channel, across which the second derivative of the tangential velocity is discontinous, and the position of this layer is determined by the requirement that the inviscid rotational flows on either side of it must satisfy the no-slip conditions on the walls.

The propagation of gravity currents in a V-shaped triangular cross-section channel: experiments and theory

2014 ◽  
Vol 754 ◽  
pp. 232-249 ◽  
Author(s):  
Marius Ungarish ◽  
Catherine A. Mériaux ◽  
Cathy B. Kurz-Besson
Keyword(s):  
Reynolds Number ◽  
Cross Section ◽  
Viscosity Coefficient ◽  
Gravity Currents ◽  
Quantitative Agreement ◽  
Flat Bottom ◽  
Special Configuration ◽  
Theoretical Predictions ◽  
Speed Of Propagation ◽  
High Reynolds

AbstractWe investigate the motion of high-Reynolds-number gravity currents (GCs) in a horizontal channel of V-shaped cross-section combining lock-exchange experiments and a theoretical model. While all previously published experiments in V-shaped channels were performed with the special configuration of the full-depth lock, we present the first part-depth experiment results. A fixed volume of saline, that was initially of length $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x_0$ and height $h_0$ in a lock and embedded in water of height $H_0$ in a long tank, was released from rest and the propagation was recorded over a distance of typically $ 30 x_0$. In all of the tested cases the current displays a slumping stage of constant speed $u_N$ over a significant distance $x_S$, followed by a self-similar stage up to the distance $x_V$, where transition to the viscous regime occurs. The new data and insights of this study elucidate the influence of the height ratio $H = H_0/h_0$ and of the initial Reynolds number ${\mathit{Re}}_0 = (g^{\prime }h_0)^{{{1/2}}} h_0/ \nu $, on the motion of the triangular GC; $g^{\prime }$ and $\nu $ are the reduced gravity and kinematic viscosity coefficient, respectively. We demonstrate that the speed of propagation $u_N$ scaled with $(g^{\prime } h_0)^{{{1/2}}}$ increases with $H$, while $x_S$ decreases with $H$, and $x_V \sim [{\mathit{Re}}_0(h_0/x_0)]^{{4/9}}$. The initial propagation in the triangle is 50 % more rapid than in a standard flat-bottom channel under similar conditions. Comparisons with theoretical predictions show good qualitative agreements and fair quantitative agreement; the major discrepancy is an overpredicted $u_N$, similar to that observed in the standard flat bottom case.

Investigation of Turbulent Mixing in a Macro-Scale Multi-Inlet Vortex Nanoprecipitation Reactor by Stereoscopic-PIV

2014 ◽  
Author(s):  
Zhenping Liu ◽  
James C. Hill ◽  
Rodney O. Fox ◽  
Michael G. Olsen
Keyword(s):  
Reynolds Number ◽  
Turbulent Mixing ◽  
Three Dimensional ◽  
Stereoscopic Particle Image Velocimetry ◽  
Vortex Center ◽  
Mean Velocity ◽  
Vortex Model ◽  
Functional Nanoparticles ◽  
Turbulent Characteristics ◽  
Macro Scale

Flash Nanoprecipitation (FNP) is a technique to produce monodisperse functional nanoparticles through rapidly mixing a saturated solution and a non-solvent. Multi-inlet vortex reactors (MIVR) have been effectively applied to FNP due to their ability to provide both rapid mixing and the flexibility of inlet flow conditions. Until recently, only micro-scale MIVRs have been demonstrated to be effective in FNP. A scaled-up MIVR could potentially generate large quantities of functional nanoparticles, giving FNP wider applicability in the industry. In the present research, turbulent mixing inside a scaled-up, macro-scale MIVR was measured by stereoscopic particle image velocimetry (SPIV). Reynolds number of this reactor is defined based on the bulk inlet velocity, ranging from 3290 to 8225. It is the first time that the three-dimensional velocity field of a MIVR was experimentally measured. The influence of Reynolds number on mean velocity becomes more linear as Reynolds number increases. An analytical vortex model was proposed to well describe the mean velocity profile. The turbulent characteristics such as turbulent kinematic energy and Reynolds stress are also presented. The wandering motion of vortex center was found to have a significant contribution to the turbulent kinetic energy of flow near the center area.

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