Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

2016 ◽  
Vol 803 ◽  
pp. 556-590 ◽  
Author(s):  
J. Feys ◽  
S. A. Maslowe
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Normal Modes ◽  
Axial Flow ◽  
Velocity Profiles ◽  
Wing Loading ◽  
Axial Velocity Component ◽  
Important Conclusion ◽  
Elliptical Instability ◽  
Loading Parameter

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.

Analysis of Three-Dimensional Viscous Flow in a Supersonic Axial Flow Compressor Rotor With Emphasis on Tip Leakage Flow

1992 ◽  
Author(s):  
G. Perrin ◽  
F. Leboeuf ◽  
W. N. Dawes
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Flow ◽  
Leakage Flow ◽  
Tip Leakage Flow ◽  
Axial Velocity Component ◽  
Tip Leakage ◽  
Axial Flow Compressor ◽  
Compressor Rotor ◽  
Flow Compressor

A three-dimensional computation has been performed for a supersonic axial flow compressor rotor by solving the Navier-Stokes equations. The results of the computation are used to analyse the tip leakage flow in more detail. As well as the global behaviour of the tip leakage vortex, the analysis focuses on the origins of this vortex. It is shown that the main source of its vorticity is the shear layer at the tip of the blade associated with the shedding of the blade loading. A separation occurs, with respect to the axial velocity component, as the jet leakage flow, crossing the clearance gap, encounters the upstream incoming flow. Although the entropy increase of this separation is low, it has a strong effect on the mixing around the leakage vortex. Overall, for this compressor and the choosen operating point, the tip leakage effects are localised near the tip wall and suction side of the blade.

scholarly journals Steady flow separation patterns in a 45 degree junction

2000 ◽  
Vol 411 ◽  
pp. 1-38 ◽  
Author(s):  
C. ROSS ETHIER ◽  
SUJATA PRAKASH ◽  
DAVID A. STEINMAN ◽  
RICHARD L. LEASK ◽  
GREGORY G. COUCH ◽  
...  
Keyword(s):  
Flow Separation ◽  
Stokes Equations ◽  
Bypass Graft ◽  
Three Dimensional ◽  
Symmetry Plane ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Secondary Flows ◽  
Measured Velocity ◽  
Curved Tube

Numerical and experimental techniques were used to study the physics of flow separation for steady internal flow in a 45° junction geometry, such as that observed between two pipes or between the downstream end of a bypass graft and an artery. The three-dimensional Navier–Stokes equations were solved using a validated finite element code, and complementary experiments were performed using the photochromic dye tracer technique. Inlet Reynolds numbers in the range 250 to 1650 were considered. An adaptive mesh refinement approach was adopted to ensure grid-independent solutions. Good agreement was observed between the numerical results and the experimentally measured velocity fields; however, the wall shear stress agreement was less satisfactory. Just distal to the ‘toe’ of the junction, axial flow separation was observed for all Reynolds numbers greater than 250. Further downstream (approximately 1.3 diameters from the toe), the axial flow again separated for Re [ges ] 450. The location and structure of axial flow separation in this geometry is controlled by secondary flows, which at sufficiently high Re create free stagnation points on the model symmetry plane. In fact, separation in this flow is best explained by a secondary flow boundary layer collision model, analogous to that proposed for flow in the entry region of a curved tube. Novel features of this flow include axial flow separation at modest Re (as compared to flow in a curved tube, where separation occurs only at much higher Re), and the existence and interaction of two distinct three-dimensional separation zones.

Tidal excitation of hydromagnetic waves and their damping in the Earth

1994 ◽  
Vol 274 ◽  
pp. 219-241 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Outer Core ◽  
Decay Rates ◽  
Wave Disturbance ◽  
Instability Mechanism ◽  
The Earth ◽  
Elliptical Instability ◽  
Hydromagnetic Waves

We examine the possibility that the Earth's outer core, as a tidally distorted fluid-filled rotating spheroid, may be the seat of an elliptical instability. The instability mechanism is described within the framework of a simple Earth-like model. The preferred forms of wave disturbance are explored and a likely growth rate supremum deduced. Estimates are made of the Ohmic and viscous decay rates of such hydromagnetic waves in the outer core. Rather than a conclusive disparity of scales, we find that typical elliptical growth rates, Ohmic decay rates and viscous decay rates all have the same order for plausible core fields and core-to-mantle conductivities. This study is all the more timely considering the recent realization that the Earth's precession may also drive similar instabilities at comparable strengths in the outer core.

A Proposal for Passive Wall Treatment Applied in High Performance Axial Flow Compressors

2020 ◽  
Author(s):  
Rubén Bruno Díaz ◽  
Jesuino Takachi Tomita ◽  
Cleverson Bringhenti ◽  
Francisco Carlos Elizio de Paula ◽  
Luiz Henrique Lindquist Whitacker
Keyword(s):  
Stokes Equations ◽  
Axial Compressor ◽  
Axial Flow ◽  
Tip Clearance ◽  
Leakage Flow ◽  
Smooth Wall ◽  
Tip Leakage Flow ◽  
Viscous Effects ◽  
Stall Margin ◽  
Tip Leakage

Abstract Numerical simulations were carried out with the purpose of investigating the effect of applying circumferential grooves at axial compressor casing passive wall treatment to enhance the stall margin and change the tip leakage flow. The tip leakage flow is pointed out as one of the main contributors to stall inception in axial compressors. Hence, it is of major importance to treat appropriately the flow in this region. Circumferential grooves have shown a good performance in enhancing the stall margin in previous researches by changing the flow path in the tip clearance region. In this work, a passive wall treatment with four circumferential grooves was applied in the transonic axial compressor NASA Rotor 37. Its effect on the axial compressor performance and the flow in the tip clearance region was analyzed and set against the results attained for the smooth wall case. A 2.63% increase in the operational range of the axial compressor running at 100%N, was achieved, when compared with the original smooth wall casing configuration. The grooves installed at compressor casing, causes an increase in the flow entropy generation due to the high viscous effects in this gap region, between the rotor tip surface and casing with grooves. These viscous effects cause a drop in the turbomachine efficiency. For the grooves configurations used in this work, an efficiency drop of 0.7% was observed, compared with the original smooth wall. All the simulations were performed based on 3D turbulent flow calculations using Reynolds Averaged Navier-Stokes equations, and the flow eddy viscosity was determined using the two-equation SST turbulence model. The details of the grooves geometrical dimensions and its implementation are described in the paper.

Annular Axial Flow-Induced Vibration of Cylindrical Shells by Flu¨gge’s Shell Theory

2006 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Makoto Katou
Keyword(s):  
Shell Theory ◽  
Stokes Equations ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Navier Stokes ◽  
Complex Eigenvalue ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Flowing Fluid ◽  
Narrow Passage

The unstable phenomena of thin cylindrical shells subjected to annular axial flow are investigated. In this paper, the analytical model is composed of an elastic axisymmetric shell and a rigid one which are arranged co-axially. Considering the fluid structure interaction between shells and fluid flowing through an annular narrow passage, the coupled equation of motion is derived using Flu¨gge’s shell theory and Navier-Stokes equations. The unstable phenomena of thin cylindrical shells are clarified by using the root locus based on the complex eigenvalue analysis. The numerical parameter studies on the shells with a freely supported end and a rigid one, and with both simply supported ends, are performed taking the dimensins of shells, the characteristics of flowing fluid so on as parameters. The influence of these parameters on the threshold of instability of the coupled vibration between thin cylindrical shells and annular axial flowing fluid are investigated and discussed.

Confinement Effects on Added Mass of Cylindrical Structures in a Potential Flow

2017 ◽  
Author(s):  
R. Capanna ◽  
G. Ricciardi ◽  
C. Eloy ◽  
E. Sarrouy
Keyword(s):  
Mechanical Behaviour ◽  
Potential Flow ◽  
Nuclear Power ◽  
Stokes Equations ◽  
Reactor Core ◽  
Axial Flow ◽  
Added Mass ◽  
Seismic Load ◽  
Slender Body Theory ◽  
Computational Costs

Efficient modelling and accurate knowledge of the mechanical behaviour of the reactor core are needed to estimate the effects of seismic excitation on a nuclear power plant. The fuel assemblies (in the reactor core) are subjected to an axial water flow which modifies their dynamical behaviour. Several fluid-structure models simulating the response of the core to a seismic load has been developed in recent years; most of them require high computational costs. The work which is presented here is a first step in order to simplify the fluid forces modelling, and thus to be able to catch the main features of the mechanical behaviour of reactor core with low computational costs. The main assumption made in this work is to consider the fluid flow as an inviscid potential flow. Thus, the flow can be described only using one scalar function (velocity potential) instead of a vector field and strongly simplifies the fluid mechanics equations, avoiding the necessity to solve Navier-Stokes equations. The pressure distribution around a cylinder is first solved in Fourier space for different values of the parameters (wavenumber, confinement size). The method is applied to a simple geometry (cylinder in an axial flow with a variable confinement) in order to test its effectiveness. The empirical model is then compared to simulations and reference works in literature. The configuration with large confinement has been solved, and results were in agreement with Slender Body Theory. The dependency on the confinement size strongly depends on the wavenumber, but in any case added mass increases as the confinement size decreases. Finally, future perspectives to extend the model to a group of cylinders and to improve the model are discussed (i.e. add viscosity to the model).

Optimization of Stacking Line and Blade Profile for Design of Axial Flow Fan Blade

2008 ◽  
Author(s):  
Abdus Samad ◽  
Ki-Sang Lee ◽  
Kwang-Yong Kim
Keyword(s):  
Objective Function ◽  
Surrogate Model ◽  
Stokes Equations ◽  
Axial Flow ◽  
Weighted Average ◽  
Optimization Procedure ◽  
Total Efficiency ◽  
Blade Profile ◽  
Axial Flow Fan ◽  
Fan Blade

This work presents a numerical optimization procedure for a low-speed axial flow fan blade with weighted average surrogate model. Reynolds-averaged Navier-Stokes equations with SST turbulence model are discretized by finite volume approximations and solved on hexahedral grids for flow analyses. The blade profile as well as stacking line is modified to enhance blade total efficiency, i.e., the objective function. Six design variables related to blade lean and blade profile are selected, and a design of experiments technique produces design points where flow analyses are performed to obtain values of the objective function. PBA model is employed as a surrogate model for optimization. A search algorithm is used to find the optimal design in the design space from the constructed surrogate model for the objective function. As a main result, the efficiency is increased effectively by the present optimization procedure.

scholarly journals Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers

2007 ◽  
Vol 585 ◽  
pp. 323-342 ◽  
Author(s):  
M. LE BARS ◽  
S. LE DIZÈS ◽  
P. LE GAL
Keyword(s):  
Growth Rates ◽  
Close Agreement ◽  
Unstable Mode ◽  
Quantitative Agreement ◽  
Fixed Rate ◽  
Elliptical Instability ◽  
Normal Mode Theory ◽  
Set Up ◽  
First Time ◽  
Selection Of

The effects of the Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical rotating containers placed on a table rotating at a fixed rate $\tilde{\Omega}^G$. For a given set-up, changing the ratio ΩG of global rotation $\tilde{\Omega}^G$ to flow rotation $\tilde{\Omega}^F$ leads to the selection of various unstable modes due to the presence of resonance bands, in close agreement with the normal-mode theory. No instability occurs when ΩG varies between −3/2 and −1/2 typically. On decreasing ΩG toward −1/2, resonance bands are first discretized for ΩG<0 and progressively overlap for −1/2 ≪ ΩG < 0. Simultaneously, the growth rates and wavenumbers of the prevalent stationary unstable mode significantly increase, in quantitative agreement with the viscous short-wavelength analysis. New complex resonances have been observed for the first time for the sphere, in addition to the standard spin-over. We argue that these results have significant implications in geo- and astrophysical contexts.

Interactional Effects of Inlet Guide Vane and Blade Angle on Hydraulic Performance Characteristics of a Submersible Axial-Flow Pump

2018 ◽  
Author(s):  
Youn-Sung Kim ◽  
Hyeon-Seok Shim ◽  
Kwang-Yong Kim
Keyword(s):  
Stokes Equations ◽  
Guide Vane ◽  
Incidence Angle ◽  
Axial Flow ◽  
Performance Characteristics ◽  
Inlet Guide Vane ◽  
Computational Domain ◽  
Blade Angle ◽  
Axial Flow Pump ◽  
Flow Pump

This study aims to evaluate effects of blade pitch and inlet guide vane (IGV) angle on the performance characteristics of a submersible axial-flow pump. According to the results of the previous study, the efficiency at the design and over-load conditions were significantly affected by the angle of IGV due to change in the incidence angle. To investigate the interactional effects of IGV and blade angle are analyzed using three-dimensional Reynolds-averaged Navier-Stokes equations with shear stress transport turbulence model. The hexahedral grids are used in the computational domain and a grid-dependency test is performed to obtain an optimal number of the grids. In this study, combinations of three different blade angles and two different IGV angles are tested. Adjusting angle of IGV increases the total pressure of the pump with a blade pitch increase, which can increase the efficiency of the pump in operating range.

Axial flow in a two-dimensional microchannel induced by a travelling temperature wave imposed at the bottom wall

2018 ◽  
Vol 848 ◽  
pp. 1040-1072 ◽  
Author(s):  
Chenguang Zhang ◽  
Harris Wong ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axial Flow ◽  
Temperature Wave ◽  
Bottom Wall ◽  
Moving Frame ◽  
Global Maximum ◽  
System Of Equations ◽  
Two Dimensional ◽  
Perturbation Solutions

Fluid flow in microchannels has wide industrial and scientific applications. Hence, it is important to explore different driving mechanisms. In this paper, we study the net transport or fluid pumping in a two-dimensional channel induced by a travelling temperature wave applied at the bottom wall. The Navier–Stokes equations with the Boussinesq approximation and the convection–diffusion heat equation are made dimensionless by the height of the channel and a velocity scale obtained by a dominant balance between buoyancy and viscous resistance in the momentum equation. The system of equations is transformed to an axial coordinate that moves with the travelling temperature wave, and we seek steady solutions in this moving frame. Four dimensionless numbers emerge from the governing equations and boundary conditions: the Reynolds number $Re$, a Reynolds number $Rc$ based on the wave speed, the Prandtl number $Pr$ and the dimensionless wavenumber $K$. The system of equations is solved by a finite-volume method and by a perturbation method in the limit $Re\rightarrow 0$. Surprisingly, the leading and first-order perturbation solutions agree well with the computed axial flow for $Re\leqslant 10^{3}$. Thus, the analytic perturbation solutions are used to study systematically the effects of $Re$, $Rc$, $Pr$ and $K$ on the dimensionless induced axial flow $Q$. We find that $Q$ varies linearly with $Re$, and $Q/Re$ versus any of the three remaining dimensionless groups always exhibits a maximum. The global maximum of $Q/Re$ in the parameter space is subsequently determined for the first time. This induced axial flow is driven solely by the Reynolds stress.

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