scholarly journals Linear stability and weakly nonlinear analysis of the flow past rotating spheres

2016 ◽  
Vol 807 ◽  
pp. 62-86 ◽  
Author(s):  
V. Citro ◽  
J. Tchoufag ◽  
D. Fabre ◽  
F. Giannetti ◽  
P. Luchini
Keyword(s):  
Three Dimensional ◽  
Base Flow ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Weakly Nonlinear ◽  
Rotating Sphere ◽  
Flow Past ◽  
The Stability ◽  
Planar Symmetry ◽  
Flow Past A Sphere

We study the flow past a sphere rotating in the transverse direction with respect to the incoming uniform flow, and particularly consider the stability features of the wake as a function of the Reynolds number $Re$ and the sphere dimensionless rotation rate $\unicode[STIX]{x1D6FA}$. Direct numerical simulations and three-dimensional global stability analyses are performed in the ranges $150\leqslant \mathit{Re}\leqslant 300$ and $0\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.2$. We first describe the base flow, computed as the steady solution of the Navier–Stokes equation, with special attention to the structure of the recirculating region and to the lift force exerted on the sphere. The stability analysis of this base flow shows the existence of two different unstable modes, which occur in different regions of the $Re/\unicode[STIX]{x1D6FA}$ parameter plane. Mode I, which exists for weak rotations ($\unicode[STIX]{x1D6FA}<0.4$), is similar to the unsteady mode existing for a non-rotating sphere. Mode II, which exists for larger rotations ($\unicode[STIX]{x1D6FA}>0.7$), is characterized by a larger frequency. Both modes preserve the planar symmetry of the base flow. We detail the structure of these eigenmodes, as well as their structural sensitivity, using adjoint methods. Considering small rotations, we then compare the numerical results with those obtained using weakly nonlinear approaches. We show that the steady bifurcation occurring for $Re>212$ for a non-rotating sphere is changed into an imperfect bifurcation, unveiling the existence of two other base-flow solutions which are always unstable.

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

1957 ◽  
Vol 2 (3) ◽  
pp. 237-262 ◽  
Author(s):  
Ian Proudman ◽  
J. R. A. Pearson
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Small Reynolds Numbers ◽  
Cylindrical Polar Coordinates ◽  
Flow Past A Sphere

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

The instability of the steady flow past spheres and disks

1993 ◽  
Vol 254 ◽  
pp. 323-344 ◽  
Author(s):  
Ramesh Natarajan ◽  
Andreas Acrivos
Keyword(s):  
Critical Reynolds Number ◽  
Unstable Mode ◽  
Three Dimensional ◽  
Circular Disk ◽  
Linear Instability ◽  
Base Flow ◽  
The Body ◽  
Incoming Flow ◽  
Flow Past ◽  
Flow Past A Sphere

We consider the instability of the steady, axisymmetric base flow past a sphere, and a circular disk (oriented broadside-on to the incoming flow). Finite-element methods are used to compute the steady axisymmetric base flows, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for the sphere and the circular disk, the first instability of the base flow is through a regular bifurcation, and the critical Reynolds number (based on the body radius) is 105 for the sphere, and 58.25 for the circular disk. In both cases, the unstable mode is non-axisymmetric with azimuthal wavenumber m = 1. These computational results are consistent with previous experimental observations (Magarvey & Bishop 1961 a, b; Nakamura 1976; Willmarth, Hawk & Harvey 1964).

Numerical investigation of transitional and weak turbulent flow past a sphere

2000 ◽  
Vol 416 ◽  
pp. 45-73 ◽  
Author(s):  
ANANIAS G. TOMBOULIDES ◽  
STEVEN A. ORSZAG
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Primary Objective ◽  
Single Frequency ◽  
Cylindrical Coordinates ◽  
Efficient Treatment ◽  
Flow Past ◽  
Planar Symmetry ◽  
Flow Past A Sphere

This work reports results of numerical simulations of viscous incompressible flow past a sphere. The primary objective is to identify transitions that occur with increasing Reynolds number, as well as their underlying physical mechanisms. The numerical method used is a mixed spectral element/Fourier spectral method developed for applications involving both Cartesian and cylindrical coordinates. In cylindrical coordinates, a formulation, based on special Jacobi-type polynomials, is used close to the axis of symmetry for the efficient treatment of the ‘pole’ problem. Spectral convergence and accuracy of the numerical formulation are verified. Many of the computations reported here were performed on parallel computers. It was found that the first transition of the flow past a sphere is a linear one and leads to a three-dimensional steady flow field with planar symmetry, i.e. it is of the ‘exchange of stability’ type, consistent with experimental observations on falling spheres and linear stability analysis results. The second transition leads to a single-frequency periodic flow with vortex shedding, which maintains the planar symmetry observed at lower Reynolds number. As the Reynolds number increases further, the planar symmetry is lost and the flow reaches a chaotic state. Small scales are first introduced in the flow by Kelvin–Helmholtz instability of the separating cylindrical shear layer; this shear layer instability is present even after the wake is rendered turbulent.

scholarly journals Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid–liquid flows

2014 ◽  
Vol 750 ◽  
pp. 464-506 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Prashant Valluri ◽  
David M. Scott ◽  
Iain Bethune ◽  
Peter D. M. Spelt
Keyword(s):  
Linear Theory ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Base Flow ◽  
Navier Stokes ◽  
Late Time ◽  
Two Dimensional ◽  
Two Phase ◽  
Weakly Nonlinear

AbstractWe consider the linear and nonlinear stability of two-phase density-matched but viscosity-contrasted fluids subject to laminar Poiseuille flow in a channel, paying particular attention to the formation of three-dimensional waves. A combination of Orr–Sommerfeld–Squire analysis (both modal and non-modal) with direct numerical simulation of the three-dimensional two-phase Navier–Stokes equations is used. For the parameter regimes under consideration, under linear theory, the most unstable waves are two-dimensional. Nevertheless, we demonstrate several mechanisms whereby three-dimensional waves enter the system, and dominate at late time. There exists a direct route, whereby three-dimensional waves are amplified by the standard linear mechanism; for certain parameter classes, such waves grow at a rate less than but comparable to that of the most dangerous two-dimensional mode. Additionally, there is a weakly nonlinear route, whereby a purely spanwise wave grows according to transient linear theory and subsequently couples to a streamwise mode in weakly nonlinear fashion. Consideration is also given to the ultimate state of these waves: persistent three-dimensional nonlinear waves are stretched and distorted by the base flow, thereby producing regimes of ligaments, ‘sheets’ or ‘interfacial turbulence’. Depending on the parameter regime, these regimes are observed either in isolation, or acting together.

Cylinder rolling on a wall at low Reynolds numbers

2011 ◽  
Vol 685 ◽  
pp. 461-494 ◽  
Author(s):  
Alain Merlen ◽  
Christophe Frankiewicz
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Around A Cylinder ◽  
Low Reynolds Numbers ◽  
Initial Location ◽  
Upstream Pressure ◽  
Small Reynolds Numbers ◽  
Onset Of Cavitation

AbstractThe flow around a cylinder rolling or sliding on a wall was investigated analytically and numerically for small Reynolds numbers, where the flow is known to be two-dimensional and steady. Both prograde and retrograde rotation were analytically solved, in the Stokes regime, giving the values of forces and torque and a complete description of the flow. However, solving Navier–Stokes equation, a rotation of the cylinder near the wall necessarily induces a cavitation bubble in the nip if the fluid is a liquid, or compressible effects, if it is a gas. Therefore, an infinite lift force is generated, disconnecting the cylinder from the wall. The flow inside this interstice was then solved under the lubrication assumptions and fully described for a completely flooded interstice. Numerical results extend the analysis to higher Reynolds number. Finally, the effect of the upstream pressure on the onset of cavitation is studied, giving the initial location of the phenomenon and the relation between the upstream pressure and the flow rate in the interstice. It is shown that the flow in the interstice must become three-dimensional when cavitation takes place.

NONLINEAR EVOLUTION OF TURBULENT COHERENT STRUCTURES IN CHANNEL FLOWS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1539-1542
Author(s):  
ZHANG LI ◽  
DENGBIN TANG ◽  
LINLIN GUO
Keyword(s):  
Coherent Structures ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Channel Flows ◽  
Navier Stokes ◽  
Theoretic Model ◽  
Compact Difference Scheme ◽  
Navier Stokes Equation ◽  
Turbulent Coherent Structures ◽  
Dimensional Coupling

The generation and the development of turbulent coherent structures in channel flows are investigated by using numerical simulation of Navier-Stokes equation and the theoretic model of turbulent coherent structures built up by the flow stability theories. The three-dimensional coupling compact difference scheme with high accuracy and resolution developed can be applied to the calculative region including points near the boundary. The results computed show nonlinear evolution process and characteristics of Reynolds stress, stream-wise vortices and span-wise vorticities, especially the nonlinear interactions between different coherent structures.

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