Rotating Shear Flows Research Articles

Accretion discs models with the β-viscosity prescription derived from laboratory experiments

We examine under which conditions one may apply, to steady state Keplerian accretion discs, the -viscosity prescription which has been derived from rotating shear flow experiments ( = R 2 ,w here is the angular velocity at radius R and is a constant of order 10 5 ; Richard & Zahn 1999). Using a vertically averaged model, we show that this law may be suitable for all three families of known systems: in young stellar objects, evolved binary stars and Active Galactic Nuclei discs (except in their outer gas pressure dominated regions where turbulence becomes hypersonic). According to the standard criterion for viscous stability, -discs are always stable throughout. Using realistic opacities and equation of state, we demonstrate that these discs are thermally unstable in the temperature domain where hydrogen recombines, when they are optically thick, and this could lead to limit cycle behavior. Radiation pressure dominated regions are thermally stable, in contrast with -discs. This results in a fully stable solution for the innermost parts of AGN discs.

The effect of rotation on the buoyant rise of magnetic flux tubes in accretion disks

The dynamics of buoyant magnetic flux tubes in thin accretion disks is studied under isothermal conditions by means of numerical simulations. The influence of rotation on the rising behavior of the flux tube is examined and the role of a weak magnetic field line twist within the tube is investigated. By employing the adaptive mesh code NIRVANA the 3D simulations have effective resolution higher than in any previous numerical work on that topic. The fate of the flux tube strongly depends on the presence or absence of rotation respective differential rotation. Rotation effectively slows down the vertical ascend of the flux tube largely as a consequence of the Coriolis force acting on the surrounding flow which, in turn, reacts upon the tube. The detailed behavior also depends on the amount of twist. In accretion disks, a weakly twisted flux tube is disrupted and its rise is halted due to the impact of the magnetic shear instability which is driven by the interaction between the background rotational shear flow and (poloidal) twist field. As a consequence, the magnetic structure is captured in the inner disk region (z < H0, H0: disk scale height) a much longer time than suggested by the buoyant time scale in a non-rotating atmosphere. Untwisted accretion disk flux tubes do not break up quickly into pieces, as was found for corresponding tubes embedded in a non-rotating environment, but retain some degree of coherence albeit the stabilizing effect of twist is missing. In general, the numerical results are in gross contradiction to what postulates a highly simplified 1D picture based on the thin flux tube approximation.

A Mechanism for Offshore Transport across the Gulf Stream

This paper explores a possible mechanism to explain an atypical phenomenon observed in the Gulf Stream between the Florida Straits and Cape Hatteras, namely, upgradient momentum transport and associated eddyto-mean energy conversion on the cyclonic shear side. The proposed mechanism is based on the results of an experimental study of free surface turbulent jets in a rotating system and on theoretical descriptions of rotating shear flows. Unlike field situations, where Coriolis effects are always present, the laboratory experiment provides a convenient means of comparing turbulence characteristics in rotating and nonrotating systems. Rotation is shown to alter turbulence eddy orientation, thus affecting the momentum transport and energy conversion processes.

The evolution of a localized vortex disturbance in external shear flows. Part 2. Comparison with experiments in rotating shear flows

The evolution of artificially generated localized disturbances in the shape of hairpin vortices, in laminar axisymmetric rotating shear flows, is investigated experimentally. The results are compared with the predictions of a theoretical model (Levinski &amp; Cohen 1995) with respect to the growth of such disturbances. Hairpin vortices were generated at the surface of the inner cylinder of an axisymmetric Couette apparatus, employing an injection–suction technique. The flow field was analysed from flow visualization using top and side views and by measurements of the mean and instantaneous velocity fields, carried out using laser Doppler anemometry and particle image velocimetry. An instability domain, within the range of base flow parameters where the flow is known to be linearly stable, was found. The marginal ratio between the angular velocities of the inner and outer cylinders beyond which the flow is stable to finite-amplitude localized disturbances agrees with the theoretical prediction based on the measured mean flow in the region of the disturbance. The dependence of the hairpin's inclination angle on the ratio between the two angular velocities is fairly well predicted by the theoretical model.

Spectral Large-Eddy Simulations and Vortex Dynamics in Turbulence.

We present a point of view of large-eddy simulations(LES)in Fourier space, where the eddy coefficients are expressed thanks to a two-point spectral closure of isotropic turbulence, the EDQNM theory. Returning to real space, this leads to models of the structure-function family(plain, selective or filtered). These models are applied with success to predict the statistical distributions and coherent-vortex dynamics for a wide variety of turbulent flows. In three-dimensional decaying isotropic turbulence, we confirm the existence of a κ4 infrared backscatter in the kinetic-energy spectrum, and predict a new κ2 law for the pressure spectrum in this range. In the mixing layer(temporal or spatial), we show how to manipulate the topology of Kelvin-Helmholtz vortices, from quasi two-dimensionality to helical pairing. The latter vortex organization is found in a backward-facing step just behind the step, and yields big staggered Λ-vortices which are carried away downstream. In a developed turbulent boundary layer, coherent vortices are hairpins generated above the low-speed streaks by a secondary Kelvin-Helmholtz instability. Afterwards, we consider LES of compressible turbulence, studied with Favre averages, and where the introduction of a macro-temperature and a macro-pressure simplifies greatly the problem. Finally, we show in rotating shear flows(free or wall bounded, axis of rotation in the spanwise direction)a universal behaviour of the mean velocity which becomes linear in certain anticyclonic regions.

Torsional flow: effect of second normal stress difference on elastic instability in a finite domain

A rotational shear flow is examined in the bounded parallel-plate geometry for a four-constant Oldroyd-type fluid which has a constant viscosity, and constant first and second normal stress coefficients. A new type of Galerkin spectral technique is introduced to solve the resulting two-dimensional stiff boundary value problem. We show that even a small second normal stress difference can lead to a significant increase (nearly 100%) in the stability of the base torsional flow. Beyond a critical Deborah number the secondary flow, in the form of travelling waves, appears to be confined between two critical radii, in qualitative agreement with the experimental results of Byars et al. (1994). The mechanism behind this instability is investigated for dilute polymer solutions.

An analysis of rotating shear flow using linear theory and DNS and LES results

The development of turbulence is investigated in the presence of a mean plane shear flow (rate S) rotating with angular velocity vector (rate Ω) perpendicular to its plane. An important motivation was generalizing the work by Lee, Kim &amp; Moin (1990) to rotating shear flow, in particular detailed comparisons of homogeneous rapid distortion theory (RDT) results and the databases of homogeneous and channel flow direct numerical simulations (DNS). Linear analysis and related RDT are used starting from the linearized equations governing the fluctuating velocity field. The parameterization based on the value of the Bradshaw–Richardson number B=R(1+R) (with R=−2Ω/S) is checked against complete linear solutions. Owing to the pressure fluctuation, the dynamics is not governed entirely by the parameter B, and the subsequent breaking of symmetry (between the R and −1 −R cases) is investigated. New analytical solutions for the ‘two-dimensional energy components’ [Escr ](l)ij =Eij(kl=0, t) (i.e. the limits at kl=0 of the one-dimensional energy spectra) are calculated by inviscid and viscous RDT, for various ratios Ω/S and both streamwise l=1 and spanwise l=3 directions. Structure effects (streak-like tendencies, dimensionality) in rotating shear flow are discussed through these quantities and more conventional second-order statistics. In order to compare in a quantitative way RDT solutions for single-point statistics with available large-eddy simulation (LES) data (Bardina, Ferziger &amp; Reynolds 1983), an ‘effective viscosity’ model (following Townsend) is used, yielding an impressive agreement.

Instability in stratified rotating shear flow along ridges

This paper examines the interaction between continuously stratified horizontal shear flow and topography. It investigates when topography whose slope changes sign (i.e., a ridge or a trench) destabilizes a shear flow. The instabilities have the form of topographic Rossby waves and rely on the background potential vorticity gradient to exist. Barotropic flow is examined in detail for general horizontal shear over stepped topography and the longwave stratified case is examined in the more specific case of linear shear flow over a top-hat ridge profile. In all cases modes are unstable over significant ranges of the relevant parameters. Stationary unstable modes are seen to correspond to points where two opposing waves are brought simultaneously to rest by the shear flow and propagating unstable modes are seen where opposing waves are brought relatively to rest. Another interpretation of the instabilities in terms of resonance between waves of oppositely signed energy is also explored. Although the greatest growth rates for instabilities occur for the fundamental mode (or external Kelvin wave) interactions, for realistic oceanographic parameters the interactions of the slower moving, higher modes are more likely to be important.

On the sensitization of turbulence models to rotation and curvature

Empirical alterations of eddy-viscosity turbulence models to account for system rotation and streamline curvature are discussed. Except in a narrow class of flows, the streamline curvature itself is an inadequate entry into a model, because it is not Galilean-invariant. We propose a measure of the extra influence on the turbulence which is invariant, fully defined in three dimensions, and unifies rotation and curvature effects. This is at the expense of involving higher derivatives than the “traditional” (non-invariant) terms do. It is closely related to an idea of Knight & Saffman [1]. Its interpretation is straightforward, at least in two dimensions, and coincides with the classical measures in a few “building-block” flows such as rotating shear flow or a curved boundary layer. It can be used in one-equation, two-equation, and similar models in conjunction with empirical functions which depend on the model; an example is given.

Torsional flow: elastic instability in a finite domain

A rotational shear flow is examined in the parallel plate geometry for the Oldroyd-B fluid. The Stokes solution is found to have eigenfunctions in an unbounded radial domain while it is unique for general boundary conditions in a finite radial domain. Critical conditions for the onset of an axisymmetric secondary flow are determined for the viscoelastic fluid, and we show that there is an almost linear relationship between the aspect ratio of the plates and the critical Deborah number for this model, especially at small values of the aspect ratio. The form of the initial secondary flow is also in agreement with experimental results obtained for a Boger fluid.

Unsteady rotating shear flow of power law fluids

We have investigated unsteady circular Couette flow for non-Newtonian fluids of power law behavior. We considered two specific problems: 1. A solid cylinder rotating in a power law fluid of infinite extent and 2. A power law fluid filling the gap between two concentric rotating cylinders. Due to the nonlinearity of non-Newtonian constitutive equations, velocity disturbances from a rotating cylinder propagate through the the fluid with finite speed. The existence of this moving velocity disturbance front was observed in the analytic similarity solutions and numerical solutions. For the problem with one cylinder two numerical methods were used to determine the velocity and shear stress distributions: 1. The Method of Lines combined with the Shooting Method, and 2. A front fixing transformation was employed and the resulting problem solved via various implicit (including Crank-Nicolson) finite diference schemes. For the two cylinder problem the velocity distribution was obtained via the Method of Lines together with a variable step-size finite difference method. For both problems the temperature distribution was obtained numerically via an implicit finite difference method. The interaction effect between the angular velocity and the thermal field was addressed.

Stability of rotating shear flows in shallow water

Asymptotic methods are used to determine the dispersion equation for disturbances of rotating parallel flows in shallow water. From this equation the unstable modes and their growth rates are determined. The solution involves seven asymptotic expansions which are matched together. The results supplement and extend those which have been obtained previously using numerical methods by Griffiths, Killworth & Stern (1982) and by Hayashi & Young (1987)

Nonlinear stability of plane rotating shear flow under three—dimensional nondivergence disturbances

Nonlinear stability criterion for plane rotating shear flow under three-dimensional nondivergence disturbances was obtained by using both variational principle and convexity estimate introduced by Arnold (1965) and Holm et al. (1985). The results obtained in this paper show that the effect of Coriolis force plays an important role in the nonlinear stability criterion, and the nonlinear stability property of the basic flow depends on both the distribution of basic states and the way the external disturbance acts on the states. The upper bound of the gradient of the mass density displacement from the equilibrium $$k^2 = \left| { \vee \left[ {\rho (\bar x,t)--\rho _e (\bar x)} \right]} \right|^2 /\left[ {\rho (\bar x,t)--\rho _e (\bar x)} \right]^2 $$ is determined by the basic states and one example was given to show the exact upper value ofk. The remarks on Blumen’s paper were also given at Section 4 of this paper.

Nonaxisymmetric instabilities in rotating shear flows - Internal gravity modes in stratified media and analogies with plane flows

The role of the internal gravity modes in mediating the growth of nonaxisymmetric instabilities is investigated by studying the instability of stratified incompressible differentially rotating fluid cylinders to global nonaxisymmetric modes. The results indicate that, in addition to a modified version of the well-known principal branch mediated by surface modes of the system (analogous to f-modes of stars), there exist unstable branches of the dispersion relation mediated by internal gravity modes of the system (similar to the g-modes of stars). These branches arise due to the interaction between the g-modes. It is shown that the maximum growth rate on one of the new branches can sometimes equal or exceed that on the principal branch, thus modifying the principal branch.

Aspects of a Rotating Shear Flow over a Mountain Ridge

Abstract The effect of a vertical incident wind shear on rotating airflow over a mountain ridge is discussed physically from a variety of perspectives. The apparent paradox that the shear reduces both the vertical displacement of fluid parcels aloft and the mountain anticyclone is resolved. The importance of meridional displacements in representing the static stability field is also demonstrated.

Motion of a sphere through a fluid in stationary homogeneous flow

We have extended our previous analysis about the motion of a particle through a fluid in elongational flow to the case in which the fluid is in arbitrary stationary homogeneous flow. Expressions for the mobility in some cases of interest, namely, rotational, elongational and simple shear flow, have been derived. The nonequilibrium contributions to such a quantity depend on the corresponding inverse penetration lengths.

Thermal convection in a rotating shear flow

The dynamics of thermal convection in a rotating shear flow was studied using a three-dimensional time-dependent numerical model of the flow. The model assumes a sheared zonal wind in the form of an atmospheric jet stream and considers convective motions imbedded in this flow. It was found that rotation plays a vital role in the dynamics. Without rotation, the convective motions extract energy and momentum from the mean zonal flow. With rotation, the convective motions feed the energy and momentum into the mean flow. The results show how the small-scale convection might influence the large-scale dynamics of rotating stars and planets.

Thermal convection in a rotating shear flow : Hathaway, D.H. and R.C.J. Somerville, 1987. Geophys. astrophys. Fluid Dynam., 38(1):43–68. Space Sci. Lab., NASA Marshall Space Flight Center, Huntsville, AL 35812, USA

Thermal convection in a rotating shear flow : Hathaway, D.H. and R.C.J. Somerville, 1987. Geophys. astrophys. Fluid Dynam., 38(1):43–68. Space Sci. Lab., NASA Marshall Space Flight Center, Huntsville, AL 35812, USA

High Reynolds Number Turbulence Model of Rotating Shear Flows

固定軸のまわりに一定角速度で回転する乱流せん断流を,レイノルズ平均法を用いて予測する際に必要な乱流モデルについて検討した.モデル定数が回転に依存しないことを主眼として,レイノルズ応力方程式の高レイノルズ数モデルを導き,回転流路内の二次元乱流境界層における実結果と比較した.その結果,回転場での経験則を全く用いずに,平均場および乱流場に対する回転の効果を正しく予測することができることが示された.

High Reynolds Number Turbulence Model of Rotating Shear Flows

A Reynolds stress closure model for rotating turbulent shear flows is developed. Special attention is paid to keeping the model constants independent of rotation. First, a general forms of the model of a Reynolds stress equation and a dissipation rate equation are derived, the only restrictions of which are high Reynolds number and incompressibility. The model equations are then applied to two-dimensional equilibrium boundary layers and the effects of Coriolis acceleration on turbulence structures are discussed. Comparisons with the experimental data and with previous results in other external force fields show that there exists a very close analogy among centrifugal buoyancy and Coriolis force fields. Finally, the model is applied to predict the two-dimensional boundary layers on rotating palane walls. Comparisons with existing data confirmed its capability of predicting mean- and turbulent quantites without employing any empirical relations in rotating fields.