scholarly journals A physical conjecture for the dipolar–multipolar dynamo transition

2019 ◽  
Vol 874 ◽  
pp. 995-1020 ◽  
Author(s):  
B. R. McDermott ◽  
P. A. Davidson
Keyword(s):  
Numerical Simulations ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Low Frequency ◽  
Rossby Number ◽  
Length Scale ◽  
Inertial Waves ◽  
Rotation Rates ◽  
Rotating Turbulence ◽  
Planetary Dynamos

In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of ${\sim}0.4$. Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value $Ro^{crit}\sim 0.4$, consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos ($Ro^{crit}\sim 0.1$), is a result of a different choice of the length scale used to define the local $Ro$. We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at $Ro^{crit}\sim 0.5$. Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for $Ro>Ro^{crit}$.

scholarly journals Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

2015 ◽  
Vol 783 ◽  
pp. 412-447 ◽  
Author(s):  
Basile Gallet
Keyword(s):  
Reynolds Number ◽  
Stokes Equation ◽  
Three Dimensional ◽  
Threshold Value ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equation ◽  
Rotating Turbulence ◽  
2D Flow ◽  
Long Time

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value$Ro_{c}(Re)$of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on$Ro_{c}(Re)$and therefore determine regions of the parameter space$(Re,Ro)$where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number$Ro^{(f)}$is lower than a Grashof-number-dependent threshold value$Ro_{c}^{(f)}(Gr)$. We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value$Ro_{\mathit{abs}}^{(f)}(Gr)$of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number$Re$and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

Rotating free-shear flows. Part 2. Numerical simulations

1995 ◽  
Vol 293 ◽  
pp. 47-80 ◽  
Author(s):  
Olivier Métais ◽  
Carlos Flores ◽  
Shinichiro Yanase ◽  
James J. Riley ◽  
Marcel Lesieur
Keyword(s):  
Solid Body ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Rossby Number ◽  
Two Dimensional ◽  
Body Rotation ◽  
Mean Flow ◽  
Hairpin Vortices ◽  
Solid Body Rotation ◽  
Rotation Rates

The three-dimensional dynamics of the coherent vortices in periodic planar mixing layers and in wakes subjected to solid-body rotation of axis parallel to the basic vorticity are investigated through direct (DNS) and large-eddy simulations (LES). Initially, the flow is forced by a weak random perturbation superposed on the basic shear, the perturbation being either quasi-two-dimensional (forced transition) or three-dimensional (natural transition). For an initial Rossby number Ro(i), based on the vorticity at the inflexion point, of small modulus, the effect of rotation is to always make the flow more two-dimensional, whatever the sense of rotation (cyclonic or anticyclonic). This is in agreement with the Taylor–Proudman theorem. In this case, the longitudinal vortices found in forced transition without rotation are suppressed.It is shown that, in a cyclonic mixing layer, rotation inhibits the growth of three-dimensional perturbations, whatever the value of the Rossby number. This inhibition exists also in the anticyclonic case for |Ro(i)| ≤ 1. At moderate anticyclonic rotation rates (Ro(i) < −1), the flow is strongly destabilized. Maximum destabilization is achieved for |Ro(i) ≈ 2.5, in good agreement with the linear-stability analysis performed by Yanase et al. (1993). The layer is then composed of strong longitudinal alternate absolute vortex tubes which are stretched by the flow and slightly inclined with respect to the streamwise direction. The vorticity thus generated is larger than in the nonrotating case. The Kelvin–Helmholtz vortices have been suppressed. The background velocity profile exhibits a long range of nearly constant shear whose vorticity exactly compensates the solid-body rotation vorticity. This is in agreement with the phenomenological theory proposed by Lesieur, Yanase & Métais (1991). As expected, the stretching is more efficient in the LES than in the DNS.A rotating wake has one side cyclonic and the other anticyclonic. For |Ro(i)| ≤ 1, the effect of rotation is to make the wake more two-dimensional. At moderate rotation rates (|Ro(i)| > 1), the cyclonic side is composed of Kármán vortices without longitudinal hairpin vortices. Karman vortices have disappeared from the anticyclonic side, which behaves like the mixing layer, with intense longitudinal absolute hairpin vortices. Thus, a moderate rotation has produced a dramatic symmetry breaking in the wake topology. Maximum destabilization is still observed for |Ro(i)| ≈ 2.5, as in the linear theory.The paper also analyses the effect of rotation on the energy transfers between the mean flow and the two-dimensional and three-dimensional components of the field.

Annular and spiral patterns in flows between rotating and stationary discs

2001 ◽  
Vol 434 ◽  
pp. 65-100 ◽  
Author(s):  
E. SERRE ◽  
E. CRESPO DEL ARCO ◽  
P. BONTOUX
Keyword(s):  
Stokes Equations ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Fourier Method ◽  
Navier Stokes ◽  
Type I ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
Rotation Rates ◽  
High Level

Different instabilities of the boundary layer flows that appear in the cavity between stationary and rotating discs are investigated using three-dimensional direct numerical simulations. The influence of curvature and confinement is studied using two geometrical configurations: (i) a cylindrical cavity including the rotation axis and (ii) an annular cavity radially confined by a shaft and a shroud. The numerical computations are based on a pseudo-spectral Chebyshev–Fourier method for solving the incompressible Navier–Stokes equations written in primitive variables. The high level accuracy of the spectral methods is imperative for the investigation of such instability structures. The basic flow is steady and of the Batchelor type. At a critical rotation rate, stationary axisymmetric and/or three-dimensional structures appear in the Bödewadt and Ekman layers while at higher rotation rates a second transition to unsteady flow is observed. All features of the transitions are documented. A comparison of the wavenumbers, frequencies, and phase velocities of the instabilities with available theoretical and experimental results shows that both type II (or A) and type I (or B) instabilities appear, depending on flow and geometric control parameters. Interesting patterns exhibiting the coexistence of circular and spiral waves are found under certain conditions.

scholarly journals The intermediate Rossby number range and two-dimensional–three-dimensional transfers in rotating decaying homogeneous turbulence

2007 ◽  
Vol 587 ◽  
pp. 139-161 ◽  
Author(s):  
LYDIA BOUROUIBA ◽  
PETER BARTELLO
Keyword(s):  
Numerical Simulations ◽  
Three Dimensional ◽  
Domain Size ◽  
Homogeneous Turbulence ◽  
Finite Domain ◽  
Rossby Number ◽  
Computational Domain ◽  
Two Dimensional ◽  
Number Range ◽  
Wide Range

Rotating homogeneous turbulence in a finite domain is studied using numerical simulations, with a particular emphasis on the interactions between the wave and zero-frequency modes. Numerical simulations of decaying homogeneous turbulence subject to a wide range ofbackground rotation rates are presented. The effect of rotation is examined in two finiteperiodic domains in order to test the effect of the size of the computational domain on the results obtained, thereby testing the accurate sampling of near-resonant interactions.We observe a non-monotonic tendency when Rossby number Ro is varied from large values to the small-Ro limit, which is robust to the change of domain size. Three rotation regimes are identified and discussed: the large-, the intermediate-, and the small-Ro regimes. The intermediate-Ro regime is characterized by a positive transfer of energy from wave modes to vortices. The three-dimensional to two-dimensional transfer reaches an initial maximum for Ro ≈ 0.2 and it is associated with a maximum skewness of vertical vorticity in favour of positive vortices. This maximum is also reached at Ro ≈ 0.2. In the intermediate range an overall reduction of vertical energy transfer is observed. Additional characteristic horizontal and vertical scales of this particular rotation regime are presented and discussed.

On the structure of inertial waves produced by an obstacle in a deep, rotating container

1979 ◽  
Vol 91 (3) ◽  
pp. 415-432 ◽  
Author(s):  
K. Stewartson ◽  
H. K. Cheng
Keyword(s):  
Asymptotic Theory ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Inviscid Flow ◽  
The Body ◽  
Transverse Motion ◽  
Rossby Number ◽  
Lee Waves ◽  
Inertial Wave ◽  
Thin Obstacle

The inviscid flow above an obstacle in slow transverse motion inside a rotating vessel is analysed to study the influence of the container depth on the basic steady flow structure. An asymptotic theory is presented for an arbitrarily small Rossby number Ro = U0/2Ωl under a fixed H = hRo/l (where Ω is the angular velocity of the container, U0 the obstacle velocity relative to the vessel, h the depth of the container, and l a body length measured transversely to the rotation axis). The equations when linearized for a thin obstacle or shallow topography take on the form of the inertial-wave equation; their solutions for non-vanishing H are obtained for obstacles of three-dimensional as well as ridge-like two-dimensional shapes. In all cases analysed, the solution possesses a bimodal structure, of which one part is column-like with a vorticity proportional to the body elevation (or ground topography). The other part is confined mainly to a region enclosing the body, extending a distance O(H½) upstream of the obstacle and behind a wedge-shaped caustic front at large distances; its contribution consists of lee waves similar to that discussed by Cheng (1977) for an infinite depth. The field associated with the lee waves is then biased on the downstream side, but there is little indication of any tendency to tilting in the sense of Hide, Ibbetson & Lighthill (1968).

scholarly journals Three-dimensional core-collapse supernova simulations of massive and rotating progenitors

2020 ◽  
Vol 494 (4) ◽  
pp. 4665-4675 ◽  
Author(s):  
Jade Powell ◽  
Bernhard Müller
Keyword(s):  
Gravitational Wave ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Low Frequency ◽  
High Mass ◽  
Core Collapse ◽  
Einstein Telescope ◽  
General Relativistic ◽  
Rapid Rotator ◽  
Helium Star

ABSTRACT We present 3D simulations of the core-collapse of massive rotating and non-rotating progenitors performed with the general relativistic neutrino hydrodynamics code coconut-fmt. The progenitor models include Wolf-Rayet stars with initial helium star masses of $39\, \mathrm{ M}_{\odot }$ and $20\, \mathrm{ M}_{\odot }$, and an $18\, \mathrm{ M}_{\odot }$ red supergiant. The $39\, \mathrm{ M}_{\odot }$ model is a rapid rotator, whereas the two other progenitors are non-rotating. Both Wolf-Rayet models produce healthy neutrino-driven explosions, whereas the red supergiant model fails to explode. By the end of the simulations, the explosion energies have already reached $1.1\times 10^{51}\, $ and $0.6\times 10^{51}\, \mathrm{erg}$ for the $39\, \mathrm{ M}_{\odot }$ and $20\, \mathrm{ M}_{\odot }$ model, respectively. They produce neutron stars of relatively high mass, but with modest kicks. Due to the alignment of the bipolar explosion geometry with the rotation axis, there is a relatively small misalignment of 30° between the spin and the kick in the rapidly rotating $39\, \mathrm{ M}_{\odot }$ model. For this model, we find that rotation significantly changes the dependence of the characteristic gravitational-wave frequency of the f-mode on the proto-neutron star parameters compared to the non-rotating case. Its gravitational-wave amplitudes would make it detectable out to almost 2 Mpc by the Einstein Telescope. The other two progenitors have considerably smaller detection distances, despite significant low-frequency emission in the most sensitive frequency band of current gravitational-wave detectors.

scholarly journals Scaling analysis and simulation of strongly stratified turbulent flows

2007 ◽  
Vol 585 ◽  
pp. 343-368 ◽  
Author(s):  
G. BRETHOUWER ◽  
P. BILLANT ◽  
E. LINDBORG ◽  
J.-M. CHOMAZ
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Small Scale ◽  
Scaling Analysis ◽  
Length Scale ◽  
Stratified Turbulence

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter $\mathcal{R} \,{=}\, \hbox{\it Re} F^2_h$. When $\mathcal{R} \,{\gg}\, 1$, viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When $\mathcal{R} \,{\ll}\, 1$, vertical viscous shearing is important so that $l_v \,{\sim}\, l_h/\hbox{\it Re}^{1/2}$ (lh is a characteristic horizontal length scale). The parameter $\cal R$ is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when $\mathcal{R} \,{\gg}\, 1$: the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being $\cal R$. The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of $\cal R$ but they tend to be smooth for $\cal R$< 1, while for $\cal R$ > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for $\cal R$ < 1 but tends to isotropy as $\cal R$ increases above unity. When $\mathcal{R}$ < 1, the horizontal and vertical energy spectra are very steep while, when $\cal R$ > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

Towards long length scale unsteady modelling in turbomachines

2003 ◽  
Vol 217 (1) ◽  
pp. 75-82 ◽  
Author(s):  
L Xu ◽  
T. P. Hynes ◽  
J. D. Denton
Keyword(s):  
Body Force ◽  
Three Dimensional ◽  
Low Frequency ◽  
High Amplitude ◽  
Length Scale ◽  
Fine Mesh ◽  
Cfd Method ◽  
Coarse Meshes ◽  
Low Amplitude ◽  
Potential Interactions

The modelling issue of long length scale unsteadiness in turbomachines is discussed in this paper and a new three-dimensional CFD method with a hierarchy of body force models is proposed. A program has been developed to deal with the long length scale problem using efficient coarse meshes and large time steps, together with body force models. The distributed viscous body force can be calculated either using a very simple viscous wall shear stress model or extracted from a ‘stand-alone’ three-dimensional single passage steady calculation using a fine mesh. By eliminating the need to compute fine viscous scales, the proposed method is several orders of magnitude more efficient than ‘standard’ fine mesh N-S unsteady calculations while maintaining respectable resolution down to the scale of blade-to-blade variation, including wake/potential interactions between blade rows. Two sample cases with high-frequency low-amplitude and low-frequency high-amplitude, respectively, are used to illustrate the accuracy of the model.

Evolution of a turbulent cloud under rotation

2014 ◽  
Vol 756 ◽  
pp. 488-509 ◽  
Author(s):  
A. Ranjan ◽  
P. A. Davidson
Keyword(s):  
Rotation Axis ◽  
Spatial Filter ◽  
Dimensionless Time ◽  
Inertial Waves ◽  
Lagrangian Particle ◽  
Rotating Turbulence ◽  
Negative Helicity ◽  
The Waves ◽  
Group Velocities ◽  
Quiescent Fluid

AbstractLocalized patches of turbulence frequently occur in geophysics, such as in the atmosphere and oceans. The effect of rotation, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\boldsymbol{\Omega}$, on such a region (a ‘turbulent cloud’) is governed by inhomogeneous dynamics. In contrast, most investigations of rotating turbulence deal with the homogeneous case, although inhomogeneous turbulence is more common in practice. In this paper, we describe the results of $512^3$ direct numerical simulations (DNS) of a turbulent cloud under rotation at three Rossby numbers ($\mathit{Ro}$), namely 0.1, 0.3 and 0.5. Using a spatial filter, fully developed homogeneous turbulence is vertically confined to the centre of a periodic box before the rotation is turned on. Energy isosurfaces show that columnar structures emerge from the cloud and grow into the adjacent quiescent fluid. Helicity is used as a diagnostic and confirms that these structures are formed by inertial waves. In particular, it is observed that structures growing parallel to the rotation axis (upwards) have negative helicity and those moving antiparallel (downwards) to the axis have positive helicity, a characteristic typical of inertial waves. Two-dimensional energy spectra of horizontal wavenumbers, $k_{\perp }$, versus dimensionless time, $2 \varOmega t$, confirm that these columnar structures are wavepackets which travel at the group velocities of inertial waves. The kinetic energy transferred from the turbulent cloud to the waves is estimated using Lagrangian particle tracking to distinguish between turbulent and ‘wave-only’ regions of space. The amount of energy transferred to waves is 40 % of the initial at $\mathit{Ro}=0.1$, while it is 16 % at $\mathit{Ro}=0.5$. In both cases the bulk of the energy eventually resides in the waves. It is evident from this observation that inertial waves can carry a significant portion of the energy away from a localized turbulent source and are therefore an efficient mechanism of energy dispersion.

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