scholarly journals Exploring the phase diagram of fully turbulent Taylor–Couette flow

2014 ◽  
Vol 761 ◽  
pp. 1-26 ◽  
Author(s):  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Siegfried Grossmann ◽  
Detlef Lohse
Keyword(s):  
Phase Diagram ◽  
Reynolds Number ◽  
Couette Flow ◽  
Parameter Space ◽  
Coriolis Force ◽  
Scaling Laws ◽  
Outer Cylinder ◽  
Rotating Cylinders ◽  
Aspect Ratios ◽  
Cylinder Rotation

AbstractDirect numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to $3\times 10^{5}$, corresponding to Taylor numbers of $\mathit{Ta}=4.6\times 10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

scholarly journals Effect of the number of vortices on the torque scaling in Taylor–Couette flow

2014 ◽  
Vol 748 ◽  
pp. 756-767 ◽  
Author(s):  
B. Martínez-Arias ◽  
J. Peixinho ◽  
O. Crumeyrolle ◽  
I. Mutabazi
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Couette Flow ◽  
Scaling Laws ◽  
Taylor Vortex ◽  
Large Radius ◽  
Aspect Ratios ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractTorque measurements in Taylor–Couette flow, with large radius ratio and large aspect ratio, over a range of velocities up to a Reynolds number of 24 000 are presented. Following a specific procedure, nine states with distinct numbers of vortices along the axis were found and the aspect ratios of the vortices were measured. The relationship between the speed and the torque for a given number of vortices is reported. In the turbulent Taylor vortex flow regime, at relatively high Reynolds number, a change in behaviour is observed corresponding to intersections of the torque–speed curves for different states. Before each intersection, the torque for a state with a larger number of vortices is higher. After each intersection, the torque for a state with a larger number of vortices is lower. The exponent, from the scaling laws of the torque, always depends on the aspect ratio of the vortices. When the Reynolds number is rescaled using the mean aspect ratio of the vortices, only a partial collapse of the exponent data is found.

Velocity profiles, flow structures and scalings in a wide-gap turbulent Taylor–Couette flow

2017 ◽  
Vol 831 ◽  
pp. 330-357 ◽  
Author(s):  
A. Froitzheim ◽  
S. Merbold ◽  
C. Egbers
Keyword(s):  
Reynolds Number ◽  
Nusselt Number ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Azimuthal Velocity ◽  
Cylinder Wall ◽  
Taylor Vortices ◽  
Wide Gap ◽  
Taylor Couette ◽  
Taylor Couette Flow

Fully turbulent Taylor–Couette flow between independently rotating cylinders is investigated experimentally in a wide-gap configuration ($\unicode[STIX]{x1D702}=0.5$) around the maximum transport of angular momentum. In that regime turbulent Taylor vortices are present inside the gap, leading to a pronounced axial dependence of the flow. To account for this dependence, we measure the radial and azimuthal velocity components in horizontal planes at different cylinder heights using particle image velocimetry. The ratio of angular velocities of the cylinder walls $\unicode[STIX]{x1D707}$, where the torque maximum appears, is located in the low counter-rotating regime ($\unicode[STIX]{x1D707}_{max}(\unicode[STIX]{x1D702}=0.5)=-0.2$). This point coincides with the smallest radial gradient of angular velocity in the bulk and the detachment of the neutral surface from the outer cylinder wall, where the azimuthal velocity component vanishes. The structure of the flow is further revealed by decomposing the flow field into its large-scale and turbulent contributions. Applying this decomposition to the kinetic energy, we can analyse the formation process of the turbulent Taylor vortices in more detail. Starting at pure inner cylinder rotation, the vortices are formed and strengthened until $\unicode[STIX]{x1D707}=-0.2$ quite continuously, while they break down rapidly for higher counter-rotation. The same picture is shown by the decomposed Nusselt number, and the range of rotation ratios, where turbulent Taylor vortices can exist, shrinks strongly in comparison to investigations at much lower shear Reynolds numbers. Moreover, we analyse the scaling of the Nusselt number and the wind Reynolds number with the shear Reynolds number, finding a communal transition at approximately $Re_{S}\approx 10^{5}$ from classical to ultimate turbulence with a transitional regime lasting at least up to $Re_{S}\geqslant 2\times 10^{5}$. Including the axial dispersion of the flow into the calculation of the wind amplitude, we can also investigate the wind Reynolds number as a function of the rotation ratio $\unicode[STIX]{x1D707}$, finding a maximum in the low counter-rotating regime slightly larger than $\unicode[STIX]{x1D707}_{max}$. Based on our study it becomes clear that the investigation of counter-rotating Taylor–Couette flows strongly requires an axial exploration of the flow.

scholarly journals Optimal Taylor–Couette flow: direct numerical simulations

2013 ◽  
Vol 719 ◽  
pp. 14-46 ◽  
Author(s):  
Rodolfo Ostilla ◽  
Richard J. A. M. Stevens ◽  
Siegfried Grossmann ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Angular Velocity ◽  
Couette Flow ◽  
Scaling Laws ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Experimental Result ◽  
Counter Rotation ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractWe numerically simulate turbulent Taylor–Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh–Bénard flow. Reynolds numbers of $R{e}_{i} = 8\times 1{0}^{3} $ and $R{e}_{o} = \pm 4\times 1{0}^{3} $ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers $Ta$ up to $1{0}^{8} $. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente Turbulent Taylor–Couette (${T}^{3} C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a= - {\omega }_{o} / {\omega }_{i} $ of about ${a}_{\mathit{opt}} = 0. 33$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of ${a}_{\mathit{opt}} = 0. 15$ for $Ta= 2. 5\times 1{0}^{7} $. An explanation for this shift is elucidated, consistent with the experimental result that ${a}_{\mathit{opt}} $ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities.

Stability of obliquely driven cavity flow

2021 ◽  
Vol 928 ◽  
Author(s):  
Pierre-Emmanuel des Boscs ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Couette Flow ◽  
Critical Reynolds Number ◽  
Cavity Flow ◽  
Basic Flow ◽  
Driven Cavity ◽  
Aspect Ratios ◽  
Driven Cavity Flow ◽  
Yaw Angle

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

scholarly journals Self-sustaining critical layer/shear layer interaction in annular Poiseuille–Couette flow at high Reynolds number

2020 ◽  
Vol 476 (2235) ◽  
pp. 20190600 ◽  
Author(s):  
Rishi Kumar ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Axial Direction ◽  
Critical Layer ◽  
Large Reynolds Number ◽  
Cylindrical Annulus ◽  
Axial Pressure Gradient ◽  
High Reynolds

The nonlinear stability of annular Poiseuille–Couette flow through a cylindrical annulus subjected to axisymmetric and helical disturbances is analysed theoretically at asymptotically large Reynolds number R based on the radius of the outer cylinder and the constant axial pressure gradient applied. The inner cylinder moves with a prescribed positive or negative velocity in the axial direction. A distinguished scaling for the disturbance size Δ =  O ( R −4/9 ) is identified at which the jump in vorticity across the fully nonlinear critical layer is in tune with that induced across a near-wall shear layer. The disturbance propagates at close to the velocity of the inner cylinder and possesses a wavelength comparable to the radius of the outer cylinder. The dynamics of the critical layer, shear layer and the Stokes layer adjacent to the stationary wall are discussed in detail. In the majority of the pipe, the disturbance is governed predominantly by inviscid dynamics with the pressure perturbation satisfying a form of Rayleigh’s equation. For a radius ratio δ in the range 0 <  δ  < 1 and a positive sliding velocity V , a numerical solution of the Rayleigh equation exists for sliding velocities in the range 0 <  V  < 1 −  δ 2  + 2 δ 2 ln δ , whereas if V  < 0, solutions exist for 1 −  δ 2  + 2ln δ  <  V  < 0. The amplitude equations for both these situations are derived analytically, and we further find that the corresponding asymptotic structures break down when the maximum value of the basic flow becomes located at the inner and outer walls, respectively.

scholarly journals Turbulence decay towards the linearly stable regime of Taylor–Couette flow

2014 ◽  
Vol 748 ◽  
Author(s):  
Rodolfo Ostilla-Mónico ◽  
Roberto Verzicco ◽  
Siegfried Grossmann ◽  
Detlef Lohse
Keyword(s):  
Reynolds Number ◽  
Accretion Disks ◽  
Outer Cylinder ◽  
Flow State ◽  
Initial State ◽  
Stable Regime ◽  
Turbulence Decay ◽  
Taylor Couette ◽  
Cylinder Rotation ◽  
Damped Oscillator

AbstractTaylor–Couette (TC) flow is used to probe the hydrodynamical (HD) stability of astrophysical accretion disks. Experimental data on the subcritical stability of TC flow are in conflict about the existence of turbulence (cf. Ji et al. (Nature, vol. 444, 2006, pp. 343–346) and Paoletti et al. (Astron. Astroph., vol. 547, 2012, A64)), with discrepancies attributed to end-plate effects. In this paper we numerically simulate TC flow with axially periodic boundary conditions to explore the existence of subcritical transitions to turbulence when no end plates are present. We start the simulations with a fully turbulent state in the unstable regime and enter the linearly stable regime by suddenly starting a (stabilizing) outer cylinder rotation. The shear Reynolds number of the turbulent initial state is up to $Re_s \lesssim 10^5$ and the radius ratio is $\eta =0.714$. The stabilization causes the system to behave as a damped oscillator and, correspondingly, the turbulence decays. The evolution of the torque and turbulent kinetic energy is analysed and the periodicity and damping of the oscillations are quantified and explained as a function of shear Reynolds number. Though the initially turbulent flow state decays, surprisingly, the system is found to absorb energy during this decay.

Flow Characteristics of Microglass Fiber Suspension in Polymeric Fluids in Spherical Gaps

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Hiroshi Yamaguchi ◽  
Xin-Rong Zhang ◽  
Xiao-Dong Niu ◽  
Yuta Ito
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Flow Characteristics ◽  
Flow Region ◽  
Fiber Suspension ◽  
Polymeric Fluids ◽  
Polymeric Fluid ◽  
Aspect Ratios ◽  
Gap Ratio ◽  
Microglass Fibers

An experimental study is carried out to investigate the effects of microglass fiber suspensions in the non-Newtonian fluids in a gap between an inner rotating sphere and an outer whole stationary sphere. In the experiments, the microglass fibers with different aspect ratios are mixed with a macromolecule polymeric fluid to obtain different suspension fluids. For comparison, a Newtonian fluid and the non-Newtonian polymeric fluid are also studied. The stationary torques of the inner sphere that the test fluids acted on are measured under conditions of various concentric spherical gaps and rotational Reynolds numbers. It is found that the polymeric fluid could be governed by the Couette flow at a gap ratio of less than 0.42 and the Reynolds number of less than 100, while the fiber-suspended polymeric fluids could expand the Couette flow region more than the Reynolds number of 100 at the same gap ratios.

scholarly journals Second-Order Phase Transition in Counter-Rotating Taylor–Couette Flow Experiment

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 58
Author(s):  
Kerstin Avila ◽  
Björn Hof
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Finite Size ◽  
Universality Class ◽  
Flow Speed ◽  
Second Order ◽  
Parallel Plates ◽  
Aspect Ratios ◽  
Entire Flow ◽  
Observation Times

In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available.

Experimental Study of Axial Slit Wall Effect on Taylor-Couette Flow

2007 ◽  
Author(s):  
Sang-Hyuk Lee ◽  
Hyoung-Bum Kim
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Vortex Flow ◽  
Outer Cylinder ◽  
Velocity Fields ◽  
Taylor Vortex ◽  
Stable Mode ◽  
Taylor Vortex Flow ◽  
Taylor Couette ◽  
Taylor Couette Flow

Taylor-Couette flow has been studied extensively and lots of variables which affect the flow instability are being reported. The wall geometry effect of Taylor-Couette flow, however, has been less studied. In this study, we investigated the effect of axial slit of outer cylinder. This kind of configuration can be easily seen in rotating machinery. Particle image velocimetry method was used to measure the velocity fields in longitudinal and latitudinal planes. The index matching method was used to avoid light refraction. The velocity fields between the slit and plain model which has the smooth wall were compared. From the experiments, both models have the same flow mode below Re = 143. The transition from circular Couette flow to plain Taylor vortex flow began at Re = 103, and the next transition to wavy vortex flow occurred at 124. The effect of slit wall appeared when the Reynolds number is larger than Re = 143. Above this Reynolds number, there was no stable mode and plain and wavy Taylor vortex flow randomly appeared.

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