Stability of Rayleigh-stable Couette flow between two differentially heated cylinders (original) (raw)
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Linear stability of cylindrical Couette flow in the convection regime
Physics of Fluids, 2005
The instability of steady circular Couette flow with radial heating across a vertically oriented annulus with a rotating inner cylinder and a stationary outer cylinder is investigated using a linear stability analysis. The convection regime base flow is developed for an infinite aspect ratio geometry and constant fluid properties with buoyancy included through the Boussinesq approximation. The base flow is characterized by a dimensionless stratification parameter ␥ that is proportional to the vertical temperature gradient. Critical stability boundaries are calculated for this assumed base flow with respect to both toroidal and helical disturbances. The numerical investigation is primarily restricted to a radius ratio of 0.6 at a Prandtl number of 100. Critical stability boundaries in Taylor-Grashof number space are presented for two values of the stratification parameter ␥ ͑4 and 13͒. The results follow the development of critical stability from Taylor cells at small Grashof numbers up to a maximum Grashof number used in this calculation of 20 000 and 80 000 for ␥ = 4 and 13, respectively. Results show that increasing the stratification parameter stabilizes the isothermal Taylor vortices, followed by a destabilization at higher azimuthal mode numbers ͑n Ͼ 0͒. The results also show that for ␥ =4 ͑close to the conduction regime͒, two modes are obtained: one is axisymmetric and the other is nonaxisymmetric. However, for the convection regime ͑large ␥͒ six asymmetric modes are obtained. Finally, the disturbance wavelength, phase speed, and spiral inclination angle are presented as a function of the critical Grashof number for the stratification parameters considered in this work.
Instability of Taylor-Couette Flow between Concentric Rotating Cylinders
The energy gradient theory is used to study the instability of Taylor-Couette flow between concentric rotating cylinders. This theory has been proposed in our previous works. In our previous studies, the energy gradient theory was demonstrated to be applicable for wall-bounded parallel flows. It was found that the critical value of the energy gradient parameter K max at turbulent transition is about 370-389 for wall-bounded parallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no turbulence occurs. In this paper, the detailed derivation for the calculation of the energy gradient parameter in the flow between concentric rotating cylinders is provided. The calculated results for the critical condition of primary instability (with semi-empirical treatment) are found to be in very good agreement with the experiments in the literature. A possible mechanism of spiral turbulence generation observed for counter-rotation of two cylinders can also be explained using the energy gradient theory. The energy gradient theory can serve to relate the condition of transition in Taylor-Couette flow to that in plane Couette flow. The latter reasonably becomes the limiting case of the former when the radii of cylinders tend to infinity. It is our contention that the energy gradient theory is possibly fairly universal for analysis of flow instability and turbulent transition, and is found valid for both pressure and shear driven flows in parallel and rotating flow configurations.
Weak temperature gradient effect on the stability of the circular Couette flow
The European Physical Journal B, 2008
We have investigated the influence of a weak radial temperature gradient in a wide gap and large aspect ratio Couette-Taylor system. The inner cylinder is rotating and can be heated or cooled, the outer cylinder is at rest and immersed in a large thermal bath. We found that a radial temperature gradient destabilizes the Couette flow leading to a pattern of traveling helicoidal vortices occurring only near the bottom of the system. The size of the pattern increases as the rotation frequency of the cylinder is increased. We have characterized the spatiotemporal properties of the pattern and we have shown that it behaves as a wall mode found in the simulation of the complex Ginzburg-Landau equation with homogeneous boundary conditions.
Stability analysis between two concentric rotating cylinders with heat and mass transfer
Heat Transfer, 2019
The stability of the liquid/vapor interface between two concentric revolving cylinders is examined. The transfer of heat/mass is allowed at the interface. Both the cylinders rotate with different angular velocities. The fluids inside the annular region are taken as incompressible and viscous. The theory of viscous potential flow analysis is used to add the viscous effects. The normal mode technique is used to calculate the growth of perturbations. If we rotate the inner cylinder, it is seen that asymmetric disturbances have a destabilizing character at the interface but the rotation of the outer cylinder has a stabilizing effect. We found that an asymmetric disturbance destabilizes the interface if the inner cylinder rotates. It is found that the arrangement gets destabilized on rotating of the inner cylinder but rotation of the outer cylinder induces stability, and the most stable case is found when the inner cylinder is stationary and the outer cylinder is rotating.
Computers & Fluids, 2014
From 28 high-order DNS computations, one investigates the formation of instabilities due to the strong competition between an azimuthal flow induced by rotation and an axial flow due to convection in a tall Taylor-Couette apparatus (C ¼ 80; g ¼ 0:8) submitted to a radial temperature gradient. One explores the richness of the transition diagram that reports seven different flow patterns appearing either as spiral rolls, wavy vortices or a combination of both depending on the Taylor and Rayleigh numbers. The partial spiral regime observed experimentally by Guillerm is not recovered at very low Rayleigh numbers. The spatio-temporal properties of the different spirals close to the threshold of the primary instability are fairly predicted and a new insight on the flow and thermal structures of the instabilities is gained from this study. Finally, the distributions of the Nusselt number against the Taylor number are established for various Rayleigh numbers.
Heat and fluid flow around two co-rotating cylinders in tandem arrangement
International Journal of Thermal Sciences, 2019
This paper discusses on the heat and fluid flow around two co-rotating cylinders in the tandem arrangement. The non-dimensional rotating speed (R.S) varies in the range of 0 ≤ R.S ≤ 4 and different non-dimensional gap spaces such as G/D = 1.5, 2.0, and 3.0 are considered between the cylinders. Computations are performed at the Reynolds number of 200 with constant Prandtl number of 7.0. It is demonstrated that rotating the cylinders deforms the recirculating regions of both upstream and downstream cylinders in which the rate of this deformation changes as a function of the R.S and G/D. On the other hand, co-rotating the cylinders shows some additional events such as the azimuthal displacement of the front stagnation points and development of the negative lift coefficient for both cylinders. It is found that the instabilities of the shear layer for both upstream and downstream cylinders are maximum at R.S = 1 and with increasing the R.S, the vortex shedding suppresses around the cylinders due to dominating the fluid rotating zone. Finally, it is revealed that at higher R.S values, a uniform Nusselt number distribution can be observed on both cylinders regardless of the gap space between the upstream and downstream cylinders.
Couette flow of two fluids between concentric cylinders
Journal of Fluid Mechanics, 1985
We consider the flow of two immiscible fluids lying between concentric cylinders when the outer cylinder is fixed and the inner one rotates. The interface is assumed to be concentric with the cylinders, and gravitational effects are neglected. We present a numerical study of the effect of different viscosities, different densities and surface tension on the linear stability of the Couette flow. Our results indicate that, with surface tension, a thin layer of the less-viscous fluid next to either cylinder is linearly stable and that it is possible to have stability with the less dense fluid lying outside.
Thermal Analysis Circular Couette Flow of Non-Newtonian Fluid with Viscous Dissipation
2018
The forced convection heat transfer in the circular couette flow of Non-Newtonian fluid is investigated when the inner cylinder is rotated at angular speed and the outer cylinder is fixed. The fluid viscosity is considered concurrently to be dependent on the temperature and shear rate. The temperature dependency of viscosity is modeled exponentially according to the Nahme law and dependence of viscosity on shear is modeled with the Carreau equation. The Viscous dissipation term is adding intricacy to the already highly interdependent set of governing motion and energy equations. The highly nonlinear governing equations are derived for the steady state base flow in the narrow gap limit. The perturbation method has been applied to obtain an approximate solution for these equations. The effect of governing parameter such as Brinkman numbers and Deborah number on the thermal stability is examined. In addition, the analysis illustrated that the Nusselt number of the outer cylinder increases as the Deborah number increases. It, although, decreases by increasing Brinkman number. The pseudoplastic fluid between concentric cylinders is heated as Brinkman number and increases due to frictional loss and it is cooled as Deborah number increases due to the fluid elasticity behavior.