Non-dimensional Rotation Rate Research Articles

Subcritical instability of finite circular Couette flow with stationary inner cylinder

The presence of endwalls in Taylor–Couette flows has far reaching effects, leading to dynamics that are qualitatively different from those associated with the idealized situation involving infinitely long cylinders. This is well known in the classical situation where the inner cylinder is rotating and the outer cylinder is stationary. The effects of endwalls in the centrifugally stable situation with stationary inner cylinder and rotating outer cylinder have not been previously considered in detail. The meridional flows induced by the endwalls lead to the formation of a thin sidewall boundary layer on the inner cylinder wall if the endwalls are rotating, or on the outer cylinder wall if they are stationary. At sufficiently high Reynolds numbers (non-dimensional rotation rate of the outer cylinder), the sidewall boundary layer has concentrated shear, the pressure gradient in the azimuthal direction (which is the streamwise direction for the boundary layer flow) is zero (the flow is axisymmetric) and the boundary layer thickness is constant. At a critical Reynolds number, the sidewall boundary layer loses stability at a subcritical Hopf bifurcation, breaking the axisymmetry of the basic state flow, and for Reynolds numbers slightly above critical, the basic state is unstable to a packet of Hopf modes with azimuthal wavenumbers clustered about the critical wavenumber. The early time evolution of the critical Hopf mode is a rotating wave whose behaviour is analogous to a Tollmien–Schlichting wave. As the Hopf modes grow with time, nonlinear interactions lead to modulations in the waves, localization of the disturbances and the evolution of concentrated streamwise vortical streaks which become very intense via vortex stretching.

EFFECT OF ROTATION RATES ON THE LAMINAR FLOW AND HEAT TRANSFER PAST A CIRCULAR CYLINDER

In this work, forced convection heat transfer past a rotating circular cylinder with a constant non-dimensional rotation rate α varying from 0 to 6 was investigated for Reynolds numbers of 20-200 and a Prandtl number of 0.7. The numerical calculations are carried out by using a finite-volume method based commercial computational fluid dynamics solver FLUENT. The successive changes in the flow pattern are studied as a function of the rotation rate. Suppression of vortex shedding occurs as the rotation rate increases (α > 2). A second kind of instability appears for higher rotation speed where a series of counter-clockwise vortices is shed in the upper shear layer. The rotation attenuates the secondary instability and increases the critical Reynolds number for the appearance of this instability. In addition, time-averaged (lift and drag coefficients and Nusselt number) results were obtained and compared with the literature data. A good agreement was obtained for both the local and averaged values.

Free stream flow and forced convection heat transfer across rotating circular cylinder in steady regime: effects of rotation, Prandtl number and thermal boundary condition

Free stream flow and forced convection heat transfer across a rotating circular cylinder have been investigated numerically for the range of Reynolds number between 5 and 40 (5 ≤ Re ≤ 40). The non-dimensional rotation rate, α is employed upto a maximum value of 5. The governing equations namely, continuity, momentum, and energy equations have been solved in 2D steady laminar flow regime for the range of Prandtl number between 0.7 and 67 (0.7 ≤ Pr ≤ 67). Uniform heat flux (UHF) and constant wall temperature (CWT) boundary conditions have been considered, separately, for the range of afore mentioned parameters. New results are presented depicting the behavior of temperature field around the cylinder rotating in a high Pr environment (Pr ≥ 33) with non-dimensional rotation rate in the range between 0 and 5 (0 ≤ α ≤ 5). It is noted that the heat transfer across the cylinder becomes increasingly independent of the thermal boundary condition i.e. UHF or CWT.

On the development of lift and drag in a rotating and translating cylinder

The two-dimensional flow around a rotating cylinder is investigated numerically using a vorticity forces formulation with the aim of analyzing quantitatively the flow structures, and their evolutions, that contribute to the lift and drag forces on the cylinder. The Reynolds number considered, based on the cylinder diameter and steady free stream speed, is Re=200, while the non-dimensional rotation rate (ratio of the surface speed and free stream speed) selected was α=1 and 3. For α=1 the wake behind the cylinder for the fully developed flow is oscillatory due to vortex shedding, and so are the lift and drag forces. For α=3 the fully developed flow is steady with constant (high) lift and (low) drag. Each of these cases is considered in two different transient problems, one with angular acceleration of the cylinder and constant speed, and the other one with translating acceleration of the cylinder and constant rotation. We characterize quantitatively the contributions of individual fluid elements (vortices) to aerodynamic forces, explaining and quantifying the mechanisms by which the lift is generated in each case. In particular, for high rotation (when α=3), we explain the relation between the mechanisms of vortex shedding suppression and those by which the lift is enhanced and the drag is almost suppressed when the fully developed flow is reached.

Three-dimensional flow past a rotating cylinder

AbstractThree-dimensional computations are carried out for a spinning cylinder placed in a uniform flow. The non-dimensional rotation rate is varied in the range $0.0\leqslant {\it\alpha}\leqslant 5.0$. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in primitive variables formulation. Linear stability analysis of the steady state shows the existence of several new unstable three-dimensional modes for $200\leqslant \mathit{Re}\leqslant 350$ and $4.0\leqslant {\it\alpha}\leqslant 5.0$. The curves of neutral stability of these modes are presented in the $\mathit{Re}{-}{\it\alpha}$ parameter space. For the flow at $\mathit{Re}=200$ and rotation rate in the ranges $0.0\leqslant {\it\alpha}\leqslant 1.91$ and $4.34\leqslant {\it\alpha}\leqslant 4.7$, the vortex shedding, earlier reported in two dimensions and commonly referred to as parallel shedding, can also exist as oblique shedding. In this mode of shedding, the vortices are inclined to the axis of the cylinder. In fact, parallel shedding is a special case of oblique shedding. It is found that the span of the cylinder plays a significant role in the time evolution of the flow. Of all the unstable eigenmodes, with varied spanwise wavenumber, only the ones whose integral number of wavelengths fit the span length of the cylinder are selected to grow. For the flow at $\mathit{Re}=200$, two steady states exist for $4.8\leqslant {\it\alpha}\leqslant 5.0$. While one of them is associated with unstable eigenmodes, the other is stable to all infinitesimal perturbations. In this regime, irrespective of the initial conditions, the fully developed flow is steady and devoid of any instabilities.

Heat Transfer Suppression in Flow Around a Rotating Circular Cylinder at High Prandtl Number

Forced convection heat transfer across a 2D circular cylinder rotating at constant speed is investigated. Numerical simulations are carried out at constant values of Reynolds number in the range of 80–160. The non-dimensional rotation rate, α, is examined up to a maximum value of 5.3 for Prandtl number (Pr) 7. At constant Reynolds number, heat transfer suppression kicks off even at lower rotation rates contrary to the systems with lower values of Pr. Moreover, the heat transfer suppression parameter in such cases saturates at almost the double value at intermediate rotation rates as compared to the systems with low Pr.

Investigation of the interaction of a turbulent impinging jet and a heated, rotating disk

The case of a turbulent round jet impinging perpendicularly onto a rotating, heated disc is investigated, in order to understand the mechanisms at the origin of the influence of rotation on the radial wall jet and associated heat transfer. The present study is based on the complementary use of an analysis of the orders of magnitude of the terms of the mean momentum and Reynolds stress transport equations, available experiments, and dedicated Reynolds-averaged Navier–Stokes computations with refined turbulence models. The Reynolds number Rej = 14 500, the orifice-to-plate distance H = 5D, where D is the jet-orifice diameter, and the four rotation rates were chosen to match the experiments of Minagawa and Obi [“Development of turbulent impinging jet on a rotating disk,” Int. J. Heat Fluid Flow 25, 759–766 (2004)] and comparisons are made with the Nusselt number distribution measured by Popiel and Boguslawski [“Local heat transfer from a rotating disk in an impinging round jet,” J. Heat Transfer 108, 357–364 (1986)], at a higher Reynolds number. The overestimation of turbulent mixing in the free-jet before the impact on the disk is detrimental to the prediction of the impingement region, in particular of the Nusselt number close to the symmetry axis, but the self-similar wall jet developing along the disk is correctly reproduced by the models. The analysis, experiments, and computations show that the rotational effect do not directly affect the outer layer, but only the inner layer of the wall jet. A noteworthy consequence is that entrainment at the outer edge of the wall jet is insensitive to rotation, which explains the dependence of the wall-jet thickness on the inverse of the non-dimensional rotation rate, observed in the experiments and the Reynolds stress model computations, but not reproduced by the eddy-viscosity models, due to the algebraic dependence to the mean flow. The analysis makes moreover possible the identification of a scenario for the appearance of rotational effects when the rotation rate is gradually increased. For weak rotation rates, the rotation-induced boundary layer appears but does not break the self-similar solution observed for the case without rotation. For intermediate rotation rates, the production of the azimuthal Reynolds stress becomes much stronger than other components, leading to a complete modification of the turbulence anisotropy which is reproduced only by Reynolds stress models. For strong rotation rates, centrifugal effects dominate, leading to an acceleration and thinning of the jet, and consequently an increase of turbulent production and heat transfer, reproduced by all the turbulence models.

Experimental evidence of new three-dimensional modes in the wake of a rotating cylinder

AbstractA recent numerical study by Rao et al. (J. Fluid Mech., vol. 717, 2013, pp. 1–29) predicted the existence of several previously unobserved linearly unstable three-dimensional modes in the wake of a spinning cylinder in cross-flow. While linear stability analysis suggests that some of these modes exist for relatively limited ranges of Reynolds numbers and rotation rates, this may not be true for fully developed nonlinear wakes. In the current paper, we present the results of water channel experiments on a rotating cylinder in cross-flow, for Reynolds numbers $200\leqslant \mathit{Re}\leqslant 275$ and non-dimensional rotation rates $0\leqslant \alpha \leqslant 2. 5$. Using particle image velocimetry and digitally post-processed hydrogen bubble flow visualizations, we confirm the existence of the predicted modes for the first time experimentally. For instance, for $\mathit{Re}= 275$ and a rotation rate of $\alpha = 1. 7$, we observe a subharmonic mode, mode C, with a spanwise wavelength of ${\lambda }_{z} / d\approx 1. 1$. On increasing the rotation rate, two modes with a wavelength of ${\lambda }_{z} / d\approx 2$ become unstable in rapid succession, termed modes D and E. Mode D grows on a shedding wake, whereas mode E consists of streamwise vortices on an otherwise steady wake. For $\alpha \gt 2. 2$, a short-wavelength mode F appears localized close to the cylinder surface with ${\lambda }_{z} / d\approx 0. 5$, which is presumably a manifestation of centrifugal instability. Unlike the other modes, mode F is a travelling wave with a spanwise frequency of ${\mathit{St}}_{3D} \approx 0. 1$. In addition to these new modes, observations on the one-sided shedding process, known as the ‘second shedding’, are reported for $\alpha = 5. 1$. Despite suggestions from the literature, this process seems to be intrinsically three-dimensional. In summary, our experiments confirm the linear predictions by Rao et al., with very good agreement of wavelengths, symmetries and the phase velocity for the travelling mode. Apart from this, these experiments examine the nonlinear saturated state of these modes and explore how the existence of multiple unstable modes can affect the selected final state. Finally, our results establish that several distinct three-dimensional instabilities exist in a relatively confined area on the $\mathit{Re}$–$\alpha $ parameter map, which could account for their non-detection previously.

Three-dimensionality in the wake of a rotating cylinder in a uniform flow

AbstractThe wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers $\mathit{Re}\leqslant 400$ and non-dimensional rotation rates of $\alpha \leqslant 2. 5$. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for $\alpha \geqslant 2. 1$ for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of $\alpha \leqslant 1$, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the three-dimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

Flow Past a Rotating Cylinder at Low and High Rotation Rates

Abstract Flow past a rotating circular cylinder is studied experimentally. The experiments are carried out in a water tunnel at Reynolds numbers of 200, 300, and 400 and nondimensional rotation rates (ratio of surface speed of the cylinder to the free stream velocity), α, varying from 0 to 5. The diagnostic is done by flow visualization using hydrogen bubble technique and quantitative measurements using a particle image velocimetry technique. We present the global view of the wake structure at the three Reynolds numbers and various rotation rates. Vortex shedding activity is observed to occur from α=0 to α~1.95, after which it is suppressed. Reynolds number is found to have a strong effect on the wake morphology near the suppression rotation rate, α=1.95. Interestingly, the vortex shedding activity again resumes in the range 4.34<α<4.70 as first discovered numerically (Mittal and Kumar, 2003, “Flow past a rotating cylinder,” J. Fluid Mech., 476, 303) for Re = 200. The shed vortices are of one sign in this range of rotation rates. Experimental evidence of this new vortex shedding mode is presented, for the first time, at α=4.45 in the newly discovered window of rotation rates, using flow visualization and particle image velocimetry measurements. Strouhal number measurements and global wake patterns agree well with the computations of Mittal and Kumar at a Reynolds number of 200.

Flow past two rotating cylinders

Flow past two uniformly rotating cylinders with the same rotation rates in a side-by-side configuration is studied experimentally. The experiments are carried out at Reynolds numbers, Re, of 100, 200, 300, 400, and 500 and nondimensional rotation rates, α, varying from 0 to 5. The spacing ratios, T/D, are 1.8, 2.5, 4.0, and 7.5. Two possibilities of rotations are considered with the cylinder surfaces in between the two cylinders moving upstream in one case (inward rotation case) and downstream in the other (outward rotation case). The diagnostics is done by flow visualization using hydrogen bubble technique and quantitative measurements using particle image velocimetry (PIV). We present, using extensive flow visualization, the global view of the wake structure at Re of 200 for various rotation rates, and two senses. Vortex shedding suppression is studied through flow visualization and/or PIV at various Re’s, T/D’s, and two senses. Vortex shedding is found to be suppressed in the inward rotation cases at all Re and T/D’s. The value of α corresponding to vortex shedding suppression, αs, in the inward rotation case is ∼2.0 for Re of 200–500 at all T/D’s. The value of αs for Re of 100 in the case of inward rotation shows an increasing trend with T/D from T/D=1.8 to 4.0 with αs changing from 1.2 to 1.7; further increase of T/D does not change αs. For outward rotation cases, vortex shedding suppression is clearly observed for Re of 100 and for all values of T/D; however, for higher Re, vortex shedding suppression is observed for T/D of 4.0 and 7.5 only. The measurements of αs in this case showed a decreasing trend with increasing T/D. Symmetry breaking is reported for inward rotation case near α=1.35 for T/D=2.5 at Re of 200 where the wake pattern changes from in-phase to antiphase mode.

Laminar flow structures from a rotating sphere: Effect of rotating axis angle

A Fourier–Chebyshev spectral collocation method is used to compute the flow past a steadily rotating sphere subjected to a uniform cross flow. The simulations are performed at Reynolds numbers Re = 100, 250 and 300, where the Reynolds number is based on freestream velocity and sphere diameter. These Reynolds numbers cover three different flow regimes for a stationary sphere, from steady axisymmetric, steady planar-symmetric to unsteady planar-symmetric. The effect of varying the rotation axis angle, α, on the laminar flow structures and time-averaged hydrodynamic forces is investigated over a range of non-dimensional rotation rates, Ω ∗. This study reveals the time-averaged flight path of a solid rotating particle. At Re = 100, the flow is always steady. The near-wake projected streamline patterns and pressure coefficient contours are topologically similar at α = 0 independent of the range of Ω ∗ considered. As α increases, the near-wake recirculation bubble is either suppressed or shifted and thus affects the pressure recovery behind the sphere. This in turn affects both the drag and lift coefficients, C D and C L . For all Ω ∗ considered, C D and C L increase as α increases and the increment is more pronounced at higher Ω ∗. At Re = 250 and 300, calculations show both steady and unsteady wake structures. The state of the wake structures depends strongly on α and Ω ∗. However, the time-averaged flow fields are qualitatively similar to Re = 100. As a result, the time averaged drag and lift coefficients, C D ¯ and C L ¯ follow the same trend as at Re = 100. The increase in Re dramatically reduce the C D ¯ compared to the same Ω ∗ and α at Re = 100. C L ¯ shows a more complicated behaviour as Re increases.

Numerical investigation of heat and fluid flow across a rotating circular cylinder maintained at constant temperature in 2-D laminar flow regime

Forced convection heat transfer across a circular cylinder rotating with a constant non-dimensional rotation rate ( α ) varying from 0 to 6 are investigated for Reynolds numbers of 20–160 and a Prandtl number of 0.7. Flow transitions is reported here for a wider range of Reynolds number and rotation rates. Heat transfer visualization technique using heatlines is implemented here, probably for the first time, in finite volume framework for the unsteady heat transfer problem in complex domain and used for heat flow analysis. Rotation can be used as a drag reduction and heat transfer suppression technique.

Turbulent Rotating Rayleigh–Benard Convection: Spatiotemporal and Statistical Study

The present study involves a 3D numerical investigation of rotating Rayleigh–Benard convection in a large aspect-ratio (8:8:1) rectangular enclosure. The rectangular cavity is rotated about a vertical axis passing through the center of the cavity. The governing equations of mass, momentum, and energy for a frame rotating with the enclosure, subject to generalized Boussinesq approximation applied to the body and centrifugal force terms, have been solved on a collocated grid using a semi-implicit finite difference technique. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Pr=0.01 and fixed Rayleigh number Ra=107 while rotational Rayleigh number Raw and Taylor number Ta are varied through nondimensional rotation rate (Ω) ranging from 0 to 104. Generation of large-scale structures is observed at low-rotation (Ω=10) rates though at higher-rotation rates (Ω=104) the increase in magnitude of Coriolis forces leads to redistribution of buoyancy-induced vertical kinetic energy to horizontal kinetic energy. This brings about inhibition of vertical fluid transport, thereby leading to reduced vertical heat transfer. The magnitude of rms velocities remains unaffected with an increase in Coriolis forces from Ω=0 to 104. An increase in rotational buoyancy (Raw), at constant rotation rate (Ω=104), on variation in Raw/Ta from 10−3 to 10−2 results in enhanced breakup of large-scale structures with a consequent decrease in rms velocities but with negligible reduction in vertical heat transport.

Rotating Rayleigh–Bénard convection under the influence of transverse magnetic field

In the present numerical study the effect of constant transverse magnetic field on convection of low Prandtl number liquid metal rotating in a cubical cavity with an aspect-ratio of 8:8:1 has been investigated. The bottom wall is heated while the top-wall is cooled and all the other walls are kept thermally insulated. The governing equations of mass, momentum, energy and magneto-hydrodynamic for a frame rotating with the enclosure, subject to Boussinesq approximation applied to gravity and centrifugal force terms, have been solved on a collocated grid using a semi-implicit finite difference method. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Pr=0.01, Rayleigh number Ra=107, and magnetic Prandtl number Pm=4.0×10-4 while Chandrasekhar number Q varies from 5.0625×104 to 1.21×106 and non-dimensional rotation rate Ω is varied from zero to 105.The increase in strength of transverse magnetic field (from Ql=5.0625×104 to Qh=1.21×106) till Q≃Ta leads to slight increase in convective heat transfer as well as formation of two-dimensional coherent structures aligned along the direction of magnetic field. For cases pertaining to Q<Ta the two-dimensionality of the flow breaks down and the rolls distort in their alignment which leads to decrease in magnitude of vertical heat transfer. For cases where Q≪Ta, the increased Coriolis forces lead to generation of large-scale circulation which forms a large cylindrical rotating column of fluid in consonance with Taylor–Proudman theorem. On increasing the strength of magnetic field the component of rms velocity in the direction of magnetic field gets suppressed while there is increase in other two components.

Investigation of coherent structures in rotating Rayleigh-Benard convection

The current work involves identification of coherent structures in rotating turbulent Rayleigh-Benard convection (RBC) in a moderately large aspect-ratio (8:8:1) rectangular enclosure. The enclosure is rotating about a vertical axis passing through its center of gravity. The incompressible Navier-Stokes and energy equations are solved in a rotating frame of reference and the resulting velocity and thermal fields are analyzed to educe coherent structures. The flow structures have been investigated at different nondimensional rotation rates ranging from ω=0 to 104 for a fixed Rayleigh number Ra=107 and Prandtl number Pr=0.01, keeping the Raw∕Ta ratio constant at 10−3 in order to stringently maintain Boussinesq approximation as well as to attain experimentally realizable rotational Rayleigh numbers. Coherent structures in the flow domain have been sought using different identification techniques, namely large eddy simulation (LES) decomposition using top-hat filter, proper orthogonal decomposition (POD) considering larger energy modes, second invariant (Q) of the velocity gradient tensor, and regions of negative λ2, the second largest eigenvalue of the tensor SikSkj+ΩikΩkj. It has been found that the coherent structures educed using POD or LES decomposition at low to moderate rotation (ω=10 to 103) show the formation of two to three large-scale rolls aligned along both horizontal directions. At higher-rotation rates corresponding to ω=104, there is a breakup of large-scale structures into multiple small-scale rolls having random spatial orientation. The thermal structures educed using both POD and large-scale LES decomposition at zero rotation show randomly rising and descending plumes that at Ω=10 coalesce to form a large cylindrical thermal plume in the core of the cavity. Further increase of rotation leads again to breakup of the cylindrical plume into multiple random plumes. Isosurfaces of Q and λ2 reveal elongated tubular roll-like structures mainly concentrated near the side-wall regions, though at higher rotation-rate (ω=104) the density of tubular structures increases significantly. There is a strong similarity of results obtained by large-scale LES decomposition and results obtained using mean plus first mode of POD decomposition. The structures educed using Q and λ2 are different in shape compared to LES or POD educed structures.

Three-Dimensional Instabilities in Flow Past a Rotating Cylinder

Flow past a spinning circular cylinder placed in a uniform stream is investigated via three-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier-Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and freestream speed of the flow is 200. The nondimensional rotation rate, α, (ratio of the surface speed and freestream speed) is 5. It is found that although the two-dimensional flow for α=5 is stable, centrifugal instabilities exist along the entire span in a three-dimensional set-up. In addition, a “no-slip” side-wall can result in separation of flow near the cylinder ends. Both these effects lead to a loss in lift and increase in drag. The end conditions and aspect ratio of the cylinder play an important role in the flow past a spinning cylinder. It is shown that the Prandtl’s limit on the maximum lift generated by a spinning cylinder in a uniform flow does not hold.

Flow past a rotating cylinder

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α &lt; 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 &lt; α &lt; 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

Shear Instability of Internal Inertia-Gravity Waves

Abstract Local shear and convective instabilities of internal inertia-gravity waves (IGW) are examined assuming a steady, plane-parallel flow with vertical profiles of horizontal velocity and static stability resembling an IGW packet in a basic state at rest, without mean vertical shear. The eigenproblem can be described in terms of a nondimensional rotation rate R = f/ω0 < 1, where f is the Coriolis parameter, ω0 is IGW intrinsic frequency, and IGW amplitude is a, such that a = 1 for convectively neutral waves. In the nonrotating case, shear instability is possible only for convectively supercritical waves, with horizontal wavevector aligned parallel or nearly parallel to the plane of IGW propagation. Transverse convection, with wavevector aligned perpendicular to the plane of IGW propagation, displays faster growth than parallel shear or convective instability at any horizontal wavenumber. For intermediate R, eigenmodes in supercritical IGW are characterized at small horizontal wavenumber k by a trans...

THE DEVELOPMENT AND LINEAR STABILITY OF ROTATING BOUNDARY LAYERS

In this study, the effect of Coriolis force on the development and linear stability of the boundary layer on a rotating flat plate is examined. The parabolized governing equations are solved using a Legendre spectral element method with a marching scheme. The shape of the velocity profile is a function of the nondimensional rotation rate Rox/Rex, where Rox is the rotation number and Rex is the Reynolds number. For a given value of Rox/Rex, the base flow is self-similar, but the shape of the velocity profile changes when Rox/Rex is varied. These results are confirmed by demonstrating that the governing partial differential equations can be reduced to one ordinary differential equation by similitude analysis. The linear stability of the boundary layer to spanwise perturbations is also considered. The non-Blasius shape of the base flow velocity profile does not have any appreciable effect on the growth rate of the vortices compared to that obtained with a Blasius profile for all Rox/Rex ≤ l×l0-4 considered. Thus, the rotational Görtler number Re1/4Rox1/2(δ/δB)3/2 (where δ and δB are the actual and Blasius boundary layer thicknesses respectively) is the appropriate parameter to describe the growth of the vortices.