Bulletin of the American Physical Society
60th Annual Meeting of the Divison of Fluid Dynamics
Volume 52, Number 12
Sunday–Tuesday, November 18–20, 2007; Salt Lake City, Utah
Session AS: Rotating Flows I |
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Chair: Keith Julien, University of Colorado, Boulder Room: Salt Palace Convention Center Ballroom EG |
Sunday, November 18, 2007 8:30AM - 8:43AM |
AS.00001: Instability in the rotating differentially heated annulus via asymptotic reduction of the Navier Stokes equations Michael Watson, Keith Julien The differentially heated rotating annulus has served as a canonical model for studying the baroclinic instability in the atmosphere for nearly 5 decades, yet numerical simulations of this experiment have difficulty reaching the low Rossby number (rapid rotation) regime. This occurs because the numerical models are based upon discretizations of the incompressible Navier Stokes equations, which necessarily resolve boundary layers and all fast dynamics. I will present a new set of governing equations for the baroclinic annulus based on an asymptotic analysis of the incompressible Navier Stokes which can be used to efficiently study the low Rossby number regime of parameter space. These equations can then be used to characterize the transition from steady to unstable dynamics in this rapid rotation limit through a linear stability analysis. [Preview Abstract] |
Sunday, November 18, 2007 8:43AM - 8:56AM |
AS.00002: Anisotropic low-wavenumber constraints on energy in rotating and stratified flows Susan Kurien, Beth Wingate, Mark Taylor Rapidly rotating, stably stratified three-dimensional inviscid flows conserve both energy and potential enstrophy. We show that in such flows, the forward cascade of potential enstrophy imposes anisotropic constraints on the wavenumber distribution of kinetic and potential energy. The horizontal kinetic energy is suppressed in the large, nearly horizontal wave modes, and should decay with the horizontal wavenumber as $k_h^{-3}$. The potential energy is suppressed in the large, nearly vertical wave modes, and should decay with the vertical wavenumber as $k_z^{-3}$ . These results augment the only other exact prediction for the scaling of energy spectra due to constraints by potential enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that in the quasi-geostrophic approximation for rotating stratified flows, the energy spectra must scale isotropically with total wavenumber as $k^{-3}$. We test our predicted scaling estimates using resolved numerical simulations of the Boussinesq equations in the relevant parameter regimes, and find reasonable agreement. [Preview Abstract] |
Sunday, November 18, 2007 8:56AM - 9:09AM |
AS.00003: Large-scale numerical simulation of rotationally constrained convection Michael Sprague, Keith Julien, Edgar Knobloch, Joseph Werne, Jeffrey Weiss Using direct numerical simulation (DNS), we investigate solutions of an asymptotically reduced system of nonlinear PDEs for rotationally constrained convection. The reduced equations filter fast inertial waves and relax the need to resolve Ekman boundary layers, which allow exploration of a parameter range inaccessible with DNS of the full Boussinesq equations. The equations are applicable to ocean deep convection, which is characterized by small Rossby number and large Rayleigh number. Previous numerical studies of the reduced equations examined upright convection where the gravity vector was anti-parallel to the rotation vector. In addition to the columnar and geostrophic-turbulence regimes, simulations revealed a third regime where Taylor columns were shielded by sleeves of opposite-signed vorticity. We here extend our numerical simulations to examine both upright and tilted convection at high Rayleigh numbers. [Preview Abstract] |
Sunday, November 18, 2007 9:09AM - 9:22AM |
AS.00004: Instabilities in the spin-up of a rotating, stratified fluid R.J. Munro, M.R. Foster, P.A. Davies Experiments were performed on a density-stratified fluid in a circular cylindrical container that is initially rotating at a steady speed, and is then spun up by increasing the rotation rate of the bottom disk of the container. The stratification is initially linear, with salt in water ($\sigma \sim 700$). Initially, a layer of well mixed fluid is observed to form above the disk (which is essentially spun-up). After $O(10\tau)$ (where $\tau=E^{-\frac{1}{2}}N^{-1}$ is the spin-up timescale based on the Ekman number and buoyancy frequency), a step-like density sub-structure is observed to develop in the linearly-stratified fluid above the mixed layer. The instability criterion based on conventional normal mode analysis is dependent on local values of the vertical and radial gradients of the background density field and zonal velocity. These quantities have been measured using a combination of zonal-plane PIV and an array of traversing conductivity probes. The standard criterion does predict that the upper layers of fluid, at mid-radii, are unstable. Further, the length scales of the perturbations predicted by this basic linear theory are an order of magnitude smaller than the step structures of the experiments. Curvature effects and non-uniform values of zonal flow and density gradient induce instabilities with larger length scales and slower growth rates, that may be examined by multiple-scale analysis. The length scales of these modes are consistent with the observations. [Preview Abstract] |
Sunday, November 18, 2007 9:22AM - 9:35AM |
AS.00005: On the stability of swirling flows in a finite pipe Shixiao Wang We study the stability mechanism of the swirling flow in a finite pipe. We first revisited the Rayleigh's linear stability theory, and build up the nonlinear theory in the framework of Hamiltonian system. We then consider the Lamb-Oseen vortex in a finite pipe with fixed flowrate condition at the boundaries. By using recently developed perturbation method of the linear operators, we analyzed the global stability equation and found the disturbance flow fields. We then conducted a study of the kinetic energy transfer mechanism between the disturbance and the base flow by using the Reynolds-Orr equation. We found that the energy transfer takes place actively at the boundaries as well as inside the flow. This is contrast to the solid body rotation flow. We further investigated Lamb-Oseen vortex in a slightly divergent pipe and showed that the internal flow has a leading role in the energy transfer mechanism. This study clarifies the relation of the Rayleigh stability and the global stability found by Wang and Rusak, and provide a basic understanding of the stability mechanism of swirling flows in a finite pipe. [Preview Abstract] |
Sunday, November 18, 2007 9:35AM - 9:48AM |
AS.00006: Gravitationally forced hyperbolic waves in a horizontal rotating cylinder Mats Nigam A thin layer of fluid, flowing axially along the inner surface of a horizontal rotating cylinder is subjected to a periodic forcing due to the gravitational acceleration. Since the frequency of the forcing lies within the critical range ($[0,\,2 \Omega]$) for which the inviscid problem is of a hyperbolic nature, the solution which in this case may be obtained on closed form displays a characteristic ``Mach wave''-like behavior. [Preview Abstract] |
Sunday, November 18, 2007 9:48AM - 10:01AM |
AS.00007: Rotating fluid cylinder submitted to a weak precession Christophe Eloy, Patrice Meunier, Romain Lagrange, Fran\c{c}ois Nadal We address experimentally and theoretically the flow inside a rotating cylinder subject to a weak precession. PIV measurements have revealed the instantaneous structure of the flow and confirmed that it is a sum of Kelvin modes forced by the precession. A linear inviscid approach predicts that the amplitude of a Kelvin mode diverges when its natural frequency resonates with the precession frequency. In this case a viscous and weakly nonlinear theory has been developed to predict correctly the finite mode amplitude. This theory has been compared to the experimental results and shows a good quantitative agreement. For low Reynolds numbers, the mode amplitude is shown to scale as the square root of the Reynolds number. When the Reynolds number is increased, the amplitude saturates at a value which scales as the precession angle power one third. A geostrophic (axisymmetric) mode is also forced as it has been observed and measured in the experiments. These results allow to fully characterize the flow inside a precessing cylinder in all regimes as long as there is no instability. [Preview Abstract] |
Sunday, November 18, 2007 10:01AM - 10:14AM |
AS.00008: Pulsatile flow through the gap between two coaxially rotating 3d cylinders: An exact solution Ali Pashaee, Nasser Fatouraee The exact solution of pulsatile pressure wave propagation in viscose flow at a rigid tube is available from the previous studies. Here a general flow system of pressure wave propagation along a gap between two coaxially rotating 3D cylinders is considered. The Navier-Stokes equations are solved analytically in this domain. Some features such as incompressibility, viscosity, pulsatility and 3D Cartesian flow are obtained from this solution. The results include the angular velocity component and radial pressure gradient which make it complex than the rigid tube flow with only one axial velocity component and one axial pressure gradient. The inner cylinder boundary condition prevents the second kind of Bessel function to be ignored in the solution as did not in the rigid tube formulation. Here the exact spatial and temporal distributions of the velocity domain, the pressure map, the wall shear stress, the volumetric flow rate and the particle trace trajectories are provided. Similar 3D transient flow systems (e.g. the oil well drilling procedure, vascular operations and medical flow image processing) can be modeled using this formulation. This solution can also be used as a bench mark test to evaluate the flow characteristic algorithms (e.g. pathline, streaklines, streamline and wall shear stress algorithms). [Preview Abstract] |
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