Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session G17: Flow Instability: Nonlinear Dynamics and Global Modes |
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Chair: Jacques Magnaudet, CNRS/IMFT Room: 205 |
Monday, November 23, 2015 8:00AM - 8:13AM |
G17.00001: A weakly nonlinear model with exact coefficients for the fluttering and spiraling motions of buoyancy-driven bodies Jacques Magnaudet, Joel Tchoufag, David Fabre Gravity/buoyancy-driven bodies moving in a slightly viscous fluid frequently follow fluttering or helical paths. Current models of such systems are largely empirical and fail to predict several of the key features of their evolution, especially close to the onset of path instability. Using a weakly nonlinear expansion of the full set of governing equations, we derive a new generic reduced-order model of this class of phenomena based on a pair of amplitude equations with exact coefficients that drive the evolution of the first pair of unstable modes. We show that the predictions of this model for the style (eg. fluttering or spiraling) and characteristics (eg. frequency and maximum inclination angle) of path oscillations compare well with various recent data for both solid disks and air bubbles. [Preview Abstract] |
Monday, November 23, 2015 8:13AM - 8:26AM |
G17.00002: Nonlinear dynamics in eccentric Taylor--Couette--Poiseuille flow Beno\^{I}t Pier, C. P. Caulfield The flow in the gap between two parallel but eccentric cylinders and driven by an axial pressure gradient and inner cylinder rotation is characterized by two geometrical parameters (radius ratio and eccentricity) and two dynamic parameters (axial and azimuthal Reynolds numbers). Such a theoretical configuration is a model for the flow between drill string and wellbore in the hydrocarbon drilling industry. The linear convective and absolute instability properties have been systematically derived in a recent study [Leclercq, Pier \& Scott, {\it J. Fluid Mech.} 2013 and 2014]. Here we address the nonlinear dynamics resulting after saturation of exponentially growing small-amplitude perturbations. By using direct numerical simulations, a range of finite-amplitude states are found and characterized: nonlinear traveling waves (an eccentric counterpart of Taylor vortices, associated with constant hydrodynamic loading on the inner cylinder), modulated nonlinear waves (with time-periodic torque and flow rate) and more irregular states. In the nonlinear regime, the hydrodynamic forces are found to depart significantly from those prevailing for the base flow, even in situations of weak linear instability. [Preview Abstract] |
Monday, November 23, 2015 8:26AM - 8:39AM |
G17.00003: Finite-amplitude solutions in rotating Hagen-Poiseuille flow Beno\^it Pier, Abhishek Kumar, Rama Govindarajan While the pipe Poiseuille base flow is linearly stable at all Reynolds numbers, a small amount of rotation of the pipe around its axis induces linear instability beyond a low critical Reynolds number $R_c \simeq 83$ [Pedley, J. Fluid Mech. 1969]. More recently [Fernandez-Feria and del Pino, Phys. Fluids 2002], this configuration has been shown to become absolutely unstable at Reynolds numbers of the same order of magnitude. Using direct numerical simulations, we investigate here finite-amplitude solutions resulting from saturation of exponentially growing small-amplitude initial perturbations. The base flow depends on two dynamical parameters (axial Reynolds number and rotation rate) and the initial perturbation is characterized by its axial wavenumber and its azimuthal mode number. The range of nonlinear waves prevailing in this configuration, the associated nonlinear dispersion relation and the spatial structure of these solutions are systematically obtained by exploring the parameter space. [Preview Abstract] |
Monday, November 23, 2015 8:39AM - 8:52AM |
G17.00004: Nonlinear interaction of stationary and travelling crossflow modes with a common critical layer Alex Amos, Xuesong Wu Laminar-turbulent transition of the three-dimensional boundary layer over a swept wing is caused by amplification of crossflow vortices. A puzzling and interesting experimental observation is that that the free-stream turbulence levels affect the development of stationary crossflow vortices. One possible explanation of this affect is that the travelling modes, which are excited by free-stream turbulence, interact nonlinearly with the stationary modes to affect their development. This interaction between modes is likely to be most effective when they share a critical level, where Rayleigh's equation becomes singular. We have shown that stationary and travelling modes having a common critical layer do exist. Their mutual nonlinear interactions are studied. The matched asymptotic expansion in conjunction with the multiple-scale method is used to derive the evolution equations for the amplitudes of the modes. The effects of the interactions on the growth of the amplitudes will be discussed, and possible self interactions and their consequence will be addressed. [Preview Abstract] |
Monday, November 23, 2015 8:52AM - 9:05AM |
G17.00005: Non-linear state selection of axially confined viscous liquid jets Alejandro Sevilla, Alejandro Mart\'inez-Calvo, Mariano Rubio-Rubio Viscous liquid jets injected at a constant flow rate vertically downwards into a gaseous atmosphere become globally unstable when the flow rate becomes smaller than a certain critical value. Previous experiments are in good agreement with a global linear stability analysis based on the leading-order one-dimensional (1D) mass and momentum conservation equations, provided that the full curvature is retained in the computations. However, linear theory cannot predict the large-time dynamics of the jet under globally unstable conditions. To that end, here we report new experiments and numerical simulations of the 1D model, showing that the unstable jet may exhibit two markedly different non-linear states in the long term: either a limit cycle featuring self-sustained oscillations without break-up, or a fully-developed dripping regime emerging after the break-up of the liquid column. A bifurcation analysis demonstrates that the length of the jet is the key parameter that controls the selection of the final state. The dependence of the critical length on the liquid viscosity, the injector radius and the liquid flow rate are also characterized in detail. [Preview Abstract] |
Monday, November 23, 2015 9:05AM - 9:18AM |
G17.00006: Empirical resolvent mode decomposition Aaron Towne, Tim Colonius, Oliver Schmidt The computation of resolvent modes is a popular method for studying the input/output behavior of fluid dynamical systems. These modes maximize the linear gain between the inputs and outputs of the system as a function of frequency and are computed via a singular decomposition of the linearized operator relating these quantities. Typically, the inputs are meant to represent the nonlinear interactions that are otherwise omitted in linear models. Here, we develop a data-based input/output methodology. The method constructs orthogonal input and output modes from ensembles of flow data that maximize the gains. The essential difference compared to traditional resolvent modes is that the empirical modes are constrained to lie within the subspace spanned by the data. The empirical modes can be shown to be equivalent to either traditional resolvent modes or proper-orthogonal-decomposition modes in appropriate limits. We demonstrate the properties and utility of the method using the complex Ginzburg-Landau equation and LES data from a Mach 0.9 turbulent jet, and compare the empirical modes to traditional resolvent modes in both cases. [Preview Abstract] |
Monday, November 23, 2015 9:18AM - 9:31AM |
G17.00007: Linear global modes in a high Reynolds number Mach 0.9 turbulent jet Oliver Schmidt, Aaron Towne, Tim Colonius A global linear stability and resolvent analysis of the mean flow from a carefully validated Mach $0.9$ turbulent jet large eddy simulation (LES) is conducted. Spatiotemporal Fourier decomposition of the simulation data reveals the presence of large scale coherent structures at small azimuthal wavenumbers. The latter wave packets appear as discrete sets of lightly dampened modes in the linear global stability analysis. Their common feature is a spatial separation into an upstream traveling acoustic perturbation in the potential core region, and a Kelvin-Helmholtz-like vortical perturbation which is advected downstream. The least stable branch of discrete modes observed at Strouhal numbers $0.38 |
Monday, November 23, 2015 9:31AM - 9:44AM |
G17.00008: Secondary instability of laminar separation bubbles in the absence of external disturbances Daniel Rodriguez, Elmer Gennaro, Leandro Souza Previous studies demonstrate that the primary instability of laminar separation bubbles (LSB) on a flat-plate in the absence of external forcing is a three-dimensional centrifugal one. This work develops a weakly non-linear expansion of the associated symmetry-breaking bifurcation, showing that it corresponds to a supercritical pitchfork bifurcation. The secondary instability of the fully 3D bifurcated LSB is then investigated by means of the temporal instability of 3D global modes, computed either as solutions of a 3D (Tri-global) eigenvalue problem, or based on a WKB approximation and the existence of local regions of absolute instability of the cross-stream planes. Both methodologies recover an amplified global oscillator, originated by the spanwise velocity gradients, that can explain the origin of the unsteadiness observed in numerical simulations of unforced LSBs with peak reversed flows below $15\%$, as the results of a secondary instability of the 3D separation bubble. [Preview Abstract] |
Monday, November 23, 2015 9:44AM - 9:57AM |
G17.00009: Stability sensitivity to gravity and base flow density modifications Kevin Chen, Geoffrey Spedding We present the novel theory of Boussinesq stability sensitivity to the gravitational force and to base flow density modifications. Given a steady-state flow with small density variations, the sensitivity of the stability eigenvalues is computed from the direct and adjoint modes of the linearized Boussinesq equations. Various combinations of the density and velocity components of these modes reveal multiple production and transport mechanisms that contribute to the eigenvalue sensitivity. This sensitivity theory is largely inspired by the study of stable density stratification, which can have seemingly contradictory effects on flow stability. On one hand, stable stratification increases the coherence and persistence of turbulent wakes; on the other hand, it can destabilize vortex structures, such as vortex pairs and rings. We present an application of the sensitivity theory to a stably density-stratified flow around a flat plate at a 90 degree angle of attack. The global mode analysis reveals lightly damped lee wave undulations, and the sensitivity theory shows that regions both immediately upstream and immediately downstream of the plate contribute most significantly to the stability sensitivity. [Preview Abstract] |
Monday, November 23, 2015 9:57AM - 10:10AM |
G17.00010: Bi-global Stability Analysis of Compressible Open Cavity Flows Yiyang Sun, Kunihiko Taira, Louis Cattafesta, Lawrence Ukeiley The effect of compressibility on stability characteristics of rectangular open cavity flows is numerically examined. In our earlier work with two-dimensional direct numerical simulation of open cavity flows, we found that increasing Mach number destabilizes the flow in the subsonic regime but stabilizes the flow in the transonic regime. To further examine the compressibility effect, linear bi-global stability analysis is performed over the same range of Mach numbers to investigate the influence of three-dimensional instabilities in flows over open cavities with length-to-depth ratios of 2 and 6. We identify dominant eigenmodes for varied Mach numbers and spanwise wavelengths with respect to two-dimensional stable and unstable steady states. Over a range of spanwise wavelengths, we reveal the growth/damp rates and frequencies of the dominant global modes. Based on the insights from the present analysis, we compare our findings from global stability analysis with our companion three-dimensional flow control experiments aimed at reducing pressure fluctuation caused by cavity flow unsteadiness. [Preview Abstract] |
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