Session GP: Shear Layer Instabilities*

Attenuation of Self-Excited Oscillations of Shear Flow Past a Perforated or Slotted Plate: Effects of Leading-Edge Ramp

Grazing flow past a perforated or slotted plate, bounded on its backside by a closed cavity, can give rise to highly coherent, self-sustained oscillations that have a wavelength much longer than the length scale of the slot or perforate. This investigation aims to suppress these purely hydrodynamic oscillations via small amplitude deflection of the inflow, induced by a ramp located at the leading-edge of the plate. Both unsteady pressure measurements and quantitative imaging of the flow structure reveal the essential features of the most effective attenuation. A central aspect is formation of a large-scale bubble involving steady entrained flow through the slots or perforates. Remarkably, attenuation can be achieved when the height of the ramp $h$ is approximately two orders of magnitude smaller than the effective length $L$ of the plate, i.e., $h/L \quad \ge $ 0.025.   

Scalar Mixing Enhancement via Streamwise Vortices and their Excited Wavy Instabilities in a Free Shear Flow

Scalar mixing enhancement for an incompressible inert fluid is considered in a spatially developing shear layer. For steady G\"{o}rtler type streamwise vortices, similarity of streamwise momentum with scalar transport problems is possible for Prandtl and Schmidt numbers unity.$^{1}$ Momentum conservation equations for the nonlinear wavy instabilities, and those for unsteady scalar transport are examined in detail. The streamwise fluctuation pressure gradient prevents similarity as for steady flow; it is estimated in terms of the fluctuation dynamical pressure and found to be much weaker than advective transport. Similarity between fluctuation streamwise velocity, temperature and concentration now becomes possible for Prandtl and Schmidt numbers unity. Behavior of scalar fluctuations is then inferred from the fluctuation streanwise momentum.$^{2}$ The wavy-instability modified heat and mass transport owing to the most amplified nonlinear sinuous mode is assessed. It is found that the nonlinearity of the wavy instabilities enhance scalar mixedness over a significant developing streamwise region well above that achieved by the steady, unmodified streamwise vortices alone. \newline $^{1}$J.T.C. Liu and A.S. Sabry, \textit{Proc. Royal Soc. London}, Ser. A \textbf{432, }1-12 (1991). \newline $^{2}$I.G. Girgis and J.T.C. Liu$^{, }$\textit{J. Fluid Mech}. \textbf{468}, 29-75 (2002).   

Transient Growth for the Linearized Navier-Stokes Equations

We analytically solve the linearized Navier-Stokes equations for streamwise-invariant sinusoidal shear flow. This is accomplished by considering the infinite dimensional system of ordinary differential equations derived via Galerkin projection onto Fourier modes, a representation which allows one to interpret the dynamics in terms of interactions between streaks and streamwise vortices. We characterize transient energy growth, a mechanism which may trigger nonlinear effects that lead to sustained turbulence, for this system. This includes calculating perturbations which give optimal initial and total energy growth, which is done numerically for large enough truncations to capture the behavior of the full system. We also numerically determine Reynolds number scalings and find optimal wavenumbers for maximum transient energy growth.   

On the stability of a free shear layer affected by a parallel magnetic field

We investigate the instability and transition to turbulence in a temporally evolving free shear layer of an electrically conducting fluid affected by an imposed parallel magnetic field. The case of low magnetic Reynolds number is considered. It has long been known that the neutral disturbances of the linear problem are three-dimensional at sufficiently strong magnetic fields. We analyze the details of this instability solving the generalized Orr-Sommerfeld equation to determine the wavenumbers, growth rates, and spatial shapes of the eigenmodes. The three-dimensional perturbations are identified as oblique waves and their properties are described. In particular, we find that at high hydrodynamic Reynolds number the effect of the strength of the magnetic field on the fastest growing perturbations is limited to increase of their oblique angle. The vertical and horizontal dimensions and the spatial shape of the waves remain unchanged. The transition to turbulence triggered by the growing oblique waves is investigated in direct numerical simulations. It is shown that initial perturbations in the form of superposition of two symmetric waves are particularly effective in inducing three-dimensionality and turbulence in the flow.   

Stability of MHD channel flow with spanwise magnetic field

We study the effect of a homogeneous magnetic field on the subcritial instability and transition to turbulence in plane Poiseuille flow of an electrically conducting fluid. The field is oriented in the spanwise direction. The transition to turbulence depends on the transient growth of non-modal linear perturbations, which are characterized by their stream- and spanwise wavenumbers. In the non-magnetic case, the most amplified perturbations are purely streamwise rolls. As the Lorentz force tends to suppress motion in the spanwise direction, the most amplified perturbations change from streamwise rolls to oblique rolls as the magnetic field increases. For sufficiently strong fields, motion in the spanwise direction is strongly damped and only the two-dimensional Tollmien-Schlichting waves aligned with the magnetic field provide transient amplification. The consequences of this damping effect on the transition to turbulence are studied through direct simulations. TB and DK are supported by the Deutsche Forschungsgemeinschaft (grant Bo 1668/2-2), and OZ by the Dept. of Energy (grant DE FG02 03ER46062). The cooperation between TU Ilmenau and UM Dearborn is supported by DAAD and NSF (grant OISE 0338713).   

Low-dimensional models of a temporally evolving free shear layer using template-based methods

Two-dimensional spatially periodic, temporally developing free shear layers are simulated and modeled. Low-dimensional models are obtained using a modified version of proper orthogonal decomposition/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. In particular, the solution is scaled at each time so that it matches a pre-selected template function, and the correct scaling is kept track of separately. Projection of incompressible Navier-Stokes equations onto the first two POD modes of the lowest spatial wavenumber gives a 2-mode model, which can describe certain single-frequency features, such as vortex roll-up, nonlinear saturation, and viscous damping. If the first two POD modes are retained for the first two spatial harmonics, the resulting 4-mode model describes more complex dynamics such as vortex merging. Phase differences between the first (symmetric) and second (asymmetric) POD modes of each wavenumber are examined, and qualitative differences are observed before and after saturation occurs.   

The dynamics and control of perturbations along a shear layer

The dynamics of small and large scale convective perturbations along a shear layer is studied using the linear and nonlinear parabolic stability equations. The predictions are compared with results from direct numerical simulations and available experimental data. The response of the flow to upstream excitations at both low and high frequencies is described. The perturbation equations are also used to derive a physically-based, reduced-order model of the flow dynamics for the open-and closed-loop control of the flow. Computed examples for the cases of a free shear layer and the shear layer behind a backward facing step are presented. The results demonstrate the effect of upstream excitation of certain sets of modes of perturbations with various magnitudes on the mean flow structure and the evolution of the perturbations. For example, it is found that exciting high frequency modes can modify the regular cascade of energy over distance along the shear layer.   

Optimal perturbation growth and bypass transition in mixing layers

Hyperbolic mixing layers are strongly unstable to the normal-mode Kelvin-Helmholtz instability (KHI). At finite amplitude, KHI billows are unstable to the hyperbolic instability, which becomes streamwise-aligned braid-centred rib vortices that trigger turbulence transition. However, the underlying linear operator is non-normal, and so there may be transient non-normal-mode perturbations. We use numerically-calculated power iteration of the linear Navier-Stokes operator and its adjoint to identify perturbations whose energy grows most rapidly over finite times, and also investigate the nonlinear evolution of these perturbations using nonlinear direct numerical simulations. If the energy of the perturbation is optimized over sufficiently long times, the optimal perturbation is unsurprisingly closely related to the KHI normal mode, and the nonlinear evolution of this perturbation exhibits only a slightly enhanced transient growth compared to the KHI. Furthermore, the long-time optimal perturbation of a time-evolving KHI billow is essentially the hyperbolic instability. However, if the perturbation growth is optimized over short times, an inherently three-dimensional non-normal mode is identified, which in our nonlinear simulations can trigger `bypass' transition without any development of the primary KH instability. The growth mechanism and structure of this mode will be discussed.   

Attenuation of Self-Sustained Cavity Oscillations via Vortex Generators

Turbulent inflow past either a free opening of a cavity, or along a perforated or slotted plate along the cavity opening, can give rise to self-sustained oscillations. At low Mach number, and in absence of wall elasticity, these oscillations may be viewed as purely hydrodynamic. Basic classes of co-rotating and counter-rotating vortex generators, located at the leading-edge of the cavity opening, have been critically evaluated as potential techniques of attenuation. High-image-density particle image velocimetry and unsteady pressure measurements lead to representations of the flow structure that are associated with maximum reduction of pressure amplitude. Moreover, global spectral analysis, based on simultaneous time records at thousands of grid points of the cinema imaging, provides further insight into the spatial patterns of the attenuated pressure amplitude. For a given configuration of vortex generator, the instability along a plate is attenuated to a smaller degree than the instability along the free cavity opening, thereby indicating it is highly robust.