### Session M20: Instability in Shear Layers

Chair: Luca Brandt, KTH Royal Institute of Technology
Room: 323

 Tuesday, November 22, 2011 8:00AM - 8:13AM M20.00001: Study of Discrete Modes Branching in High-Speed Boundary Layers Anatoli Tumin , Yuli Lifshitz , David Degani The branching of discrete modes in high-speed boundary layers is investigated using the Parabolized Stability Equations (PSE). The fast and slow discrete modes associated with the fast and slow acoustic modes, respectively, are considered in high-speed boundary layers over adiabatic and cooled walls. Whereas the conventional Linear Stability Theory approach leads to singular behavior in the vicinity of the fast mode synchronization with the entropy and vorticity modes, the PSE results do not reveal singular behavior of the solution and are consistent with the available Direct Numerical Simulations of perturbations in high-speed boundary layers. Also, the PSE results do not reveal a singular behavior at the point of synchronism of the slow and fast discrete modes. Tuesday, November 22, 2011 8:13AM - 8:26AM M20.00002: Enhancement of thermal fluctuations in Plane Couette Flow Jose M. Ortiz de Zarate , Jan V. Sengers Mode-coupling phenomena in systems outside equilibrium generically cause an enhancement of thermal fluctuations. These enhancements can be studied by Landau's fluctuating hydrodynamics. Here we present a detailed study for the case of plane Couette flow based on stochastically forced Orr-Sommerfeld and Squire equations. The forcing arises from random contributions to the stress tensor due to the stochastic nature of molecular collisions. This intrinsic stochastic forcing is then amplified by mode- coupling mechanisms associated with the shear flow. We discuss the different coupling mechanisms, the most important one being the direct coupling between fluctuations of the wall-normal velocity and vorticity. The most pronounced effect is amplification of wall-normal vorticity fluctuations with a spanwise modulation at dimensionless wave numbers $q_\parallel$ around 1.5. Tuesday, November 22, 2011 8:26AM - 8:39AM M20.00003: Vortex-wave interactions/self-sustained processes in shear layers Philip Hall It was shown by Hall and Sherwin (2010) that the so-called lower branch self-sustained processes which have been found in the last decade are finite Reynolds numbers versions of the vortex-wave interactions described in a number of papers in the early 1990's by Hall and Smith. Here the corresponding structures are developed in natural convection and a much simplified set of interaction equations are derived. The states are shown to be subharmonic with respect to the spanwise variable and their stability is discussed. The relevance of these states to Couette flow is discussed. Tuesday, November 22, 2011 8:39AM - 8:52AM M20.00004: Optimal disturbances in shearing and swirling flows Conor Daly Over the past twenty years transient energy density growth of linearly stable disturbances has shown to be the likely instigator for transition to turbulence in parallel shear flows. In this vein, optimal linear perturbations are calculated for two flows which have a mixture of forces acting on the fluid body. These are; rotating plane Couette flow (RPCF), which combines pressure-driven shear and swirl, and cylindrical Couette-Poiseuille flow (CCPF), which combines pressure-driven and Couette shear. Contours are presented of the maximum achievable linear transient growth, $G$, over the full range of wavenumbers within the linearly stable parameter regimes. Reference is made to experimental works on each flow and we examine the role that optimal disturbances have in the different transition phenomena that are observed. It is found that the contours of $G$ fall qualitatively alongside the points of transition in the two flows, in support of the notion that large linear transient growth can act a precursor to transition. Despite the combination of effects acting on each fluid, transition in both flows falls in the range $10^2 [Preview Abstract] Tuesday, November 22, 2011 8:52AM - 9:05AM M20.00005: Transversal motion and flow structure of fully nonlinear streaks in a laminar boundary layer Juan Angel Martin , Carlos Martel Typical streak computations present in the literature correspond to linear streaks or to small amplitude nonlinear streaks computed using DNS or nonlinear PSE. We use the Reduced Navier-Stokes (RNS) equations to compute the streamwise evolution of fully non-linear streaks with high amplitude in a laminar flat plate boundary layer. The RNS formulation provides Reynolds number independent solutions that are asymptotically exact in the limit$Re \gg 1\$, it requires much less computational effort than DNS, and it does not have the consistency and convergence problems of the PSE. We present various streak computations to show that the flow configuration changes substantially when the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, that end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results. Tuesday, November 22, 2011 9:05AM - 9:18AM M20.00006: Numerical study of instability in a laminar plane wall jet Lars-Uve Schrader , Catherine Mavriplis , Luca Brandt A two-dimensional jet released parallel to a wall is called a plane wall jet. This flow type plays an important role in cooling of surfaces, e.g. in laptop processors. Developed laminar wall jets are nearly self-similar in the streamwise direction. Their instability was investigated by linear stability theory (LST) in previous studies. In this presentation, we report a direct numerical simulation (DNS) study of the linear spatial evolution of the wall jet instabilities and compare our DNS results with findings from LST. We find large discrepancies of the linear disturbance growth rates and attribute this to the incompatibility of the parallel flow assumption of LST with the rapidly evolving plane wall jet. A correction to the instability map is suggested, where especially the stable hole'' (a patch of stable conditions within the unstable region) turns out to be significantly larger than known so far. We also computed optimal initial disturbances using a method based on the adjoint Navier-Stokes equations. These disturbances evolve into traveling wave packets with maximum possible disturbance kinetic energy. Tuesday, November 22, 2011 9:18AM - 9:31AM M20.00007: Non-linear single-wave Kelvin-Helmholtz instability in a channel Annagrazia Orazzo , Gennaro Coppola , Luigi de Luca A stratified viscous gas-liquid two-phase flow confined in a horizontal channel is studied, surface tension effects being included. Contrary to previous papers of literature, where a parallel flow configuration is classically analyzed with plug-velocity profile in both fluids, here the flow is spatially developing starting from a plug-plug profile at the channel entrance. The sudden change of interface boundary condition produces the flow development and the emergence of a solitary Kelvin-Helmholtz wave, whose formation and evolution, inherently non linear, are studied through numerical simulations based on the Volume of Fluid (VOF) technique. The amplitude growth rate and the propagation velocity of the wave at early instants agree closely with the predictions of a straightforward model. Later times simulations show the wave break-up in small droplets. Tuesday, November 22, 2011 9:31AM - 9:44AM M20.00008: A nonlinear variational approach to triggering transition in plane Couette flow S.M.E. Rabin , C.P. Caulfield , R.R. Kerswell The study of the stability of shear flows has a long history dating back more than a hundred years. Understanding how and when turbulence emerges in such flows is highly significant to many processes studied throughout engineering. A feature of turbulent flows are that they have significantly higher kinetic energy than those that remain laminar. As a consequence research has focused on optimizing kinetic energy at a specific target time, initially for the linearized Navier Stokes equations (Butler \& Farrell 1992), and more recently for the full Navier Stokes equations (Pringle \& Kerwell 2010, Cherubini et al 2010). The belief is that by achieving high energies turbulence can be triggered. An alternative theory is that optimizing time averaged dissipation is more effective at triggering turbulence (Monokrousos et al 2011). In this study we optimize kinetic energy growth over all initial states and all target times for a given initial kinetic energy in order to reveal the minimal seed'' for turbulence (the disturbance of lowest kinetic energy which can trigger turbulence). We present results for a geometry originally considered by Butler \& Farrell (1992) and also compare our minimal seed prediction with that made recently by Monokrousos et al (2011). Tuesday, November 22, 2011 9:44AM - 9:57AM M20.00009: Dynamics of vorticity defects in layered stratified shear flows C.P. Caulfield , A. Roy , N.J. Balmforth Layered stratified flows, where relatively deep regions of weak stratification are separated by thinner interfacial layers of substantially stronger density gradient are commonly observed in nature. If such flows are subjected to vertical shear, it is well-known that a wide range of qualitatively different instabilities may develop. For example, the three-layer, two interface case is susceptible to a Taylor'' instability which, although superficially similar to the classic Kelvin-Helmholtz instability, is actually qualitatively different in its growth mechanism. The investigation of the nonlinear dynamics of this instability, and to a lesser extent the single-interface Holmboe'' instability, has proved difficult, as the need to resolve the associated sharp density gradients places heavy demands on the required numerical resolutions for simulation. However, we show that it is possible to gain insight into the key nonlinear dynamics of such layered stratified shear flows by generalizing a reduced matched asymptotic vorticity defect'' model (N. J. Balmforth et al. {\sl J. Fluid Mech.} {\bf 333}, 197 [1997]) to include the dynamical effects of density variations. We particularly focus on investigating the finite amplitude structure of the saturated primary Taylor instability, and the properties of the secondary instabilities to which Taylor and Holmboe instabilities are susceptible. Tuesday, November 22, 2011 9:57AM - 10:10AM M20.00010: Laminar streak enhancement using streamwise grooves Carlos Martel , Juan \'Angel Mart\'In Laminar streak promotion in a flat plate boundary layer results in an increase of the stability of the Tollmien-Schlichting waves with respect to that of the 2D Blasius profile. This stabilization delays the laminar-turbulent transition, increasing the laminar phase of the flow. The stabilization effect is stronger for higher streak amplitudes, and therefore simple ways of generating high amplitude stable streaks are sought to be used as boundary layer flow control methods. In a recent experiment [Tallamelli \& Franson,AIAA 2010-4291] high amplitude stable steady streaks have been produced using Miniature Vortex Generators (MGVs), where one array of MGVs is used to excite the streak and a second array is used downstream to enhance their amplitude. In this presentation we numerically explore the possibility of enhancing the streaks using a different passive mechanism: streamwise grooves carved in the plate. We will present some numerical simulations for different values of the spanwise period of the streaks and of the grooves, and we will show the combinations that provide maximum streak amplitude.