Bulletin of the American Physical Society
2005 58th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 20–22, 2005; Chicago, IL
Session EQ: Boundary Layer Instabilities I |
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Chair: Anatoli Tumin, University of Arizona Room: Hilton Chicago Stevens 2 |
Sunday, November 20, 2005 4:10PM - 4:23PM |
EQ.00001: Energy thresholds in the asymptotic suction boundary layer Ori Levin, Niklas Davidsson, Dan Henningson Energy thresholds for transition to turbulence in the asymptotic suction boundary layer are calculated by means of temporal direct numerical simulations. Three well known transition scenarios are investigated: oblique transition, the growth and breakdown of streaks triggered by streamwise vortices, and the development of random noise. Linear disturbance simulations and stability diagnostics are performed for a base flow consisting of the asymptotic suction boundary layer and a streak to investigate the most amplified streamwise wavenumber of the secondary instability. The scenarios are found to trigger transition by similar mechanisms as obtained for other flows. Transition at the lowest initial energy is provided by the oblique wave scenario for the considered Reynolds numbers $\mathrm{Re=500}$, 800 and 1200. The Reynolds number dependence on the energy thresholds are determined for each scenario. The threshold scales like $\mathrm{Re}^{-2.6}$ for oblique transition and like $\mathrm{Re}^{-2.1}$ for transition initiated by streamwise vortices and random noise, indicating that oblique transition has the lowest energy threshold also for larger Reynolds numbers. [Preview Abstract] |
Sunday, November 20, 2005 4:23PM - 4:36PM |
EQ.00002: Fundamental resonance induced by a 3D shallow roughness Igor Paula, Marcello Medeiros, Werner W\"{u}rz, Marcio Mendonca The present work consists in a experimental study of the effect that a shallow in 3D roughness element has on the evolution of a 2D Tollmien-Schlichting wave in a Blasius boundary layer. The experiments were carried out in the laminar wind tunnel of the Universit\"{a}t Stuttgart. The TS source and the roughness element were mounted in an airfoil model. The roughness used in these experiments was a cylindrical element. The height of the roughness was slowly oscillated. The oscillation frequency was approximately 1500times lower than the frequency of the studied TS waves. The synchronization of all equipments permitted the using of ensemble average techniques. Thus, time signal from hot-wire was averaged and the results corresponding to different roughness heights were obtained by windowing of this time series. Two amplitudes of TS waves were studied and the results are reported. The experimental growth rates of the 2D and 3D modes were measured. The results show a strong amplification in a small spanwise wave number range. The growth rates of this wave numbers were compared with the secondary instability theory provided by a PSE solver. The comparison shows that the oblique modes selected are in agreement with the most unstable ones predicted by theory. [Preview Abstract] |
Sunday, November 20, 2005 4:36PM - 4:49PM |
EQ.00003: Transient growth on streaks Luca Brandt, Jerome Hoepffner, Dan Henningson The behaviour of linear perturbations developing on boundary layer streamwise streaks assumed steady and spanwise periodic is investigated for streak amplitudes below or at the onset of the inflectional secondary instability. The input velocity fields leading to an output flow of maximum possible energy at a given time are first computed. It is found that significant transient growth may occur for both sinuous and varicose modes. The transient growth is larger for sinuous modes, it increases with the Reynolds number and it is already relevant at amplitudes well below the threshold for the onset of secondary instabilities. The optimal initial condition consists of velocity perturbations tilted upstream from the wall. The optimal response is still localized in the areas of largest shear but it is tilted in the flow direction. The largest velocity component of the optimal disturbance is the spanwise whereas the optimal response is strongest in its streamwise velocity component. To quantify the realizability of this growth process in noisy situations a stochastic approach is followed. A receptivity coefficient is defined by relating the variance of the environmental disturbance to that of the optimal flow response at a later time. [Preview Abstract] |
Sunday, November 20, 2005 4:49PM - 5:02PM |
EQ.00004: DNS for Turbulent Spot Formation of Wavepacket Evolution in a Blasius Boundary Layer Xijing Zhao, Khoon Seng Yeo, Zhengyi Wang The spatial temporal development of the pulse-initiated wavepacket in a Blasius boundary layer was studies by DNS from linear stage up to formation of the turbulent spot and allowed us to extract more intricate details of the turbulent spot than could generally be deduced from experiments. The wavepacked is evolving in turn by the appearance of triad resonance, streaky structures, onset of mean flow distortion and rapid intensification of frequency and wavenumber spectra, and formation of the turbulent spot. The incipient turbulent spot contains low frequency, low spanwise wavenumber fluctuations in streamwise velocity and high frequency harmonics in vertical and spanwise velocities with selective frequencies up to 7 times the fundamental frequency. The vertical component of disturbance plays an important role in the onset of turbulent spots by transporting energy from the outer region into the inner region. Four-and five-wave resonances also appear to play a non-trivial role. The vortex core and wavelet analyses were used to identify vertical structures and showed the incipient turbulent spot is derived from spatial amalgamation of a multitude of small-scale vertical patches. Preliminary simulations have been conducted for non-linearly evolving wavepackets over compliant panels and indicated this can affect and suppress the nonlinear growth of wavepacket disturbances. [Preview Abstract] |
Sunday, November 20, 2005 5:02PM - 5:15PM |
EQ.00005: Nonlinear growth (and breakdown) of disturbances in developing Hagen Poiseuille flow Peter Duck The effect of disturbances on developing Hagen-Poiseuille flow is investigated for large Reynolds numbers. In this developing region (which is long - ${\cal{O}}$(Reynolds number) $\times$ the pipe diameter), assuming uniform flow at the inlet, then boundary layers develop on the walls of the pipe, which then eventually merge downstream. Boundary layers are known to be susceptible to three-dimensional algebraic growth in the streamwise direction. In this study, non-axisymmetric disturbances are imposed on the developing flow through two alternative mechanisms: (i) the imposition of eigenstates at the pipe inlet and (ii) by means of forcing the azimuthal velocity on the pipe wall. Fully nonlinear, steady disturbances (which are known in the linear context to be the most \lq dangerous') are considered. If the disturbance amplitude is sufficiently large, a solution \lq breakdown' is observed, associated with a rapid growth of the high-order azimuthal modes, suggesting a possible and alternative mechanism for pipe-flow transition. Comparison is also made with the analogous effect on planar (Blasius-type) boundary layers. [Preview Abstract] |
Sunday, November 20, 2005 5:15PM - 5:28PM |
EQ.00006: Streamwise non-normality in boundary-layer instabilities Uwe Ehrenstein , Fran\c{c}ois Gallaire Temporal linear stability modes depending on two space directions are computed for a two-dimensional unstable boundary-layer flow along the flat plate. The linearized Navier-Stokes system is discretized using Chebyshev-collocation in both the streamwise and wall-normal direction and the resulting eigenvalue problem is solved by means of a Krylov-Arnoldi method. It is shown, that for appropriate inflow and outflow conditions the spatial structure of each individual temporally stable mode is reminiscent of the spatial exponential growth of perturbations along the flat plate, as predicted by local stability analyses. An optimal temporal growth analysis is performed and it is demonstrated that an appropriate superposition of a moderate number of non-normal temporal modes gives rise to a spatially localized wave packet, starting at inflow and exhibiting transient temporal growth when evolving downstream along the plate. This wave packet is shown to be in qualitative agreement with the convectively unstable disturbance observed when solving the full Navier-Stokes equations for an equivalent initial condition. This confirms that a transient cooperation of a finite number of non-normal temporal modes reproduces real-flow convective instabilities, which opens new possibilities of model reduction in open flow problems. [Preview Abstract] |
Sunday, November 20, 2005 5:28PM - 5:41PM |
EQ.00007: Application of exact solutions of the Bussman-M\"unz equation John Russell The {\sc Bussman-M\"unz} (B-M) equation is the differential equation of classical instability of the asymptotic-suction boundary layer. The B-M equation has a nonzero coefficient of the third-derivative term but is otherwise similar to the {\sc Orr-Sommerfeld} (O-S) equation. In 1950 and 1970 {\sc D.~Grohne} and {\sc P.~Baldwin} found integral representations of fundamental systems of exact solutions of the O-S and B-M equations, respectively. The present author has reexpresseed these and other solutions of the B-M equation in terms of the $G$-function of {\sc C.S. Meijer}, the result being a symmetric system of seven solutions (three of dominant-recessive type, three of balanced type, and one of well-balanced type). The present talk will present new results that include three exact connection formulas, each of which expresses the linear dependence of a subset of the symmetric system of seven solutions. The results also include application of the $G$-functions to computation of familiar quantities such as the critical {\sc Reynolds} number in the temporal instability problem. If one evaluates the integral representations of the exact solutions by the method of steepest descents the form of the integration path depends upon the independent variable and undergoes a major qualitative change as the latter crosses a {\sc Stokes} line. The present talk will furnish illustrations of such changes [Preview Abstract] |
Sunday, November 20, 2005 5:41PM - 5:54PM |
EQ.00008: Secondary instability in pipe flow: optimal non-axisymmetric base-flow deviations Guy Ben-Dov, Jacob Cohen The temporal growth of disturbances developing in pipe Poiseuille flow, which has been modified by primary helical axially-independent finite-amplitude initial disturbances, is analyzed. Such finite disturbances in the developed parabolic profile may occur as a result of transient growth amplifications. The optimal modification is defined as the primary non-axisymmetric base-flow deviation, with a specific amplitude norm, that yields the maximum growth rate for the secondary disturbances. Optimal modifications are computed by a variational technique. Unstable modes are found to exist for very small values of the primary disturbances amplitudes at relatively small Reynolds numbers. The optimal base-flow deviations are localized in a narrow radial range, implying that an inviscid instability mechanism is responsible for the evolution of the secondary disturbances. [Preview Abstract] |
Sunday, November 20, 2005 5:54PM - 6:07PM |
EQ.00009: Linear Stability Analysis of a Channel Flow with Porous Walls Nils Tilton, Luca Cortelezzi This study is motivated by the extensive use of wall-transpiration in numerical studies related to inhibition and control of wall-turbulence. In general, wall-transpiration has been implemented by providing the wall-normal velocity and imposing a no-slip condition on the wall-tangential velocity. Physically, however, the pores cannot be infinitesimally small and, consequently, it is important to address how the presence of the pores affects the slip velocity at the wall and the stability of the boundary layer. Moreover, our work is motivated by the existence of only few studies on the linear stability of channels with porous walls. Our study considers a parallel-plate channel with porous walls such that a longitudinal pressure gradient induces a laminar flow in both the open channel region and the porous walls. Simplified counterparts to the Orr-Sommerfeld and Squire equations are derived for the porous regions that are valid for small permeablities. The linear stability analysis takes account of the coupling between the three disturbance fields through boundary conditions recently derived by Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer, Vol. 38, 1995, pp 2635-2646). The resulting Orr-Sommerfeld spectrum and eigenfunctions reduce to those for Poiseuille flow as the permeability of the walls tends to zero, but are altered for greater values. We discuss symmetrical flows where parameters at both porous walls are identical as well as skewed flows where parameters at the two walls differ. [Preview Abstract] |
Sunday, November 20, 2005 6:07PM - 6:20PM |
EQ.00010: Leaky waves in boundary layer flow Jan Pralits, Paolo Luchini Linear stability analysis of boundary layer flow is traditionally performed by solving the Orr-Sommerfeld equation (OSE), either in a temporal or a spatial framework. The mode structure of the OSE is in both cases composed of a finite number of discrete modes which decay at infinity in the wall- normal direction $y$, and a continuous spectrum of propagating modes behaving as $\exp(\pm \mathrm{i} k y)$ when $y\rightarrow\infty$, with real $k$. A peculiarity of this structure is that the number of discrete modes changes with the Reynolds number, $Re$. They indeed seem to disappear behind the continuous spectrum at certain $Re$. This phenomenon is here investigated by studying the response of the Blasius boundary layer forced instantaneously in space and time. Since the solution of the forced and homogeneous Laplace-transformed problem both depend on the free-stream boundary conditions, it is shown here that a suitable change of variables can remove the branch cut in the Laplace plane. As a result, integration of the inverse Laplace transform along the two sides of the branch cut, which gives rise to the continuous spectrum, can be replaced by a sum of residues corresponding to an additional set of discrete eigenvalues. These new modes grow at infinity in the $y$ direction, and are analogous to the {\em leaky waves} found in the theory of optical waveguides, i.e. optical fibers, which are attenuated in the direction of the waveguide but grow unbounded in the direction perpendicular to it. [Preview Abstract] |
Sunday, November 20, 2005 6:20PM - 6:33PM |
EQ.00011: Rayleigh Instability in Supersonic Compression Ramp Flow Kevin Cassel, Danny Bockenfeld The supersonic flow past a compression ramp with ramp angle of $O(Re^{-1/4})$ is governed by the supersonic triple-deck formulation. For scaled ramp angles $\alpha \geq 3.9$ Cassel, Ruban \& Walker (1995)\footnote{Cassel, Ruban \& Walker, {\it J. Fluid Mech.} {\bf 300}, 265--286 (1995).} have found that the triple-deck flow is unstable to long-wave Rayleigh (LWR) modes, which have wavelengths shorter than the $O(Re^{-3/8})$ streamwise length scale of the triple-deck region, but larger than the $O(Re^{-5/8})$ vertical extent of the lower deck. The LWR instability is manifest in the unsteady triple-deck calculations as an absolute instability in the form of a wave packet. In the present investigation, the possible presence of a Rayleigh instability is investigated in the supersonic compression ramp flow. Rayleigh modes, which are of the same order as the $O(Re^{-5/8})$ viscous lower deck, are not admitted in the triple-deck formulation due to the additional physics that is required in the $O(Re^{-5/8}) \times O(Re^{-5/8})$ Rayleigh region. However, the Rayleigh instability has a faster growth rate than the LWR instability and would be expected to dominate in solutions of the full Navier-Stokes equations for the compression ramp flow. The stability problem consists of solving the triple-deck formulation for the base flow and the Rayleigh equation for the perturbations to this base flow. Results for ramp angles up to $\alpha=5.5$ show that for all cases that are unstable to LWR modes, i.e.\ that contain inflectional velocity profiles, the flow is also unstable to Rayleigh modes. [Preview Abstract] |
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