Session EN: Boundary Layer Instabilities II

Linear Stability Analysis of Couette Flow with a Porous Wall

It is well known that plane Couette flow in a channel with perfectly smooth, impermeable walls is linearly stable for all Reynolds numbers. Little attention has been given in literature to the stability of plane Couette flow when at least one of the walls is porous. In this study, we consider a channel delimited by an impermeable moving wall, which drives the flow, and a stationary, rigid, homogeneous, isotropic, porous block. We perform a three-dimensional linear stability analysis of the fully developed laminar flow in both the channel and the porous block. We restrict the study to sufficiently small permeabilities in order to neglect inertial effects in the porous flow. We solve the coupled linear stability problem, arising from the adjacent channel and porous flows, using a spectral collocation technique. The linear stability analysis takes account of the coupling between the two disturbance fields through boundary conditions recently derived by Ochoa-Tapia and Whitaker ({\it Int. J. Heat Mass Transfer}, {\bf 38}, 1995). We find that Couette flow over a permeable wall is no longer absolutely stable. While the critical Reynolds number tends to infinity as the permeability tends to zero, it decreases drastically for higher permeabilities. We also find a new channel mode and new class of modes in the porous region. We compare and discuss these results in terms of the recently published results of a three-dimensional linear stability analysis of a channel flow with porous walls (Tilton and Cortelezzi, {\it Phys. Fluids} {\bf 18}, 051702, 2006).   

Laminar-Turbulent Transition: A Hysteresis Curve of Two Critical Reynolds Numbers in Pipe Flow

A laminar-turbulent transition model (DFD 2004) has been constructed for pipe flows: (1) Natural transition occurs in the entrance region, and (2) Entrance shape determines a critical Reynolds number Rc. To verify the model, we have carried out experiments similar to Reynolds's color-dye experiment with 5 bellmouth entrances and a straight pipe. Then, we observed the following: (i) two different types of Rc exist, Rc1 from laminar to turbulent and Rc2 from turbulent to laminar, and (ii) the ratio of bellmouth diameter BD to pipe diameter D affects the values of Rc1 and Rc2. For each entrance, Rc1 has a maximum value Rc1(max) and Rc2 has a minimum value Rc2(min). When overlapping the two curves of Rc1(max) and Rc2(min) against BD/D, a hysteresis curve is confirmed. All Rc values exist inside this hysteresis curve. Consequently, Rc takes a minimum value Rc(min) of approximately 2000 when BD/D is at a minimum, i.e., at BD/D = 1, Rc(min) = Rc1(max) = Rc2(min) = 2000. Regarding Reynolds's Rc of 12,830, we observed Rc1(max) of approximately 13,000 at BD/D above 1.54. Therefore, the model has been partly verified.   

Receptivity and Transient Growth of Disturbances Generated by Random Distributed Roughness

Recent experiments on the receptivity of transient disturbances to 3D roughness have utilized spanwise-periodic arrays of cylindrical roughness elements. To connect these experiments with more realistic situations, quasi-random rough surfaces are generated numerically and manufactured using rapid-prototyping technology. Measurements of the resulting disturbances are obtained in a Blasius boundary layer using hotwire anemometry for freestream velocities of 7.5, 9.3 and 11.3 m/s; these values correspond to roughness-based Reynolds numbers of \textit{Re}$_{k}$ = 164, 227 and 301. The \textit{Re}$_{k}$ = 301 case is observed to be supercritical with transition occurring approximately 180 mm downstream of the roughness elements. Measurements over the roughness surface are made for the \textit{Re}$_{k}$ = 227 case in order to study the distributed receptivity effect of the roughness elements. Evidence of transient growth is apparent in the behavior of the disturbance energy and comparisons are made between spanwise power spectra computed from the disturbance profiles and the roughness elements.   

An Experimental Study of the Receptivity of a Supersonic Laminar Boundary Layer

The prediction of transition and its sensitivity to various disturbances has a major impact on system performance. It is also probably true that the least well understood aspect of the onset of transition in a supersonic boundary layer is the receptivity of the boundary layer to free stream acoustic disturbances and free stream turbulence (vorticity and entropy fluctuations). Techniques have been developed (for this short run time) which have enabled the rejection of the `naturally occurring' boundary layer response to other free stream disturbances. An x-t diagram has been created from the experimental results. The diagram and the corresponding detailed collapse of the wave packet measurements clearly show an oblique acoustic wave in the free stream traveling at 1003m/s (reference temperature 297.65K) and amplifying boundary layer wave-packets which travel at 392m/sec and 424m/sec. The x-t diagram also shows a surprising sensitivity in the receptivity to the actual free stream disturbance. On the x-t diagram, the amplifying boundary layer wave-packets are traced back to the leading edge of the boundary layer. It is found that the instability waves are driven by the followed packets in the free stream, whose wave vector is different from that of the leading packet. Further studies show that the followed packets have wave lengths in the spanwise direction that match those of the instability waves at down stream locations.   

Analysis of receptivity in wall-bounded shear flow by Schmidt decomposition of the resolvent

The receptivity of linearized Poiseuille flow to physically realistic, temporally and spatially constant or periodic forcing is investigated using a Schmidt decomposition of the resolvent. The orthogonal modes rank the body forcings that yield the highest disturbance energy gain. The concepts arising have a close relationship to frequency domain transfer function analysis methods known in the control systems literature. It is proposed that the processes leading to dominant structures identified by this procedure will be present in turbulent flow, where the required forcing shapes are provided by a nonlinear feedback process.   

Transition to Turbulence of Flow Due to a Vibrating Quartz Fork in Classical and Quantum Fluids

Flow due to a commercially available vibrating quartz fork at various temperatures and pressures is studied in gaseous helium, He I, and He II, over wide range of temperature and pressure. On increasing the driving force, the flow changes in character from laminar (characterized by a linear velocity versus drive dependence) to turbulent (characterized by a square root velocity versus drive dependence). In classical fluids, we characterize this transition by a critical Reynolds number, $Re_c=U_{cr}\delta/\nu$, where $U_{cr}$ is the critical velocity, $\nu$ stands for the kinematic viscosity, $\delta=\sqrt{2\nu/\omega}$ is the viscous penetration depth and $\omega$ is the angular frequency of oscillations. This scaling is continuous through the lambda-transition and effective kinematic viscosity for He II is extracted from the temperature dependence of the transition data. $U_{cr}$ of order 10~cm/s observed in He II agree with those found with other vibrating objects such as spheres, wires or grids, as well as with available numerical simulations of vortex motion in an applied ac flow.   

Hydrodynamics of an oscillating sphere in water

We have studied the flow patterns and damping of a one inch steel ball oscillating in water. The suspension was a 128 cm copper wire which allowed electrical connection to the water bath providing visualization by means of the Baker (thymol blue) electrolytic technique. The ball could be set into motion by means of a linear motor arranged to oscillate in the horizontal direction at the top of the suspension. Alternatively the bob could be set in motion and allowed to decay freely. The range of Reynolds numbers based on the maximum velocity ranged from 376 to 820 and the Carpenter-Keulegan number from 1.46 to 3.20. The period of oscillation was 2.5 sec. For Reynolds numbers up to about 380 we observed a boundary layer on the ball with a suggestion of a laminar wake spreading from the equator in the direction of oscillation. At higher Reynolds numbers around 550 we began to see periodic structure developing on the wake. By Re=726 it is clear the disturbances are a series of vortex rings which form on the rear of the sphere during an oscillation, and leapfrog over the sphere and propagate away when the direction of oscillation is reversed. The significance of this experiment for quantum turbulence will be discussed.