Thomas C. Corke

RESONANT GROWTH OF THREE-DIMENSIONAL MODES IN TRANSITIONING BLASIUS BOUNDARY LAYERS

By carefully controlled phase-coupled input of simultaneous two- and three-dimensional disturbances, the nonlinear evolution and breakdown of the laminar flow in a boundary layer was examined. This involved the generation of plane Tollmien-Schlichting waves and pairs of oblique waves so as to promote near-resonance conditions which have been theoretically shown to lead to the rapid development of three-dimensionality in unstable boundary layers

Mode selection and resonant phase locking in unstable axisymmetric jets

This paper presents experimental results on the nonlinear phase locking present in the resonant growth of unstable modes in the shear layer of an axisymmetric jet. The initial instability modes scale with the exiting shear layer and grow convectively with downstream distance. Because of the special condition at the exit lip of the jet, the initial growth of modes is very sensitive to local unsteady pressure fields. A part of the unsteady field is stochastic in nature. To a larger extent, the pressure field at the lip of the jet contains the imprint of the downstream-developing instability modes, in particular the first unstable axisymmetric mode and its subharmonic. These are felt at the lip of the jet as a result of the energetic processes of the first vortex rollup and vortex pairing. As a result, a resonant feedback exists which under special conditions makes the initial region of this flow in some sense absolutely unstable. The features of this process are brought out by the normalized crossbispectrum or cross-bicoherence between the instantaneous unsteady pressure at the lip of the jet and velocity time series measured at the same azimuthal position for different downstream locations. These give a measure of the nonlinear phase locking between the principle modes and their sum and difference modes. Analysis of these show a perfect nonlinear phase locking at the fundamental axisymmetric and subharmonic frequencies between the pressure field at the lip and the velocity field at the downstream locations corresponding to the energy saturations of the fundamental and subharmonic modes. This resonance process can be suppressed or enhanced by low-amplitude axisymmetric mode forcing at the natural preferred frequency of slightly detuned cases. Contrasted to this is the behaviour of the fundamental m = 1 helical mode. This mode was found to have the same spatial growth rate as the axisymmetric mode and a streamwise frequency approximately 20 % higher, in agreement with theoretical predictions. However, short-time spectral estimates showed that these two fundamental modes do not exist at the same time or space. This suggests that each is a basin of attraction which suppresses the existence of the other. The apparent non-deterministic switching observed between these modes is probably the result of the response of the jet to stochastic input of axisymmetric or non-axisymmetric disturbances. This scenario may lead to a lowdimensional temporal model based on the interaction between these two modes which captures most of the early random nature seen in our experiments.

Stationary travelling cross-flow mode interactions on a rotating disk

This work involves the study of the development of Type 1 stationary and travelling cross-flow modes in the three-dimensional boundary layer over a rotating disk. In order to control the characteristics of the stationary modes, we utilized organized patterns of roughness which were applied to the disk surface. These consisted of ink dots which were equally spaced in the azimuthal direction at a fixed radius in order to enhance particular azimuthal wavenumbers. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Velocity fluctuation time series were decomposed into the components corresponding to the stationary and travelling modes using the instantaneous disk position as a reference. Their development was documented through the linear and nonlinear stages leading to turbulence. The linear stage agreed well with linear stability predictions for both modes. In the nonlinear stage we documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which lead to the growth of a low-azimuthal-number, stationary mode. This mode had the largest amplitude and appeared to dominate transition

Boundary layer receptivity to free-stream sound on parabolic bodies

We use a numerical approach to study the receptivity of the boundaryn layer flow over a slender body with a leading edge of finite radius of curvature to smalln streamwise velocity fluctuations of a given frequency. The body of interest is a parabolan in order to exclude jumps in curvature, which are known sites of receptivity andn which occur on elliptic leading edges matched to finite-thickness at plates. The infinitesimallyn thin flat plate is the limiting solution for the parabola as the nose radiusn of curvature goes to zero. The formulation of the problem allows the two-dimensionaln unsteady Navier–Stokes equations in stream function and vorticity form ton be converted to two steady systems of equations describing the basic (nonlinear) flow and then perturbation (linear) flow. The results for the basic flow are in excellent agreementn with those in the literature. As expected, the perturbation flow was found to be a combinationn of an unsteady Stokes flow and Orr–Sommerfeld modes. To separate these,n the unsteady Stokes flow was solved separately and subtracted from the total perturbationn flow. We found agreement with the streamwise wavelengths and locations of Branchesn I and II of the linear stability neutral growth curve for Tollmien–Schlichtingn waves. The results showed an increase in the leading-edge receptivity with decreasing nosen radius, with the maximum occurring for an infinitely sharp flat plate. The receptivityn coefficient was also found to increase with angle of attack. These results were inn qualitative agreement with the asymptotic analysis of Hammerton & Kerschen (1992).n Good quantitative agreement was also found with the recent numerical resultsn of Fuciarelli (1997), and the experimental results of Saric, Wei & Rasmussen (1994).

THREE-DIMENSIONAL-MODE RESONANCE IN FAR WAKES

This work is aimed at understanding mechanisms which govern the growth of secondary three-dimensional modes of a particular type which feed from a resonant energy exchange with the primary Karman instability in two-dimensional wakes. Our approach was to introduce controlled time-periodic three-dimensional (oblique) wave pairs of equal but opposite sign, simultaneously with a two-dimensional wave. The waves were introduced by an array of v -component-producing elements on the top and bottom surfaces of the body. These were formed by metallized electrodes which were vapour deposited onto a piezoelectrically active polymer wrapped around the surface. The amplitudes, streamwise and spanwise wavenumbers, and initial phase difference are all individually controllable. The initial work focused on a fundamental/subharmonic interaction, and the dependence on spanwise wave-number. The results include mode eigenfunction modulus and phase distributions in space, and stream functions for the phase-reconstructed flow field. Analysis of these shows that such a resonance mechanism exists and its features can account for characteristic changes associated with the growth of three-dimensional structures in the wake of two-dimensional bodies.

High Mach Number Leading-Edge Flow Separation Control Using AC DBD Plasma Actuators

Wind tunnel experiments were conducted to quantify the e↵ectiveness of alternating current dielectric barrier discharge flow control actuators to suppress leading-edge stall on a NASA energy e cient transport airfoil at compressible freestream speeds. The objective of this research was to increase lift, reduce drag, and improve the stall characteristics of the supercritical airfoil near stall by flow reattachment at relatively high Mach and Reynolds numbers. In addition, the e↵ect of unsteady (or duty cycle) operation on these aerodynamic quantities was also investigated. The experiments were conducted at the University of Notre Dame Mach 0.6 Wind Tunnel for a range of Mach numbers between 0.1 and 0.4 with an airfoil model of chord 30.48 cm at atmospheric conditions corresponding to a Reynolds number range of 560, 000 through 2, 260, 000. Lift and drag forces, as well as the quarter chord moments were measured directly by a sting which reacted on load cells and torque sensors on the outside of the 0.91⇥0.91 m wind tunnel test section. Two leading-edges of the airfoil were fabricated. The first was covered in a Kapton dielectric film of 0.127 mm and had a 7 μm copper electrode, and the second was a thick-dielectric Macor with a copper tape exposed (76 μm thick) electrode. A high voltage AC signal was applied to electrodes for the flow control case. The results show that the plasma actuators were e↵ective at reattaching the leading-edge separated flow as evidenced by the increase in maximum lift coe cient and stall angle. In the post stalled regime, the lift was dramatically increased, by as much as 90%. Drag in the stalled regime was reduced by as much as 28% and the nose down pitching moment was reduced by as much as 40%. Pressure taps on the suction surface confirmed flow reattachment as evidenced by the return of a pressure peak near the leading-edge and better pressure recovery aft of the leading-edge when the active flow control was enabled. Time-averaged PIV confirmed the airflow following the airfoil surface closely. The experiment also showed that lift was increased the most in deep stall when the plasma actuator was operated unsteady with a reduced frequency of unity, whereas in light stall steady operation was preferred. Overall, both AC DBD plasma actuator designs were able to increase the maximum lift coe cient and stall angle of attack for the full range of Mach numbers, with the thick-dielectric Macor leading-edge performing better at Mach 0.4.

Resonant growth of three-dimensional modes in Falkner–Skan boundary layers with adverse pressure gradients

This work documents the spatial development of a triad of instability waves consisting of a plane TS mode and a pair of oblique modes with equal-opposite wave angles which are undergoing subharmonic transition in Falkner-Skan boundary layers with adverse pressure gradients. The motivation for this study is that for wings with zero or moderate sweep angles, transition is most likely to occur in the adverse pressure gradient region past the maximum thickness point and, starting with low initial amplitudes, subharmonic mode transition is expected to be the predominant mechanism for the first growth of of three-dimensional modes. The experiment follows that of Corke & Mangano (1989) in which the disturbances to produce the triad of waves are introduced by a spanwise array of heating wires located near Branch I. The initial conditions are carefully controlled. These include the initial amplitudes, frequencies, relative phase and oblique wave angles. The basic flow consisted of a Falkner-Skan (Hartree) boundary layer with a dimensionless pressure gradient parameter in the range -0.06 ≤ β H ≤ -0.09. The frequency of the TS wave was selected to be near the most amplified based on linear theory. The frequency of the oblique waves was the subharmonic of the TS frequency. The oblique wave angles were set to give the largest secondary growth ( 60°). Compared to similar conditions in a Blasius boundary layer, the adverse pressure gradient was observed to lead to an extra rapid growth of the two- and three-dimensional modes. In this there was a relatively larger maximum amplitude of the fundamental mode and considerably shortened amplitude saturation region compared to zero pressure gradient cases. Analysis of these results includes frequency spectra, the wall-normal distributions of each mode amplitude, and mean velocity profiles. Finally, the streamwise amplitude development is compared with the amplitude model from the nonlinear critical layer analysis of Goldstein & Lee (1992).

Cross-Flow Instability with Periodic Distributed Roughness

This work involves the study of the sensitivity of the Type 1 cross-flow instability to patterned distributed roughness, and the evolution of these modes to turbulence. The basic flow consists of the 3-D boundary layer over a rotating disk. The roughness is made up of ink dots which were applied to the surface of the disk. The pattern of dots consisted of an equally spaced array in the azimuthal direction, at a fixed radius. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Both subcritical and supercritical radii were explored. The dots were accurately placed using the computer controlled traversing mechanism which was equipped with an inking pen. The diameter and heights of the dots are 1.6mm and 0.06mm (left( {hsqrt {{w mathord{left/ {vphantom {w v}} right. kern-nulldelimiterspace} v}} = 0.16} right)) respectively. The azimuthal number of dots was intended to excite a particular azimuthal mode number of cross-flow modes. By this, uniform stationary modes could be excited so that we could more accurately separate fluctuations in velocity time series due to these, from modes which were travelling with respect to the disk rotation frame. The velocity time series were simultaneously measured at two radial positions. The instability azimuthal wave number, β, was found to change in response to the azimuthal dot number, α. The result was that the local angles of the cross-flow waves changed to keep the most amplified radial wave number, a. We documented the linear growth stage for stationary and travelling modes. We also documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which lead to the growth of low azimuthal number stationary mode. This mode had the largest amplitude and appeared to dominate transition.

Tip Clearance Flow Control in a Linear Turbine Cascade using Plasma Actuation

This research examines the use of passive and active on-blade flow control to reduce the unwanted losses associated with the blade tip clearance flow in a stationary, rectilinear turbine cascade. A SDBD plasma actuator and a passive partial suction-side squealer were tested over a Reynolds number range from 5.3×10 4 to 1.03×10 5 at a fixed tip clearance of 2.18 percent of axial chord. The pla sma actuator was designed to mimic the beneficial effects of the suction-side squealer ti p, while removing the negative aspects of the passive squealer design, including blade degradation flow recircul ation or potential blade-wall contact. The flowfield was documented with five-hole-probe measurements at 1 axial chord downstream of the test blade and within the clearance using wall pressure taps located on the endwall opposite the blade tip. These tests allowed the loss associated with the flow and the change in this loss with a pplied flow control to be recorded. The plasma actuator caused an improvement in the downstream flow, with a reduction in the total pressure loss coefficient within the tip leakage vortex ranging between 2% to 12%, depending on Reynolds number, while the passive squealer showed a change of approximately 40%. On the endwall within the clearance, the plasma actuator generated a 19% peak increase in wall static pressure while the passive squealer caused a maximum increase of 52%. These results show that the plasma actuator was able to favorably mitigate the adverse effects of the tip clearance flow in a similar manner as the squealer tip, withou t the drawbacks of the passive squealer method. Although less effective than the squealer tip, the positive results of the plasma actuator show that this type of flow control is suitable as a means of reducing the tip clearan ce flow loss.

Control of a Turbine Tip Leakage Vortex Using Casing Vortex Generators

An experiment was conducted in a linear cascade of Pak-B blades to simulate the ∞ow in the tip-gap region of a low pressure turbine blade row, and investigate the sensitivity of the ∞ow to casing-mounted passive and active vortex generators. The inlet Reynolds number was 5£10 5 corresponding to an inlet Mach number of 0.2 and an exit Mach number of 0.3. The gap-to-chord ratio was 5%. The ∞ow was documented using blade-tip static pressure measurements and downstream total pressure loss coe‐cients. Additionally, surface ∞ow visualization was performed on the cascade end-wall for a greater understanding of the gap∞ow behavior. These measurement techniques were used to investigate the response of the ∞ow to passive ∞ow control using vortex generators located on the cascade end-wall. The vortex generators were designed to produce vorticity of opposite sign of the tip-leakage vortex. Vortex generators that were of the height and roughly half the height of the clearance gap were investigated. Additionally, two placements of the vortex generator were investigated: on the wall directly across from the trailing edge of the blade and approximately 0.25cx upstream of the trailing edge. All investigated placements and sizes of the passive vortex generators reduced the total pressure loss associated with the tip leakage vortex. A 25% reduction was achieved with both heights of the vortex generators when placed at the upstream location. When placed at the trailing edge, the shorter vortex generator resulted in a 15% reduction of total pressure loss associated with the tip leakage vortex, and the taller vortex generator resulted in a 20% reduction. The sensitivity of the ∞ow to the passive devices was used to determine placement and design of a plasma actuator on the end-wall as an active ∞ow control mechanism. Three plasma actuators were investigated, one of which resulted in a 7% decrease in total pressure loss associated with the tip leakage vortex.