Parabolized stability equations (PSE) have opened new avenues to the analysis of the streamwise growth of linear and nonlinear disturbances in slowly varying shear flows such as boundary layers, jets, and far wakes. Growth mechanisms include both algebraic transient growth and exponential growth through primary and higher instabilities. In contrast to the eigensolutions of traditional linear stability equations, PSE solutions incorporate inhomogeneous initial and boundary conditions as do numerical solutions of the Navier-Stokes equations, but they can be obtained at modest computational expense. PSE codes have developed into a convenient tool to analyze basic mechanisms in boundary-layer flows. The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods promise improved design capabilities for laminar flow control systems.

Parabolized stability equations

Pressure fluctuations are an important ingredient in turbulence, e.g. in the pressure strain terms which redistribute turbulence among the different fluctuating velocity components. The variation o ...

Pressure statistics and their scaling in high-Reynolds-number turbulent boundary layers

Detailed velocity measurements were made along a flat plate subject to the same dimensionless pressure gradient as the suction side of a modern low-pressure turbine airfoil. Reynolds numbers based on wetted plate length and nominal exit velocity were varied from 50,000 to 300,000, covering cruise to takeoff conditions. Low and high inlet free-stream turbulence intensities (0.2% and 7%) were set using passive grids. The location of boundary-layer separation does not depend strongly on the free-stream turbulence level or Reynolds number, as long as the boundary layer remains non-turbulent prior to separation. Strong acceleration prevents transition on the upstream part of the plate in all cases. Both free-stream turbulence and Reynolds number have strong effects on transition in the adverse pressure gradient region. Under low free-stream turbulence conditions transition is induced by instability waves in the shear layer of the separation bubble. Reattachment generally occurs at the transition start. At Re = 50,000 the separation bubble does not close before the trailing edge of the modeled airfoil. At higher Re, transition moves upstream, and the boundary layer reattaches. With high free-stream turbulence levels, transition appears to occur in a bypass mode, similar to that in attached boundary layers. Transition moves upstream, resulting in shorter separation regions. At Re above 200,000, transition begins before separation. Mean velocity, turbulence and intermittency profiles are presented.

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Measurements in Separated and Transitional Boundary Layers Under Low-Pressure Turbine Airfoil Conditions

Excitation of Gortler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Gortler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Gortler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Gortler vortices are excited. The amplitude of the Gortler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for GΛ ≫ 1 to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x* ~ ΛRΛG−2/3Λ, where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛGΛ−2/5. This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Gortler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. Gortler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2Λ.

Excitation of steady and unsteady Gortler vortices by free-stream vortical disturbances

A compressible countercurrent shear layer was investigated experimentally by establishing reverse flow around the perimeter of a supersonic jet. Measurements demonstrate that spatial growth rates of the countercurrent shear layer significantly exceed those of the classical coflowing layer at comparable density ratios and levels of compressibility. Experiments also reveal the presence of coherent three-dimensional structures in the countercurrent shear layer at convective Mach numbers where similar structures are not present in coflowing layers. It is argued that these kinematic differences are responsible for the enhanced diffusion of the shear layer with counterflow. The spatio-temporal theory is used to examine the connection between the experimental observations and the existence of a transition from convective to absolute instability in high-speed shear layers.

The effect of counterflow on the development of compressible shear layers

An experimental investigation of the transition process on flat-plate and concave curved-wall boundary layers for various free-stream turbulence levels was performed. Results show that for transition of a flat-plate, the two forms of boundary layer behavior, identified as laminar-like and turbulent-like, cannot be thought of as separate Blasius and fully-turbulent profiles, respectively. Thus, simple transition models in which the desired quantity is assumed to be an average, weighted on intermittency, of the theoretical laminar and fully turbulent values is not expected to be successful. Deviation of the flow identified as laminar-like from theoretical laminar behavior is shown to be due to recovery after the passage of a turbulent spot, while deviation of the flow identified as turbulent-like from the full-turbulent values is thought to be due to incomplete establishment of the fully-turbulent power spectral distribution. Turbulent Prandtl numbers for the transitional flow, computed from measured shear stress, turbulent heat flux and mean velocity and temperature profiles, were less than unity. For the curved-wall case with low free-stream turbulence intensity, the existence of Gortler vortices on the concave wall within both laminar and turbulent flows was established using liquid crystal visualization and spanwise velocity and temperature traverses. Transition was found to occur via a vortex breakdown mode. The vortex wavelength was quite irregular in both the laminar and turbulent flows, but the vortices were stable in time and space. The upwash was found to be more unstable, with higher levels of u' and u'v', and lower skin friction coefficients and shape factors. Turbulent Prandtl numbers, measured using a triple-wire probe, were found to be near unity for all post-transitional profiles, indicating no gross violation of Reynolds analogy. No evidence of streamwise vortices was seen in the high turbulence intensity case.

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Free-stream turbulence and concave curvature effects on heated, transitional boundary layers

This study was undertaken to examine the mechanisms which produce the large entrainment measured near the exit of thermal plasma torches. A research facility was constructed to examine low density jet behavior under similar dimensionless conditions as those produced by thermal plasma spray torches; the Reynolds number based on jet diameter and average properties was 1000, and the ratio of jet to ambient density was 0.07. This very low density jet produced organized vortex structures which were partially responsible for the rapid entrainment of external air. The formation of these organized structures could be disrupted by introducing turbulence, but the rapid entrainment process was not significantly affected. The structure of the jet produced by a commercial plasma torch was examined and compared to the low density research jet. At low gas flow rates the plasma jet also displayed the formation of coherent vortex structures, the passage frequency of which compared favorably with that measured in the low density research jet. At higher gas flow rates the shear layer of the plasma jet rapidly broke down producing relatively small scale turbulence. Visualizations of the hot plasma core were compared against measurements of the torch voltage fluctuations caused by arc instabilities. At low flow rates the arc voltage fluctuations were quite low and the plume was very steady. At higher flow rates the arc voltage fluctuations increased and produced “surging” and “whipping” in the hot potential core. It is believed that this low frequency unsteadiness is partially responsible for the rapid entrainment measured in plasma torches.

Mixing in plasma and low density jets

The purpose of this paper is to convey information and experience gained in the use of a rotating slanted hot-wire probe to measure the complete Reynolds stress tensor and the mean velocity vector The technique is limited to nearly one-dimensional flows with turbulence intensities less than 10% This paper extends the description of the method and discusses potential problems and difficulties one my encounter with this technique In particular, the use of an appropriate least-squares technique (singular value decomposition) is suggested as this gives a more accurate fitting than simply using a matrix inversion of the normal equations The covariance matrix is used to determine the slant angle and rotation positions of the probe most favorable to the least squares fitting and provides a measure of the uncertainties in the solution for the Reynolds stresses The technique presented was used to determine the anisotropy of a turbulent flow with no mean shear The results compare well with cross-wire measurements

S. Russ

Papers

Free-stream turbulence and concave curvature effects on heated, transitional boundary layers

Mixing in plasma and low density jets

On the rotating, slanted, hot-wire technique