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

▪ Abstract The recent progress in three-dimensional boundary-layer stability and transition is reviewed. The material focuses on the crossflow instability that leads to transition on swept wings and rotating disks. Following a brief overview of instability mechanisms and the crossflow problem, a summary of the important findings of the 1990s is given.

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Stability and transition of three-dimensional boundary layers

The current understanding of boundary-layer receptivity to external acoustic and vortical disturbances is reviewed. Recent advances in theoretical modeling, numerical simulations, and experiments are discussed. It is shown that aspects of the theory have been validated and that the mechanisms by which freestream disturbances provide the initial conditions for unstable waves are better understood. Challenges remain, however, particularly with respect to freestream turbulence

Boundary-layer receptivity to freestream disturbances

Advances in global linear instability analysis of nonparallel and three-dimensional flows

The objective of this review is to provide a critical evaluation of linear stability theory for wall-bounded shear flows, with an emphasis on results, not techniques. The results deal with applications to different flowfields related to aircraft systems. Because the flight condition is usually a low-disturbance environment, stability plays an important role in the transition process. Linear stability theory is not a new topic, so emphasis is placed on the most recent accomplishments.

Linear Stability Theory Applied to Boundary Layers

Boundary-layer transition-to-turbulence studies are conducted in the Arizona State University Unsteady Wind Tunnel on a 45-deg swept airfoil. The pressure gradient is designed so that the initial stability characteristics are purely crossflow dominated. Flow-visualization and hot-wire measurements show that the development of the crossflow vortices is influenced by roughness near the attachment line. Comparisons of transition location are made between a painted surface (distributed 9-μm peaks and valleys on the surface), a machine-polished surface (0.5-μm rms finish), and a hand-polished surface (0.25-μm rms finish). Then isolated 6-μm roughness elements are placed near the attachment line on the airfoil surface under conditions of the final polish (0.25-μm rms). These elements create an enhanced packet of stationary crossflow waves, which results in localized early transition. The diameter, height, and location of these roughness elements are varied in a systematic manner. Spanwise hot-wire measurements are taken behind the roughness element to document the enhanced vortices. These scans are made at several different chord locations to examine vortex growth

Effect of Isolated Micron-Sized Roughness on Transition in Swept-Wing Flows

An extensive computational investigation of a generic high-lift configuration comprising a wing and a half-span flap reveals details of the mean flow field for flap deflections of 29 and 39 degrees. The computational effort involves solutions of the thin layer form of the Reynolds Averaged Navier-Stokes(RANS) equations. For both flap deflections, the steady results show the presence of a dualvortex system; a strong vortex forming on the lower portion of the flap side edge and a weaker one forming near the edge on the flap top surface. Downstream, the vortex on the flap side edge grows and eventually merges with the vortex on the flap top surface. Comparison of on- and off-surface flow quantities with the experimental measurements of Radeztsky, Singer and Khorrami (AIAA Paper 98-0700) show remarkable agreement. For the 39 degree flap deflection, the calculation also reveals the occurrence of a vortex breakdown, which is corroborated by 5-hole probe velocity measurements performed in the Quiet Flow Facility at NASA Langley. The presence of the vortex breakdown significantly alters the flow field near the side edge.

Reynolds-averaged Navier-Stokes computations of a flap-side-edge flowfield

Detailed flow measurements were performed with a five-hole probe in the flap side-edge region of a wing with a half-span flap. The experiments were conducted in the NASA-Langley Quiet Flow Facility as a part of an extensive experimental and computational investigation of airframe-related noise sources. Basic flow properties were verified using static pressure taps, pressure-sensitive paint, and oil-flow visualization. Detailed five-hole probe surveys were conducted in a set of planes at ten streamwise stations for flap deflections of 29 and 39 degrees. These measurements show the development of a two-vortex system at the flap side edge. The vortex path and strength are determined, and the process of vortex merging is documented in detail in the mid-chord region. At the higher flap deflection angle, vortex bursting is observed, which may be a factor in the observed higher noise levels for this condition. Results are compared with Reynolds-averaged NavierStokes calculations of a similar configuration (AIAA Paper 98-0768). These measurements and calculations form the basis for instability modeling and unsteady calculations which will ultimately lead to an understanding of flap side-edge noise sources.

Detailed measurements of a flap side-edge flow field

An experimental study of receptivity was carried out using an acoustical disturbance in the freestream. The receptivity was enhanced by using a uniform two-dimensional roughness strip (tape). The roughness strip generated the local adjustment in the flow needed to couple the long-wavelength sound wave with the short-wavelength T-S wave. The method proved to be highly sensitive, with slight changes in the forcing frequency or in the height of the 2D roughness element having a strong effect on the amplitude of the observed T-S wave.

Boundary-layer receptivity of sound with roughness

Stability experiments are conducted in the ASU Unsteady Wind Tunnel on a 45° swept airfoil. The surface of the is polished to 0.25 μm rms. Under these conditions, natural stationary crossflow vortices are not measurable. This state is used to measure roughness-induced stationary crossflow. Spanwise arrays of 70–150 μm roughness elements are introduced near the attachment line. Detailed hot-wire measurements are taken to document the growth of these vortices. The data clearly show that linear stability theory does not accurately predict the growth rates of stationary crossflow waves under these conditions.

Ronald H. Radeztsky

Papers

Effect of Isolated Micron-Sized Roughness on Transition in Swept-Wing Flows

Reynolds-averaged Navier-Stokes computations of a flap-side-edge flowfield

Detailed measurements of a flap side-edge flow field

Boundary-layer receptivity of sound with roughness

Development of Stationary Crossflow Vortices on a Swept Wing