Spatially developing secondary instabilities in compressible swept airfoil boundary layers

Abstract

Two-dimensional eigenvalue analysis is used on a massive scale to study spatial instabilities of compressible shear flows with two inhomogeneous directions. The main focus of the study is crossflow dominated swept-wing boundary layers although the methodology can also be applied to study other type of flows, such as the attachment-line flow. Certain unique aspects of formulating a spatial, two-dimensional eigenvalue problem for the secondary instability of finite amplitude crossflow vortices are discussed, namely, fixing the spatial growth direction unambiguously through a non-orthogonal formulation of the linearized disturbance equations. A primary test case used for parameter study corresponds to the low-speed, NLF-0415(b) airfoil configuration as tested in the ASU Unsteady Wind Tunnel, wherein a spanwise periodic array of roughness elements was placed near the leading edge in order to excite stationary crossflow modes with a specified fundamental wavelength. The two classes of flow conditions selected for this analysis include those for which the roughness array spacing corresponds to either the naturally dominant crossflow wavelength, or a subcritical wavelength that serves to reduce the growth of the naturally excited dominant crossflow modes. Numerical predictions are compared with the measured database, both as indirect validation for the spatial instability analysis and to provide a basis for comparison with a higher Reynolds number, supersonic swept-wing configuration. Application of the eigenvalue analysis to the supersonic configuration reveals that a broad spectrum of stationary crossflow modes can sustain sufficiently strong secondary instabilities as to potentially cause transition over this configuration. Implications of this finding for transition control in swept wing boundary layers are examined.

Wake Instabilities behind Discrete Roughness Elements in High Speed Boundary Layers

“…Since the spanwise and wall normal length scales of the boundary layer streaks within the wake are comparable, the modified boundary layer flow has a strongly inhomogeneous character in both y and z directions. Therefore, its stability characteristics are more appropriately studied by solving an eigenvalue problem based on two-dimensional partial differential equations (PDEs), 24,25 rather than using the conventional linear stability analysis, which is based on the assumption of basic state inhomogeneity in a single spatial coordinate (viz., the surface normal direction). The spatial stability of the computed streaks was examined at three selected streamwise locations in the roughness wake.…”

Stability Analysis for HIFiRE Experiments

“…15(b)). Of course, because of the rapid azimuthal variations associated with the primary vortex roll-up near the leeward line, accurate predictions of instability evolution along this plane may well require an advanced stability analysis based on partial differential equations 23,24 . The crossflow instability becomes important in characterizing the transition mechanism on the surface of the elliptic cone away from the attachment line.…”

Experimental study of stability and transition of supersonic boundary layer on swept wing at mach number 2-4

“…Note that the excitation of several modes of secondary instability was detected theoretically in [5,9]. Modes with the peak growth rate at frequencies about 100 kHz, 1.05 MHz, and 970 kHz were detected in [5] at M = 3.5, and with frequencies about 1 MHz were detected in [9] at M = 2.4. Unfortunately, a hot-wire anemometer cannot measure the perturbations with frequencies around 1 MHz.…”

“…An attempt to predict the transition with accounts for all major stages was made theoretically by Choudhari et al [8]. Numerical studies of the secondary instability, the transition prediction, and control for swept wing supersonic boundary layers were made by Li and Choudhari in [9], where the experimental con¦guration of [10] was modeled. Experiments [10] on the passive §ow control by a distributed roughness were made at Mach number M = 2.4 on 73 degree swept wing with the thicknessto-chord ratio of 4%.…”