Earlier theoretical work on the boundary-layer receptivity problem utilized the triple-deck framework, and typically produced only the leading-order asymptotic result. The applicability of these predictions was limited to the generation of Tollmien–Schlichting-type (viscosity-conditioned) instabilities and rather high values of an appropriate Reynolds number. Generalizing the concepts behind the asymptotic theory of Goldstein and Ruban, the classical Orr–Sommerfeld theory is utilized to predict the receptivity due to small-amplitude surface nonuniformities. This approach accounts for the finite Reynolds-number effects, and can also be extended easily to problems involving other types of instabilities. It is illustrated here for the case of the Tollmien–Schlichting wave generation in a Blasius boundary layer, due to the interaction of a free-stream acoustic wave with a region of short-scale variation in one of the surface boundary conditions. The type of surface disturbances examined include regions of short-scale variations in wall suction, wall admittance, and wall geometry (roughness). Results from the finite Reynolds-number approach are compared in detail with previous asymptotic predictions, as well as the available experimental data.
Abstract. A study of instabilities in incompressible boundary-layer ow on a at plate is conducted by spatial direct numerical simulation DNS of the Navier-Stokes equations. Here, the DNS results are used to critically evaluate the results obtained using parabolized stability equations PSE theory and to study mechanisms associated with breakdown from laminar to turbulent o w. Three test cases are considered: two-dimensional TollmienSchlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-ow distortion component of the 2-D test problem. This discrepancy is attributed to far-eld boundarycondition di erences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic ow and a variation of the disturbance frequency result in better agreement with the experiments.
The economic and environmental benefits of laminar flow technology via reduced fuel burn of subsonic and supersonic aircraft cannot be realized without minimizing the uncertainty in drag prediction in general and transition prediction in particular. Transition research under NASA's Aeronautical Sciences Project seeks to develop a validated set of variable fidelity prediction tools with known strengths and limitations, so as to enable "sufficiently" accurate transition prediction and practical transition control for future vehicle concepts. This paper provides a summary of research activities targeting selected gaps in high-fidelity transition prediction, specifically those related to the receptivity and laminar breakdown phases of crossflow induced transition in a subsonic swept-wing boundary layer. The results of direct numerical simulations are used to obtain an enhanced understanding of the laminar breakdown region as well as to validate reduced order prediction methods. Nomenclature A= amplitude of crossflow instability mode or secondary instability mode, measured in terms of peak chordwise velocity perturbation and normalized with respect to freestream velocity c = wing chord length normal to the leading edge C = coupling coefficient for receptivity D = diameter of cylindrical roughness element f = frequency of instability oscillations (Hz) F = forcing function representing external disturbance environment F = geometry factor from receptivity theory G = gain factor associated with instability amplification i = (-1) 1/2 k = grid index along wall-normal direction M = freestream Mach number N = N-factor (i.e., logarithmic amplification ratio) of linear crossflow instability or secondary instability q = arbitrary flow variable R = transfer function associated with receptivity Re c = Reynolds number based on wing chord t = time (u,v,w) = Cartesian velocity components aligned with (x,y,z) axes x = chordwise coordinate in direction perpendicular to leading edge y = Cartesian coordinate normal to x-y plane z = spanwise coordinate, i.e., the coordinate parallel to airfoil leading edge α = chordwise wavenumber 1 2 β = spanwise wavenumber Γ = wing sweep angle λ = spanwise wavelength of crossflow instability λ w = chordwise wavelength of wall waviness Λ = efficiency function for localized receptivity ξ = dummy integration variable in chordwise direction π = normalized mode shape of instability wave θ = phase function φ = normalized mode shape of instability wave ω = radian frequency of instability wave Subscripts c = crossflow direction i = initial or reference location (typically chosen to be lower branch neutral station for primary instability and inflow location of x/c=0.502 for secondary instability) init = initial or reference location (used interchangeably with subscript i) ins = instability wave lb = lower branch neutral station for the instability mode of interest L = linear amplification phase N = nonlinear amplification phase r = roughness w = wall τ = edge streamline direction Superscripts + = wall units
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