Triple-deck structure

Abstract

The asymptotic foundation of two-dimensional steady triple-deck theory is reviewed. The total flow is assumed to be the sum of a basic-flow component Q( x, y) governed by the boundary-layer equations and a disturbance component q( x, y), resulting from a sudden streamwise change (localized disturbance), such as that caused by a hump or dip, a suction or blowing discontinuity, a heating or cooling discontinuity, a trailing edge etc. Substituting the total flow Q + q into the steady two-dimensional Navier-Stokes equations and subtracting the basic-flow quantitites yields nonlinear equations describing the disturbance quantities q. The disturbance quantities are assumed to vary with the streamwise scale X = ( x − 1)Re α , where x = x ∗¦x r ∗, x r ∗ is the center of the localized disturbance, Re = U ∞ ∗x r ∗/v , U ∞ ∗ is the freestream velocity and v is the kinematic viscosity of the fluid. Introducing the stretching transformation Y = yRe β , β > 0, and scaling ψ as ψ = ψRe − χ , where ψ is the streamfunction of the disturbance, we find that there are three distinguished limits when 0 < α < 1 2 and hence three sets of least degenerate problems, resulting in a triple-deck structure. These limits correspond to β = α, β = 1 2 and β = 1 2 + 1 3 α . Matching the pressure in these decks demands that α = 3 8 and leads to the triple-deck structure. The upper- and middle-deck problems are linear, whereas the lower-deck problem is linear when χ > 3 4 and nonlinear when χ ⩽ 3 4 . When α = 1 2 , there are two distinguished limits corresponding to β = 1 2 and 2 3 . The upper-deck problem is linear, whereas the lower-deck problem is linear when χ > 5 6 and nonlinear when χ ⩽ 5 6 . The theory is self-consistent and does not depend on the problem being investigated. Application of these theories to the linear three-dimensional compressible stability of two-dimensional compressible boundary layers is described and limitations of the triple-deck theory are discussed.