Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances

Abstract

The three-dimensional, algebraically growing instability of a Blasius boundary layer is studied in the nonlinear regime, employing a nonparallel model based on boundary layer scalings. Adjoint-based optimization is used to determine the "optimal" steady leading-edge excitation that provides the maximum energy growth for a given initial energy. Like in the linear case, the largest transient growth is found for inlet streamwise vortices, that yield streamwise streaks downstream. Two different definitions of growth are employed, providing qualitatively similar results, although the spanwise wavenumbers of optimal growth differ by up to 20% in the two cases. The wavelength of the most amplified optimal disturbance increases with the initial amplitude. For large input amplitudes, significant deformations of the mean velocity field are found; in such cases it is reasonable to expect that nonlinear streaks may break down through a secondary instability.  2005 Elsevier SAS. All rights reserved.

Theoretical and Computational Studies of Stability, Transition and Flow Control in High-Speed Flows

“…However, in the incompressible case and for Re --c, [LucO0] observed that the choice 'uin = 0, Pin = 0, vin and Win related by the continuity equation guarantees the maximum gain in an input-output fashion (in the incompressible case the number of independent initial conditions is two; see also [LB98,LucOO,ZBL06]). This choice also corresponds to the physical mechanism, observed in transitional boundary layer flows, known as the lift-up effect [Lan80], according to which streamwise vortices lift low-momentum flow up (from the wall) and push down highmomentum flow causing streaks that eventually break down to turbulence.…”

“…After employing the equation of state for the basic flow and for the perturbation, and observing that in the limit e -0 v and w can be neglected (Reynolds-independent approach, see [LucO]), the norm reads The initial condition for the compressible boundary-layer equations is not arbitrary, but only three of the five variables can be imposed at xin [Tin65]. However, in the incompressible case and for Re --+ o, it was observed that the choice uin = 0, pin = 0, vin and win related by the continuity equation corresponds to the maximum gain in an input-output fashion [LucO0] (in the incompressible case the number of independent initial conditions is two; see also [LucOO,ZBL06,LB98]). This choice also corresponds to the physical mechanism, observed in transitional boundary-layer flows, known as the lift-up effect [Lan80], according to which streamwise vortices lift low momentum flow up (from the wall) and push down high momentum flow causing streaks that eventually break down to turbulence.…”

“…On the contrary, when the adjoint equations are derived from a constrained optimization (as in our case) only if the objective function includes exclusively quantities at the boundaries of the domain then their form is the same as those derived from an inner product equality. In fact, if we try to maximize the integral of the energy over the whole domain (as opposed to the outlet energy only), a source term arises in the adjoint equations [CLOO, ZLB04,ZBL06]. As opposed to the continuous version of the Lagrange multiplier approach, less numerous are the examples where this technique is applied directly to the discrete equations [LB98, LB01, LucO0, CLOO, ZLB04, ZBL06].…”

“…The use of the discrete approach has several advantages among which the necessity of an "ad hoc" adjoint code is avoided and a foolproof test is available by comparing the results of the direct and adjoint calculation, which must match up to machine accuracy for any step size and not only in the limit of step size tending to zero [LucO0, ZBL06,LB98].…”

“…= 0.45 (scaled with 1 = V_L/U , v being the kinematic viscosity, U, the freestream velocity and L the longitudinal distance from the leading edge to the location where output energy is maximized). For the spatial problem, [ZBL06] computed the optimal perturbations in the nonlinear case.…”