On the temporal linear stability of the asymptotic suction boundary layer

Abstract

A temporal linear stability analysis of the asymptotic suction boundary layer is presented. For this, the Orr–Sommerfeld equation is solved in terms of generalized hypergeometric functions. Together with the corresponding boundary conditions, an algebraic eigenvalue problem is formulated. Thereof we derive the temporal continuous spectrum yielding a rather distinct spectrum if, for example, compared to the one from the Blasius solution. A second key result is that the discrete spectrum in the limits α→0,that is, small streamwise wave numbers, and Re→∞ is only present in the distinguished limit Re α=O(1). This results in a degenerated Orr–Sommerfeld equation and the expanded algebraic eigenvalue problem poses a lower limit of (Re α)min≈0.841 91. We show that this lower bound corresponds to a maximum extension of the viscous eigenfunction in the wall-normal direction. The full algebraic eigenvalue problem is numerically solved for the temporal case up to Re=6.0×106. Besides the further refined critical values αcr=0.155 46, ωcr=0.023 297, Recr=54 378.620 32, discrete spectra and eigenfunctions are examined and ω=ωr+iωi is the complex frequency. In particular, eigenvalue spectra are investigated with regard to their behavior due to a variation of the Reynolds number and the wave number, respectively, and only A-modes according to the definition of Mack [“A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer,” J. Fluid Mech. 73, 497–520 (1976)] were identified. From these, three different classes of eigenfunctions of the wall-normal disturbance velocity are presented. Finally, we find that the inviscid part of the eigenfunctions is dominant in wall-normal direction and only propagates in streamwise direction, while the viscous part is limited to the vicinity of the wall and propagates toward it in an almost perpendicular direction.