Algebraic/transcendental disturbance growth behind a row of roughness elements

Abstract

This paper is a continuation of the work begun in Goldstein et al. (J. Fluid Mech., vol. 644, 2010, p. 123), who constructed an asymptotic high-Reynolds-number solution for the flow over a spanwise periodic array of relatively small roughness elements with (spanwise) separation and plan form dimensions of the order of the local boundary-layer thickness. While that paper concentrated on the linear problem, here the focus is on the case where the flow is nonlinear in the immediate vicinity of the roughness with emphasis on the intermediate wake region corresponding to streamwise distances that are large in comparison with the roughness dimension, but small in comparison with the distance between the roughness array and the leading edge. An analytical O(h 2 ) asymptotic solution is obtained for the limiting case of a small roughness height parameter h. These weakly nonlinear results show that the spanwise variable component of the wall-pressure perturbation decays as x −5/3 ln x when x → ∞ (where x denotes the streamwise distance scaled on the roughness dimension), but the corresponding component of the streamwise velocity perturbation (i.e. the wake velocity) exhibits an O(x 1/3 ln x) algebraic/transcendental growth in the main boundary layer. Numerical solutions for h = O(1) demonstrate that the wake velocity perturbation for the fully nonlinear case grows in the same manner as the weakly nonlinear prediction -which is considerably different from the strictly linear result obtained in Goldstein et al. (2010).

Nonlinear wakes behind a row of elongated roughness elements

“…The resulting flow is formally nonlinear in the intermediate wake region, where the streamwise distance is large compared to the roughness dimensions but small compared to the downstream distance from the leading edge, as well as in the far wake region where the streamwise length scale is of the order of the downstream distance from the leading edge. In contrast, the flow perturbations in both of these wake regions were strictly linear in the earlier work by Goldstein et al (2010Goldstein et al ( , 2011. This is an important difference because the nonlinear wake flow in the present case provides an appropriate basic state for studying the secondary instability and eventual breakdown into turbulence.…”

On the Growth of Görtler Vortices Excited by Distributed Roughness Elements

“…Since the equations are nonlinear, the convergence was achieved by a relaxation method; this was done by including time derivate terms in the equations that were converged to zero, using an appropriate preconditioning technique applied to the continuity equation. The linearized boundary region equations (13)(14)(15)(16) were solved using a similar scheme as described in GSDC.…”

“…It was shown that the flow in the proximity of the roughness elements decays algebraically over a very short streamwise distance behind the roughness elements, and then breaks down in the downstream region. In a following article, Goldstein et al 16 extended the analysis to second order and showed that the downstream wakes exhibit algebraic/transcendental growth and that the location of growth onset moves closer to the roughness elements with increasing roughness height.…”

On the Effect of Small Steps on Pre-transitional Streamwise Elongated Streaks

“…The primary effects of surface imperfections can be characterized by their impact on receptivity and linear amplification when the transition occurs via receptivity and linear growth followed by a sequence of nonlinear effects (Goldstein et al 11 ). The surface imperfections may introduce substantial changes in the boundary layer instability mechanisms, such as the onset of unsteady vortex shedding in a low-speed boundary layer (Acarlar and Smith; 1 Klebanoff et al, 20 Goldstein et al 11,12 ). Prior experimental measurements (e.g., White et al, 34,35 Reshotko 24 ) showed that surface imperfections have the potential to accelerate or delay the transition to turbulence, depending on the shape and size.…”

“…Two step configurations of forward and backward facing types, and three 'streak strengths' (quantified by the difference between the maximum and minimum of the high-and low-speed regions, respectively) are considered. We numerically solve the boundary region equations subject to upstream boundary conditions as derived in Goldstein et al 11,12 The generalized Rayleigh pressure equation is solved as an eigenvalue problem to determine the growth rates associated with the secondary instabilities. It is found that as the height of the step is increased beyond 0.5 of boundary layer displacement thickness, the growth rate reduces at the peak streamwise wavenumber of the instability.…”