TRANSIENT GROWTH: EXPERIMENTS, DNS AND THEORY

Transient growth of linearly stable disturbances is believed to play an important role in the subcritical transition of laminar boundary layers and the self-sustained nature of boundary layer fluctuations in a fully turbulent flow. Prior work on transient growth has focused on identifying the optimum initial disturbances that result in maximum transient growth. This paper addresses the companion issue of receptivity of those disturbances, the mechanism that determines the actual magnitudes of transient growth that are realized in a given physical situation. A synergistic combination of experimental, computational, and theoretical approaches is used to quantify the flow receptivity to surface roughness in a Blasius boundary layer. Results reveal the non-optimality of the transient growth factors involved as well as the sensitive dependence of flow perturbations to the geometric characteristics of the roughness distribution. Direct numerical simulations (DNS) are compared in detail with experimental results, results obtained from linear receptivity theory and optimal disturbance calculations. DNS shows good agreement with the experimental results. Differences between the linear theory and DNS are attributed to nonlinear receptivity mechanisms. Results also support the proposal by Fransson et al. (2004) that disagreement between optimal disturbances and experiments/DNS may be attributed to differences involving the wall normal location of the streamwise vortex initiating the transient growth. INTRODUCTION The transient growth phenomenon refers to an algebraic amplification of small-amplitude disturbances prior to an exponential decay farther downstream. Transient growth has been proposed as a likely mechanism behind laminar-turbulent transition scenarios that cannot be explained by the classical paradigm of hydrodynamic instabilities. Transient growth is also believed to play an important role in the self-generation of turbulence in fully turbulent wall shear flows (Butler and Farrell 1993; Chernyshenko and Baig 2005). Physically, the occurrence of transient growth can be explained by Landahl’s (1980) “lift-up” mechanism, where a pair of stable, counter-rotating, streamwise-oriented vortices transfers momentum across the boundary layer, creating a significantly stronger streamwise velocity perturbation. The mathematical foundation of transient growth has been described by Schmid and Henningson (2001) and Reshotko (2001). Transient growth arises because the linearized disturbance equations are not self-adjoint and, therefore, have nonorthogonal eigenmodes. In boundary layers, the correct representation of disturbances is the superposition of the discrete and continuous eigenmodes of the Orr–Sommerfeld (OS) equation. In classical linear stability analysis, the most amplified discrete eigenmode represents the dominant disturbance observed in an experiment. However, even in the subcritical region where the TS waves are damped, a suitable mixture of continuous spectrum modes (each of which has a different decay rate) can exhibit temporary algebraic growth before the exponential decay manifests itself. A particular combination of the modes will form an initial disturbance that experiences the maximum amount of growth. Such a disturbance is called the optimal disturbance (Farrell 1988) and many investigators have contributed to the modelling of spatially growing optimal disturbances (Andersson et al. 1999; Luchini 2000; Tumin and Reshotko 2001). For a laminar Blasius boundary layer, the optimal initial disturbance has been shown to consist of an array of stationary streamwise vortices with a dimensionless spanwise wavenumber of 0.45. Andersson et al. (1999) also found that the measured evolution of low-frequency boundary-layer disturbances excited via freestream turbulence is in agreement with the optimal growth theory. On the other hand, there is a significant mismatch between the optimal growth theory and measured (i.e. realizable) disturbances due to controlled surface roughness. White (2002) finds that roughness-induced disturbances show suboptimal behavior. Fransson et al. (2004) confirm this observation and attribute the mismatch to differences between the initial disturbance profiles used in optimal disturbance studies and those induced by surface roughness. The former corresponds to a streamwise vortex that is not confined to the boundary layer. Unlike the case of freestream turbulence, however, disturbances generated by low amplitude surface roughness are mostly confined to the boundary layer region. Transient disturbances are extremely sensitive to initial disturbance conditions as these determine the spectrum of modes that make up the disturbance and, hence, its algebraic growth rate. Since the maximum disturbance energy attained downstream of the source is mostly specified by the algebraic growth rate, understanding the receptivity mechanism that determines the initial disturbance condition is crucial. The objective of this work is to summarize the recent progress in experimental, computational and theoretical investigations of roughness-induced transient growth, with an emphasis on detailed comparisons between the respective findings for specific roughness configurations. Remaining challenges to assessing the relevance of the optimal growth theory to roughness effects on boundary-layer transition are also outlined. EXPERIMENTS White and coworkers (White and Ergin 2003; White et al. 2005; Ergin and White 2005) at Case Western Reserve University (Case) have conducted a number of experiments on the behavior of roughness-induced transient disturbances. Present efforts are focused on generating disturbance data that is suitable for comparison with DNS and theoretical models. A brief description of the experimental setup and data analysis procedure is given below. Measurements are obtained in the boundary layer of a flat plate with an elliptical leading edge and disturbances are generated by a spanwise array of cylindrical roughness elements. Roughness arrays provide considerable flexibility in varying the controlled disturbance inputs, via variations in the height, spacing and diameter of the roughness elements. Various averaging techniques, such as spatial phase-locked averaging, can be utilized to improve the signal-to-noise characteristics during data analysis. Throughout this paper, k denotes the roughness height, λk is the spanwise roughness spacing (19 mm), xk is the array’s streamwise location relative to the physical leading edge (300 mm), and Rek is the roughness Reynolds number, U(k)k/ν. Hotwire velocity measurements are performed in planes perpendicular to the streamwise flow. An illustration of the experimental setup is given in Figure 1. The main data analysis technique is to decompose the kinetic energy associated with the steady disturbance into spanwise wavelength components by performing a spanwise Fourier transform, computing the energy of each and examining the downstream evolution of these energies. The energy is defined by