Disturbance amplification in boundary layers over thin wall films
Physics of Fluids
Turbulence and skin friction modification in channel flow with streamwise-aligned superhydrophobic surface texture Physics of Fluids 26, 095102 (2014); https://doi.org/10.1063/1.4894064 Shear sheltering and the continuous spectrum of the Orr-Sommerfeld equation In single-fluid boundary layers, streaks can amplify at sub-critical Reynolds numbers and initiate early transition to turbulence. Introducing a wall film of different viscosities can appreciably alter the stability of the base flow and,
... in particular, the transient growth of the perturbation streaks. The formalism of seminorms is used to identify optimal disturbances which maximize the kinetic energy in the two-fluid flow. An examination of optimal growth over a range of viscosity ratios of the film relative to the outer flow reveals three distinct regimes of amplification, each associated with a particular combination of the eigenfunctions. In order to elucidate the underlying amplification mechanisms, a model problem is formulated: An initial value problem is solved using an eigenfunction expansion and is used to compute the evolution of pairs of eigenfunctions. By appropriately selecting the pair, the initial value problem qualitatively reproduces the temporal evolution of the optimal disturbance, and provides an unambiguous explanation of the dynamics. Two regimes of transient growth are attributed to the evolution of the interface mode along with free-stream vortical modes; the third regime is due to the evolution of the interface and a discrete mode. The results demonstrate that a lower-viscosity film can effectively reduce the efficacy of the lift-up mechanism and, as a result, transient growth of disturbances. However, another mechanism of amplification of wall-normal vorticity arises due to the deformation of the two-fluid interface and becomes dominant below a critical viscosity ratio. C 2016 AIP Publishing LLC. [http://dx. Saha, Page, and Zaki Phys. Fluids 28, 024108 (2016) to weaken the pre-transitional boundary-layer streaks. This work therefore bridges two strands of research: instability of two-fluid flows and transient growth analysis in single-fluid boundary layers. A. Two-fluid modal instabilities In the context of two-fluid shear flows, research has primarily focused on discrete instability waves. In addition to the conventional Tollmien-Schlichting waves of the outer flow, three groups of discrete instability modes have been identified: the "soft" mode, 4 "H" mode, 5 and the high-Reynolds-number instability. 6 The soft mode was established in the pioneering work by Yih. 4 The long-wavelength instability arises from the viscosity mismatch across the two-fluid interface and can exist at any Reynolds number. However, this mode becomes stable when the lower layer is thinner and less viscous compared to the upper layer. 7,8 Unlike the soft mode, the "H mode" discovered by Hooper and Boyd 5 is unstable independent of the viscosity of the inner fluid. It is a short wavelength instability and is also due to the viscosity jump at the interface. There is no experimental evidence of this mode since a nominal level of surface tension is sufficient to stabilize this mechanism. 9 The final instability exists at high Reynolds numbers, when the kinematic viscosity of the wall-bounded fluid is less than that of the upper fluid. 6 The wavelength of the instability is on the order of the film thickness and is unaffected by surface tension. The instability derives energy largely from the Reynolds stress in the lower fluid and has its origin in a viscous layer near the wall. Recent studies have examined the spatio-temporal character of these two-fluid instabilities and their evolution in the non-linear regime. For example, Valluri et al. 10 demonstrated that channel flow with a thin wall-film is convectively unstable and that the amplifying soft mode leads to the formation of ligaments. On the other hand, the non-linear development of the instability waves for a thick wall film results in the formation of slug structures. 11 A combination of both twoand three-dimensional instabilities in channel flow can produce ligaments, sheets, or "interfacial turbulence" in different regions of the parameter space. 12 For spatially developing two-fluid flows, the non-linear parabolized stability equations have been formulated and applied to study intermodal energy transfer and the distortion to the mean flow. 13, 14 In the present study, we focus on the stabilizing influence of a wall film on the outer boundary layer. Exponential instabilities can be avoided at moderate Reynolds numbers and when the wall film is less viscous with a nominal level of surface tension at the interface. As a result, linear disturbances can only exhibit transient amplification. Transient growth analyses are well established in the context of stability of single-fluid boundary layers and have received some attention in the context of subcritical two-fluid channel flows.     They have also been applied to unstable configurations, such as core-annular pipe flow, 19 two-fluid mixing layers, 20 and round viscous jets. 21 The literature has not, however, addressed the two-fluid boundary-layer configuration which is analyzed herein.