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Register a new userAnalytical Model of the Time Developing Turbulent Boundary Layer
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We present an analytical model for the time-developing turbulent boundary layer (TD-TBL) over a flat plate. The model provides explicit formulae for the temporal behavior of the wall-shear stress and both the temporal and spatial distributions of the mean streamwise velocity, the turbulence kinetic energy and Reynolds shear stress. The resulting profiles are in good agreement with the DNS results of spatially-developing turbulent boundary layers at momentum thickness Reynolds number equal to 1430 and 2900. Our analytical model is, to the best of our knowledge, the first of its kind for TD-TBL.
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This fluid dynamics video submitted to the Gallery of Fluid motion shows a turbulent boundary layer developing under a 5 metre-long flat plate towed through water. A stationary imaging system provides a unique view of the developing boundary layer as it would form over the hull of a ship or fuselage of an aircraft. The towed plate permits visualisation of the zero-pressure-gradient turbulent boundary layer as it develops from the trip to a high Reynolds number state ($Re_tau approx 3000$). An evolving large-scale coherent structure will appear almost stationary in this frame of reference. The visualisations provide an unique view of the evolution of fundamental processes in the boundary layer (such as interfacial bulging, entrainment, vortical motions, etc.). In the more traditional laboratory frame of reference, in which fluid passes over a stationary body, it is difficult to observe the full evolution and lifetime of turbulent coherent structures. An equivalent experiment in a wind/water-tunnel would require a camera and laser that moves with the flow, effectively `chasing eddies as they advect downstream.
Turbulent boundary layers exhibit a universal structure which nevertheless is rather complex, being composed of a viscous sub-layer, a buffer zone, and a turbulent log-law region. In this letter we present a simple analytic model of turbulent boundary layers which culminates in explicit formulae for the profiles of the mean velocity, the kinetic energy and the Reynolds stress as a function of the distance from the wall. The resulting profiles are in close quantitative agreement with measurements over the entire structure of the boundary layer, without any need of re-fitting in the different zones.
Recent progress in understanding subcritical transition to turbulence is based on the concept of the edge, the manifold separating the basins of attraction of the laminar and the turbulent state. Originally developed in numerical studies of parallel shear flows with a linearly stable base flow, this concept is adapted here to the case of a spatially developing Blasius boundary layer. Longer time horizons fundamentally change the nature of the problem due to the loss of stability of the base flow due to Tollmien--Schlichting (TS) waves. We demonstrate, using a moving box technique, that efficient long-time tracking of edge trajectories is possible for the parameter range relevant to bypass transition, even if the asymptotic state itself remains out of reach. The flow along the edge trajectory features streak switching observed for the first time in the Blasius boundary layer. At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories. In this situation we suggest a reinterpretation of the edge as a manifold dividing the state space between the two main types of boundary layer transition, i.e. bypass transition and classical transition.
The present numerical investigation uses well-resolved large-eddy simulations to study the low-frequency unsteady motions observed in shock-wave/turbulent-boundary-layer interactions. Details about the numerical aspects of the simulations and the subsequent data analysis can be found in three papers by the authors (Theo. Comput. Fluid Dyn., 23:79--107 (2009); Shock Waves, 19(6):469--478 (2009) and J. of Fluid Mech. (2011)). The fluid dynamics video illustrates the complexity of the interaction between a Mach 2.3 supersonic turbulent boundary layer and an oblique shock wave generated by a 8-degree wedge angle. The first part of the video highlights the propagation of disturbances along the reflected shock due to the direct perturbation of the shock foot by turbulence structures from the upstream boundary layer. The second part of the video describes the observed block-like back-and-forth motions of the reflected shock, focusing on timescales about two orders of magnitude longer than the ones shown in the first part of video. This gives a visual impression of the broadband and energetically-significant peak in the wall-pressure spectrum at low frequencies. The background blue-white colouring represents the temperature field (with white corresponding to hot) and one can clearly appreciate why such low-frequency shock motions can lead to reduced fatigue lifetimes and is detrimental to aeronautical applications.
The aim of this paper is to substantiate the importance of non-normality of shear flow linear operators and its consequence -- the non-modal dynamics of the perturbations -- in the formation of acoustic wave output of time-developing free shear/mixing layers. Initially, the linear transient dynamics of spatial Fourier harmonics is considered in a 3D homentropic parallel unbounded inviscid constant shear flow which can model the central/body part of the shear layer. The non-modal approach allows to capture the only linear mechanism of the acoustic wave generation -- textit{the linear vortex-wave mode coupling induced by the shear flow non-normality}. We describe the specific/key features of the generation process that should leave traces on the further dynamics of the generated waves. Thereafter, the results of direct numerical simulations of compressible turbulent time-developing mixing layers for a moderate convective Mach number (specifically, $M_c=0.7$) and simulation boxes $(L_x,L_y,L_z)$ with fixed streamwise and shearwise lengths ($L_x=100, L_y=200$) and different streamwise-spanwise aspect ratios ($L_x/L_z=0.5,1,2$) are presented. The simulations identify the origin of the acoustic wave output: the dominance of a emph{linear generation process of acoustic waves in the shear layer core region, induced by the flow non-normality}, observable in the near field of acoustic waves emitted by the flow.
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