Nonlinear unsteady disturbances generated by the interaction of free-stream vorticity with a laminar boundary layer

As a contribution towards understanding the impact of free-stream perturbations on laminar-to-turbulent boundary layer transition, we calculate the signature of unsteady disturbances engendered by the interaction of free-stream vortical fluctuations with a laminar boundary layer over a flat and a curved plate. We concentrate on low-frequency perturbations which, in the case of a flat plate, induce strong streamwise-elongated components of the boundary-layer signature, known as Klebanoff modes or streaks. In boundary layers over suitably curved concave walls, Klebanoff modes are expected to develop into Gortler vortices. The generation and nonlinear evolution of the induced perturbations, which acquire an O(1) magnitude, are described on a self-consistent and first-principle basis using the mathematical framework of the nonlinear unsteady boundary-region equations (NUBREs), subject to appropriate upstream and far-field boundary conditions. The nonlinear response of a compressible flat-plate boundary layer to free-stream vorticity is investigated first. The problem is governed by the compressible NUBREs, which are derived herein for the first time. The free-stream flow is studied by including the boundary-layer displacement effect and the solution is matched asymptotically with the boundary-layer flow. The nonlinear interactions inside the boundary layer drive an unsteady two-dimensional flow of acoustic nature in the outer inviscid region through the displacement effect. Analytical solutions are derived by exploiting the well-known analogy with the flow over a thin oscillating airfoil, which is used herein for the first time to study unsteady boundary layers. In the subsonic regime the perturbation is felt from the plate in all directions, while at supersonic speeds the disturbance only propagates within the dihedron defined by the Mach line. Numerical computations are performed for carefully chosen parameters that characterize three practical applications: turbomachinery systems, supersonic flight conditions and wind-tunnel experiments. The results show that nonlinearity plays a marked stabilizing role on the velocity and temperature streaks, and this is found to be the case for low-disturbance environment such as flight conditions. Increasing the free-stream Mach number inhibits the kinematic fluctuations but enhances the thermal streaks, relative to the free-stream velocity and temperature respectively, and the overall effect of nonlinearity becomes weaker. An abrupt deviation of the nonlinear solution from the linear one is observed in the case pertaining to a supersonic wind tunnel. Large-amplitude thermal streaks and the strong abrupt stabilizing effect of nonlinearity are two new features of supersonic flows. In the second part of the thesis, the generation and nonlinear development of unsteady Gortler vortices in an incompressible boundary layer over a concave plate is studied. The centrifugal force caused by the concavity of the wall is included in the incompressible NUBREs. The results show that the stabilizing effect on nonlinearity is significantly intensified in the presence of centrifugal forces. Sufficiently downstream the nonlinear vortices generated at different free-stream turbulence levels Tu are stabilized to the same amplitude, suggesting that the initial intensity of the forcing becomes unimportant. At low Tu the perturbation undergoes a quasi-exponential growth with the growth rate being enhanced for lower frequencies and more curved plates. At higher Tu, in the typical range of turbomachinery applications, the Gortler vortices do not exhibit an exponential growth as nonlinearity saturates rapidly, and the wall curvature does not influence the boundary-layer response. Good quantitative agreement with direct numerical simulations and experimental data is obtained

Dynamics and proliferation of turbulent stripes in plane-Poiseuille and plane-Couette flows

The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes are central to transition, and recent studies proposed an analogy to directed percolation where the stripes' proliferation is ultimately responsible for turbulence to become sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure and shear driven planar flows, respectively plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar-turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off / shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes.Comment: To be published in the Journal of Fluid Mechanics. 26 pages with 15 figures including the appendi

Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence

Data-driven dimensionality reduction methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for the pressure-driven flow in a rectangular channel by formulating a continuous symmetry reduction method that eliminates the translations simultaneously in the streamwise and spanwise directions. As an application, we consider turbulence in a minimal flow unit at a Reynolds number (based on the centerline velocity and half-channel height) Re = 2000 and compute the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding data windows of varying durations. SRDMD of channel flow reveals episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.Comment: 10 pages, 6 figure

Discontinuous transition to shear flow turbulence

Depending on the type of flow the transition to turbulence can take one of two forms, either turbulence arises from a sequence of instabilities, or from the spatial proliferation of transiently chaotic domains, a process analogous to directed percolation. Both scenarios are inherently continuous and hence the transformation from ordered laminar to fully turbulent fluid motion is only accomplished gradually with flow speed. Here we show that these established transition types do not account for the more general setting of shear flows subject to body forces. By attenuating spatial coupling and energy transfer, spatio-temporal intermittency is suppressed and with forcing amplitude the transition becomes increasingly sharp and eventually discontinuous. We argue that the suppression of the continuous range and the approach towards a first order, discontinuous scenario applies to a wide range of situations where in addition to shear, flows are subject to e.g. gravitational, centrifugal or electromagnetic forces

Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer

The nonlinear response of a compressible boundary layer to unsteady free-stream vortical fluctuations of the convected-gust type is investigated theoretically and numerically. The free-stream Mach number is assumed to be of O(1) and the effects of compressibility, including aerodynamic heating and heat transfer at the wall, are taken into account. Attention is focused on low-frequency perturbations, which induce strong streamwise-elongated components of the boundary-layer disturbances, known as streaks or Klebanoff modes. The amplitude of the disturbances is intense enough for nonlinear interactions to occur within the boundary layer. The generation and nonlinear evolution of the streaks, which acquire an O(1) magnitude, are described on a self-consistent and first-principle basis using the mathematical framework of the nonlinear unsteady compressible boundary-region equations, which are derived herein for the first time. The free-stream flow is studied by including the boundary-layer displacement effect and the solution is matched asymptotically with the boundary-layer flow. The nonlinear interactions inside the boundary layer drive an unsteady two-dimensional flow of acoustic nature in the outer inviscid region through the displacement effect. A close analogy with the flow over a thin oscillating airfoil is exploited to find analytical solutions. This analogy has been widely employed to investigate steady flows over boundary layers, but is considered herein for the first time for unsteady boundary layers. In the subsonic regime the perturbation is felt from the plate in all directions, while at supersonic speeds the disturbance only propagates within the dihedron defined by the Mach line. Numerical computations are performed for carefully chosen parameters that characterize three practical applications: turbomachinery systems, supersonic flight conditions and wind tunnel experiments. The results show that nonlinearity plays a marked stabilizing role on the velocity and temperature streaks, and this is found to be the case for low-disturbance environments such as flight conditions. Increasing the free-stream Mach number inhibits the kinematic fluctuations but enhances the thermal streaks, relative to the free-stream velocity and temperature respectively, and the overall effect of nonlinearity becomes weaker. An abrupt deviation of the nonlinear solution from the linear one is observed in the case pertaining to a supersonic wind tunnel. Large-amplitude thermal streaks and the strong abrupt stabilizing effect of nonlinearity are two new features of supersonic flows. The present study provides an accurate signature of nonlinear streaks in compressible boundary layers, which is indispensable for the secondary instability analysis of unsteady streaky boundary-layer flows

Growth and wall-transpiration control of nonlinear unsteady Gortler vortices forced by free-stream vortical disturbances

The generation, nonlinear evolution, and wall-transpiration control of unsteady Gortler vortices in ¨ an incompressible boundary layer over a concave plate is studied theoretically and numerically. Gortler rolls are initiated and driven by free-stream vortical perturbations of which only the low- ¨ frequency components are considered because they penetrate the most into the boundary layer. The formation and development of the disturbances are governed by the nonlinear unsteady boundaryregion equations with the centrifugal force included. These equations are subject to appropriate initial and outer boundary conditions, which account for the influence of the upstream and free-stream forcing in a rigorous and mutually consistent manner. Numerical solutions show that the stabilizing effect on nonlinearity, which also occurs in flat-plate boundary layers, is significantly enhanced in the presence of centrifugal forces. Sufficiently downstream, the nonlinear vortices excited at different free-stream turbulence intensities Tu saturate at the same level, proving that the initial amplitude of the forcing becomes unimportant. At low Tu, the disturbance exhibits a quasi-exponential growth with the growth rate being intensified for more curved plates and for lower frequencies. At higher Tu, in the typical range of turbomachinery applications, the Gortler vortices do not undergo a modal stage ¨ as nonlinearity saturates rapidly, and the wall curvature does not affect the boundary-layer response. Good quantitative agreement with data from direct numerical simulations and experiments is obtained. Steady spanwise-uniform and spanwise-modulated zero-mass-flow-rate wall transpiration is shown to attenuate the growth of the Gortler vortices significantly. A novel modified version of the Fukagata- ¨ Iwamoto-Kasagi identity, used for the first time to study a transitional flow, reveals which terms in the streamwise momentum balance are mostly affected by the wall transpiration, thus offering insight into the increased nonlinear growth of the wall-shear stress

Dynamics and proliferation of turbulent stripes in plane-Poiseuille and plane-Couette flows

The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes is central to transition, and recent studies proposed an analogy to directed percolation where the stripes’ proliferation is ultimately responsible for the turbulence becoming sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure- and shear-driven planar flows, respectively, plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar–turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off/shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes

Stabilising pipe flow by a baffle designed using energy stability

Abstract EPSR

Effects of streaky structures on the instability of supersonic boundary layers

Streaky structures in the boundary layers are often generated by surface roughness elements and/or free-stream turbulence, and are known to have significant effects on boundary-layer instability. In this paper, we investigate the impact of two forms of streaks on the instability of supersonic boundary layers. The first concerns the streaks generated by an array of spanwise periodic and streamwise elongated surface roughness elements, and our interest is how these streaks influence the lower-branch viscous first modes, whose characteristic wavelength and frequency are on the classical triple-deck scales. By adapting the triple-deck theory in the incompressible regime to the supersonic one, we first derived a simplified system which allows for efficient calculation of the streaks. The asymptotic analysis simplifies a bi-global eigenvalue problem to a one-dimensional problem in the spanwise direction, showing that the instability is controlled at leading order solely by the spanwise-dependent wall shear. In the fundamental configuration, the streaks stabilize first modes at low frequencies but destabilize the high-frequency ones. In the subharmonic configuration, the streaks generally destabilize the first mode across the entire frequency band. Importantly, the spanwise even modes are of radiating nature, i.e. they emit acoustic waves spontaneously to the far field. Streaks of the second form are generated by low-frequency vortical disturbances representing free-stream turbulence. They alter the flow in the entire layer and their effects on instability are investigated by solving the inviscid bi-global eigenvalue problem. Different from the incompressible case, a multitude of compressible instability modes exists, of which the dominant mode is an inviscid instability associated with the spanwise shear. In addition, there exists a separate branch of instability modes that have smaller growth rates but are spontaneously radiating