Two major problems in viscoelastic turbulence, the effects of polymers on the laminar-turbulent transition dynamics and the origin of the maximum drag reduction asymptote, can be both better understood in the regime near the margin of turbulence. In the first part of this thesis, direct numerical simulation trajectories initiated from the edge state are used to follow its unstable manifold into the turbulent basin. In Newtonian flow, the growth of turbulence starts with the intensification of velocity streaks and a sharp rise in the Reynolds shear stress. It is followed by a quick breakdown into high-intensity small-scale fluctuations before entering the core of turbulence. Adding drag-reducing polymers does not affect the initial growth of turbulence but stabilizes the primary streak-vortex structure, which help the flow circumvent the breakdown stage. Throughout the process, polymers act in reaction to the growing turbulence and do not drive the instability. This part not only reveals the transition dynamics into turbulence but also presents a comprehensive view of the bursting stage observed in the near-wall self-sustaining cycle, which starts as the flow leaves hibernating turbulence and is redirected towards the turbulent basin by the unstable manifold of the edge state. On the other hand, this thesis also discusses the effects of polymer addition on the laminar-turbulent transition in extended domains. Localized turbulent spot can be clearly observed in the large box, and this turbulent region will spread as well as tend to “split” but finally fill up the whole domain before it is separated. Polymers do not affect the flow dynamics until the burst. Similarly, vortex structures rapidly break down into small scales after the first bursting of Reynolds shear stress, but polymer additives depress this process. The thesis offers a clear and comprehensive overview of the transition into turbulence in the presence of drag-reducing polymers. Future work remains in two major directions. The first is to pinpoint the flow states responsible for the quantitative origin of the universal upper limit of drag reduction observed in experiments. The second is to determine the role, if any, of elasticity-driven instabilities in the transition. / Thesis / Master of Applied Science (MASc) / Turbulence exists everywhere and can be observed in most fluid flows occurring in nature. To reduce the energy consumption, frictional resistance in the turbulence must be considered in fluid transportation. It has been known since the 1940s that a small amount of long-chain polymer additives can dramatically reduce such drag. The mechanism of drag reduction has attracted extensive attention. Two problems of particular interest are the upper limit of drag reduction (termed maximum drag reduction) and the polymer effects on the laminar-turbulent transition. In this thesis, full transient trajectories from marginal turbulent states towards sustained turbulence in both Newtonian and polymeric flows are monitored by direct numerical simulations. It is observed that polymer additives do not affect the initial growth of turbulence but prevent flows from breaking into strong but small-scale fluctuations afterwards. In a more extended domain, turbulence starts as localized spots which spread across the channel. Adding polymers changes the dynamics of turbulence propagation as well. In addition to the aforementioned problems, this study also sheds lights on the so-called bursting events intermittent surges in turbulent activities observed in experiments.

This thesis deals with the instabilities of the incompressible boundary-layer flow thatis induced by a disk rotating in otherwise still fluid. The results presented include bothwork in the linear and nonlinear regime and are derived from direct numerical sim-ulations (DNS). Comparisons are made both to theoretical and experimental resultsproviding new insights into the transition route to turbulence. The simulation codeNek5000 has been chosen for the DNS using a spectral-element method (SEM) witha high-order discretization, and the results were obtained through large-scale paral-lel simulations. The known similarity solution of the Navier–Stokes equations for therotating-disk flow, also called the von K ́arm ́an rotating-disk flow, is reproduced by theDNS. With the addition of modelled small simulated roughnesses on the disk surface,convective instabilities appear and data from the linear region in the DNS are anal-ysed and compared with experimental and theoretical data, all corresponding verywell. A theoretical analysis is also presented using a local linear-stability approach,where two stability solvers have been developed based on earlier work. Furthermore,the impulse response of the rotating-disk boundary layer is investigated using DNS.The local response is known to be absolutely unstable and the global response, onthe contrary, is stable if the edge of the disk is assumed to be at radius infinity. Herecomparisons with a finite domain using various boundary conditions give a globalbehaviour that can be both linearly stable and unstable, however always nonlinearlyunstable. The global frequency of the flow is found to be determined by the Rey-nolds number at the confinement of the domain, either by the edge (linear case) or bythe turbulence appearance (nonlinear case). Moreover, secondary instabilities on topof the convective instabilities induced by roughness elements were investigated andfound to be globally unstable. This behaviour agrees well with the experimental flowand acts at a smaller radial distance than the primary global instability. The sharpline corresponding to transition to turbulence seen in experiments of the rotating diskcan thus be explained by the secondary global instability. Finally, turbulence datawere compared with experiments and investigated thoroughly. / <p>QC 20170203</p>

This thesis deals with numerical simulation of turbulentflows in geometrically simple cases. Both plane and cylindricalgeometries are used. The simplicity of the geometry allows theuse of spectral methods which yield a very high accuracy usingrelatively few grid points. A spectral method for planegeometries is implemented on a parallel computer. Thetransitional Reynolds number for plane Couette flow is verifiedto be about 360, in accordance with earlier findings. TurbulentCouette flow at twice the transitional Reynolds number isstudied and the findings of large scale structures in earlierstudies of Couette flow are substantiated. These largestructures are shown to be of limited extent and give anintegral length scale of six half channel heights, or abouteight times larger than in pressure-driven channel flow.Despite this, they contain only about 10 \% of the turbulentenergy. This is demonstrated by applying a very smallstabilising rotation, which almost eliminates the largestructures. A comparison of the Reynolds stress budget is madewith a boundary layer flow, and it is shown that the near-wallvalues in Couette flow are comparable with high-Reynolds numberboundary layer flow. A new spectrally accurate algorithm isdeveloped and implemented for cylindrical geometries andverified by studying the evolution of eigenmodes for both pipeflow and annular pipe flow. This algorithm is a generalisationof the algorithm used in the plane channel geometry. It usesFourier transforms in two homogeneous directions and Chebyshevpolynomials in the third, wall-normal, direction. TheNavier--Stokes equations are solved with a velocity-vorticityformulation, thereby avoiding the difficulty of solving for thepressure. The time advancement scheme used is a mixedimplicit/explicit second order scheme. The coupling between twovelocity components, arising from the cylindrical coordinates,is treated by introducing two new components and solving forthem, instead of the original velocity components. TheChebyshev integration method and the Chebyshev tau method isboth implemented and compared for the pipe flow case.

Experimental findings show that transition from laminar toturbulent ow may occur also if the exponentially growingperturbations, eigensolutions to the linearised disturbanceequations, are damped. An alternative non-modal growthmechanism has been recently identi fied, also based on thelinear approximation. This consists of the transient growth ofstreamwise elongated disturbances, with regions of positive andnegative streamwise velocity alternating in the spanwisedirection, called streaks. These perturbation are seen toappear in boundary layers exposed to signi ficant levels offree-stream turbulence. The effect of the streaks on thestability and transition of the Blasius boundary layer isinvestigated in this thesis. The analysis considers the steadyspanwise-periodic streaks arising from the nonlinear evolutionof the initial disturbances leading to the maximum transientenergy growth. In the absence of streaks, the Blasius pro filesupports the viscous exponential growth of theTollmien-Schlichting waves. It is found that increasing thestreak amplitude these two-dimensional unstable waves evolveinto three-dimensional spanwiseperiodic waves which are lessunstable. The latter can be completely stabilised above athreshold amplitude. Further increasing the streak amplitude,the boundary layer is again unstable. The new instability is ofdifferent character, being driven by the inectional pro filesassociated with the spanwise modulated ow. In particular, it isshown that, for the particular class of steady streaksconsidered, the most ampli fied modes are antisymmetric andlead to spanwise oscillations of the low-speed streak (sinuousscenario). The transition of the streak is then characterisedby the appearance of quasi-streamwise vorticesfollowing themeandering of the streak. Simulations of a boundary layer subjected to high levels offree-stream turbulence have been performed. The receptivity ofthe boundary layer to the external perturbation is studied indetail. It is shown that two mechanisms are active, a linearand a nonlinear one, and their relative importance isdiscussed. The breakdown of the unsteady asymmetric streaksforming in the boundary layer under free-stream turbulence isshown to be characterised by structures similar to thoseobserved both in the sinuous breakdown of steady streaks and inthe varicose scenario, with the former being the mostfrequently observed. <b>Keywords:</b>Fluid mechanics, laminar-turbulent transition,boundary layer ow, transient growth, streamwise streaks,lift-up effect, receptivity, free-stream turbulence, secondaryinstability, Direct Numerical Simulation.

Bypass transition, i.e. transition of a boundary layer at subcritical Reynolds numbers, has been studied. Fundamental studies of the phenomenon as such have been performed side by side with experiments aimed at controlling, i.e. delaying, transition. The experiments have been performed in three different flow facilities, two with air as the working fluid (a plane channel flow and a wind-tunnel) and one with water (a water channel). From the water channel data the well known low-speed streaks appearing in a boundary layer under a turbulent free stream are found to be correlated with upward motion in the boundary layer. The streaks are found to scale in proportion to the boundary-layer thickness in both the streamwise and wall-normal directions. The streamwise length is around hundred boundary-layer thicknesses. It is found that the secondary instability of the streaks grows slower for disturbances consisting of less than four wavelengths, as compared to continuous wavetrains. Elongated low-speed structures are controlled, first in the plane channel flow and then by a reactive system in the wind-tunnel. In the channel, the breakdown of generated streaks is delayed by applying localized suction under the regions of low velocity. Measurements of the disturbance environment withand without control applied show that both the growth of the secondary instability and its spreading in the spanwise direction are reduced when applying the control. In order to be successful, the control has to be applied to a narrow region (about 1/10th of a streak width) around the position of minimum velocity. The reactive system in the windtunnel, comprising four upstream sensors and four suction ports downstream, inhibits the growth of the amplitude of the streaks for a certain distance downstream of the suction ports. After the inhibited growth the disturbances start to grow again and far downstream the streak amplitude returns to close to the uncontrolled values. / QC 20100527

<p>This thesis deals with numerical simulation of turbulentflows in geometrically simple cases. Both plane and cylindricalgeometries are used. The simplicity of the geometry allows theuse of spectral methods which yield a very high accuracy usingrelatively few grid points. A spectral method for planegeometries is implemented on a parallel computer. Thetransitional Reynolds number for plane Couette flow is verifiedto be about 360, in accordance with earlier findings. TurbulentCouette flow at twice the transitional Reynolds number isstudied and the findings of large scale structures in earlierstudies of Couette flow are substantiated. These largestructures are shown to be of limited extent and give anintegral length scale of six half channel heights, or abouteight times larger than in pressure-driven channel flow.Despite this, they contain only about 10 \% of the turbulentenergy. This is demonstrated by applying a very smallstabilising rotation, which almost eliminates the largestructures. A comparison of the Reynolds stress budget is madewith a boundary layer flow, and it is shown that the near-wallvalues in Couette flow are comparable with high-Reynolds numberboundary layer flow. A new spectrally accurate algorithm isdeveloped and implemented for cylindrical geometries andverified by studying the evolution of eigenmodes for both pipeflow and annular pipe flow. This algorithm is a generalisationof the algorithm used in the plane channel geometry. It usesFourier transforms in two homogeneous directions and Chebyshevpolynomials in the third, wall-normal, direction. TheNavier--Stokes equations are solved with a velocity-vorticityformulation, thereby avoiding the difficulty of solving for thepressure. The time advancement scheme used is a mixedimplicit/explicit second order scheme. The coupling between twovelocity components, arising from the cylindrical coordinates,is treated by introducing two new components and solving forthem, instead of the original velocity components. TheChebyshev integration method and the Chebyshev tau method isboth implemented and compared for the pipe flow case.</p>

<p>Experimental findings show that transition from laminar toturbulent ow may occur also if the exponentially growingperturbations, eigensolutions to the linearised disturbanceequations, are damped. An alternative non-modal growthmechanism has been recently identi fied, also based on thelinear approximation. This consists of the transient growth ofstreamwise elongated disturbances, with regions of positive andnegative streamwise velocity alternating in the spanwisedirection, called streaks. These perturbation are seen toappear in boundary layers exposed to signi ficant levels offree-stream turbulence. The effect of the streaks on thestability and transition of the Blasius boundary layer isinvestigated in this thesis. The analysis considers the steadyspanwise-periodic streaks arising from the nonlinear evolutionof the initial disturbances leading to the maximum transientenergy growth. In the absence of streaks, the Blasius pro filesupports the viscous exponential growth of theTollmien-Schlichting waves. It is found that increasing thestreak amplitude these two-dimensional unstable waves evolveinto three-dimensional spanwiseperiodic waves which are lessunstable. The latter can be completely stabilised above athreshold amplitude. Further increasing the streak amplitude,the boundary layer is again unstable. The new instability is ofdifferent character, being driven by the inectional pro filesassociated with the spanwise modulated ow. In particular, it isshown that, for the particular class of steady streaksconsidered, the most ampli fied modes are antisymmetric andlead to spanwise oscillations of the low-speed streak (sinuousscenario). The transition of the streak is then characterisedby the appearance of quasi-streamwise vorticesfollowing themeandering of the streak.</p><p>Simulations of a boundary layer subjected to high levels offree-stream turbulence have been performed. The receptivity ofthe boundary layer to the external perturbation is studied indetail. It is shown that two mechanisms are active, a linearand a nonlinear one, and their relative importance isdiscussed. The breakdown of the unsteady asymmetric streaksforming in the boundary layer under free-stream turbulence isshown to be characterised by structures similar to thoseobserved both in the sinuous breakdown of steady streaks and inthe varicose scenario, with the former being the mostfrequently observed.</p><p><b>Keywords:</b>Fluid mechanics, laminar-turbulent transition,boundary layer ow, transient growth, streamwise streaks,lift-up effect, receptivity, free-stream turbulence, secondaryinstability, Direct Numerical Simulation.</p>

The rotating-disk boundary layer is not only a simpler model for the study of cross-flow instability than swept-wing boundary layers but also a useful simplification of many industrial-flow applications where rotating configurations are present. For the rotating disk, it has been suggested that a local absolute instability, leading to a global instability, is responsible for the small variation in the observed laminar-turbulent transition Reynolds number however the exact nature of the transition is still not fully understood. This thesis aims to clarify certain aspects of the transition process. Furthermore, the thesis considers the turbulent rotating-disk boundary layer, as an example of a class of three-dimensional turbulent boundary-layer flows. The rotating-disk boundary layer has been investigated in an experimental apparatus designed for low vibration levels and with a polished glass disk that gave a smooth surface. The apparatus provided a low-disturbance environment and velocity measurements of the azimuthal component were made with a single hot-wire probe. A new way to present data in the form of a probability density function (PDF) map of the azimuthal fluctuation velocity, which gives clear insights into the laminar-turbulent transition region, has been proposed. Measurements performed with various disk-edge conditions and edge Reynolds numbers showed that neither of these conditions a↵ect the transition process significantly, and the Reynolds number for the onset of transition was observed to be highly reproducible. Laminar-turbulent transition for a ‘clean’ disk was compared with that for a disk with roughness elements located upstream of the critical Reynolds number for absolute instability. This showed that, even with minute surface roughness elements, strong convectively unstable stationary disturbances were excited. In this case, breakdown of the flow occurred before reaching the absolutely unstable region, i.e. through a convectively unstable route. For the rough disk, the breakdown location was shown to depend on the amplitude of individual stationary vortices. In contrast, for the smooth (clean-disk) condition, the amplitude of the stationary vortices did not fix the breakdown location, which instead was fixed by a well-defined Reynolds number. Furthermore, for the clean-disk case, travelling disturbances have been observed at the onset of nonlinearity, and the associated disturbance profile is in good agreement with the eigenfunction of the critical absolute instability. Finally, the turbulent boundary layer on the rotating disk has been investigated. The azimuthal friction velocity was directly measured from the azimuthal velocity profile in the viscous sublayer and the velocity statistics, normalized by the inner scale, are presented. The characteristics of this three-dimensional turbulent boundary-layer flow have been compared with those for the two-dimensional flow over a flat plate and close to the wall they are found to be quite similar but with rather large differences in the outer region. / <p>QC 20150119</p>

In this thesis the orientation of macro-sized fibres in turbulent flows is studied, as well as the effect of nano-sized fibrils on hydrodynamic stability. The focus lies on enabling processes for new materials where cellulose is the main constituent. When fibres (or any elongated particles) are added to a fluid, the complexity of the flow-problem increases. The fluid flow will influence the rotation of the fibres, and therefore also effect the overall fibre orientation. Exactly how the fibres rotate depends to a large extent on the mean velocity gradient in the flow. In addition, when fibres are added to a suspending fluid, the total stress in the suspension will increase, resulting in an increased apparent viscosity. The increase in stress is related to the direction of deformation in relation to the orientation of the particle, i.e. whether the deformation happens along the long or short axis of the fibre. The increase in stress, which in most cases is not constant neither in time nor space, will in turn influence the flow. This thesis starts off with the orientation and spatial distribution of fibres in the turbulent flow down an inclined plate. By varying fibre and flow parameters it is discovered that the main parameter controlling the orientation distribution is the aspect ratio of the fibres, with only minor influences from the other parameters. Moreover, the fibres are found to agglomerate into streamwise streaks. A new method to quantify this agglomeration is developed, taking care of the problems that arise due to the low concentration in the experiments. It is found that streakiness, i.e. the tendency to agglomerate in streaks, varies with Reynolds number. Going from fibre orientation to flow dynamics of fibre suspensions, the influence of cellulose nanofibrils (CNF) on laminar/turbulent transition is investigated in three different setups, namely plane channel flow, curved-rotating channel flow, and the flow in a flow focusing device. This last flow case is selected since it is can be used for assembly of CNF based materials. In the plane channel flow, the addition of CNF delays the transition more than predicted from measured viscosities while in the curved-rotating channel the opposite effect is discovered. This is qualitatively confirmed by linear stability analyses. Moreover, a transient growth analysis in the plane channel reveals an increase in streamwise wavenumber with increasing concentration of CNF. In the flow focusing device, i.e. at the intersection of three inlets and one outlet, the transition is found to mainly depend on the Reynolds number of the side flow. Recirculation zones forming downstream of two sharp corners are hypothesised to be the cause of the transition. With that in mind, the two corners are given a larger radius in an attempt to stabilise the flow. However, if anything, the flow seems to become unstable at a smaller Reynolds number, indicating that the separation bubble is not the sole cause of the transition. The choice of fluid in the core flow is found to have no effect on the stability, neither when using fluids with different viscosities nor when a non-Newtonian CNF dispersion was used. Thus, Newtonian model fluids can be used when studying the flow dynamics in this type of device. As a proof of concept, a flow focusing device is used to produce a continuous film from CNF. The fibrils are believed to be aligned due to the extensional flow created in the setup, resulting in a transparent film, with an estimated thickness of 1 um. / <p>QC 20141003</p>

In this study, numerical investigations of stability and transition problems were performed for 2D compressible boundary layers over a flat plate in adiabatic wall condition. Emphasis was placed on linear stability theory. The mathematical formulation for 3D boundary layers with oblique waves including detailed theoretical information was followed by use of the numerical techniques for the solution of resulting differential system of the instability problem, consequently an eigenvalue problem. First, two-dimensional sinusoidal disturbances were analyzed at various Mach numbers including the subsonic, transonic, supersonic and even hypersonic flow speeds. In this case, the second mode (acoustic mode), namely the Mack mode, and its behavior with the increasing Mach number were visualized. The results were then compared with the available data in literature concluding with good agreements. Secondly, similar analysis was carried out for oblique waves. Here, not only the effect of flow speed but also the effect of wave orientation was demonstrated. For this purpose, instability problem was solved for several wave angles at each Mach number in the range of M=0 and M=5. In this respect, the angle at which the waves were most unstable was also obtained at each investigated flow speed. The resultant stability diagrams corresponding to M=4 and higher Mach numbers for which both first and the second modes appear revealed that plane waves were more stable than oblique waves for the Tollmien-Schlichting mode, however, this was the opposite for the acoustic mode where oblique waves were observed to be more stable. As a final step, estimation of the transition location was handled for the most unstable wave condition. Smith-Van Ingen transition method was applied as the prediction device. The results representing the influence of Mach number on transition Reynolds number were then compared with the experimental data as well as the numerical ones in literature ending up with very good agreements.