Andersen, Morten

Ved at benytte en koordinatfri formulering af helisk symmetri frem for mere traditionelle definitioner baseret på koordinater, diskuterer vi grundlæggende egenskaber for heliske vektor felter og sammenligner resultater fra litteraturen. For inviscid væske, hvor et hastighedsfelt er genereret af en sum af heliske hvirveltråde med samme pitch, benytter vi de opnåede resultater til at konstruere et kort bevis for, at hastighedsfeltet er helisk. Vi diskuterer strømfuntionens indflydelse på topologien af strømlinjerne for inviscid, helisk flow. På baggrund af dette laver vi en udførlig undersøgelse af topologien af et hastighedsfelt genereret af en helisk hvirveltrad i en idealvæske. Det klassiske udtryk for strømfunktionen opnået af Hardin (Phys. Fluids 25, 1982) indeholder en uendelig sum af modificerede Bessel funktioner. Ved at benytte metoden beskrevet af Okulov (Russ. J. Eng. Thermophys. 5, 1995) opnår vi en approksimation på lukket form, der er væsentlig lettere at analysere. Kritiske punkter for strømfunktionen kan findes som nulpunkterne af en enkelt reel funktion af en variabel, og vi viser, at tre forskellige toplogier kan forekomme, afhængig af en enkelt dimensionsløs parameter. Ved at inkludere den selvinducerede hastighed på hvirveltråden via ’den lokaliserede induktionstilnærmelse’ bliver strømfunktionen lettere modificeret og en ekstra parameter introduceres. Nu fremkommer to nye topologier men der er højst to kritiske punkter for en vilkårlig kombination af parametrene. Analysen af den lukkede form er lovende for at undersøge mere komplekse strømninger med helisk symmetri, for eksempel flere heliske hvirveltråde inde i en cylinder, hvilket har industriel relevans. Vi skifter fokus og undersøger dannelse, forsvinden og vekselvirkning af hvirvler for todimensionelle strømninger. En hvirvel strømmer over en væg, hvilket forårsager et viskøst respons nær væggen, hvilket genererer en ny hvirvelstruktur. Problemet undersøges numerisk ved brug af kode udviklet af Prof. M. Thompson og hans gruppe på Monash Universitet, Australien. Vi undersøger også problemet analytisk ved brug af teori for normalformer. Det er ikke en simpel opgave at definere en hvirvel på en måde, der tillader studiet af dannelse og forsvinden af hvirvler. Vi undersøger tre fornuftige valg, vorticitetsekstrema, strømlinjecentre i et koordinatsystem, hvor væggen står stille, og strømlinjecentre i et koordinatsystem, der bevæger sig med konstant hastighed dikteret af hastigheden af en punkthvirvel over en væg i inviscid væske. Der er ingen grund til at forvente ens resultater af de tre hvirveldefinitioner a priori. Studiet er dog kraftigt motiveret af undersøgelserne af Kudela og Malecha (Fluid Dyn. Res. 41, 2009) som finder god overensstemmelse mellem vorticiteten og strømlinjer for koordinatsystemet, hvor væggen står stille. For små Re ; ses ingen nye hvirvler. Hvirveldannelse sker for tilstrækkelig stor Re ; for alle de anvendte hvirveldefinitioner. Den nye hvirvel løfter den oprindelige hvirvel længere væk fra væggen samt bremser dens horisontale bevægelse. I systemet, hvor væggen står stille, sker hvirveldannelse gennem en karakteristik ”8-tals” bifurkation. I det andet koordinatsystem er der ingen toplogisk ændring, der indikerer, hvornår en hvirvel har forladt grænselaget. Derimod er der bemærkelsesværdig god overensstemmelse mellem strømlinjer og vorticitetskonturer selv for kortlivede hvirvler tæt på væggen. Tilgangen med normalformer afslører ingen simple sammenhænge mellem strømlinjetopologien og vorticitetstoplogien. Kun for et ikke-simpel degenereret kritisk punkt på væggen kan en bifurkation forekomme både i strømlinjetopologien og vorticitetstoplogien. Strømlinjebifurkationerne i denne normal form indeholder den nedre del af ”8-talsbifurkationen”, der ses i simuleringerne. Lighederne og forskellene i strømlinjetopologierne for de to koordinatsystemer er velbeskrevet med normalforms teori.Vi udleder kravet, u _ r! = 0, for perfekt matchende strømlinjer og vorticitetskonturer. Dette er opfyldt når Navier - Stokes ligningerne og varmeligningen har samme løsninger.Til slut fokuserer vi på superposition af to rotationsinvariante hvirvler i R2. Topologien af vorticitetskonturerne og strømlinjerne er begge bestemt af nulpunkterne af en reel funktion af en variabel. For det kanoniske eksempel med to Gaussiske hvirvler findes tre parametre. Tre strukturelt stabile topologier observeres. For strømlinjerne er to af topologierne velkendte for den tilsvarende situation med to punkthvirvler, når singulariteterne opfattes som centre. Den sidste topologi er et enkelt center som er konsistent med et kraftfulde resultat bevist af Gallay &amp; Wayne (Comm. In Math. Phys. 255, 2005). Tilfældet med tre kritiske punkter for strømlinjerne er en delmængde af tre kritiske punkterne for vorticiteten. Dette forklarer observationen i simuleringerne af hvirveldannelse nær en væg. En ny hvirvel, der er drevet væk fra væggen forsvinder grundet viskositet. Dette ses først, når man betragter strømlinjedefinitionen, mens hvirvlen er mere robust ud fra vorticetsdefinitionen.Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature. For inviscid flow where a velocity field is generated by a sum of helical vortex filaments with same pitch we use the established results to prove briefly that the velocity field is helical. We discuss the role of the stream function for the topology of the streamlines in incompressible, helical flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Phys. Fluids 25, 1982) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Russ. J. Eng. Thermophys. 5, 1995) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. Including the self-induced velocity on the vortex filament by the localised induction approximation the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of the parameters. The analysis of the closed form show promise for analysing more complex flow with helical symmetry e.g. multiple helical vortex filaments inside a cylinder which has industrial relevance. We then change focus and study creation, destruction and interaction of vortices in two dimensional flow. A vortex is advected above a wall causing a viscous response near the wall which generates a new vortex structure. The problem is studied numerically relying on the code developed by Prof. M. Thompson and his group at Monash University, Australia. We also investigate the problem analytically using normal form theory. It is not a simple task to define a vortex in a proper way that allow the study of creation and destruction of vortices. We investigate three sound choices: the vorticity extrema, the streamline centers in a coordinate system with zero wall speed and the streamline centers in a frame moving with constant velocity as predicted by a point vortex above a wall in inviscid fluid. There is no reason to a priori expect equivalent results of the three vortex definitions. However, the study is mainly motivated by the findings of Kudela &amp; Malecha (Fluid Dyn. Res. 41, 2009) who find good agreement between the vorticity and streamlines in the fixed wall system. For small Re no new vortices are observed. Creation of a vortex occurs for sufficiently large Re for all the applied vortex definitions. The new vortex alters the generating vortex motion by slowing its horizontal motion and lifting it further from the wall. In the fixed wall system vortex eruption happens through a characteristic ’figure 8’ bifurcation. Considering the other coordinate system there is no topological change indicating when a vortex has left the boundary layer. However, here there is remarkable good agreement between streamlines and the vorticity contours even for short-lived vortices close to the wall. The normal form approach does not reveal simple connections between the streamline topology and the vorticity contour topology. Only for a non simple degenerate on wall critical point may a bifurcation occur in both the streamlines and the vorticity contours. The streamline bifurcations in this normal form contain the lower part of the ’figure 8’ bifurcation observed in numerics. The similarities and differences of the streamlines in the two different coordinate systems are well described by normal form theory. We derive the criterion, u⋅∇ ω = 0, for exactly matching contours of the vorticity contours and streamlines. This is fulfilled when the Navier - Stokes equations and the heat equation have identical solutions. Finally we focus on the superposition of two rotational invariant vortices in R2. The topology of the streamlines and the topology of the vorticity conturs are determined by the zeros of a single real function. For the canonical example of two Gaussian vortices three parameters exist. Three structurally stable topologies are observed. For the streamlines two of the topologies are well known for the corresponding situation of two point vortices when the singularities are treated as centers. The last topology is a single center which is consistent with the powerful result on the long time behaviour proved by Gallay &amp; Wayne (Comm. in Math. Phys. 255, 2005). The case of three critical points of the streamlines is a subset of three critical points of the vorticity. This explains an observation in the simulations of vortex generation near a wall. Here, a long living erupted vortex disappears due to viscosity. This happens first considering the streamlines while being more robust when considering the vorticity formulation.</p

Online Research Database In Technology

Topology of streamlines and vorticity contours for two - dimensional flows - DTU Orbit(09/11/2017) Topology of streamlines and vorticity contours for two - dimensional flowsConsidering a coordinate-free formulation of helical symmetry rather than more traditional definitions based oncoordinates, we discuss basic properties of helical vector fields and compare results from the literature. For inviscid flowwhere a velocity field is generated by a sum of helical vortex filaments with same pitch we use the established results toprove briefly that the velocity field is helical. We discuss the role of the stream function for the topology of the streamlinesin incompressible, helical flows. On this basis, we perform a comprehensive study of the topology of the flow fieldgenerated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin(Phys. Fluids 25, 1982) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Russ. J.Eng. Thermophys. 5, 1995) we obtain a closed-form approximation which is considerably easier to analyse. Critical pointsof the stream function can be found from the zeroes of a single real function of one variable, and we show that threedifferent flow topologies can occur, depending on a single dimensionless parameter. Including the self-induced velocity onthe vortex filament by the localised induction approximation the stream function is slightly modified and an extra parameteris introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combinationof the parameters. The analysis of the closed form show promise for analysing more complex flow with helical symmetrye.g. multiple helical vortex filaments inside a cylinder which has industrial relevance.We then change focus and study creation, destruction and interaction of vortices in two dimensional flow. A vortex isadvected above a wall causing a viscous response near the wall which generates a new vortex structure. The problem isstudied numerically relying on the code developed by Prof. M. Thompson and his group at Monash University, Australia.We also investigate the problem analytically using normal form theory. It is not a simple task to define a vortex in a properway that allow the study of creation and destruction of vortices. We investigate three sound choices: the vorticity extrema,the streamline centers in a coordinate system with zero wall speed and the streamline centers in a frame moving withconstant velocity as predicted by a point vortex above a wall in inviscid fluid. There is no reason to a priori expectequivalent results of the three vortex definitions. However, the study is mainly motivated by the findings of Kudela &Malecha (Fluid Dyn. Res. 41, 2009) who find good agreement between the vorticity and streamlines in the fixed wallsystem. For small Re no new vortices are observed. Creation of a vortex occurs for sufficiently large Re for all the appliedvortex definitions. The new vortex alters the generating vortex motion by slowing its horizontal motion and lifting it furtherfrom the wall. In the fixed wall system vortex eruption happens through a characteristic ’figure 8’ bifurcation. Consideringthe other coordinate system there is no topological change indicating when a vortex has left the boundary layer. However,here there is remarkable good agreement between streamlines and the vorticity contours even for short-lived vorticesclose to the wall.The normal form approach does not reveal simple connections between the streamline topology and the vorticity contourtopology. Only for a non simple degenerate on wall critical point may a bifurcation occur in both the streamlines and thevorticity contours. The streamline bifurcations in this normal form contain the lower part of the ’figure 8’ bifurcationobserved in numerics. The similarities and differences of the streamlines in the two different coordinate systems are welldescribed by normal form theory.We derive the criterion, u⋅∇ ω = 0, for exactly matching contours of the vorticity contours and streamlines. This is fulfilledwhen the Navier - Stokes equations and the heat equation have identical solutions.Finally we focus on the superposition of two rotational invariant vortices in R2. The topology of the streamlines and thetopology of the vorticity conturs are determined by the zeros of a single real function. For the canonical example of twoGaussian vortices three parameters exist. Three structurally stable topologies are observed. For the streamlines two of thetopologies are well known for the corresponding situation of two point vortices when the singularities are treated ascenters. The last topology is a single center which is consistent with the powerful result on the long time behaviour provedby Gallay & Wayne (Comm. in Math. Phys. 255, 2005). The case of three critical points of the streamlines is a subset ofthree critical points of the vorticity. This explains an observation in the simulations of vortex generation near a wall. Here, along living erupted vortex disappears due to viscosity. This happens first considering the streamlines while being morerobust when considering the vorticity formulation. General informationState: PublishedOrganisations: Department of Applied Mathematics and Computer Science , Mathematics Authors: Andersen, M. (Intern), Brøns, M. (Intern)Number of pages: 150Publication date: 2013 Publication informationPlace of publication: Kgs. LyngbyPublisher: Technical University of Denmark (DTU)Original language: English Series: DTU Compute PHD-2013Number: 314ISSN: 0909-3192Main Research Area: Technical/natural sciencesElectronic versions: phd314_Andersen_M.pdf Publication: Research › Ph.D. thesis – Annual report year: 2014 

Topology of streamlines and vorticity contours for two - dimensional flows

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Topology of streamlines and vorticity contours for two - dimensional flows

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