DOCSLIB.ORG

Fundamentals of Vortex Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fundamentals of Vortex Dynamics Fundamentals of vortex dynamics Andrew David Gilbert, Mathematics Department, University of Exeter, UK Leverhulme Trust Research Fellowship - 2019 Motivation: vortices and more vortices • wing-tip vortices (NASA) • draining Lake Texoma, USA Turbulence • Leonardo da Vinci’s sketch • vortices in turbulence simulations • vortices in quantum turbulence Instabilities • von Karman vortex street • Crow instability • Kelvin-Helmholtz instability • Widnall vortex ring instability or or Dt⇢ + ⇢ u =0, D ⇢ + ⇢ u =0(2.25), (2.25) r · t r · which captures nicely the ideawhich that the captures density nicely of a the fluid idea element that the changes, density by of a fluid element changes, by D ⇢or, decreasing if the divergence of the fluid flow u > 0, and increasing if t Dt⇢, decreasing if ther divergence· of the fluid flow u > 0, and increasing if u < 0. u <D0.t⇢ + ⇢ u =0, r · (2.25) r · or r · r · whichor captures nicely the ideaor that the density of a fluid element changes, by Dt⇢ + ⇢ u =0, (2.25) Dt⇢, decreasing if the divergenceD ⇢ + of⇢ theu fluid=0r ,· flow Dtu⇢ +>⇢0, andu =0(2.25) increasing, if (2.25) 2.5 Navier–Stokeswhich captures equation nicely again2.5 the Navier–Stokest idea thatr · the equation density againr of· a fluidr element· changes, by uwhich< 0. captures nicely the ideawhich that captures the density nicely of the a fluid idea element that the changes, density of by a fluid element changes, by r · Dt⇢, decreasing if theD divergence⇢, decreasing of the if the fluid divergence flow ofu the> 0, fluid and flow increasingu > 0, if and increasing if Dt⇢, decreasing if the divergencet of the fluid flow u > r0,· and increasing ifr · u < 0. We nowu < have0. the Navier–Stokesr · equation (2.14) and the continuity equation We now haveru <· the0. Navier–Stokesr equation· (2.14) and the continuity equation (2.23),2.5 amountingr Navier–Stokes· to 4 equations equation(2.23), in 3-d again amounting space, for to 5 4 quantities, equations in the 3-d components space, for 5 quantities, the components u of u, p and ⇢. The system would need to be completed by an equation of ui of u,2.5p and2.5 Navier–Stokes⇢ Navier–Stokes. The system equation would equationi 2.5 again need Navier–Stokes again to be completed equation by again an equation of state, for example linking pressurestate, to for density. example linking pressure to density. We now have the Navier–Stokes equation (2.14) and the continuity equation We will however adopt the simplest fluid dynamical system in which to under- We willWe howeverWe now now have adopt have the the Navier–Stokes the simplest Navier–StokesWe fluid now equation havedynamical equation the (2.14) Navier–Stokes system and (2.14) the in and continuity which equation the to continuity under- equation (2.14) and equation the continuity equation (2.23), amounting to 4 equationsstand vortex in 3-d dynamics space, and for topological 5 quantities, aspects, the components taking the dynamic viscosity stand vortex(2.23),(2.23), dynamics amounting amounting and to 4 topological equations to 4(2.23), equations in amounting aspects, 3-d space, in 3-d taking to for space, 4 5 equations quantities, the for dynamic 5 quantities, in the 3-d components space, viscosity the for components 5 quantities, the components ui of u, p and ⇢. The systemµ and woulddensity need⇢ to be to constant be completed (and uniform by an in equation space). Then of the continuity µ and densityu of u, ⇢p toand be⇢. constant The system (andui of wouldu uniform, p needand ⇢ in to. The space). be completed system Then would by the an need continuity equation to be completed of by an equation of state,i forui of exampleu, p and linking⇢. Theequation pressure system expresses would to density. need that to the be fluid completed flow u is by divergenceless, an equation of and with this we equationstate, expresses for example that linkingthe fluid pressurestate, flow foru to exampleis density. divergenceless, linking pressure and with to density. this we state, for example linkingcan also pressure rewrite the to density. Navier–Stokes equation as: can also rewrite the Navier–Stokes equation as: WeWe will will however however adopt adopt the the simplestWe simplest will however fluid fluid dynamical adopt dynamical the system simplest system in which fluid in dynamical to which under- to system under- in which to under- We will however adopt the simplest fluid dynamical system in which2 to under- standstand vortex vortex dynamics dynamics and andstand topological topological vortexDtu aspects, dynamics=2 aspects,@tu taking+ andu taking topological theu = dynamic thep dynamic+ aspects, viscosity⌫ u, taking viscosity theu dynamic=0. viscosity(2.26) standDtu vortex= @tu + dynamicsu uNavier-Stokes= and topologicalp + ⌫ u, aspects,and· rEuleru taking=0r. equations the dynamic(2.26)r viscosityr · µ and density ⇢ to be· constantr µ andr (and density uniformr⇢ to in be space). constantr · Then (and the uniform continuity in space). Then the continuity µ and density ⇢ to be constantHere we have (and divided uniform through in space). by the Then constant the⇢ continuityand replaced p/⇢ by p (to equationµ and expresses density that⇢ to the beequation constant fluid flow expresses (andu is divergenceless, uniform that the in fluid space). and flow with Thenu is this divergenceless, the we continuity and with this we Hereequation we have expresses divided through thatavoid the by the fluid too constant many flow symbols!).u⇢isand divergenceless, replaced The kinematicp/⇢ by and viscosityp (to with⌫ this= µ/⇢ weis a constant, and canequation also rewrite expresses the Navier–Stokes that•canTake the also constant fluid equation rewrite flow density, the as:u Navier–Stokes isconstant divergenceless, viscosity, equation incompressible and as: with this flow we and write avoidcan too also many rewrite symbols!). the Navier–Stokes Thewe kinematic usually refer viscosity equation to this⌫ as: as= justµ/⇢ theis a viscosity constant, in and what follows. can also rewrite the Navier–Stokes equation as: 2 we usually refer to this as just the viscosityD intu what= @2tu follows.+ u u = p + ⌫ u, u =0. (2.26) Dtu = @tu + u u = p + ⌫ u, u =0. (2.26) · r r r 2 2r· r· r r r · Dtu =D@tutu=+@tuu + uu =u = p +p⌫+ ⌫u, u, u u=0=0. (2.26)(2.26) Here we have divided through2.6Here· Eulerr by we· ther have equationr constant dividedr for⇢r ideal throughandr flow replaced byr ther·p/· constant⇢ by p (to⇢ and replaced p/⇢ by p (to 2.6 Euler equation for ideal flow Hereavoid weHere too have we many divided have symbols!). divided through• Theavoidwith through kinematic kinematic too by many the by the viscosityconstantviscosity symbols!). constant ⌫ =⇢ The µ/and ⇢ ⇢ kinematicand isand replaced a replaced constant,replacing viscosityp/ andp/ ⇢ ⇢ by ⌫byby= p pµ/ for(to(to⇢ convenienceis a constant, and we usually refer to this as just the viscosity in what follows. avoidwe too usuallyavoid many too refer many symbols!). to this symbols!). asThe just The Euler the Thekinematic viscosity equation kinematic in viscosity is what the viscosity case follows.⌫ of=⌫ (2.26)=µ/µ/⇢ is⇢ withoutis a a constant, constant, viscosity and and⌫ =0,andforms Thewe Euler usuallywe equation usually refer is to refer the this case to asthe this• of justEuler basis as (2.26) thejust equation of withoutmuch viscosity the viscosityfor of ideal our viscosity in discussion flow what in what ⌫ follows.=0,andforms follows.below: (highly singular limit) the basis2.6 of much Euler equation of our discussion for ideal2.6 flow below: Euler equation for ideal flow Dtu = @tu + u u = p, u =0. (2.27) 2.6 Euler equation for ideal flow · r r r · 2.6 Euler equationDtu = @tu for+ idealu u flow= p, u =0. (2.27) The Euler equation is theSetting·The caser Euler of⌫ =0in(2.26)isahighlysingularlimitas (2.26)r equation withoutr is· the viscosity case of⌫ (2.26)=0,andforms without viscosity⌫ multiplies⌫ =0,andforms the highest Settingthe⌫ =0in(2.26)isahighlysingularlimitas basis of much of ourderivatives discussionthe basis below: in of the much equation! of our⌫ multiplies discussion A fluid with below: the⌫ highest=0isalsodescribedasanideral The Euler equation is the case of (2.26) without viscosity ⌫ =0,andforms derivatives in the equation! Afluid fluid—thereisnofrictionandnodissipationofenergy.TheEulerequation with ⌫ =0isalsodescribedasanideral The Euler equation is the case of (2.26)D withouttu = @tu + viscosityu u =⌫ =0,andformsp, u =0. (2.27) the basis ofD muchtu = of@cantu our+ beu discussion writtenu = in below: thep, attractiveu =0 form·.r r (2.27)r · fluidthe—thereisnofrictionandnodissipationofenergy.TheEulerequation basis of much of our discussion· r below:r r · can be written in the attractive formSetting ⌫ =0in(2.26)isahighlysingularlimitas⌫ multiplies the highest Setting ⌫ =0in(2.26)isahighlysingularlimitasDtu = @tu + u u = p,⌫@ multipliesu = u u =0! the. highestP, (2.27) (2.28) derivatives in· r the equation!r t A fluidr · ⇥ withr⌫ =0isalsodescribedasanideral derivatives in theDtu equation!= @tu A+ fluidu withu =⌫ =0isalsodescribedasanp, u =0. ideral (2.27) fluidSetting—thereisnofrictionandnodissipationofenergy.TheEulerequation⌫ =0in(2.26)isahighlysingularlimitas@t(usingufluid= u vector—thereisnofrictionandnodissipationofenergy.TheEulerequation·!r calculusP,r identityr (2.5)),· ⌫ wheremultiplies(2.28) the Bernoulli the highest function is can be⇥ writtenr in the attractive form Settingcanderivatives be⌫ written=0in(2.26)isahighlysingularlimitas in in the the attractive equation! form A fluid with ⌫ =0isalsodescribedasan⌫ multiplies the highestideral (using vector calculus
Load more

Recommended publications
  • On the Effect of Pipe Boundary Layer Growth on the Formation of a Laminar Vortex Ring Generated by a Piston/Cylinder Arrangement
    Theoret. Comput. Fluid Dynamics (2002) 15: 303–316 Theoretical and Computational Fluid Dynamics Springer-Verlag 2002 On the Effect of Pipe Boundary Layer Growth on the Formation of a Laminar Vortex Ring Generated by a Piston/Cylinder Arrangement Michael Shusser and Morteza Gharib Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A. Moshe Rosenfeld Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Kamran Mohseni Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309-0429, U.S.A. Communicated by R.D. Moser Received 19 January 2000 and accepted 17 August 2001 Abstract. The growth of a boundary layer at the nozzle wall during laminar vortex ring formation by a nozzle flow generator (piston/cylinder arrangement) is analysed theoretically and numerically and used for modelling the formation of real vortex rings. The predictions of the model are in good agreement with previous experimental and numerical results. 1. Introduction Vortex rings are usually generated in the laboratory by the motion of a piston pushing a column of fluid of length L through an orifice or nozzle of diameter D. This results in a separation of the boundary layer at the edge of the orifice or nozzle and its subsequent spiral roll-up. The piston/cylinder arrangement has been extensively used to address the problem of vortex ring forma- tion (Shariff and Leonard, 1992; Lim and Nickels, 1995). Recently Gharib et al. (1998) in their experimental study of vortex ring formation addressed the question of the largest circulation that a vortex ring can attain, by increasing L/D while keeping the average piston velocity fixed.
    [Show full text]
  • Correlation Between Vortex Ring Formation and Mitral Annulus Dynamics During Ventricular Rapid Filling
    ASAIO Journal 2007 Biomedical Engineering Correlation Between Vortex Ring Formation and Mitral Annulus Dynamics During Ventricular Rapid Filling ARASH KHERADVAR,*† MICHELE MILANO,† AND MORTEZA GHARIB*† One of the most important fluid phenomena observed in the process of vortex ring formation and its influence on dynamics left ventricle during diastole is the presence of vortex rings of the left heart have not been elucidated so far. that develop with a strong jet entering through the mitral The presence of vortex rings that develop during cardiac valve. The present study is focused on the rapid filling phase diastole was initially recognized by in vitro visualization of the of diastole, during which the left ventricle expands and re- ventricular flow5,6 and subsequently confirmed by analyses ceives blood through the fully open mitral valve. The atrio- based on color Doppler mapping7,8 and magnetic resonance ventricular system during the rapid filling phase was emu- imaging.9,10 During diastole, when the left ventricle is filling lated experimentally with a simplified mechanical model in with the blood jet from the atrium, the ventricle expands, and which the relevant pressure decay and the dimension of as a result, the atrioventricular plane moves in the opposite mitral annulus approximate the physiologic and pathologic direction with respect to blood flow. Little has been done so far values. Digital particle image velocimetry measurements to understand if any correlation exists between this movement were correlated with the force measurements on the mitral and the formation process of the vortex ring created by the annulus plane to analyze the relation between flow and the blood jet.
    [Show full text]
  • Analysis of Vortex Ring Formation in the Heart Chamber by Instantaneous Vortex Deviation Based on Convolutional Neural Network
    E3S Web of Conferences 271, 03009 (2021) https://doi.org/10.1051/e3sconf/202127103009 ICEPE 2021 Analysis of vortex ring formation in the heart chamber by instantaneous vortex deviation based on convolutional neural network Ke Yang1, Shiqian Wu2,*, and Kelvin K.L. Wong3 1 School of Machinery and Automation, Wuhan University of Science and Technology, Wuhan, China 2 School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan, China 3 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China Abstract: The formation of vortex rings during the left ventricle (LV) filling is an optimized mechanism for blood transport, and the vorticity is an important measure of a healthy heart and LV. There is a relationship between abnormal diastolic vortex structure and impaired LV, and hence vortex identification is vital for understanding the underlying physical mechanism of blood flow. However, due to lack of quantitative methods, defining, computing and mapping the left ventricular vortices has not been rigorously studied previously. In this paper, a novel method of vortex detection based on the convolutional neural network (CNN) is created, which enables determination of the boundary of vortex and integrates the local and global flow fields. We have used the CNN-based vortex identification and vector flow mapping (VFM) to quantify left ventricular vorticity. In the clinical application of our methodology to healthy subjects and uremic patients, we find differences in the strength and position of the vortices between healthy and patients with uremia cardiomyopathy. Our results can accurately indicate the role of vortex formation in intra- cardiac flow, and provide new insights into the blood flow within the heart structure.
    [Show full text]
  • An Experimental Study of a Turbulent Vortex Ring
    J. Fluid Mech. (1990), vol. 211, pp. 243-283 243 Printed in Great Britain An experimental study of a turbulent vortex ring By ARI GLEZERl DONALD COLES2 AND Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 8572 1, USA California Institute of Technology, Pasadena, CA 91125, USA (Received 18 July 1987 and in revised form 12 July 1989) A turbulent vortex ring having a relatively thin core is formed in water by a momentary jet discharge from an orifice in a submerged plate. The necessary impulse is provided by a pressurized reservoir and is controlled by a fast programmable solenoid valve. The main aim of the research is to verify the similarity properties of the mean flow, as defined by ensemble averaging, and to find the distribution of mean vorticity, turbulent energy, and other quantities in the appropriate non-steady similarity coordinates. The velocity field of the vortex is measured for numerous realizations with the aid of a two-channel tracking laser-Doppler velocimeter. The problem of dispersion in the trajectories of the individual rings is overcome by development of a signature-recognition technique in two variables. It is found that the turbulence intensity is largest near the vortex core and that at least the radial component is not negligible in the near wake. The slow growth of the ring structure is controlled by a slight excess of entrainment over de-entrainment. An important inference is that the growth process and the process of turbulence production probably involve secondary vortices wrapped around the core in azimuthal planes.
    [Show full text]
  • Vortex-Ring-Induced Stratified Mixing
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Apollo Under consideration for publication in J. Fluid Mech. 1 1 Vortex-Ring-Induced Stratified Mixing: 2 Mixing Model 3 Jason Olsthoorny, and Stuart B. Dalziel 4 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 5 Centre for Mathematical Sciences, Wilberforce Road, Cambridge UK, CB3 0WA 6 (Received ?; revised ?; accepted ?. - To be entered by editorial office) 7 The study of vortex-ring-induced mixing has been significant for understanding stratified 8 turbulent mixing in the absence of a mean flow. Renewed interest in this topic has 9 prompted the development of a one-dimensional model for the evolution of a stratified 10 system in the context of isolated mixing events. This model is compared to numerical 11 simulations and physical experiments of vortex rings interacting with a stratification. 12 Qualitative agreement between the evolution of the density profiles is observed, along 13 with close quantitative agreement of the mixing efficiency. This model highlights the key 14 dynamical features of such isolated mixing events. 15 1. Introduction 16 Understanding the mixing produced by turbulent motion in a stratified environment 17 remains elusive. Such mixing is particularly relevant in an oceanographic context (Ivey 18 et al. 2008). The energy cascade found in turbulent flows results in a large range of length 19 scales, which complicates the analysis. However, Turner (1968), while examining grid- 20 generated stratified turbulence (with no mean flow), argued that most of the mixing of the 21 density field was generated by independent localized mixing events, resulting from large- 22 scale turbulent eddies impacting the density interface.
    [Show full text]
  • Vortex Ring Mixing in the Left Ventricle of the Human Heart
    Toger et al. Journal of Cardiovascular Magnetic Resonance 2013, 15(Suppl 1):E27 http://www.jcmr-online.com/content/15/S1/E27 POSTERPRESENTATION Open Access Vortex ring mixing in the left ventricle of the human heart Johannes Toger1,2*, Mikael Kanski1, Marcus Carlsson1, Sándor J Kovács3, Gustaf Söderlind2, Hakan Arheden1, Einar Heiberg1 From 16th Annual SCMR Scientific Sessions San Francisco, CA, USA. 31 January - 3 February 2013 Background During rapid filling of the left ventricle, a vortex ring forms downstream from the mitral valve. Previous experi- ments in water tanks have shown that vortex ring forma- tion is an optimized method for fluid transport. The rotation of the vortex ring leads to mixing of the inflowing blood and blood that was already in the ventricle (Figure 1). In water tanks, the amount of mixing decreases with increasing vortex formation ratio (VFR), a dimensionless parameter relating inflowing volume to the annulus dia- meter (Figure 2). However, the flow and anatomy of the left ventricle is more complex which may affect this estab- lished relationship. Therefore, we aimed to investigate if the relationship between VFR and mixing ratio demon- strated in water tank experiments holds in the human heart. Methods Nine healthy volunteers and four patients with dilated ischemic cardiomyopathy underwent cardiovascular mag- netic resonance including 4D PC-MRI. Particle tracing was used to track blood flowing into the left ventricle dur- ing rapid filling (Figure 1). Lagrangian Coherent Structures (LCS) were used to define the boundary of the vortex ring. The entrained volume was defined as the volume of blood from the ventricle (Figure 1, colored blue) inside the vor- tex ring (Figure 1B, yellow line).
    [Show full text]
  • Vortex Dynamics 1
    VORTEX DYNAMICS 1. Introduction A vortex is commonly associated with the rotating motion of fluid around a common centerline. It is de¯ned by the vorticity in the fluid, which measures the rate of local fluid rotation. Typically, the fluid circulates around the vortex, the speed increases as the vortex is approached and the pressure decreases. Vor- tices arise in nature and technology in a large range of sizes as illustrated by the examples given in Table 1. The next section presents some of the mathematical background necessary to understand vortex formation and evolution. Section 3 describes sample flows, including important instabilities and reconnection pro- cesses. Section 4 presents some of the numerical methods used to simulate these flows. Vortex Diameter superfluid vortices 10¡8 cm (= 1A)º trailing vortex of Boeing 727 1{2 m dust devils 1{10 m tornadoes 10{500 m hurricanes 100{2000 km Jupiter's Red Spot 25,000 km spiral galaxies thousands of light years TABLE 1: Sample vortices and typical sizes. 2. Background Let D be a region in 3D space containing a fluid, and let x =(x; y; z)T be a point in D. The fluid motion is described by its velocity u(x;t)=u(x;t)i + v(x;t)j+w(x;t)k, and depends on the fluid density ½(x;t), temperature T (x;t), gravitational ¯eld g and other external forces possibly acting on it. The fluid vorticity is de¯ned by ! = r£u. The vorticity measures the local fluid rotation about an axis, as can be seen by expanding the velocity near x = x0, 1 2 u(x)=u(x0)+D(x0)(x ¡ x0)+ !(x0) £ (x ¡ x0)+O(jx ¡ x0j ) (1) 2 where " # ux uy uz 1 T v v v D(x0)= (ru + ru ); ru = x y z : (2) 2 wx wy wz The ¯rst term u(x0) corresponds to translation: all fluid particles move with constant velocity u(x0).
    [Show full text]
  • 1 Theoretical Models of Vortex Rings
    1 Theoretical Models of Vortex Rings 1.1 Vortex Rings Systematic scientific studies of vortex rings have been performed since the middle of the 19th century. Originally they were stimulated by the work of Helmholtz, who founded the theory of vortex motions of fluids and also carried out a series of experimental observations of vortex ring motion (Helmholtz 1858). Experiments with the motion of air vortex rings colored with smoke and vortex rings in fluids visualized with dye showed that a vortex ring is a toroidal volume of vortical fluid moving in a surrounding medium at an approximately constant translational speed perpendicular to the ring plane (Fig. 1.1). The fluid motion is axisymmetric, and the vector of vorticity (rotor of velocity ω = rotu ) in the torus is directed along the circles concentric with the circular axis of the torus. A certain volume of the fluid which embraces the ring and looks like an ellipsoid flattened along the direc- tion of motion is moving together with the toroidal vortex ring. This enclosed vol- ume of the fluid is called vortex atmosphere. Inside the vortex atmosphere the fluid is circulating along the closed streamlines encompassing the toroidal core of the vortex. Motion of the fluid surrounding the vortex atmosphere resembles a pat- tern of flow without separation past a corresponding solid body (Fig. 1.2). This qualitative picture of the flow structure around a vortex ring was described by Reynolds back in 1876 (Reynolds 1876). Later for a long period of time studies of vortex rings were mostly developed theoretically within the framework of an in- viscid (ideal) fluid.
    [Show full text]
  • Vortex Ring Propagation in Confined Spheroidal Domains and Applications to Cardiac Flows
    VORTEX RING PROPAGATION IN CONFINED SPHEROIDAL DOMAINS AND APPLICATIONS TO CARDIAC FLOWS By MILAD SAMAEE Bachelor of Science in Mechanical Engineering Khaje Nasir Toosi University of Technology Tehran, Iran 2008 Master of Science in Biomedical Engineering Amirkabir University of Technology Tehran, Iran 2011 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2019 ii VORTEX RING PROPAGATION IN CONFINED SPHEROIDAL DOMAINS AND APPLICATIONS TO CARDIAC FLOWS Dissertation Approved: Dr. Arvind Santhanakrishnan Dissertation Adviser Dr. Jamey Jacob Dr. Khaled Sallam Dr. Sundar Madihally Dr. Yu Feng ii Name: MILAD SAMAEE Date of Degree: DECEMBER, 2019 Title of Study: VORTEX RING PROPAGATION IN CONFINED SPHEROIDAL DOMAINS AND APPLICATIONS TO CARDIAC FLOWS Major Field: MECHANICAL ENGINEERING Abstract: Vortex ring formation is observed in the human cardiac left ventricle (LV) during diastole. Numerous studies to date have examined the intraventricular filling vortex for potential use in the diagnosis of cardiac dysfunction. However, a systematic understanding of the effects of LV size on vortex ring properties is currently unavailable. In diastolic dysfunction (LVDD), heart muscles can thicken and decrease the internal volume of the LV. On the other hand, by progressing diastolic dysfunction, the transmitral inflow changes and the impact of inflow altering on vortex ring properties is also unknown. In contrast to the considerable body of research on vortex ring propagation in semi-infinite domains and interaction with walls, only a limited number of studies have examined confined vortex ring dynamics. The latter studies only considered radially confined domains, which are a simplification to the combined radial and axial confinement observed in spheroidal domains such as the LV.
    [Show full text]
  • Coherent Structure Dynamics in Jets from Irregular Shaped Nozzles
    15th Australasian Fluid Mechanics Conference The University of Sydney, Sydney, Australia 13-17 December 2004 Coherent Structure Dynamics in Jets from Irregular Shaped Nozzles J. Hart1,2, J. Naser1,2 and P. Witt 1,3 1Cooperative Research Centre for Clean Power from Lignite 2School of Engineering & Science, Swinburne University of Technology, VIC, 3122 AUSTRALIA 3CSIRO, Division of Minerals, Clayton, VIC, 3169, AUSTRALIA Abstract nozzle exit to introduce extra turbulence has been used to Computational Fluid Dynamics was used to model spatially increase mixing of rectangular jets with their surroundings [2]. developing, initially laminar, gas jets issuing from nozzles of The high rates of wear caused by coal particles make this regular and irregular cross-section, using the Large Eddy impractical in coal-burner nozzles. This paper proposes that by Simulation (LES) technique. Nozzles were based on bluff jets combining the properties of circular and rectangular jets the used in tangentially fired lignite-burning furnaces. Validation of mixing characteristics may be altered, and these characteristics the LES based model was achieved by comparing predicted jet might then be used to selectively enhance or suppress mixing and entrainment rates with published experimental and numerical spreading rates. In order to investigate this, coherent structure data. Large-scale coherent turbulent structures including ring and dynamics were modelled in jets that are geometrically and braid vortices and other stream wise structures are predicted by hydrodynamically similar to burner jets, with novel semi-circular, the LES model. The shape of the ring vortices formed was found semi-rectangular nozzles, hereafter referred to as hybrid nozzles. to directly relate to the nozzle geometry.
    [Show full text]
  • Lagrangian Coherent Structures in Vortex Ring Formation
    Lagrangian Coherent Structures in Vortex Ring Formation Master’s Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Braxton N. Harter, B.S. Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2019 Thesis Committee: Dr. James W. Gregory, Advisor Dr. Mohammad Samimy Dr. Matthew H. McCrink Copyrighted by Braxton N. Harter 2019 Abstract Vortex rings are among the only consistent natural feature to achieve effective transport of mass, momentum, and energy in fluids. For most scenarios where each of these transport mechanisms are utilized, efficiency of the vortex generator is parameterized by how much vorticity is entrained by the created vortex ring—the more vorticity that a vortex ring entrains during the formation process, the more efficient each of the transport mechanisms will be. They cannot grow indefinitely, however. Vortex ring formation is known to be limited by a dimensionless ratio that compares the length of an ejected slug of fluid L to its diameter D. This ratio is termed formation time, Ţ = L/D, and has a limiting value of approximately 4. The upper limit of ring growth has been justified by the Kelvin-Benjamin variational principle, which states that vortex ring formation is limited due to an energy deficit of the vortex generator in comparison to the translating vortex ring. This principle forms an elegant analytical theory for predicting vortex ring pinch-off; however, a key limitation of findings reported in the literature is the inability to accurately quantify vortex characteristics early in the formation process (especially before pinch-off).
    [Show full text]
  • Chemically Reactive Turbulent Vortex Rings H
    Chemically reactive turbulent vortex rings H. Johari Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 (ReGeived 2 November 1993; accepted 22 June 1995) Employing an aqueous acid-base reaction, the minimum mixing rate of turbulent vortex rings was investigated in a water tank. Vortex rings were generated by a simple apparatus with a cylindrical geometry. The released fluid surrounding the vortex core mixed very rapidly when compared with the fluid in the toroidal core. Moreover, the fluid within the core did not mix uniformly in the azimuthal direction. The normalized distance a vortex ring must travel, in order to completely mix with the ambient fluid to a specific volumetric ratio, depends on the aspect ratio of the generating cylinder. Scaling arguments are presented that relate the above distance to the spreading rate and the generating apparatus parameters. Due to the very small net entrainment rate of vortex rings, the detrainment of core material cannot be ignored when the mixing rate of the core is considered. © 1995 American Institute of Physics. I. INTRODUCTION other free shear flows, is dependent on the generating appa­ ratus geometry as well as the Reynolds number. 11 It has also A vortex ring is created when a fluid parcel is impul­ been reported that at far downstream locations, a turbulent sively discharged from a circular opening into a quiescent vortex ring may undergo a transformation that changes the environment. Vortex rings have been studied for more than a spreading rate.11•12 At present there does not exist a method century primarily due to their ability to transport mass and of predicting the spreading rate of a turbulent vortex ring, momentum for a relatively long distance with little mixing.
    [Show full text]