Circulation and Vorticity
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Energy, Vorticity and Enstrophy Conserving Mimetic Spectral Method for the Euler Equation
Master of Science Thesis Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Copyright ⃝c D.J.D. de Ruijter, BSc All rights reserved. Delft University Of Technology Department Of Aerodynamics, Wind Energy, Flight Performance & Propulsion The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled \Energy, vorticity and enstro- phy conserving mimetic spectral method for the Euler equation" by D.J.D. de Ruijter, BSc in partial fulfillment of the requirements for the degree of Master of Science. Dated: 5 September 2013 Head of department: prof. dr. F. Scarano Supervisor: dr. ir. M.I. Gerritsma Reader: dr. ir. A.H. van Zuijlen Reader: P.J. Pinto Rebelo, MSc Summary The behaviour of an inviscid, constant density fluid on which no body forces act, may be modelled by the two-dimensional incompressible Euler equations, a non-linear system of partial differential equations. If a fluid whose behaviour is described by these equations, is confined to a space where no fluid flows in or out, the kinetic energy, vorticity integral and enstrophy integral within that space remain constant in time. Solving the Euler equations accompanied by appropriate boundary and initial conditions may be done analytically, but more often than not, no analytical solution is available. -
Lecture 5: Saturn, Neptune, … A. P. Ingersoll [email protected] 1
Lecture 5: Saturn, Neptune, … A. P. Ingersoll [email protected] 1. A lot a new material from the Cassini spacecraft, which has been in orbit around Saturn for four years. This talk will have a lot of pictures, not so much models. 2. Uranus on the left, Neptune on the right. Uranus spins on its side. The obliquity is 98 degrees, which means the poles receive more sunlight than the equator. It also means the sun is almost overhead at the pole during summer solstice. Despite these extreme changes in the distribution of sunlight, Uranus is a banded planet. The rotation dominates the winds and cloud structure. 3. Saturn is covered by a layer of clouds and haze, so it is hard to see the features. Storms are less frequent on Saturn than on Jupiter; it is not just that clouds and haze makes the storms less visible. Saturn is a less active planet. Nevertheless the winds are stronger than the winds of Jupiter. 4. The thickness of the atmosphere is inversely proportional to gravity. The zero of altitude is where the pressure is 100 mbar. Uranus and Neptune are so cold that CH forms clouds. The clouds of NH3 , NH4SH, and H2O form at much deeper levels and have not been detected. 5. Surprising fact: The winds increase as you move outward in the solar system. Why? My theory is that power/area is smaller, small‐scale turbulence is weaker, dissipation is less, and the winds are stronger. Jupiter looks more turbulent, and it has the smallest winds. -
Lecture 18 Ocean General Circulation Modeling
Lecture 18 Ocean General Circulation Modeling 9.1 The equations of motion: Navier-Stokes The governing equations for a real fluid are the Navier-Stokes equations (con servation of linear momentum and mass mass) along with conservation of salt, conservation of heat (the first law of thermodynamics) and an equation of state. However, these equations support fast acoustic modes and involve nonlinearities in many terms that makes solving them both difficult and ex pensive and particularly ill suited for long time scale calculations. Instead we make a series of approximations to simplify the Navier-Stokes equations to yield the “primitive equations” which are the basis of most general circu lations models. In a rotating frame of reference and in the absence of sources and sinks of mass or salt the Navier-Stokes equations are @ �~v + �~v~v + 2�~ �~v + g�kˆ + p = ~ρ (9.1) t r · ^ r r · @ � + �~v = 0 (9.2) t r · @ �S + �S~v = 0 (9.3) t r · 1 @t �ζ + �ζ~v = ω (9.4) r · cpS r · F � = �(ζ; S; p) (9.5) Where � is the fluid density, ~v is the velocity, p is the pressure, S is the salinity and ζ is the potential temperature which add up to seven dependent variables. 115 12.950 Atmospheric and Oceanic Modeling, Spring '04 116 The constants are �~ the rotation vector of the sphere, g the gravitational acceleration and cp the specific heat capacity at constant pressure. ~ρ is the stress tensor and ω are non-advective heat fluxes (such as heat exchange across the sea-surface).F 9.2 Acoustic modes Notice that there is no prognostic equation for pressure, p, but there are two equations for density, �; one prognostic and one diagnostic. -
ATMOSPHERIC and OCEANIC FLUID DYNAMICS Fundamentals and Large-Scale Circulation
ATMOSPHERIC AND OCEANIC FLUID DYNAMICS Fundamentals and Large-Scale Circulation Geoffrey K. Vallis Contents Preface xi Part I FUNDAMENTALS OF GEOPHYSICAL FLUID DYNAMICS 1 1 Equations of Motion 3 1.1 Time Derivatives for Fluids 3 1.2 The Mass Continuity Equation 7 1.3 The Momentum Equation 11 1.4 The Equation of State 14 1.5 The Thermodynamic Equation 16 1.6 Sound Waves 29 1.7 Compressible and Incompressible Flow 31 1.8 * More Thermodynamics of Liquids 33 1.9 The Energy Budget 39 1.10 An Introduction to Non-Dimensionalization and Scaling 43 2 Effects of Rotation and Stratification 51 2.1 Equations in a Rotating Frame 51 2.2 Equations of Motion in Spherical Coordinates 55 2.3 Cartesian Approximations: The Tangent Plane 66 2.4 The Boussinesq Approximation 68 2.5 The Anelastic Approximation 74 2.6 Changing Vertical Coordinate 78 2.7 Hydrostatic Balance 80 2.8 Geostrophic and Thermal Wind Balance 85 2.9 Static Instability and the Parcel Method 92 2.10 Gravity Waves 98 v vi Contents 2.11 * Acoustic-Gravity Waves in an Ideal Gas 100 2.12 The Ekman Layer 104 3 Shallow Water Systems and Isentropic Coordinates 123 3.1 Dynamics of a Single, Shallow Layer 123 3.2 Reduced Gravity Equations 129 3.3 Multi-Layer Shallow Water Equations 131 3.4 Geostrophic Balance and Thermal wind 134 3.5 Form Drag 135 3.6 Conservation Properties of Shallow Water Systems 136 3.7 Shallow Water Waves 140 3.8 Geostrophic Adjustment 144 3.9 Isentropic Coordinates 152 3.10 Available Potential Energy 155 4 Vorticity and Potential Vorticity 165 4.1 Vorticity and Circulation 165 -
An Overview of Impellers, Velocity Profile and Reactor Design
An Overview of Impellers, Velocity Profile and Reactor Design Praveen Patel1, Pranay Vaidya1, Gurmeet Singh2 1Indian Institute of Technology Bombay, India 1Indian Oil Corporation Limited, R&D Centre Faridabad Abstract: This paper presents a simulation operation and hence in estimating the approach to develop a model for understanding efficiency of the operating system. The the mixing phenomenon in a stirred vessel. The involved processes can be analysed to optimize mixing in the vessel is important for effective the products of a technology, through defining chemical reaction, heat transfer, mass transfer the model with adequate parameters. and phase homogeneity. In some cases, it is Reactors are always important parts of any very difficult to obtain experimental working industry, and for a detailed study using information and it takes a long time to collect several parameters many readings are to be the necessary data. Such problems can be taken and the system has to be observed for all solved using computational fluid dynamics adversities of environment to correct for any (CFD) model, which is less time consuming, inexpensive and has the capability to visualize skewness of values. As such huge amount of required data, it takes a long time to collect the real system in three dimensions. enough to build a model. In some cases, also it As reactor constructions and impeller is difficult to obtain experimental information configurations were identified as the potent also. Pilot plant experiments can be considered variables that could affect the macromixing as an option, but this conventional method has phenomenon and hydrodynamics, these been left far behind by the advent of variables were modelled. -
Vorticity and Strain Analysis Using Mohr Diagrams FLOW
Journal of Structural Geology, Vol, 10, No. 7, pp. 755 to 763, 1988 0191-8141/88 $03.00 + 0.00 Printed in Great Britain © 1988 Pergamon Press plc Vorticity and strain analysis using Mohr diagrams CEES W. PASSCHIER and JANOS L. URAI Instituut voor Aardwetenschappen, P.O. Box 80021,3508TA, Utrecht, The Netherlands (Received 16 December 1987; accepted in revisedform 9 May 1988) Abstract--Fabric elements in naturally deformed rocks are usually of a highly variable nature, and measurements contain a high degree of uncertainty. Calculation of general deformation parameters such as finite strain, volume change or the vorticity number of the flow can be difficult with such data. We present an application of the Mohr diagram for stretch which can be used with poorly constrained data on stretch and rotation of lines to construct the best fit to the position gradient tensor; this tensor describes all deformation parameters. The method has been tested on a slate specimen, yielding a kinematic vorticity number of 0.8 _+ 0.1. INTRODUCTION Two recent papers have suggested ways to determine this number in naturally deformed rocks (Ghosh 1987, ONE of the aims in structural geology is the reconstruc- Passchier 1988). These methods rely on the recognition tion of finite deformation parameters and the deforma- that certain fabric elements in naturally deformed rocks tion history for small volumes of rock from the geometry have a memory for sense of shear and flow vorticity and orientation of fabric elements. Such data can then number. Some examples are rotated porphyroblast- be used for reconstruction of large-scale deformation foliation systems (Ghosh 1987); sets of folded and patterns and eventually in tracing regional tectonics. -
Numerical Study of a Mixing Layer in Spatial and Temporal Development of a Stratified Flow M. Biage, M. S. Fernandes & P. Lo
Transactions on Engineering Sciences vol 18, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533 Numerical study of a mixing layer in spatial and temporal development of a stratified flow M. Biage, M. S. Fernandes & P. Lopes Jr Department of Mechanical Engineering Center of Exact Science and Tecnology Federal University ofUberldndia, UFU- MG-Brazil [email protected] Abstract The two-dimensional plane shear layers for high Reynolds numbers are simulated using the transient explicit Spectral Collocation method. The fast and slow air streams are submitted to different densities. A temporally growing mixing layer, developing from a hyperbolic tangent velocity profile to which is superposed an infinitesimal white-noise perturbation over the initial condition, is studied. In the same way, a spatially growing mixing layer is also study. In this case, the flow in the inlet of the domain is perturbed by a time dependent white noise of small amplitude. The structure of the vorticity and the density are visualized for high Reynolds number for both cases studied. Broad-band kinetic energy and temperature spectra develop after the first pairing are observed. The results showed that similar conclusions are obtained for the temporal and spatial mixing layers. The spreading rate of the layer in particular is in good agreement with the works presented in the literature. 1 Introduction The intensely turbulent region formed at the boundary between two parallel fluid streams of different velocity has been studied for many years. This kind of flow is called of mixing layer. The mixing layer of a newtonian fluid, non-reactive and compressible flow, practically, is one of the most simple turbulent flow. -
Scaling the Vorticity Equation
Vorticity Equation in Isobaric Coordinates To obtain a version of the vorticity equation in pressure coordinates, we follow the same procedure as we used to obtain the z-coordinate version: ∂ ∂ [y-component momentum equation] − [x-component momentum equation] ∂x ∂y Using the p-coordinate form of the momentum equations, this is: ∂ ⎡∂v ∂v ∂v ∂v ∂Φ ⎤ ∂ ⎡∂u ∂u ∂u ∂u ∂Φ ⎤ ⎢ + u + v +ω + fu = − ⎥ − ⎢ + u + v +ω − fv = − ⎥ ∂x ⎣ ∂t ∂x ∂y ∂p ∂y ⎦ ∂y ⎣ ∂t ∂x ∂y ∂p ∂x ⎦ which yields: d ⎛ ∂u ∂v ⎞ ⎛ ∂ω ∂u ∂ω ∂v ⎞ vorticity equation in ()()ζ p + f = − ζ p + f ⎜ + ⎟ − ⎜ − ⎟ dt ⎝ ∂x ∂y ⎠ p ⎝ ∂y ∂p ∂x ∂p ⎠ isobaric coordinates Scaling The Vorticity Equation Starting with the z-coordinate form of the vorticity equation, we begin by expanding the total derivative and retaining only nonzero terms: ∂ζ ∂ζ ∂ζ ∂ζ ∂f ⎛ ∂u ∂v ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞ 1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞ + u + v + w + v = − ζ + f ⎜ + ⎟ − ⎜ − ⎟ + ⎜ − ⎟ ()⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ∂t ∂x ∂y ∂z ∂y ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠ Next, we will evaluate the order of magnitude of each of these terms. Our goal is to simplify the equation by retaining only those terms that are important for large-scale midlatitude weather systems. 1 Scaling Quantities U = horizontal velocity scale W = vertical velocity scale L = length scale H = depth scale δP = horizontal pressure fluctuation ρ = mean density δρ/ρ = fractional density fluctuation T = time scale (advective) = L/U f0 = Coriolis parameter β = “beta” parameter Values of Scaling Quantities (midlatitude large-scale motions) U 10 m s-1 W 10-2 m s-1 L 106 m H 104 m δP (horizontal) 103 Pa ρ 1 kg m-3 δρ/ρ 10-2 T 105 s -4 -1 f0 10 s β 10-11 m-1 s-1 2 ∂ζ ∂ζ ∂ζ ∂ζ ∂f ⎛ ∂u ∂v ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞ 1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞ + u + v + w + v = − ζ + f ⎜ + ⎟ − ⎜ − ⎟ + ⎜ − ⎟ ()⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ∂t ∂x ∂y ∂z ∂y ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠ ∂ζ ∂ζ ∂ζ U 2 , u , v ~ ~ 10−10 s−2 ∂t ∂x ∂y L2 ∂ζ WU These inequalities appear because w ~ ~ 10−11 s−2 the two terms may partially offset ∂z HL one another. -
Ductile Deformation - Concepts of Finite Strain
327 Ductile deformation - Concepts of finite strain Deformation includes any process that results in a change in shape, size or location of a body. A solid body subjected to external forces tends to move or change its displacement. These displacements can involve four distinct component patterns: - 1) A body is forced to change its position; it undergoes translation. - 2) A body is forced to change its orientation; it undergoes rotation. - 3) A body is forced to change size; it undergoes dilation. - 4) A body is forced to change shape; it undergoes distortion. These movement components are often described in terms of slip or flow. The distinction is scale- dependent, slip describing movement on a discrete plane, whereas flow is a penetrative movement that involves the whole of the rock. The four basic movements may be combined. - During rigid body deformation, rocks are translated and/or rotated but the original size and shape are preserved. - If instead of moving, the body absorbs some or all the forces, it becomes stressed. The forces then cause particle displacement within the body so that the body changes its shape and/or size; it becomes deformed. Deformation describes the complete transformation from the initial to the final geometry and location of a body. Deformation produces discontinuities in brittle rocks. In ductile rocks, deformation is macroscopically continuous, distributed within the mass of the rock. Instead, brittle deformation essentially involves relative movements between undeformed (but displaced) blocks. Finite strain jpb, 2019 328 Strain describes the non-rigid body deformation, i.e. the amount of movement caused by stresses between parts of a body. -
Chapter 7 Balanced Flow
Chapter 7 Balanced flow In Chapter 6 we derived the equations that govern the evolution of the at- mosphere and ocean, setting our discussion on a sound theoretical footing. However, these equations describe myriad phenomena, many of which are not central to our discussion of the large-scale circulation of the atmosphere and ocean. In this chapter, therefore, we focus on a subset of possible motions known as ‘balanced flows’ which are relevant to the general circulation. We have already seen that large-scale flow in the atmosphere and ocean is hydrostatically balanced in the vertical in the sense that gravitational and pressure gradient forces balance one another, rather than inducing accelera- tions. It turns out that the atmosphere and ocean are also close to balance in the horizontal, in the sense that Coriolis forces are balanced by horizon- tal pressure gradients in what is known as ‘geostrophic motion’ – from the Greek: ‘geo’ for ‘earth’, ‘strophe’ for ‘turning’. In this Chapter we describe how the rather peculiar and counter-intuitive properties of the geostrophic motion of a homogeneous fluid are encapsulated in the ‘Taylor-Proudman theorem’ which expresses in mathematical form the ‘stiffness’ imparted to a fluid by rotation. This stiffness property will be repeatedly applied in later chapters to come to some understanding of the large-scale circulation of the atmosphere and ocean. We go on to discuss how the Taylor-Proudman theo- rem is modified in a fluid in which the density is not homogeneous but varies from place to place, deriving the ‘thermal wind equation’. Finally we dis- cuss so-called ‘ageostrophic flow’ motion, which is not in geostrophic balance but is modified by friction in regions where the atmosphere and ocean rubs against solid boundaries or at the atmosphere-ocean interface. -
Vortex Stretching in Incompressible and Compressible Fluids
Vortex stretching in incompressible and compressible fluids Esteban G. Tabak, Fluid Dynamics II, Spring 2002 1 Introduction The primitive form of the incompressible Euler equations is given by du P = ut + u · ∇u = −∇ + gz (1) dt ρ ∇ · u = 0 (2) representing conservation of momentum and mass respectively. Here u is the velocity vector, P the pressure, ρ the constant density, g the acceleration of gravity and z the vertical coordinate. In this form, the physical meaning of the equations is very clear and intuitive. An alternative formulation may be written in terms of the vorticity vector ω = ∇× u , (3) namely dω = ω + u · ∇ω = (ω · ∇)u (4) dt t where u is determined from ω nonlocally, through the solution of the elliptic system given by (2) and (3). A similar formulation applies to smooth isentropic compressible flows, if one replaces the vorticity ω in (4) by ω/ρ. This formulation is very convenient for many theoretical purposes, as well as for better understanding a variety of fluid phenomena. At an intuitive level, it reflects the fact that rotation is a fundamental element of fluid flow, as exem- plified by hurricanes, tornados and the swirling of water near a bathtub sink. Its derivation from the primitive form of the equations, however, often relies on complex vector identities, which render obscure the intuitive meaning of (4). My purpose here is to present a more intuitive derivation, which follows the tra- ditional physical wisdom of looking for integral principles first, and only then deriving their corresponding differential form. The integral principles associ- ated to (4) are conservation of mass, circulation–angular momentum (Kelvin’s theorem) and vortex filaments (Helmholtz’ theorem). -
Studies on Blood Viscosity During Extracorporeal Circulation
Nagoya ]. med. Sci. 31: 25-50, 1967. STUDIES ON BLOOD VISCOSITY DURING EXTRACORPOREAL CIRCULATION HrsASHI NAGASHIMA 1st Department of Surgery, Nagoya University School of Medicine (Director: Prof Yoshio Hashimoto) ABSTRACT Blood viscosity was studied during extracorporeal circulation by means of a cone in cone viscometer. This rotational viscometer provides sixteen kinds of shear rate, ranging from 0.05 to 250.2 sec-1 • Merits and demerits of the equipment were described comparing with the capillary viscometer. In experimental study it was well demonstrated that the whole blood viscosity showed the shear rate dependency at all hematocrit levels. The influence of the temperature change on the whole blood viscosity was clearly seen when hemato· crit was high and shear rate low. The plasma viscosity was 1.8 cp. at 37°C and Newtonianlike behavior was observed at the same temperature. ·In the hypothermic condition plasma showed shear rate dependency. Viscosity of 10% LMWD was over 2.2 fold as high as that of plasma. The increase of COz content resulted in a constant increase in whole blood viscosity and the increased whole blood viscosity after C02 insufflation, rapidly decreased returning to a little higher level than that in untreated group within 3 minutes. Clinical data were obtained from 27 patients who underwent hypothermic hemodilution perfusion. In the cyanotic group, w hole blood was much more viscous than in the non cyanotic. During bypass, hemodilution had greater influence upon whole blood viscosity than hypothermia, but it went inversely upon the plasma viscosity. The whole blood viscosity was more dependent on dilution rate than amount of diluent in mljkg.