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Lagrangian and Eulerian Representations of Fluid Flow: Part I, Kinematics and the Equations of Motion James F. Price MS 29, Clark Laboratory Woods Hole Oceanographic Institution Woods Hole, MA, 02543
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  Lagrangian and Eulerian Representations of Fluid Flow: Part I,Kinematics and the Equations of Motion James F. PriceMS 29, Clark LaboratoryWoods Hole OceanographicInstitutionWoods Hole, MA, 02543 http://www.whoi.edu/science/PO/people/jprice  jprice@whoi.eduVersion 7.4September 13, 2005 Summary:  This essay introduces the two methods that are commonly used to describe fluid flow, byobserving the trajectories of parcels that are carried along with the flow or by observingthe fluid velocity atfixed positions. These yield what are commonly termed Lagrangian and Eulerian descriptions. Lagrangianmethods are often the most efficient way to sample a fluid domain and the physical conservation laws areinherently Lagrangian since they apply to specific material parcels rather than pointsin space. It happens,though, that the Lagrangian equations of motion applied to a continuum are quite difficult, and thus almostall of the theory (forward calculation)in fluid dynamics is developed withinthe Eulerian system. Euleriansolutionsmay be used to calculate Lagrangian properties, e.g., parcel trajectories, which is often a valuablestep in the description of an Eulerian solution. Transformation to and from Lagrangian and Eulerian systems— the central theme of this essay — is thus the foundationof most theory in fluid dynamics and is a routinepart of many investigations.The transformation of the Lagrangian conservationlaws into the Eulerian equations of motion requiresthree key results. (1) The first is dubbed the Fundamental Principle of Kinematics; the velocity at a givenpositionand time (the Eulerian velocity) is identicallythe velocity of the parcel (the Lagrangian velocity)that occupies that positionat that time. (2) The material time derivative relates the time rate of changeobserved followinga moving parcel to the time rate of change and advective rate of change observed at afixed position; D . /= Dt   D @. /=@ t   C  V   r  . / . (3) And finally, the time derivative of an integral over amoving fluid volume can be transformed intofield coordinatesby means of the Reynolds Transport Theorem.1  Once an Eulerian velocity field has been found, it is more or less straightforwardto computeLagrangian properties, e.g., parcel trajectories, which are often of practical interest. The FPK assures that theinstantaneousEulerian and Lagrangian velocities are identically equal. However, when averaging orintegrating takes place, then the Eulerian mean velocity and the Lagrangian mean velocity are not equal,except in the degenerate case of spatially uniform flows. If the dominant flow phenomenon is wavelike, thentheir difference may be understood as Stokes drift, a correlation between displacement and velocitydifferences.In an Eulerian system the local (at a point) effect of transport by the fluid flow is represented by theadvective rate of change, V   r  . / , the product of an unknown velocity and the first partial derivative of anunknown field variable. This nonlinearityleads to much of the interestingand most of the challengingphenomenon of fluid flows. We can put some useful boundsupon what advection alone can do. For variablesthat can be written in conservation form, e.g., mass and momentum, advection alone can not be a net sourceor sink when integrated over a closed or infinite domain. Advection represents the transport of fluidproperties at a definite rate and direction, that of the fluid velocity, so that parcel trajectories are thecharacteristics of the advection equation. Advection by a nonuniform velocity may cause importantdeformation of a fluid parcel, and it may also cause rotation, an analog of angular momentum, and thatfollows a particularly simple and useful conservation law. Cover page graphic:  SOFAR float trajectories (green worms) and horizontal velocity measured by a currentmeter (black vector) during the Local Dynamics Experiment conducted in the Sargasso Sea. Click on thefigure to start an animation. The float trajectories are five-day segments, and the current vector is scaledsimilarly. The northeast to southwestoscillationseen here appears to be a barotropic Rossby wave; see Price,J. F. and H. T. Rossby, ’Observationsof a barotropic planetary wave in the western North Atlantic’,  J. Marine Res. ,  40 , 543-558, 1982. An analysis of the potentialvorticity balance of this motion is in Section6.4.4. These data and much more are available online from http://ortelius.whoi.edu/  Some animations of theextensive float data archive from the North Atlantic are athttp://www.phys.ocean.dal.ca/ lukeman/projects/argo/. Contents 1 Kinematics of Fluid Flow. 4 1.1 Physical properties of material; how are fluids different from solids? . . . . . . . . . . . . . . 41.1.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Shear deformation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Two ways to observe fluid flow and the Fundamental Principle of Kinematics . . . . . . . . . 101.4 About this essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 The Lagrangian (or Material) Coordinate System. 14 2  32.1 Material coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Forces on a parcel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Stress components in an ideal fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Stress components in a viscousfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 The Lagrangian equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 The Eulerian (or Field) Coordinate System. 25 3.1 The material derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 The Eulerian equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Momentum conservation;Newton’sSecond Law. . . . . . . . . . . . . . . . . . . . . 313.3.3 Energy conservation;the First Law of Thermodynamics. . . . . . . . . . . . . . . . . 323.3.4 State Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Remarks on the Eulerian balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Depictions of Fluid Flow. 35 4.1 Trajectories, or pathlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Streaklines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Eulerian to Lagrangian Transformation by Approximate Methods. 38 5.1 Tracking parcels around a steady vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Tracking parcels in gravity waves; Stokes drift . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Aspects of Advection. 46 6.1 Modes of an advection equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Fluxes in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2.1 Global conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2.2 Control volume budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 Advection of a finite parcel; the Cauchy-Stokes Theorem . . . . . . . . . . . . . . . . . . . . 586.4.1 The rotation rate tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.2 The strain rate tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.3 The Cauchy-Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4.4 Rotation and divergence of a geophysicalflow; the potential vorticity . . . . . . . . . 67 7 Concluding Remarks. 71  1 KINEMATICS OFFLUID FLOW.  4 1 Kinematics of Fluid Flow. The broad aim of this essay is to introduce and exercise a few of the concepts and mathematical tools thatmake up the foundation of fluid mechanics. Fluid dynamics is a vast subject, encompassing widely diversematerials and phenomenon. This essay emphasizes aspects of fluid dynamics that are relevant to thegeophysicalflows of what one might term ordinary fluids, air and water, that make up the Earth’s fluidenvironment. The physicsthat govern geophysicalflow is codified by the conservation laws of classicalmechanics; conservation of mass, (linear) momentum, angular momentum and energy. TheLagrangian/Euleriantheme of this essay followsfrom the question, How can we apply these conservationlaws to the analysis or prediction fluid flow?In principle the answer is straightforward;first we erect a coordinate system that is suitable fordescribing a fluid flow, and then we derive the mathematical form of the conservation laws that correspond tothat system. The definition of a coordinate system is a matter of choice, and the issues to be considered aremore in the realm of kinematics than of physics. However, as we will describe in this introductorysection,the kinematics of a fluid flow are certainly dependent upon the physicalproperties of the fluid (reviewed inSection 1.1), and, the kinematics of even the smallest and simplest fluid flow is likely to be quite complex;fully three-dimensional and time-dependent flows are common rather than exceptional (Section 1.2). Thuskinematics is at the nub of what makes fluid mechanics challenging, and specifically, requires that thedescription of fluid flows be in terms of fields (beginningin Section 1.3). 1 ; 2 1.1 Physical properties of material; how are fluids different from solids? For most purposes of classical fluid dynamics, fluids such as air and water can be idealized as an infinitelydivisiblecontinuum withinwhich the pressure,  P  , and the velocity, V  , temperature,  T  , are in principledefinable at every point in space. 3 The molecular makeup of the fluid will be studiouslyignored, and thecrucially important physicalproperties of a fluid, e.g., its mass density,  , its heat capacity,  Cp , amongothers, must be provided from outside of a continuum theory, Table (1).The space occupied by the material will be called the domain. Solids are materials that have a more orless definite or intrinsicshape, and will not conform to their domain under normal conditions. Fluids (gasesand liquids)have no intrinsicshape or preferred configuration. Gases are fluids that will completely fill theirdomain (or container) and liquidsare fluids that form a free surface in the presence of an acceleration field,i.e., gravity.An important property of any material is its response to an applied force, Fig. (1). If the force on the 1 Footnotes provide references, extensions or qualifications of material discussed in the main text, along with a few homework assignments. They may be skippedon first reading. 2 An excellent web page that surveys the wide range of fluid mechanicsis http://physics.about.com/cs/fluiddynamics/  3 Readers are presumed to have a college-level background in physics and multivariable calculus and to be familiar with basicphysical concepts such as pressure and velocity, Newton’s laws of mechanics and the ideal gas laws. We will review the definitionswhen we require an especially sharp or distinct meaning.
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