DOCSLIB.ORG

Euler Equations (Fluid Dynamics) - Wikipedia

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Euler Equations (Fluid Dynamics) - Wikipedia Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... Euler equations (fluid dynamics) From Wikipedia, the free encyclopedia In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity.[1] In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the Potential flow around a wing. flow velocity is a solenoidal field, or using another appropriate energy This incompressible potential equation respectively (the simplest form for Euler equations being the flow satisfies the Euler conservation of the specific entropy). Historically, only the equations for the special case incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy of zero vorticity. equation – of the more general compressible equations together as "the Euler equations".[2] From the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field (i.e. in the limit of high Froude number). In fact, like any Cauchy equation, the Euler equations originally formulated in convective form (also called usually "Lagrangian form", but this name is not self-explanatory and historically wrong, so it will be avoided) can also be put in the "conservation form" (also called usually "Eulerian form", but also this name is not self-explanatory and is historically wrong, so it will be avoided here). The conservation form emphasizes the mathematical interpretation of the equations as conservation equations through a control volume fixed in space, and is the most important for these equations also from a numerical point of view. The convective form emphasizes changes to the state in a frame of reference moving with the fluid. Contents 1 History 2 Incompressible Euler equations with constant and uniform density 2.1 Properties 2.2 Nondimensionalisation 2.3 Conservation form 2.4 Spatial dimensions 3 Incompressible Euler equations 3.1 Conservation form 3.2 Conservation variables 4 Euler equations 4.1 Incompressible constraint 4.2 Enthalpy conservation 4.3 Thermodynamic systems 4.4 Conservation form 5 Quasilinear form and characteristic equations 5.1 Characteristic equations 5.2 Waves in 1D inviscid, nonconductive thermodynamic fluid 5.3 Compressibility and sound speed 5.3.1 Ideal gas 1 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... 6 Bernoulli's theorems for steady inviscid flow 6.1 Incompressible case and Lamb's form 6.2 Compressible case 6.3 Friedman form and Crocco form 7 Discontinuities 7.1 Rankine–Hugoniot equations 7.2 Finite volume form 8 Constraints 8.1 Ideal polytropic gas 8.2 Steady flow in material coordinates 8.2.1 Streamline curvature theorem 9 Exact solutions 10 See also 11 Notes 12 Further reading History The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Academie des Sciences de Berlin in 1757 (in this article Euler actually published only the general form of the continuity equation and the momentum equation;[3] the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816. During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector.[4] Incompressible Euler equations with constant and uniform density In convective form (i.e. the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are:[5] Incompressible Euler equations with constant and uniform density (convective or Lagrangian form) where: is the flow velocity vector, with components in a N-dimensional space , denotes the material derivative in time, 2 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... denotes the scalar product, is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation), and is the convective operator, is the specific (with the sense of per unit mass) thermodynamic work, the internal source term. represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on. The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become: In fact for a flow with uniform density the following identity holds: where is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not casual, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to: So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density being analyzed is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevancy. The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing scalar components, where is the physical dimension of the space of interest). In 3D for example and the and vectors are explicitly and . Flow velocity and pressure are the so-called physical variables.[1] These equations may be expressed in subscript notation: where the and subscripts label the N-dimensional space components. These equations may be more succinctly expressed using Einstein notation: 3 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... where the and subscripts label the N-dimensional space components, and ; is the Kroenecker delta. In 3D and the and vectors are explicitly and , and matched indices imply a sum over those indices and and . Properties Although Euler first presented these equations in 1755, many fundamental questions about them remain unanswered. In three space dimensions it is not even known whether solutions of the equations are defined for all time or if they form singularities.[6] Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: In the one dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers equation: This is a model equation giving many insights on Euler equations. Nondimensionalisation In order to make the equations dimensionless, a characteristic length , and a characteristic velocity , need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained: and of the field unit vector: Substitution of these inversed relations in Euler equations, defining the Froude number, yields 4 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... (omitting the * at apix): Incompressible Euler equations with constant and uniform density (nondimensional form) Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory. Conservation form The conservation form emphasizes the mathematical properties of Euler equations,
Load more

Recommended publications
  • Che 253M Experiment No. 2 COMPRESSIBLE GAS FLOW
    Rev. 8/15 AD/GW ChE 253M Experiment No. 2 COMPRESSIBLE GAS FLOW The objective of this experiment is to familiarize the student with methods for measurement of compressible gas flow and to study gas flow under subsonic and supersonic flow conditions. The experiment is divided into three distinct parts: (1) Calibration and determination of the critical pressure ratio for a critical flow nozzle under supersonic flow conditions (2) Calculation of the discharge coefficient and Reynolds number for an orifice under subsonic (non- choked) flow conditions and (3) Determination of the orifice constants and mass discharge from a pressurized tank in a dynamic bleed down experiment under (choked) flow conditions. The experimental set up consists of a 100 psig air source branched into two manifolds: the first used for parts (1) and (2) and the second for part (3). The first manifold contains a critical flow nozzle, a NIST-calibrated in-line digital mass flow meter, and an orifice meter, all connected in series with copper piping. The second manifold contains a strain-gauge pressure transducer and a stainless steel tank, which can be pressurized and subsequently bled via a number of attached orifices. A number of NIST-calibrated digital hand held manometers are also used for measuring pressure in all 3 parts of this experiment. Assorted pressure regulators, manual valves, and pressure gauges are present on both manifolds and you are expected to familiarize yourself with the process flow, and know how to operate them to carry out the experiment. A process flow diagram plus handouts outlining the theory of operation of these devices are attached.
    [Show full text]
  • Laws of Similarity in Fluid Mechanics 21
    Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimen- sional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists.
    [Show full text]
  • Chapter 4: Immersed Body Flow [Pp
    MECH 3492 Fluid Mechanics and Applications Univ. of Manitoba Fall Term, 2017 Chapter 4: Immersed Body Flow [pp. 445-459 (8e), or 374-386 (9e)] Dr. Bing-Chen Wang Dept. of Mechanical Engineering Univ. of Manitoba, Winnipeg, MB, R3T 5V6 When a viscous fluid flow passes a solid body (fully-immersed in the fluid), the body experiences a net force, F, which can be decomposed into two components: a drag force F , which is parallel to the flow direction, and • D a lift force F , which is perpendicular to the flow direction. • L The drag coefficient CD and lift coefficient CL are defined as follows: FD FL CD = 1 2 and CL = 1 2 , (112) 2 ρU A 2 ρU Ap respectively. Here, U is the free-stream velocity, A is the “wetted area” (total surface area in contact with fluid), and Ap is the “planform area” (maximum projected area of an object such as a wing). In the remainder of this section, we focus our attention on the drag forces. As discussed previously, there are two types of drag forces acting on a solid body immersed in a viscous flow: friction drag (also called “viscous drag”), due to the wall friction shear stress exerted on the • surface of a solid body; pressure drag (also called “form drag”), due to the difference in the pressure exerted on the front • and rear surfaces of a solid body. The friction drag and pressure drag on a finite immersed body are defined as FD,vis = τwdA and FD, pres = pdA , (113) ZA ZA Streamwise component respectively.
    [Show full text]
  • Aerodynamics Material - Taylor & Francis
    CopyrightAerodynamics material - Taylor & Francis ______________________________________________________________________ 257 Aerodynamics Symbol List Symbol Definition Units a speed of sound ⁄ a speed of sound at sea level ⁄ A area aspect ratio ‐‐‐‐‐‐‐‐ b wing span c chord length c Copyrightmean aerodynamic material chord- Taylor & Francis specific heat at constant pressure of air · root chord tip chord specific heat at constant volume of air · / quarter chord total drag coefficient ‐‐‐‐‐‐‐‐ , induced drag coefficient ‐‐‐‐‐‐‐‐ , parasite drag coefficient ‐‐‐‐‐‐‐‐ , wave drag coefficient ‐‐‐‐‐‐‐‐ local skin friction coefficient ‐‐‐‐‐‐‐‐ lift coefficient ‐‐‐‐‐‐‐‐ , compressible lift coefficient ‐‐‐‐‐‐‐‐ compressible moment ‐‐‐‐‐‐‐‐ , coefficient , pitching moment coefficient ‐‐‐‐‐‐‐‐ , rolling moment coefficient ‐‐‐‐‐‐‐‐ , yawing moment coefficient ‐‐‐‐‐‐‐‐ ______________________________________________________________________ 258 Aerodynamics Aerodynamics Symbol List (cont.) Symbol Definition Units pressure coefficient ‐‐‐‐‐‐‐‐ compressible pressure ‐‐‐‐‐‐‐‐ , coefficient , critical pressure coefficient ‐‐‐‐‐‐‐‐ , supersonic pressure coefficient ‐‐‐‐‐‐‐‐ D total drag induced drag Copyright material - Taylor & Francis parasite drag e span efficiency factor ‐‐‐‐‐‐‐‐ L lift pitching moment · rolling moment · yawing moment · M mach number ‐‐‐‐‐‐‐‐ critical mach number ‐‐‐‐‐‐‐‐ free stream mach number ‐‐‐‐‐‐‐‐ P static pressure ⁄ total pressure ⁄ free stream pressure ⁄ q dynamic pressure ⁄ R
    [Show full text]
  • A Semi-Hydrostatic Theory of Gravity-Dominated Compressible Flow
    Generated using version 3.2 of the official AMS LATEX template 1 A semi-hydrostatic theory of gravity-dominated compressible flow 2 Thomas Dubos ∗ IPSL-Laboratoire de Météorologie Dynamique, Ecole Polytechnique, Palaiseau, France Fabrice voitus CNRM-Groupe d’étude de l’Atmosphere Metéorologique, Météo-France, Toulouse, France ∗Corresponding author address: Thomas Dubos, LMD, École Polytechnique, 91128 Palaiseau, France. E-mail: [email protected] 1 3 Abstract 4 From Hamilton’s least action principle, compressible equations of motion with density diag- 5 nosed from potential temperature through hydrostatic balance are derived. Slaving density 6 to potential temperature suppresses the degrees of freedom supporting the propagation of 7 acoustic waves and results in a sound-proof system. The linear normal modes and dispersion 8 relationship for an isothermal state of rest on f- and β- planes are accurate from hydrostatic 9 to non-hydrostatic scales, except for deep internal gravity waves. Especially the Lamb wave 10 and long Rossby waves are not distorted, unlike with anelastic or pseudo-incompressible 11 systems. 12 Compared to similar equations derived by Arakawa and Konor (2009), the semi-hydrostatic 13 system derived here possesses an additional term in the horizontal momentum budget. This 14 term is an apparent force resulting from the vertical coordinate not being the actual height 15 of an air parcel, but its hydrostatic height, i.e. the hypothetical height it would have after 16 the atmospheric column it belongs to has reached hydrostatic balance through adiabatic 17 vertical displacements of air parcels. The Lagrange multiplier λ introduced in Hamilton’s 18 principle to slave density to potential temperature is identified as the non-hydrostatic ver- 19 tical displacement, i.e.
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • Literature Review
    SECTION VII GLOSSARY OF TERMS SECTION VII: TABLE OF CONTENTS 7. GLOSSARY OF TERMS..................................................................... VII - 1 7.1 CFD Glossary .......................................................................................... VII - 1 7.2. Particle Tracking Glossary.................................................................... VII - 3 Section VII – Glossary of Terms Page VII - 1 7. GLOSSARY OF TERMS 7.1 CFD Glossary Advection: The process by which a quantity of fluid is transferred from one point to another due to the movement of the fluid. Boundary condition(s): either: A set of conditions that define the physical problem. or: A plane at which a known solution is applied to the governing equations. Boundary layer: A very narrow region next to a solid object in a moving fluid, and containing high gradients in velocity. CFD: Computational Fluid Dynamics. The study of the behavior of fluids using computers to solve the equations that govern fluid flow. Clustering: Increasing the number of grid points in a region to better resolve a geometric or flow feature. Increasing the local grid resolution. Continuum: Having properties that vary continuously with position. The air in a room can be thought of as a continuum because any cube of air will behave much like any other chosen cube of air. Convection: A similar term to Advection but is a more generic description of the Advection process. Convergence: Convergence is achieved when the imbalances in the governing equations fall below an acceptably low level during the solution process. Diffusion: The process by which a quantity spreads from one point to another due to the existence of a gradient in that variable. Diffusion, molecular: The spreading of a quantity due to molecular interactions within the fluid.
    [Show full text]
  • Introduction to Compressible Computational Fluid Dynamics James S
    Introduction to Compressible Computational Fluid Dynamics James S. Sochacki Department of Mathematics James Madison University [email protected] Abstract This document is intended as an introduction to computational fluid dynamics at the upper undergraduate level. It is assumed that the student has had courses through three dimensional calculus and some computer programming experience with numer- ical algorithms. A course in differential equations is recommended. This document is intended to be used by undergraduate instructors and students to gain an under- standing of computational fluid dynamics. The document can be used in a classroom or research environment at the undergraduate level. The idea of this work is to have the students use the modules to discover properties of the equations and then relate this to the physics of fluid dynamics. Many issues, such as rarefactions and shocks are left out of the discussion because the intent is to have the students discover these concepts and then study them with the instructor. The document is used in part of the undergraduate MATH 365 - Computation Fluid Dynamics course at James Madi- son University (JMU) and is part of the joint NSF Grant between JMU and North Carolina Central University (NCCU): A Collaborative Computational Sciences Pro- gram. This document introduces the full three-dimensional Navier Stokes equations. As- sumptions to these equations are made to derive equations that are accessible to un- dergraduates with the above prerequisites. These equations are approximated using finite difference methods. The development of the equations and finite difference methods are contained in this document. Software modules and their corresponding documentation in Fortran 90, Maple and Matlab can be downloaded from the web- site: http://www.math.jmu.edu/~jim/compressible.html.
    [Show full text]
  • Chapter 4: Immersed Body Flow [Pp
    MECH 3492 Fluid Mechanics and Applications Univ. of Manitoba Fall Term, 2017 Chapter 4: Immersed Body Flow [pp. 445-459 (8e), or 374-386 (9e)] Dr. Bing-Chen Wang Dept. of Mechanical Engineering Univ. of Manitoba, Winnipeg, MB, R3T 5V6 When a viscous fluid flow passes a solid body (fully-immersed in the fluid), the body experiences a net force, F, which can be decomposed into two components: a drag force F , which is parallel to the flow direction, and • D a lift force F , which is perpendicular to the flow direction. • L The drag coefficient CD and lift coefficient CL are defined as follows: FD FL CD = 1 2 and CL = 1 2 , (112) 2 ρU A 2 ρU Ap respectively. Here, U is the free-stream velocity, A is the “wetted area” (total surface area in contact with fluid), and Ap is the “planform area” (maximum projected area of an object such as a wing). In the remainder of this section, we focus our attention on the drag forces. As discussed previously, there are two types of drag forces acting on a solid body immersed in a viscous flow: friction drag (also called “viscous drag”), due to the wall friction shear stress exerted on the • surface of a solid body; pressure drag (also called “form drag”), due to the difference in the pressure exerted on the front • and rear surfaces of a solid body. The friction drag and pressure drag on a finite immersed body are defined as FD,vis = τwdA and FD, pres = pdA , (113) ZA ZA Streamwise component respectively.
    [Show full text]
  • Part III: the Viscous Flow
    Why does an airfoil drag: the viscous problem – André Deperrois – March 2019 Rev. 1.1 © Navier-Stokes equations Inviscid fluid Time averaged turbulence CFD « RANS » Reynolds Averaged Euler’s equations Reynolds equations Navier-stokes solvers irrotational flow Viscosity models, uniform 3d Boundary Layer eq. pressure in BL thickness, Prandlt Potential flow mixing length hypothesis. 2d BL equations Time independent, incompressible flow Laplace’s equation 1d BL Integral 2d BL differential equations equations 2d mixed empirical + theoretical 2d, 3d turbulence and transition models 2d viscous results interpolation The inviscid flow around an airfoil Favourable pressure gradient, the flo! a""elerates ro# $ero at the leading edge%s stagnation point& Adverse pressure gradient, the low decelerates way from the surface, the flow free tends asymptotically towards the stream air freestream uniform flow flow inviscid ◀—▶ “laminar”, The boundary layer way from the surface, the fluid’s velocity tends !ue to viscosity, the asymptotically towards the tangential velocity at the velocity field of an ideal inviscid contact of the foil is " free flow around an airfoil$ stream air flow (magnified scale) The boundary layer is defined as the flow between the foil’s surface and the thic%ness where the fluid#s velocity reaches &&' or &&$(' of the inviscid flow’s velocity. The viscous flow around an airfoil at low Reynolds number Favourable pressure gradient, the low a""elerates ro# $ero at the leading edge%s stagnation point& Adverse pressure gradient, the low decelerates +n adverse pressure gradients, the laminar separation bubble forms. The flow goes flow separates. The velocity close to the progressively turbulent inside the bubble$ surface goes negative.
    [Show full text]
  • Compressible Flow
    94 c 2004 Faith A. Morrison, all rights reserved. Compressible Fluids Faith A. Morrison Associate Professor of Chemical Engineering Michigan Technological University November 4, 2004 Chemical engineering is mostly concerned with incompressible flows in pipes, reactors, mixers, and other process equipment. Gases may be modeled as incompressible fluids in both microscopic and macroscopic calculations as long as the pressure changes are less than about 20% of the mean pressure (Geankoplis, Denn). The friction-factor/Reynolds-number correlation for incompressible fluids is found to apply to compressible fluids in this regime of pressure variation (Perry and Chilton, Denn). Compressible flow is important in selected application, however, including high-speed flow of gasses in pipes, through nozzles, in tur- bines, and especially in relief valves. We include here a brief discussion of issues related to the flow of compressible fluids; for more information the reader is encouraged to consult the literature. A compressible fluid is one in which the fluid density changes when it is subjected to high pressure-gradients. For gasses, changes in density are accompanied by changes in temperature, and this complicates considerably the analysis of compressible flow. The key difference between compressible and incompressible flow is the way that forces are transmitted through the fluid. Consider the flow of water in a straw. When a thirsty child applies suction to one end of a straw submerged in water, the water moves - both the water close to her mouth moves and the water at the far end moves towards the lower- pressure area created in the mouth. Likewise, in a long, completely filled piping system, if a pump is turned on at one end, the water will immediately begin to flow out of the other end of the pipe.
    [Show full text]
  • Incompressible Irrotational Flow
    Incompressible irrotational flow Enrique Ortega [email protected] Rotation of a fluid element As seen in M1_2, the arbitrary motion of a fluid element can be decomposed into • A translation or displacement due to the velocity. • A deformation (due to extensional and shear strains) mainly related to viscous and compressibility effects. • A rotation (of solid body type) measured through the midpoint of the diagonal of the fluid element. The rate of rotation is defined as the angular velocity. The latter is related to the vorticity of the flow through: d 2 V (1) dt Important: for an incompressible, inviscid flow, the momentum equations show that the vorticity for each fluid element remains constant (see pp. 17 of M1_3). Note that w is positive in the 2 – Irrotational flow counterclockwise sense Irrotational and rotational flow ij 0 According to the Prandtl’s boundary layer concept, thedomaininatypical(high-Re) aerodynamic problem at low can be divided into outer and inner flow regions under the following considerations: Extracted from [1]. • In the outer region (away from the body) the flow is considered inviscid and irrotational (viscous contributions vanish in the momentum equations and =0 due to farfield vorticity conservation). • In the inner region the viscous effects are confined to a very thin layer close to the body (vorticity is created at the boundary layer by viscous stresses) and a thin wake extending downstream (vorticity must be convected with the flow). Under these hypotheses, it is assumed that the disturbance of the outer flow, caused by the body and the thin boundary layer around it, is about the same caused by the body alone.
    [Show full text]