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工程流体力学(英文版)第三章.pdf.pdf

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工程 流体力学 英文 第三 PDF
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Chapter 3 Concepts and Basic Equation of ?Fluids in motion ( one ?dimension, ideal fluids) ?? ?? ?? ?? ???????????????????????????? ???????????????????????????? ???????????????????????????? ???????????????????????????? ?? ?? ?? ??Contents 1. Methods to Study Fluids in Motion 2. Flow Classification 3. Pathline ( ?? ?? ?? ?? ?? ?? ?? ?? ) and Streamline ( ???? ???? ???? ???? ) 4. Streamtube ( ???? ???? ???? ???? ) and Discharge ( ???? ???? ???? ???? ) 5. Continuity Equation for Steady Flow in a ?Conduit ? 6. Motion Differential Equation for Ideal 1-D ?Flow ?water flows around the ?ship. Three-dimensional Flow:A pathline is the trace after a single particle ?travels in a field of flow over a period of time. (1).Definition Kinescope1 Kinescope2 ?? ?? ?? ?? 3.3 Pathline ( ???? ???? ???? ???? ) and Streamline ( ???? ???? ???? ???? ) 1. Pathline (2) ?? Equation of Pathline u ?? v ?? w are functions of both time t and space ?(x ?? y ?? z). Here t is an independent variable dy dx dz u v w dt = = =A Streamline is a curve that show the ?direction of a number of particles at the ?at the same instant of time. ?The curve indicates the velocity ?vectors of any points occupying on the ?streamline. Kinescope 2. Streamline 1. Definition a V ?? ? b V ?? ? c V ?? ? d V ?? ? e V ?? ?2 ?? ?? ?? ?? Equation of Streamline Select point A in streamline, ds is a differential arc length, u is the ?velocity at point A ds dxi dyj dzk = + + ? ? ? ? ds u A V ui vj wk = + + ? ? ? ? Directional cosine between velocity ?vector and coordinates cos( , ) v V y V = ? cos( , ) u V x V = ? cos( , ) w V z V = ? Directional cosine between ds and coordinates cos( , ) dx ds x ds = ? cos( , ) dy ds y ds = ? cos( , ) dz ds z ds = ?// ds V ? ? dy dx dz u v w = = ——Streamline equation So, velocity vector is tangent to streamline cos( , ) cos( , ) ds x V x = ? ? cos( , ) cos( , ) ds y V y = ? ? cos( , ) cos( , ) ds z V z = ? ? u dx V ds = v dy V ds = w dz V ds = u v w V dx dy dz ds = = =3 ?? ?? ?? ?? Character of Streamline b ?? At the same instant of time, streamlines ?can not intersect. c ?? Streamline can¢t be a folding line, ?but a smooth curve. ?U 2 L 1 L 2 U 1 d. In steady flow, streamlines ?and ?pathlines coincide. m t2 Stream line Path ?Line ?t1 m t1 m t3 ?t3 ?t2 (b)Unsteady ?Flow a ?? streamline:many fluid particles, one instant of time (pathline: a fluid particle, a period of time)1 u x = + v y = - Example ?? ?? Find the streamline equation of point ?? 1 ?? 2 ?? . Solution ?? d d x y u v = so ?? d d 1 x y x y = + - integrate it : ( ) ln 1 ln ln x y C + =- + or ?( ) 1 x y C + = the streamline equation of point ?? 1 ?? 2 ?? : ( ) 1 4 x y + = The velocity of a 2-d flow field is :?? ?? ?? ?? 3.4 Streamtube ( ???? ???? ???? ???? ) and Discharge ( ???? ???? ???? ???? ) Consider a closed curve (not streamline) in the flow field, ?then draw streamlines through every point on it, so as to ?form a tube-shaping space whose walls are streamlines. This ?tube is called the streamtube. Fluid ?fulling the stream tube is called the tube flow and ?the ?limit of a tube flow is a streamline. Stream tube and tube flow 1 ?? ?? ?? ?? Streamtube ?? ?? ?? ?? ( ???? ???? ???? ???? ) ?2 ?? ?? ?? ?? Tube Flow: ?( ???? ???? ???? ???? )The section is perpendicular to the direction of fluid ?flow (such as pipe flow and channel flow). ?1 1 2 2 Cross Section 4.1 ?Total Flow Analysis Method If the walls of the stream tube are extended to the flow ?field boundary, the fluid flow within the boundary is the ?Total Flow. 3 ?? ?? ?? ?? Total Flow ?? ?? ?? ?? ( ???? ???? ???? ???? ) 4 ?? ?? ?? ?? Cross Section ?? ?? ?? ?? ( ???????? ???????? ???????? ???????? )Amount of fluid pass through a cross section (such as the ?section in ?the channel or pipe) per unit time. Volume discharge ?? ?? ?? ?? m 3 /s ?? ?? ?? ?? Mass discharge ?? ?? ?? ?? kg/s ?? ?? ?? ?? Weight discharge ?? ?? ?? ?? N/s ?? ?? ?? ?? 5. Discharge ?? ?? ?? ?? ( ???? ???? ???? ???? ) A Q VdA = ? m A Q VdA = ? r g A Q gVdA = ? r6. ? Mean Velocity ( ???????? ???????? ???????? ???????? ) The velocities of points on the same cross section in ?the total flow are different, so usually an average ?velocity is used instead of the real velocity over the ?cross section, this average velocity is called the mean ?velocity. Q V A =1 ?? What are streamline and pathline ?? What are their differences ?? 2 ?? What are streamline and pathline ¢s ?characters ?? 3 ?? Is there any streamlines in real flow ?? What is purpose of ?introducing the ?concept of streamline ?? Questions Streamlines are curves that show the mean direction of a number of particles at ?the same instant of time; curves are tangent to the velocity vectors of any points ?occupying on the streamline. Pathline is the trace after a single particle travels in a field of flow over a period ?of time. a ?? At the same instant of time, streamlines can not intersect. b ?? Streamline can¢t be a folding line, it must be a smooth curve. c ?? Streamline cluster density reflects the amount of velocity. (Dense ?streamlines mean large velocity; while sparse streamlines mean small velocity.) No. It is convenient to analyze the ?fluid flow and define the trends of fluid ?flow.5 ?? What are the characters of steady flow ?? 6 ?? What are the research objects of Eulerian Description and Lagrangian Description ?of Motion ?? In engineering, which description ?of motion is utilized? Steady flow is the flow whose hydraulic motion factors don¢t change with time; In steady flow, streamline and pathline coincide, and local acceleration is equal to zero. 0 t A = ? ? ? The research object of Eulerian Description is flow field and the research ?object of Lagrangian Description ?of Motion is fluid particles; In engineering, ?Eulerian Description is used widely.?? ?? ?? ?? 3.5 Continuity Equation for Steady Flow ?in a Conduit ?(1-D) ? ?? ?? ?? ?? ???????????????????????? ???????????????????????? ???????????????????????? ???????????????????????? ) 1.All the parameters do not vary across the cross-sections 1-D flow: 1 ?? ?? ?? ?? 1 ?? ?? ?? ?? 2 ?? ?? ?? ?? 2 ?? ?? ?? ?? s ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? 2.Only the mean parameters of the cross-sections are ?considered. or:?? ?? ?? ?? 3.5 Continuity Equation for Steady Flow ?in a Conduit ?(1-D) ? ?? ?? ?? ?? ???????????????????????? ???????????????????????? ???????????????????????? ???????????????????????? ) Consider a control volume and the conditions below: ?? 1 ?? No fluid can leave or enter the control volume through the ?tube wall ?? ?? 2 ?? Fluid is a continuum, and there is no gap in the tube flow ?? ?? 3 ?? Ignore the possibility that the mass turns to energy. 1 ?? ?? ?? ?? 1 ?? ?? ?? ?? 2 ?? ?? ?? ?? 2 ?? ?? ?? ?? s ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ??1. Select a control volume 1122 in a flow field volume ?? W left surface A1 (cross section): A 1 , V 1 , ? ? ? ? r 1 right surface A2 (cross section): A 2 , V 2 , ? ? ? ? 1 V 2 V 1 ?? ?? ?? ?? 1 ?? ?? ?? ?? 2 ?? ?? ?? ?? 2 ?? ?? ?? ?? s ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? r 2 ?? ?? ?? ?? 3.5 ?Continuity Equation for Steady Flow ?in a Conduit ?(1-D) ? ( ???????????????????????? ???????????????????????? ???????????????????????? ???????????????????????? )Based on the law of conservation of mass, there is 1 ?? ?? ?? ?? 1 ?? ?? ?? ?? 2 ?? ?? ?? ?? 2 ?? ?? ?? ?? s ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? r ?? ?? V ?? ?? A ?? ?? ?? ?? ?? ?? ?? ?? ?? 1 2 1 1 1 2 2 2 A A ? V dA V dA d? t r r r ? - = ? ? ? ? left surface flow in (dt): 1 1 1 1 A V dA dt r ? ? right surface flow out (dt): 2 2 2 2 A V dA dt r ? ? Mass change (dt): ? dt d? t r ? ? ? ?Physical meaning ?? ?? ?? ?? The net mass discharge entering the control volume is equal to the mass increased in unit time due to the change in density. Fit for ?? ?? ?? ?? Steady flow, unsteady flow, compressible and incompressible fluid, ideal fluid and real fluid. 1 2 1 1 1 2 2 2 A A ? V dA V dA d? t r r r ? - = ? ? ? ? § Continuity equation in total flow (general form) 1 2 1 1 1 2 2 2 A A ? V dA V dA d? t r r r ? - = ? ? ? ? If density do not vary across the inlet and outlet areas:Fit for ?? ?? ?? ?? All steady flows within solid boundary, including compressible ?and incompressible fluid, ideal fluid and real fluid. 0 t r = ? ? § Continuity Equation in Steady Total Flow For steady flow: Thus: A VdA V Q A A ? = = 1 2 1 2 1 2 V A V A r r = 1 2 1 1 1 2 2 2 0 A A V dA V dA r r - = ? ? 1 1 2 2 1 2 V A V A r r = 1 2 1 1 1 2 2 2 A A ? VdA V dA d? t r r r ? - = ? ? ? ?§ Continuity Equation in Incompressible Total Flow Physical meaning ?? ?? ?? ?? For the incompressible fluid, mean velocity is inverse proportional ?to the cross-section area. ?\ 1 1 2 2 V A V A = or 1 2 Q Q = Fit for ?? ?? ?? ?? Incompressible fluid, including steady and unsteady flow, ideal ?and real fluid. For incompressible fluid: 1 2 1 1 2 2 0 A A V dA V dA - = ? ? Const r = 1 2 1 1 1 2 2 2 A A ? VdA V dA d? t r r r ? - = ? ? ? ?§ Continuity Equation in Branching Total Flow 1 1 Q 1 2 2 3 3 Q 2 Q 3 junction 1 1 1 2 2 2 3 3 3 V A V A V A = + r r r 1 1 2 2 3 3 V A V A V A = + Continuity equation for junction ?? ?? ?? ?? ( ) ( ) in out VA VA r r = ? ? ( ) ( ) in out VA VA = ? ?

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