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Unformatted text preview: 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 1 Chapter 3 Bernoulli Equation
3.1 Flow Patterns: Streamlines, Pathlines, Streaklines
1) A streamline , is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamlines around an airfoil (left) and a car (right) 2) A pathline is the actual path traveled by a given fluid particle. An illustration of pathline (left) and an example of pathlines, motion of water induced by surface waves (right) 3) A streakline is the locus of particles which have earlier passed through a particular point. An illustration of streakline (left) and an example of streaklines, flow past a fullsized streamlined vehicle in the GM aerodynamics laboratory wind tunnel, and 18ft by 34ft test section facilility by a 4000hp, 43ftdiameter fan (right) 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 2 Note: 1. For steady flow, all 3 coincide. 2. For unsteady flow, pattern changes with time, whereas pathlines and streaklines are generated as the passage of time Streamline: 0 which upon expansion yields the By definition we must have equation of the streamlines for a given time ̂ ̂ ̂ ̂ 0 , , where = integration parameter. So, if ( , , ) are known, integration with rewith I.C. ( , , , ) at 0, and then eliminating spect to starting at provides streamlines. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 3 Pathline: The path line is defined by integration of the relationship between velocity and displacement. Integrate , , Streakline: with respect to using I.C. ( , , , ) then eliminate . To find the streakline, use the integrated result for the pathline retaining time as a parameter. Now, find the integration constant which causes the path. Then eliminate . line to pass through ( , , ) for a sequence of time 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 4 3.2 Streamline Coordinates
Equations of fluid mechanics can be expressed in different coordinate systems, which are chosen for convenience, e.g., application of boundary conditions: Cartesian ( , , ) or orthogonal curvilinear (e.g., , , ) or nonorthogonal curvilinear. A natural coordinate system is streamline coordinates ( , , ℓ); however, difficult to use since solution to flow problem (V) must be known to solve for steamlines. For streamline coordinates, since V is tangent to there is only one velocity component. V where 0 by definition. , , , Figure 4.8 Streamline coordinate system for twodimensional flow. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 5 The acceleration is V where, V V V V V For V , V V Thus, (1) V 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 6 Figure 4.9 Relationship between the unit vector along the streamline, , and the radius of curvature of the streamline, Space increment: Note: 1 Time increment: 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 7 From (1), θ
⁄ or with , where, = local = local in ̂ direction in direction due to spatial gradient of V = convective i.e. convergence /divergence = convective due to curvature of : inward centrifugal acceleration, i.e. towards center of curvature. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 8 3.3 Bernoulli Equation
Consider the small fluid particle of size by in the plane of the figure and normal to the figure as shown in the freebody diagram below. For steady flow, the components of Newton’s second law along the streamline and normal directions can be written as following: 1) Along a streamline ∑ where, V V sin 2 V Thus, V V V sin
1st order Taylor Series 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 9 sin sin → change in speed due to and (i.e. along ) 2) Normal to a streamline ∑ where, V V cos 2 V
1st order Taylor Series
2 Thus, V
2 2 V cos V cos
cos → streamline curvature is due to and (i.e. along ) 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 10 In a vector form: (Euler equation) or Steady flow, = constant, equation 0 2 Steady flow, = constant, equation For curved streamlines (= constant for static fluid) decreases in the rection, i.e. towards the local center of curvature. di It should be emphasized that the Bernoulli equation is restricted to the following: • • • • inviscid flow steady flow incompressible flow flow along a streamline Note that if in addition to the flow being inviscid it is also irrotational, i.e. V = 0, the Bernoulli constant is same for all , rotation of fluid = = vorticity = as will be shown later. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 11 3.4 Physical interpretation of Bernoulli equation
Integration of the equation of motion to give the Bernoulli equation actually corresponds to the workenergy principle often used in the study of dynamics. This principle results from a general integration of the equations of motion for an object in a very similar to that done for the fluid particle. With certain assumptions, a statement of the workenergy principle may be written as follows: The work done on a particle by all forces acting on the particle is equal to the change of the kinetic energy of the particle. The Bernoulli equation is a mathematical statement of this principle. In fact, an alternate method of deriving the Bernoulli equation is to use the first and second laws of thermodynamics (the energy and entropy equations), rather than Newton’s second law. With the approach restrictions, the general energy equation reduces to the Bernoulli equation. An alternate but equivalent form of the Bernoulli equation is 2 along a streamline. Pressure head: Velocity head: Elevation head: The Bernoulli equation states that the sum of the pressure head, the velocity head, and the elevation head is constant along a streamline. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 12 3.5 Static, Stagnation, Dynamic, and Total Pressure
1 2 along a streamline. Static pressure: Dynamic pressure: Hydrostatic pressure: Stagnation points on bodies in flowing fluids. Stagnation pressure: (assuming elevation effects are negligible) where and are the pressure and velocity of the fluid upstream of stagnation point. At stagnation point, fluid velocity becomes zero and all of the kinetic energy converts into a pressure rize. Total pressure: At stagnation point = 0, (along a streamline) ∴ Stagnation pressure > static pressure by upstream . 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 13 The Pitotstatic tube (left) and typical Pitotstatic tube designs (right). Typical pressure distribution along a Pitotstatic tube. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 14 3.6 Applications of Bernoulli Equation
1) Stagnation Tube 2 2 2 2 , 0 2 Limited by length of tube and need for free surface reference 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 15 2) Pitot Tube Note: Δz1 = elevation difference between 1 and 3 Δz2 = elevation difference between 2 and 4 i.e., p1 = p3 for z3 – z1 small and p2 = p4 for z4z2 small 2 2 2 where, 0 and = piezometric head 2 from manometer or pressure gage For gas flow Δ ⁄ Δ
Note: 2Δ In general, using Δz = z1 – z2 small = 0 for pressure transducer. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 16 3) Simplified form of the continuity equation Steady flow into and out of a tank Obtained from the following intuitive arguments: Volume flow rate: Mass flow rate: Conservation of mass requires
Note: 0 For incompressible flow , we have for system. RTT: or 0 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 17 4) Volume Rate of Flow (flowrate, discharge) 1. Crosssectional area oriented normal to velocity vector (simple case where = constant: constant: = volume flux = [m/s × m2 = m3/s] Similarly the mass flux = 2. General case V V cos V Average velocity: 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 18 Example: At low velocities the flow through a long circular tube, i.e. pipe, has a parabolic velocity distribution (actually paraboloid of revolution). 1 where, = centerline velocity a) find and V 2 where, 2 , 2 and not , 1 2 1 2 2 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 19 5) Flowrate measurement Various flow meters are governed by the Bernoulli and continuity equations. Typical devices for measuring flowrate in pipes. Three commonly used types of flow meters are illustrated: the orifice meter, the nozzle meter, and the Venturi meter. The operation of each is based on the same physical principles—an increase in velocity causes a decrease in pressure. The difference between them is a matter of cost, accuracy, and how closely their actual operation obeys the idealized flow assumptions. ), steady, inviscid, and incomWe assume the flow is horizontal ( pressible between points (1) and (2). The Bernoulli equation becomes: 1 2 1 2 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 20 1 2 2Δ 1 1 / 1 If we assume the velocity profiles are uniform at sections (1) and (2), the continuity equation can be written as: is the small ( ) flow area at section (2). Combination of these where two equations results in the following theoretical flowrate 2 1 ⁄ assumed vena contracta = 0, i.e., no viscous effects. Otherwise, 2 1 ⁄ where = contraction coefficient 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 21 Vertical flow from a tank Application of Bernoulli equation between points (1) and (2) on the streamline shown gives 1 2 Since , 0, 0, 0, 1 2 2 2 1 2 0, we have Bernoulli equation between points (1) and (5) gives 2 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 22 A smooth, wellcontoured nozzle (left) and a sharp corner (right) The velocity profile of the left nozzle is not uniform due to differences in elevation, but in general and we can safely use the centerline velocity, , as a reasonable “average velocity.” For the right nozzle with a sharp corner, will be less than . This phenomenon, called a vena contracta effect, is a result of the inability of the fluid to turn the sharp 90° corner. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 23 Figure 3.14 Typical flow patterns and contraction coefficients The vena contracta effect is a function of the geometry of the outlet. Some typical configurations are shown in Fig. 3.14 along with typical values of the expe⁄ , where and are the rimentally obtained contraction coefficient, areas of the jet a the vena contracta and the area of the hole, respectively. 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 24 Other flow meters based on the Bernoulli equation are used to measure flowrates in open channels such as flumes and irrigation ditches. Two of these devices, the sluice gate and the sharpcrested weir, are discussed below under the assumption of steady, inviscid, incompressible flow. Sluice gate geometry We apply the Bernoulli and continuity equations between points on the free surfaces at (1) and (2) to give: 1 2 and 1 2 With the fact that 2 1 In the limit of , then 0: ⁄
⁄ 2 : 2 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 25 Rectangular, sharpcrested weir geometry For such devices the flowrate of liquid over the top of the weir plate is dependent on the weir height, , the width of the channel, , and the head, , of the water above the top of the weir. Between points (1) and (2) the pressure and gravitational fields cause the fluid to accelerate from velocity to velocity . At , while at (2) the pressure is essentially atmospheric, (1) the pressure is 0. Across the curved streamlines directly above the top of the weir plate (section a–a), the pressure changes from atmospheric on the top surface to some maximum value within the fluid stream and then to atmospheric again at the bottom surface. For now, we will take a very simple approach and assume that the weir flow is similar in many respects to an orificetype flow with a free streamline. In this instance we would expect the average velocity across the top of the weir to be and the flow area for this rectangular weir to be proporproportional to 2 tional to . Hence, it follows that 2 i.e., 2 2 = proportionality coefficient 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 26 3.7 Energy grade line (EGL) and hydraulic grade line (HGL)
This part will be covered later at Chapter 5. 3.8 Limitations of Bernoulli Equation
Assumptions used in the derivation Bernoulli Equation: (1) Inviscid (2) Incompressible (3) Steady (4) Conservative body force 1) Compressibility Effects: The Bernoulli equation can be modified for compressible flows. A simple, although specialized, case of compressible flow occurs when the temperature of a perfect gas remains constant along the streamline—isothermal flow. Thus, we consider , where is constant (In general, , , and will vary). An equation similar to the Bernoulli equation can be obtained for isentropic flow of a perfect gas. For steady, inviscid, isothermal flow, Bernoulli equation becomes 1 2 The constant of integration is easily evaluated if location on the streamline. The result is 2 ln 2 , , and are known at some 57:020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2009 Chapter 3 27 2) Unsteady Effects: The Bernoulli equation can be modified for unsteady flows. With the inclusion of the unsteady effect ( ⁄ 0) the following is obtained: 0 (along a streamline) For incompressible flow this can be easily integrated between points (1) and (2) to give (along a streamline) 3) Rotational Effects Care must be used in applying the Bernoulli equation across streamlines. If the flow is “irrotational” (i.e., the fluid particles do not “spin” as they move), it is appropriate to use the Bernoulli equation across streamlines. However, if the flow is “rotational” (fluid particles “spin”), use of the Bernoulli equation is restricted to flow along a streamline. 4) Other Restrictions Another restriction on the Bernoulli equation is that the flow is inviscid. The Bernoulli equation is actually a first integral of Newton's second law along a streamline. This general integration was possible because, in the absence of viscous effects, the fluid system considered was a conservative system. The total energy of the system remains constant. If viscous effects are important the system is nonconservative and energy losses occur. A more detailed analysis is needed for these cases. The Bernoulli equation is not valid for flows that involve pumps or turbines. The final basic restriction on use of the Bernoulli equation is that there are no mechanical devices (pumps or turbines) in the system between the two points along the streamline for which the equation is applied. These devices represent sources or sinks of energy. Since the Bernoulli equation is actually one form of the energy equation, it must be altered to include pumps or turbines, if these are present. ...
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This note was uploaded on 02/15/2010 for the course MECHANICS 1 taught by Professor Fredstern during the Spring '06 term at 東京国際大学.
 Spring '06
 FredStern
 mechanics

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