Basics of fluid flow Types of flow Fluid Ideal/Real Compressible/Incompressible Flow Steady/Unsteady Uniform/Non-uniform Laminar/Turbulent Pressure/Gravity (free surface) 1
Basics of fluid flow (Chapter 4) Basics of fluid flow, kinematics Statics Mechanics Kinematics Dynamics Kinetics Kinematics: deals with motion apart from considerations of mass, force or energy 2
Basics of fluid flow Path lines, streamlines and streak lines Path line: the trajectory that a fluid particle would make as it moves around with the flow Streamline: line that shows the flow direction, local velocity vector is tangent to the streamline at every point along the line at that instant 3
Basics of fluid flow Types of flow Steady flow: all fluid/flow properties at any point in the flow do not change with time; however, conditions may be different at different points. For steady flows: Uniform flow: at every point in the flow, the velocity (in both magnitude and direction) is identical at any given instant. 4
Basics of fluid flow One -, two -, and three- dimensional flows This is the most general 3-D flow: The flow is classified as 2-D if: The flow can be viewed as 1-D if: 5
Partial derivative Differentiating a function of more than one variable with respect to a particular variable, with the other variables kept constant: the notation f/ t means the partial derivative of the function f with respect to t f/ t : partial derivative df/dt : total derivative For more info: http://apollo.lsc.vsc.edu/classes/met380/fingerhuts_notes/driv.pdf 6
Basics of fluid flow Velocity and Acceleration, 4.12 a t a n Convective (spatial) acceleration Local (temporal) acceleration 7
Basics of fluid flow Flow rate and Mean velocity Flow rate: the rate at which fluid crosses a known surface volume flow rate mass flow rate The volume flow rate passing through the element of area da (in yz plane) is dq = u(cosθ)da=uda volume flow rate is equal to the magnitude of the mean velocity multiplied by the flow area at right angles to the direction of the mean velocity 8
Basics of fluid flow Flow rate and Mean velocity The volume flow rate passing through the element of area da is dq = u da =uda the local time mean velocity, u, will vary across the section for real fluid Q uda A AV m uda AV Q 9 A
Basics of fluid flow Reynolds Transport Theorem & Continuity AV 1 1 A2V 2 Q 10
FIGURE 5-24 Copyright The McGraw-Hill Companies, Inc. Bernoulli s Equation (Energy per unit weight)
Energy in Steady Flow (Chapter 5) Energies of a Flowing Fluid (Euler s Equation) Kinetic Energy 1/2mV 2 V 2 /2g Unit: L Potential Energy Wz z (Energy per unit weight) Pressure Head p = γh p/γ 12
Derivation of the Bernoulli Equation Steady flow: The forces acting on a fluid particle along a streamline. Steady, incompressible flow: Bernoulli equation The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible. The Bernoulli equation between any two points on the same streamline: 13
Energy in Steady Flow (Chapter 5) Bernoulli s Equation Unit: L (Energy per unit weight) Piezometric pressure Basic assumptions: Inviscid & incompressible fluid Steady flow Applies along a streamline No energy added or removed from the fluid along the streamline 14
FIGURE 5-22 Copyright The McGraw-Hill Companies, Inc.
Energy in Steady Flow, Pipe Flow V p/γ V 2 /2g p/γ Pitot Tube (Measures stagnation pressure) Free stream dynamic pressure Free stream static pressure Bring moving water to a halt, and it'll drive a column of water up to exactly the height from which water would flow to gain that velocity. 16
Energy in Steady Flow, Free surface flow Pitot Tube (Measures stagnation pressure) V V 2 /2g Bring moving water to a halt, and it'll drive a column of water up to exactly the height from which water would flow to gain that velocity. 17
Example: Bernoulli s principle, Pitot Tube http://www.youtube.com/watch?v=dk39ffdwq_e 18
Example: Water Discharge from a Large Tank Example: Spraying Water into the Air 19
Hydraulic grade line (HGL), P/ g + z The line that represents the sum of the static pressure and the elevation heads. Energy grade line (EGL), P/ g + V 2 /2g + z The line that represents the total head of the fluid. Dynamic head, V 2 /2g The difference between the heights of EGL and HGL. The hydraulic grade line (HGL) and the energy grade line (EGL) for free discharge from a reservoir through a horizontal pipe with a diffuser. 20
Energy in Steady Flow Stagnation pressure, ideal fluid (5.4) V 1 2 V 1 = V, p 2 is the stagnation pressure 21
Energy in Steady Flow General Energy Equation, steady flow, incompressible fluid For an incompressible fluid with γ = const. and α =1: 22
Energy in Steady Flow General Energy Equation, steady flow, incompressible fluid Real fluid For an incompressible fluid with γ = const. and α =1: If there is no machine between points 1 and 2: If head loss is neglected: Ideal fluid 23
Energy in Steady Flow Power considerations in fluid flow, Derivation of Power Equation Power: P = (Force) x (Velocity) Power: P = Energy / Time P = FV (F =ΔpA) P = (Energy/Weight) x (Weight/Time) P = (ΔpA)V (Δp = γh) P = (γha)v (Q = AV) head (h) γq P = γhq P = h γq P = ΔpQ 24
Energy in Steady Flow Power considerations in fluid flow, Units of Power P = γhq http://www.waterencyclopedia.com/po-re/pumps-traditional.html Power in BG units Horsepower = P = γhq/550 ( [Q] = cfs, [h] = ft, [γ] = pcf ) Power in SI units Kilowatts = P = γhq/1000 ( [Q] = m 3 /s, [h] = m, [γ] = N/m 3 ) P = γhq P: power put into flow by a pump, then h = h pump P: power lost because of friction, then h = h L Pump efficiency, η = (power output) / (power input) 25