Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
FT I 2015 Alda Simões INTERACTION WITH SURFACES Boundary Layer Development of a velocity distribution Theory of the boundary layer 2 Ludwig Prandtl • Wall: null velocity (no-slip) • Ludwig Prandtl (1904): for a fluid flowing, the velocity is disturbed by the solid surface only on a thin layer near the surface (boundary layer). • The boundary layer develops whenever a flow is disturbed by the presence of a surface. FT I Alda Simões 2 3 Development of a boundary layer FT I Alda Simões 3 Growth of a boundary layer (BL) 4 Magnified scale FT I Alda Simões 4 Turbulent boundary layer 5 FT I Alda Simões 5 Reynolds number for the boundary layer 6 • Thickness of the boundary layer increases from the leading edge along the flow direction. • Region close to the leading edge: laminar boundary layer • Reynolds: 𝑅𝑒𝐵𝐿 = 𝜌𝑣∞ 𝑥 𝜇 • After a certain length, the boundary layer becomes turbulent. Transition depends on: – – – – – Velocity of the flow Length of the solid surface Properties of the fluid Turbulence out of the BL Rouphness of the surface. FT I Alda Simões 6 Thickness of the BL 7 Where does the BL end (at what distance from the surface, δ)? Where v 0.99 v x l 1/ 2 x Laminar: 5 * 1/ 5 Turbulent: 0,37 * x Laminar-turbulent transition: ReBL = 3,5 x 105 - 3x 106 FT I Alda Simões 7 Boundary layer 8 Formation and growth of a viscous sub-layer inside the turbulent BL. Viscous sub-layer: velocity gradient dvx/dy is large: Viscous forces predominate. Newton’s law of viscosity applies. Turbulent region: turbulent transport of momentum predominates, by eddies. Local velocity becomes nearly constant inside the BL. Between the two, a buffer zone develops, in which the two transport mechanisms act. FT I Alda Simões 8 Development of a velocity profile in a pipe 9 The BL gowing in the entry section of a pipe determines the flow regime along the pipe. 9 FT I Alda Simões FT I 2015 Alda Simões FLOW EXTERNAL TO OBJECTS Outer flow 11 Airplanes, rockets, ships, submarines, cars, trucks, motorcycles, bridges, buildings, wind turbines, undersea cables, offshore platforms… FT I Alda Simões Outer flow 12 Separation operations: Sedimentation operations (solid in a gas or liquid), extraction of polutants, washing of gases (gas bubbles going through a liquid) FT I Alda Simões Outer flow 13 Flow around an immersed sphere Forces acting on an immersed particle Drag coefficient Stokes Law Terminal velocity of a sphere falling under the effect of gravity. FT I Alda Simões Boyant force 14 Forces acting at rest, Fs: Gravity and boyant force FI z Fg Buoyancy (Arquimedes, century III B.C.): A body immersed or floating in a fluid at rest is subjected to a force acting upwards, which is equal to the weight of the fluid that is displaced by the body. F FI - Fg fl - c Vg FT I Alda Simões 14 Falling shere 15 Dynamic force, Fk : force resulting from the relative velocity. FI z How does the velocity of a falling object vary? FT I Alda Simões 15 Sheres: Stokes’ Law 16 Laminar flow of viscous fluids around objects. Stokes’s law for flow around spheres, with low Re: 𝐹𝑘 = 3𝜋𝜇𝐷𝑣∞ Components: Tangential direction Normal direction FT I Alda Simões xz zz v z v x z x v z v z v z 2 z z z 16 17 http://faraday.fc.up.pt/Faraday/Recursos/multimedia/stokes_anim.gif FT I Alda Simões 17 Terminal velocity of a falling sphere 18 Sphere falling in a viscous fluid at rest, with uniform velocity: FI Fk FI Fk - Fg 0 4 3 4 3 R fl g 6 R v - R s g 0 3 3 gD s fl 2 v FT I Alda Simões 18 Fg • Viscometers • Sedimentation 18 Reynolds number immersed objects 19 The drag coefficient depends on the Reynolds number, defined for the object-fluid system: 𝑣∞ 𝜌𝐿 𝑅𝑒𝑃 = 𝜇 FT I Alda Simões 𝑣∞ : velocity of approach L: reference length , : properties of the fluid 19 Projected area 20 Sphere: 𝜋𝐷2 𝐴𝑃 = 4 Cylinder: Cross flow 𝐴𝑃 = 𝐿𝐷 Axial flow 𝜋𝐷2 𝐴𝑃 = 4 FT I Alda Simões 20 Drag Coefficient 21 Objects 𝐷𝑟𝑎𝑔 𝑓𝑜𝑟𝑐𝑒 𝐹𝑘 𝐶𝑤 = = 𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 × 𝐾 𝐴𝑝 𝐾 Cw: Drag coefficient, Re < 0,1: Creeping flow 24 𝐶𝑤 = 𝑅𝑒𝑝 FT I Alda Simões 21 Exemple 22 Determine the uniform velocity for a plastic sphere falling freely in a liquid at rest. D= 5 cm; (plastic) =1,1 g cm-3 Liquid: =0,98 g cm-3 ; viscosity= 90 cP? Assume creeping flow. R: 182 cm s-1 FT I Alda Simões 22 Drag coefficient vs. Friction factor 23 Objects 𝐷𝑟𝑎𝑔 𝑓𝑜𝑟𝑐𝑒 𝐶𝑤 = 𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 × 𝐾 𝐹𝑘 𝐶𝑤 = 𝐴𝑝 𝐾 Re < 0,1: creeping flow (spheres) 24 𝐶𝑤 = 𝑅𝑒𝑝 FT I Alda Simões Flow in ducts 𝑓= 𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙 𝑊𝑒𝑡 𝑎𝑟𝑒𝑎 × 𝐾 𝑓= 𝐹𝜏 𝐴𝑤𝑒𝑡 𝐾 Re < 2100, laminar flow 16 𝑓= 𝑅𝑒 23 Drag Coefficient diagram 24 Drag coefficient FT I Alda Simões 24 Drag coefficient values at high Re 25 CW,1 CW,2 FT I Alda Simões 25 Cw values 26 FT I Alda Simões 26 Example 27 A chimney 20 m high and with D= 2 m suffers the action of lateral wind with 80 km/h. Determine the lateral force that the chimney suffers. R: FT I Alda Simões 27 Flow pattern vs. Re 28 FT I Alda Simões 28 29 FT I Alda Simões 29 Fluxo em torno de objectos 30 FT I Alda Simões 30 Drag coefficient for sheres 31 Fk Fg FI 4 3 Fk R g s fl 3 4 3 R g s fl Fk Cw 3 1 Ap K 2 R fl v2 2 4 Dg s fl Cw 2 fl 3v FT I Alda Simões FI Fk Fg 31 Calculation of terminal velocity 32 In the absence of data for calculation of Re: • Iterative method • Graphical method Iterative method: • • • • • • FT I Alda Simões We assume a starting velocity v∞ Calculate Re Enter the diagrama and extract Cw Re-calculate v∞ Use v∞ in the subsequent iteration Stop the process when two identical readings are made in the diagram 32 Graphical method 33 Non-dimensional friction group: 3 gD fl 2 4 Cw Re esf fl 2 3 log Cw log Cw Re2 2 log Re Plot this correlation in the diagram Determine the interseption of the straight line with the Cw curve. FT I Alda Simões 33 Determination of diameter of a shere 34 Non-dimensional friction group: 3 gD fl 2 4 Cw Re esf fl 2 3 log Cw log Cw Re2 2 log Re Plot this correlation in the diagram Determine the interseption of the straight line with the Cw curve. FT I Alda Simões 34 35 A drop of water approximately sherical and with a diameter of 0,5 mm falls in air at rest. The distance made is sufficiently long for the velocity to reach uniform movement. What is the terminal velocity of the drop, assuming constant dimensions? . ρair= 1,21 kg/m3, μair = 0,0181 mPas. FT I Alda Simões 35