Question

Problem 1 (40%) A thin rectangular plate is 6 meters long and 0.8 meter wide. The plate is submerged (both sides) and held stationary in a stream of water (T = 10°C) that has a velocity of 0.5 meters/second. a) (5%) What is the boundary layer thickness at a distance x = 0.5m downstream of the leading edge? b) (5%) What is the shear stress on the plate at the same point? c) (10%) What is the friction drag on the plate? (the plate is oriented in the flow as shown in the first figure) d) (10%) The plate is now rotated 90 degrees, still parallel to the flow, but with side of length B along the flow. What is the friction drag in this case? e) (10%) The plate is now repositioned normal to the flow. What is the drag force acting on the plate? Uo = 0.5 m/s B = 0.8m Uo 0.5 m/s 1 L = 6m L = 6m Questions a-c Question d B = 0.8m

Answer 1

solution: V.-0.5 m/s B = 0.8m Lebon below (Taken from Properties of water at T = 10°C are given teatboole" Fluid Mechanics by Y. Cengel, etal". P= 999.7 kg/m3, M = 1.307 X103 kgims (a) Reynolds number Rex for water is given by Rex= Pux = 3.8244x1024 ggg. 7x0.5x2 1.3078103 5 = 3.8246x10x0.5 Rex at x = 0.5m S 1.9122 X10 Rexso.sm since is laminar. flow Boundary layer thickness for laminar flow is given by Rex < 5x105, s= 5 .: S SE 5x0.5 (1.9122410

S=0.00572m will It S =8 (5) shear stress on a plate at 2:0.5m Shear stress laminar for flow To= 0.3320 (4) Rex :: to 0.332x1.307X10° X x(US) / 1.91224105 To=0:1897 Pa

(c) Friction drag in the plate: Reynolds number at the end of the plate y Turbulent Laminar REL ggg.7x0.5x6 1:30 7X102 = 22.95 x105 ra 수 the plate. flow turns turbulent over the length of Let for be the position where Laminar boundary layer begins to transition to runsulent. 5x105 = 999.7*0.5x.xu = Xur= 1.307 m 1.307 X10-3 Hence needs to evaluate drag Wefficient Cof 0.45 1700 (legio Re,) 2.58 Reu 1700 0.455 Logvo (22.95 X105) 22.55x105 CD f = 0.00385 0.00074 :. Coti -0.00311 Total friction in the plate For = CDf a fe U²bL

= 3.059 X los fot = 0.00311* x999.7 * 10.5/80.8x6 Fof = 1.865 N (d) frictim drag When plate rotated goo. Effective Length of the plate is 0.8m. critical Length of plate is 1.307m. Hence, flow is Laminat at x=0.8m ggg.7x0.580.8 (Fof) Lam -Rex=0.8m 1-307x103 /Rex 0.666x 6x (0.5%*0.8 x 999.7 0.66485022 :.(For more 13.059 X 105 ...(Fof) lam = 1.440N

(1) Drag force normied to the plate 6 Plate 10.8 1 Uozo.5 mis Drag force acts in a direction opposite to the flow of motion water stream. of fo= Co A pV²/2 where uz upstream relocity suo = 7:5 7.5 } Co= 1.2 6 L 0 08 - fy = 1-2 X 6x0.88 999.7x (0:572/2 fo=719.8N