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White F.M., “Viscous Fluid Flow” Chapter 4, Exercise 8 8. Homework. Due date: 20.11.2015 HW1 (3p): Flow straighteners consist of arrays of narrow ducts placed in a flow to remove swirl and other transverse (secondary) velocities. One element can be idealized as a box with thin sides as shown in Fig. 1. Using laminar flat-plate theory, derive a formula for the pressure loss ∆p across an N × N bundle of such boxes, and apply it to compute ∆p, if the dimension of each element are given as: a = 10mm and L = 100mm; and that the properties of the fluid are given as: ρ = 1.23kg/m3 and µ = 0.0183 · 10−3 Pa · s. Use U∞ = 10 m/s. Figure 1: An idealized flow straightener element HW2 (3p): A thin equilateral triangle plate is immersed parallel to a 6 m/s stream of air at +20o C and at 1 atm, as shown in the figure below. Assuming laminar flow, estimate the friction drag of this plate. 1 White F.M., “Viscous Fluid Flow” Chapter 4, Exercise 8 Demo: Air flows over a smooth 1m × 1m flat plate affecting only one side of the plate. The free-stream velocity of the flow is 10m/s, temperature 20o C and pressure 1bar. Calculate the boundary layer thickness at the trailing edge and find an expression for the shear stress at the plate surface. What is magnitude and direction of the force that the flow exerts on the plate? Solution: Using U = 10m/s, L = 1m and ν = 15 × 10−6 m2 /s, the Reynolds number at the trailing edge is Re = UνL ≈ 666 667. The limit of transition is Re ≈ 106 , so the flow is laminar and we can apply the Blasius solution. The boundary layer thickness at the trailing edge can be calculated from eq. 4-49 in White 5.0 x 5.0 × 1 δ=√ =q = 6.1 mm 10×1 Rex −5 (1) 1.5×10 Using equation 4-52, the shear stress can be written as τW = 0.332ρU 2 √ . Rex (2) The force exerted on the plate can be calculated by integrating the shear stress over the plate Z 1 Z 1 1/2 3/2 1/2 Fx = W τW dx = 0.332W ρ U µ x−1/2 dx 0 0 = 0.664W ρ = 0.098 N. The force acts on the direction of the flow. 2 1/2 U 3/2 1/2 µ (3)