(b) In turbulent flows contrast the approaches that are used to represent the velocity profile within the boundary layer and comment on any limitations that each may offer.
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- Air at 20C flows at 1 m/s between two parallel flat plates spaced 5 cm apart. Estimate the distance from the entrance to the point at which the hydrodynamic boundary layers meet.What are the physical and quantitative evidence of turbulence in fluid flow?(a) What is boundary layer flow and what are the key assumptions used in simple analytical solutions. What are the outputs from simple solutions and to what practical configurations do they apply most closely and why.
- An approximation for the boundary-layer shape in is the formula u(y) - U sin 0 sys d where U is the stream velocity far from the wall and d is the boundary layer thickness, as in Fig. If the fluid is helium at 20°C and 1 atm, and if U = 10.8 m/s and 8= 3 cm, use the formula to (a) estimate the wall shear stress Tw in Pa, and (b) find the position in the boundary layer where t is one-half of Tw. -- y = 6 u(y)Consider laminar flow entering a tube. Before the fluid enters, the velocity profile is constant and equal to to. The tube diameter is D and radius R. You are interested in estimating how much distance the fluid must travel into the tube before the classic Poiseuille parabolic velocity profile is established - this distance is called the "entrance length" or Zent- Using boundary layer theory, which of the following would be a reasonable estimate? (Hint: assume as flow enters the tube, you can model the growing viscous boundary layer as laminar flow over a flat plate, and note that the De where v = Reynolds number for a tube is Rep %3D Zent = (Rep) Zent = D 2 (Rep) Zent = 2. Rep - Rep O Zent R-Rep %3D O None of the above.Fluid from a large reservoir at temperature T 0 fl ows into acircular pipe of radius R . The pipe walls are wound with anelectric resistance coil that delivers heat to the fl uid at a rateq w (energy per unit wall area). If we wish to analyze thisproblem by using the full continuity, Navier-Stokes, andenergy equations, what are the proper boundary conditionsfor the analysis?
- (b) In two dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reach the value of zero at y = 6. Based on Table 2 and setting given to you; () Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method ; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2 : Equation of Velocity Profile Setting Equation wU = 3(y/8)/2 – (y/8j?/2An isothermal long cylinder with square (side of 0.4 m and diagonal of 0.5657 m) cross-section is placed in a flow with a velocity of 1 m/s as shown. The properties of the fluid are: kinematic viscosity is 1.38 x 10-5 m²/s, thermal conductivity is 0.024 W/m.°C, and Prandtl number is 0.73. What is the average Nusselt number for the flow? 1253 a Figure 1: Flow configuration for Problem 6. The dimensions are: a=0.4 m, b=0.5657 m(b) In two-dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reach the value of zero at y = 8. Based on Table 2 and setting given to you; (i) Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method ; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2: Equation of Velocity Profile Equation u/U = 3(y/S)/2 – (y/8)³/2
- Using von Karman momentum integral, derive boundary layer height 8, boundary layer displacement thickness d, boundary layer momentum thickness 0, wall shear stress To, local skin friction coefficient c, and total drag coefficient C, for turbulent boundary layer flow with power law constant, n = 5. Discuss by comparing your answers to turbulent boundary layer flow with power law constant, n = 7. Take the empirical wall shear stress: To = 0.0204pU 2 %3D SU 1/4Water (T= 10 C, , k=0.644 W/m.C, u = 5.54× 10^-7 m2/s, Pr=3.55), flow over plate 1m x 1m , with velocity 2 m/s. the plate surfące temperature is 90 C. Give the answer for the following |-the flow tyPe 2-total het transfer 3-theavargeheat transfer cafficient 4- Nu ?(b) In two-dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reaches the value of zero at y = 8. Based on Table 2 and setting given to you; (i) Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2: Equation of Velocity Profile Equation u/U = 3(y/8)/2 – (y/8)³/2 Setting 2