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# HW 2 - 2 Calculate the local Mach number and total...

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Homework 2: 1-D Isentropic Flow Handed out 1/27/11; Due 2/3/11 1. Use a binomial expansion to show that, for Mach numbers much less than unity, the stagnation pressure definition p o = p (1 + γ - 1 2 M 2 ) γ γ - 1 reduces to the Bernoulli definition p o = p + 1 2 ρU 2 , where U is the velocity magnitude. 2. Show that in an adiabatic expansion of a flow from a reservoir, the maximum velocity that may be reached is given by: u 2 max = 2 h o What are the corresponding values of T and M? Interpret this result. 3. Prove, using calculus, that the maximum ˙ m A for isentropic, 1-D flow with area change occurs for a Mach number equal to unity. 4.At a given point in the high-speed flow over an airplane wing, the local Mach number, pressure, and temperature are 0.7, 0.9 atm, and 250 K, respectively. Calculate the values of p o , T o , p * , T * , and a * at this point. 5. At a given point in a supersonic wind tunnel, the pressure and temperature are 5e4 N/m 2 and 200 K, respectively. The total pressure at this point is 1.5e6 N/m
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Unformatted text preview: 2 . Calculate the local Mach number and total temperature. 6. Show that for a perfect gas, M = u a o [1-γ-1 2 ( u a o ) 2 ]-1 / 2 Then show that for Mach numbers much less than unity, this reduces to M = u a o [1 + γ-1 4 ( u a o ) 2 ] 7. Air ±ows isentropically from atmosphere at a pressure of 1 atm through two di²erent ducts, each of uniform cross section. In ducts A and B, the static pressures are 85 × 10 3 N/m 2 and 36 × 10 3 N/m 2 , respectively. The atmospheric temperature is 15.0 deg. C. Using tables , Fnd the Mach numbers, static temperatures, and velocities in A and B. 8. Air ±ows isentropically from atmospheric conditions (p = 1 atm, T = 300 K) into a duct with a cross-sectional area of 200 cm 2 . At one point in the duct, the Mach number is 1.6. Calculate the static pressure, velocity, static temperature at this location, along with the mass ±ow rate. 1...
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