Question

4. [25 pts] The pressure coefficient is defined as Cp = 1 Con = P — Poo POU Assuming incompressible flow, use the incompressible form of Bernoulli's equation to show that the value of Cp at a stagnation point is given by Cjosp = 1. Now assume isentropic compressible flow of a gas with constant specific heats and derive the following relation for the value of the pressure coefficient at a stagnation point: 2 sp /M% poo M. ! 2 (Po y - 1 y/(1-1) Come ve erg - 1) = xx [(a+"=1m3)*** - 1] 1 M n 2 (Note that this relation is not valid for supersonic Mach numbers, since there would be a shock wave in front of a stagnation point, and the flow would then no longer be isentropic.) Determine the freestream Mach number for which the value of Cp at the stagnation point deviates from the incompressible value by (i) 1%, (ii) 5% and (iii) 10%.

Answer 1

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Solution The pressure coefficient is given beg- CP= P-Poo inc Poo on? Assumptions Incomprecible flow show that cose = 1 Bernoulli's equation for incompressible, ideal fluid along stream line is given by- between pointy and Assum horizontal bipe, 200=2. If head fluid is come to rest isen tropically , velocity of fluid is converted into pressure head. u=0 I = Poo + Vos² R. Poo = I Poe (bo² Coefficient cp= P. o pressure - Soola and from equation pressure coefficient for incompressible flow-

6 Assumption - I sentropic Compressible flow 2- Constant specific heat When moving fluid is decelerated isentropically o reach zero speed , then the termodynamic State is referred to as stagnation state. Then simplified from of energy equation for Steady, one-dimensional How with no weat addition across two region, and of a contool volume is given by - for a caterically from equ ② perfect gas h=ept Cpti + t = ca 124 42 ? State ① refers to any as bitory state and State ④ refers to stag naction state. cp Too + ² = cp To we know that cp = &R put value of up in equation ④ Coo 2Too FR Y- 292 Where a= a= acoustic Velocity JYR Too

10 where Moo = mach Number Moo = Sco For an isentropic flow, the mody namic relation is given by- from equation & prevure coefficient is given by- put value of Plpos from equation 4 we know that for isen tropic process eup - reus = luc e @ from ideal gasen" ( 1 ) Por = YRT 2= PRTO we know that - = a = YRTa - 6

from equ . & (B) =q2 = r boo YRTO Soo But value of boo in equation ® apsp = 20 ( 14 Il mon) ] Cosp = paint (1+ y mother