Handout 18

MAE 3302 Aero dynamics of Incompressible Flow

Homework 5 Solutions

Answers to Study Questions

1. Why is the energy equation not needed in the study of incompressible ow?

ans: For constant density, there is no exchange b etween the mechanical pressure p o-

tential and kinetic energy and the internal thermal energy. As a result the mass and

conservation equations are sucient to solve for the velo city and pressure.

2. What is the Bernoulli equation? When is it valid?

ans: The Bernoulli equation says that the pressure plus the kinetic energy p +

2

1=2 U  is constant along a streamline. It is valid for steady, incompressible, inviscid

0

ow.

3. What is the interpretation of the Bernoulli equation from an energy conservation stand-

p oint?

ans: The sum of the pressure p otential energy and kinetic energy remains xed  rst

law of thermo dynamics for this simple system.

4. When is the Bernoulli constant a true constant? do es not vary in space

ans: When the uid prop erties are uniform do not vary in space far upstream.

5. What three conditions must be met b efore the Bernoulli equation Eq. 3.15 is valid?

i.e. what three principal assumptions are made in the derivation of the Bernoulli

equation?

ans: Incompressible, steady, inviscid ow.

6. Why is the pressure lower that the ambientvalue at the throat of a subsonic venturi?

ans: dictates an increase in velo city at the throat; Bernoulli's

equation then implies a decrease in pressure there.

7. How can aventuri be used to measure airsp eed?

ans: Airsp eed is prop ortional to the square ro ot of the pressure drop between the

inlet and throat.

8. Prop ellers achieve maximum eciency when they result in a small pressure rise dis-

tributed over a large area i.e. large prop eller turning slowly. Do es this fact help to

explain the classical wind tunnel design shown in Fig. 3.8? Explain your answer.

ans: The venturi allows the velo city in the test section to b e high while the velo city

at the fan is relatively low.

9. What is meantby static pressure?

ans: The usual thermo dynamic pressure. 1

10. What is meantby dynamic pressure?

2

ans: The quantity1=2 U , which is the increase in the pressure over the free-stream

0

1

value that would be realized at a stagnation p oint.

11. What is meantby stagnation pressure sometimes referred to as total pressure?

ans: The pressure that is achieved at a stagnation p oint.

12. Which comp onent of pressure is constant along a streamline in an incompressible,

inviscid, steady ow? How do you know this?

2

ans: The stagnation pressure. Bernoulli's equation says p +1=2 U is constant.

1 0

1

13. Nearly all aircraft use a pitot-static tub e to measure airsp eed. How do es this device

work? Will it still work if the ight is greater than 0.3? Explain.

ans: The airsp eed is prop ortional to the square ro ot of the di erence between the

stagnation and static pressures. The pitot tub e can be used for M > 0:3, but the

equation relating the airsp eed to the pressure di erence is more complicated.

14. What is the advantage of working in terms of the pressure co ecient as opp osed to

the pressure itself ?

ans: The pressure co ecient is non-dimensional and thus do es not dep end explicitly

on the free stream values of pressure,density, velo cityorbody size.

15. How is the pressure co ecient related to the velo city for steady, incompressible, invis-

cid ow?

ans: Through the non-dimensional form of the Bernoulli equation: C = 1

p

2

U=U  .

1

16. What value do es the pressure co ecient take at a stagnation p oint for steady, incom-

pressible, inviscid ow? What value do es it take in the free-stream?

ans: C =1 at a stagnation p oint, C = 0 in the free-stream.

p p

17. When is the pressure co ecient negative?

ans: The pressure co ecient is negative when the lo cal velo city is greater that the

free stream sp eed.

18. What condition must the velo city eld satisfy if the ow is incompressible? Which

conservation lawdoes this condition come from?

~

ans: of velo city is zero rU = 0. This result comes from the mass

conservation law.

19. The velo city p otential will satisfy Laplace's equation provided what two conditions are

met?

ans: The ow is irrotational and incompressible.

20. What additional condition must be met if the streamfunction is to satisfy Laplace's

equation also?

ans: The ow must be two-dimensional.

21. What simpli cation in the solution to aero dynamics arises from the fact that Laplace's

equation is linear?

ans: Elementary solutions maybe sup erimp osed to yield more complex solutions. 2

22. What b oundary condition must the velo city eld satisfy at in nity?

ans: The velo city must revert to the free-stream condition.

23. What b oundary condition must be satis ed by an inviscid uid at a solid surface? Is

this condition ever met in nature? Why do esn't the inconsistency of this b oundary

condition lead to serious problems in aero dynamics?

ans: The velo city must be parallel to the surface. This condition is not met in

nature since viscosity ensures that the ow is brought to rest at a solid surface. The

inviscid b oundary condition is an approximation where the body surface is translated

to the edge of the b oundary layer, where the ow tangency condition is valid. For high

Reynolds numb ers this approximation is acceptable since the b oundary layers are thin

compared with the body thickness.

24. What is meant by an elementary ow solution? How can the elementary solutions be

used to solve problems in aero dynamics?

ans: It is a solution to Laplace's equation that results in a relatively simple ow pat-

tern. Elementary solutions are sup erimp osed together to give more complex solutions.

25. What are the velo city p otential and streamfunction for a uniform ow in the x direc-

tion?

ans:  = U y and  = U x.

1 1 3