Flow Over Immersed Bodies: Boundary Layer, Compressible Flow, Lift and Drag

1. Subject Name: FLUID MECHANICS Subject Code:10ME36B Prepared By: R Punith Department: Aeronautical Engineering Date: 1-11-2014

2. Flow over immersed bodies • Boundary layer concept • Introduction to Compressible Flow • Velocity of sound • Mach number

3. Lift and Drag • Shear stress and pressure integrated over the surface of a body create force • Drag: force component in the direction of upstream velocity • Lift: force normal to upstream velocity (might have 2 components in general case)

4. Flow past an object • Dimensionless numbers involved • for external flow: Re>100 dominated by inertia, Re<1 – by viscosity

5. Flow past an object Character of the steady, viscous flow past a circular cylinder: (a) low Reynolds number flow, (b) moderate Reynolds number flow, (c) large Reynolds number flow.

6. Boundary layer characteristics • For large enough Reynolds number flow can be divided into boundary region where viscous effect are important and outside region where liquid can be treated as inviscid

7. Laminar/Turbulent transition • Near the leading edge of a flat plate, the boundary layer flow is laminar. • If the plate is long enough, the flow becomes turbulent, with random, irregular mixing. A similar phenomenon occurs at the interface of two fluids moving with different speeds.

8. Boundary layer characteristics • Boundary layer thickness Boundary layer displacement thickness: Boundary layer momentum thickness (defined in terms of momentum flux):

9. Prandtl/Blasius boundary layer solution Let’s consider flow over large thin plate: • approximations: • than: • boundary conditions:

10. Prandtl/Blasius boundary layer solution • as dimensionless velocity profile should be similar regardless of location: • dimensionless similarity variable • stream function

11. Drag on a flat plate • Drag on a flat plate is related to the momentum deficit within the boundary layer • Drag and shear stress can be calculated just by assuming some velocity profile in the boundary layer

12. Speed of Sound 1] The speed of any wave depends upon the properties of the medium through which the wave is traveling. Typically there are two essential types of properties which effect wave speed - inertial properties and elastic properties. 2] The density of a medium is an example of an inertial property. The greater the inertia (i.e., mass density) of individual particles of the medium, the less responsive they will be to the interactions between neighboring particles and the slower the wave. If all other factors are equal (and seldom is it that simple), a sound wave will travel faster in a less dense material than a more dense material. Thus, a sound wave will travel nearly three times faster in Helium as it will in air; this is mostly due to the lower mass of Helium particles as compared to air particles. 3] Elastic properties are those properties related to the tendency of a material to either maintain its shape and not deform whenever a force or stress is applied to it. A material such as steel will

13. experience a very small deformation of shape (and dimension) when a stress is applied to it. Steel is a rigid material with a high elasticity. On the other hand, a material such as a rubber band is highly flexible; when a force is applied to stretch the rubber band, it deforms or changes its shape readily. A small stress on the rubber band causes a large deformation. Steel is considered to be a stiff or rigid material, whereas a rubber band is considered a flexible material. At the particle level, a stiff or rigid material is characterized by atoms and/or molecules with strong attractions for each other. When a force is applied in an attempt to stretch or deform the material, its strong particle interactions prevent this deformation and help the material maintain its shape. Rigid materials such as steel are considered to have a high elasticity (elastic modulus is the technical term). The phase of matter has a tremendous impact upon the elastic properties of the medium. In general, solids have the strongest interactions between particles, followed by liquids and then gases. For this reason,

14. longitudinal sound waves travel faster in solids than they do in liquids than they do in gases. Even though the inertial factor may favor gases, the elastic factor has a greater influence on the speed of a wave, thus yielding this general pattern: 4] The speed of a sound wave in air depends upon the properties of the air, namely the temperature and the pressure. The pressure of air (like any gas) will effect the mass density of the air (an inertial property) and the temperature will effect the strength of the particle interactions (an elastic property). At normal atmospheric pressure, the temperature dependence of the speed of a sound wave through air at T=20C is a = 343 m/s

15. Speed of sound in compressible flow and incompressible flow 1] incompressible flow * theoretically but ; actually in common liquid = order of 1,500 m/s fluid velocity can be produced in liquid * An entire region of flow is able to sense instantaneously the motion of an object through it. 2] compressible flow * * At high speeds, the streamlines are affected only a short distance ahead, which means that they approach the object at a steeper angle.

17. 2.3 Velocity of Sound 1./ Sound Wave Formation

18. Speed of Sound for a Perfect Gas The speed of sound is the speed at which a pressure disturbance of small amplitude travels through a fluid. 1] derivation 1) assumptions ; steady, one dimensional flow 2) control volume, force diagram 3) continuity equation

19. Momentum Equation

20. Sound wave formation process = isentropic process so In liquids and solids, changes in pressure generally produce only small changes in temperature. So in this case; Speed of sound for a perfect gas from relation for a perfect gas undergoing isentropic process

23. Ahead of the pressure pulse Behind the pressure pulse Moving Wave Entering CV Leaving CV Stationary Wave

25. Speed of Sound for a Substance that is not a Perfect Gas 1. Compressibility ; a measure of relative volume change with pressure for a given process 2. Speed of sound

26. Subsonic and Supersonic Flows 1. Sound Speeds Generated by a Disturbance Traveling at Subsonic and Supersonic Speeds

27. Mach Wave 1. Definition of Mach Wave * very weak pressure wave due to a point source moving with speed greater than the speed of sound. * infinitesimal deflection of the stream due to the body(=infinitesimal pressure disturbance) when M>1. 2. Mechanism of Wave Consider a small solid body moving relative to a gas. In order for the gas to pass smoothly over the body, disturbances tend to be propagated ahead of the body to warn the gas of the approach of the body, i.e., because the pressure at the surface of the body is greater than in the surrounding gas, pressure waves spread out from the body. Since these pressure waves are very weak except in the immediate vicinity of the body, they effectively move outward at speed of sound. 3. Spread of waves from a point source of disturbance(point source moving in compressible fluid)

28. Mach angle(Mach cone's vertex angle) • definition • Mach angle - Mach number; • greater Mach number → smaller Mach angle • maximum great Mach angle when M=1