MECH 221 FLUID MECHANICS (Fall 06/07) REVIEW

1. MECH 221 FLUID MECHANICS(Fall 06/07)REVIEW

2. What Have You Learnt? • Fluid Statics • Fluids in Motions • Kinematics of Fluid Motion • Integral and Differential Forms of Equations of Motion • Dimensional Analysis • Inviscid Flows • Boundary Layer Flows • Flows in Pipes • Open Channel Flows On coming week lectures

3. Fluid Statics • It is to calculate the fluid pressure when the fluid is no moving • Shear stress is due to relative motion of fluid, so no shear stress and only normal stress (Pressure) acting on the fluid • The fluid pressure is only due to body force, Gravitational Force

4. z g y 0 x Fluid Statics • Fluid pressure will increase when the position of the fluid become deeper, we have following equation:

5. Fluid Statics • Total force acting on the surface become:

6. Fluid In Motion (Inviscid Flow) • 2 sets equations for solving fluid motion problems • Conservation of Mass • Conservation of Momentum

7. Fluid In Motion (Inviscid Flow) • By invoking the continuity equation, the momentum equation becomes Euler’s equation of motion • Bernoulli equation is a special form of the Euler’s equation along a streamline Along streamline incompressible flow

8. Fluid In Motion (Inviscid Flow) • A conical plug is used to regulate the air flow from the pipe. The air leaves the edge of the cone with a uniform thickness of 0.02m. If viscous effects are negligible and the flowrate is 0.05m3/s, determine the pressure within the pipe.

9. Fluid In Motion (Inviscid Flow) • Procedure: • Choose the reference point • From the Bernoulli equation • P, V, Z all are unknowns • For same horizontal level, Z1=Z2 • Flowrate conservation • Q=AV

10. Fluid In Motion (Inviscid Flow) • From the Bernoulli equation,

11. Fluid In Motion (Inviscid Flow) • From flowrate conservation,

12. Fluid In Motion (Inviscid Flow) • Sub. into the Bernoulli equation,

13. Fluid In Motion (Viscous Flow) • In the mentioned fluid motion is inviscid flows, only pressure forces act on the fluid since the viscous forces (stress) were neglected • With the viscous stress, the total stress on the fluid is the sum of pressure stress ( ) and viscous stress ( ) given by:

14. Fluid In Motion (Viscous Flow) • The substitution of the viscous stress into the momentum equations leads to: • These equations are also named as the Navier-Stokes equations

15. Dimensional Analysis • The objective of dimensional analysis is to obtain the key non-dimensional parameters that govern the physical phenomena of flows • After the dimensional analysis or normalization of the complicated Navier-Stokes equations (steady flow), the non-dimensional parameters are identified • The equations are reduced to simple equation and solvable analytically under certain conditions

16. Dimensional Analysis • By using proper scales, the variables, velocity (u), pressure (p) and length (L) are normalized to obtain the non-dimensional variables, which are order one

17. Dimensional Analysis • For simplicity consider the case where the gravitational force has no consequence to the dynamic of the flow, the Navier-Stokes equations becomes When Re >> 1

18. Inviscid Flow Vs. Boundary Layer Flow where is the viscous diffusion length in an advection time interval of . • Here, measures the time required for fluid travel a distance L.

19. Inviscid Flow Vs. Boundary Layer Flow • When , inertia force is much greater than viscous force, i.e., the viscous diffusion distance is much less than the length L. • Viscous force is unimportant in the flow region of , but can become very important in the region of near the solid boundary. • This flow region near the solid boundary is called an boundary layer as first illustrated by Prandtl.

20. Potential flow Boundary layer flow U Inviscid Flow Vs. Boundary Layer Flow • Flow in the region outside the boundary layer where viscous force is negligible is inviscid. The inviscid flow is also called the potential flow.

21. Inviscid Flow • Inviscid flow implies that the viscous effect is negligible. The governing equations are Continuity equation and Euler equation. • We introduce a potential function, which is automatically satisfy the continuity equation

22. Inviscid Flow • The continuity equation becomes Laplace equation. The flow is described by Laplace equation is called potential flow • For 2D potential flows, a stream function (x,y)can also be defined together with (x,y)

23. Inviscid Flow • If 1 and 2 are two potential flows, the sum =(1+2) also constitutes a potential flow • We can combine certain basic solutions to obtain more complicated solution + =

24. Inviscid Flow Basic Potential Flows Combined Potential Flows

25. Inviscid Flow For stagnation flow,

26. Boundary Layer Flow • The thin layer adjacent to a solid boundary is called the boundary layer and the flow inside the layer is called the boundary layer flow • Inside the thin layer the velocity of the fluid increases from zero at the wall (no slip) to the full value of corresponding potential flow.

27. Boundary Layer Flow • There exists a leading edge for all external flows. The boundary layer flow developing from leading edge is laminar

28. Boundary Layer Flow • When we normalize the governing equations with Re underneath the viscous term and resolve the variables of y and v inside the boundary flow, the non-dimensional normalized variables are selected: V be the scale of v in the boundary layer L isviscous diffusion layer near the wall (boundary layer)

29. Boundary Layer Flow • These results in the boundary layer equations that in dimensional form are given by: Continuity: X-momentum: Y-momentum:

30. (*) Boundary Layer Flow • A boundary layer flow is similar and its velocity profile as normalized by U depends only on the normalized distance from the wall: i.e.,

31. (**) Boundary Layer Flow • By introduce a stream function • The boundary layer equation in term of the similarity variables becomes:

32. Boundary Layer Flow • After we solve this ordinary equation, we obtain a solution of • We first find the value of by Equ. (*) based on coordinate of x and y, then find out the value of by checking the solution table in the reference. Finally the u at x and y is calculated by Equ. (**) • Therefore, we obtain following results:

33. Boundary Layer Flow • Laminar boundary layer flow can become unstable and evolve to turbulent boundary layer flow at down stream. This process is called transition

34. Boundary Layer Flow • Under typical flow conditions, transition usually occurs at a Reynolds number of 5 x 105 • Velocity profile of turbulent boundary layer flows is unsteady • A good approximation to the mean velocity profile for turbulent boundary layer is the empirical 1/7 power-law profile given by

35. Boundary Layer Flow • For turbulent boundary layer, empirically we have

36. Boundary Layer Flow • The net force, F, acting on the body • The resultant force, F, can be decomposed into parallel and perpendicular components. The component parallel to the direction of motion is called the drag, D, and the component perpendicular to the direction of motion is called the lift, L.

37. Boundary Layer Flow • If i is the unit vector in the body motion direction, then magnitude of drag FD becomes: • For two-dimensional flows, we can denotes j as the unit vector normal to the flow direction, FL is the magnitude of lift and is determined by: Friction Drag Pressure Drag

38. Boundary Layer Flow • The drag coefficient defined as • For uniform flow passing a flat plate and no pressure gradient is zero and no flow separation, : Laminar Friction Drag Turbulent Friction Drag

39. Boundary Layer Flow • The pressure drag is usually associated with flow separation which provide the pressure difference between the front and rear faces of the body • For low velocity flows passing a sphere of diameter D, the drag coefficient then is expressed as: