1. Fluid Mechanics FLOWING FLUIDS Engineering Fluid Mechanics 8/E by Crowe, Elger, and Roberson Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

2. Streamlines & Flow Patterns Flow Pattern: Construction of streamlines showing the flow direction Streamlines (light blue): Local velocity vector is tangent to the streamline at every point along the line at a single instant. Flow through an opening in a tank & over an airfoil section.

3. Streamline & Pathline Streamline: line drawn through flow field such that local velocity vector is tangent at every point at that instant – Tells direction of velocity vector – Does not directly indicate magnitude of velocity Pathline: shows the movement of a particle over time ► In unsteady flow, all can be distinct lines. ► The latter two tells us the history of flow as the former indicates the current flow pattern.

4. Examples... Predicted streamline pattern over the Volvo ECC prototype. Pathlines of floating particles.

5. TYPES OF FLOW Uniform: Velocity is constant along a streamline (Streamlines are straight and parallel) Non-uniform: Velocity changes along a streamline (Streamlines are curved and/or not parallel) Express velocity V = V(s,t) Vortex flow

6. Steady: streamline patterns are not changing over time Unsteady: velocity at a point on a streamline changes over time Flow patterns can tell you whether flow is uniform or non-uniform, but not steady vs. unsteady… Why? Because streamlines are only instantaneous representation of the flow velocity. TYPES OF FLOW

7. LAMINAR & TURBULENT FLOW (a)Experiment to illustrate the type of the flow (b) Typical dye streaks for different cases (a) (b)

8. Engineering Fluid Mechanics 8/E by Crowe, Elger, and Roberson Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. LAMINAR & TURBULENT FLOW

9. DIMENSIONALITY OF FLOW FLIED → Characterized by the number of spatial dimensions needed to describe velocity field. 1-D flow: Axisymmetric uniform flow in a circular duct 2-D flow: Uniform flow in a square duct 3-D flow: Uniform flow in an expanding square duct

10. FLOW ACCELERATION (rate of change of velocity with time ) Consider a fluid particle moving along a pathline... There are two components of acceleration: Tangential to pathline a t : the time-dependent acceleration related to change in speed. Normal to pathline a n : the centripetal acceleration related to motion along a curved pathline.

11. Flow Acceleration Local acceleration – occurs when flow is unsteady (direction or magnitude is changing with respect to time) Convective acceleration – occurs when flow is nonuniform (acceleration can depend on position in a flow field) Local acceleration – occurs when flow is unsteady Centripetal acceleration – occurs when the pathline is curved (normal to the pathline & directed toward the center of rotation)

12. Example: Convective Acceleration The nozzle shown below is 0.5 meters long. Find the convective acceleration at x = 0.25 m. The equation describing velocity variation is provided below.

13. Problem 4.17:

14. Problem 4.17: (Solution)

15. Example: