CEE 262A H YDRODYNAMICS

1. CEE 262A HYDRODYNAMICS Lecture 3 Kinematics Part I

2. Definitions, conventions & concepts • Motion of fluid is typically described by velocity Dimensionality Steady or Unsteady • Given above there are two frames of reference for describing this motion Lagrangian “moving reference frame” Eulerian “stationary reference frame” • Pathline • Focus on behavior of particular particles as they move with the flow • Focus on behavior of group of particles at a particular point

3. v v v Steady flow y x Streamlines • Individual particles must travel on paths whose tangent is always in direction of the fluid velocity at the point. In steady flows, (Lagrangian) path lines are the same as (Eulerian) streamlines.

4. Lagrangian vs. Eulerian frames of reference X2 t1 t0 Lagrangian particle path X1 * Following individual particle as it moves along path… At t = t0 position vector is located at Any flow variable can be expressed as following particle position which can be expressed as

5. Eulerian * Concentrating on what happens at spatial point Any flow variable can be expressed as Local time-rate of change: Local spatial gradient: This only describes local change at point in Eulerian description! Material derivative “translates” Lagrangian concept to Eulerian language.

6. Consider ; • As particle moves distance d in time dt Substitute (2) (1) and dt Local rate of change at a point Advective change past Material Derivative (Substantial or Particle) -- (1) • If increments are associated with following a specific particle whose velocity components are such that -- (2) -- (3)

7. n s Vector Notation: ESN: e.g. if Along ‘Streamlines’: Magnitude of

8. nozzle nozzle t0 t1 a b c d e a b c d Pathlines, Streaklines & Streamlines Pathlines: Line joining positions of particle “a” at successive times Streaklines: Line joining all particles (a, b, c, d, e) at a particular instant of time Sreamlines: Trajectories that at an instant of time are tangent to the direction of flow at each and every point in the flowfield

9. Streamtubes • No flow can pass through a streamline because velocity is always tangent to the line. • Concept of streamlines being “solid” surfaces forming “tubes” of flow and isolating “tubes of flow” from one another.

10. No flux ds s c Calculation of streamlines and pathlines Streamline By definition: (i) Pathline

11. Example 1: Stagnation point flow x3

12. Stagnation-point velocity field: (a) Calculate streamlines Cleverly chosen integration constant

14. (b) Calculate pathlines

15. Example 2: a (more complicated) velocity field: in a surface gravity wave: Stream/streak/path lines are completely different.

16. X2 X2 X1 X1 t0 t1 X2 X1 t1 Relative Motion near a Point (1) Basic Motions (a) Translation (b) Rotation • No change in dimensions of control volume

17. X2 X2 t0 t1 X1 X1 (c) Straining (need for stress): Linear (Dilatation) – Volumetric Expansion/Contraction (d) Angular Straining – No volume change X2 t1 X1 Note: All motion except pure translation involves relative motion of points in a fluid

18. P’ x2 x2 P O’ O t0 t1 x1 x1 General motion of two points: Consider two such points in a flow, O with velocity And P with velocity moving to O’ and P’ respectively in time dt

19. Therefore, after time O’: P’: Taylor series expansion of Relative motion of two points depends on the velocity gradient, , a 2nd-order tensor. O() means “order of” = “proportional to” to first order -- (A)

20. ={ rate of strain tensor} + {rate of rotation tensor} (2) Decomposition of Motion “…Any tensor can be represented as the sum of a symmetric part and an anti-symmetric part…” Note: (i) Symmetry about diagonal (ii) 6 unique terms Linear & angular straining

21. Note: (i) Anti-symmetry about diagonal (ii) 3 unique terms (r12, r13, r23) Rotation Terms in

22. Let’s check this assertion about rij • The recipe: • m = i and l =j • l = i and m = j • gives

23. Interpretation Relative velocity due to deformation of fluid element Relative velocity due to rotation of element at rate 1/2

24. General result: = 2 x {Local rotation rate of fluid elements) x3 Consider solid body rotation about x2 axis with angular velocity a x1 x3 Simple examples: u3(x1) u1(x3) x1

25. x3 x1 Consider the flow What happens to the box? t0 t1 t2 It is flattened and stretched

26. (3) Types of motion and deformation . (i) Pure Translation X2 t1 t0 X1 (ii) Linear Deformation - Dilatation X2 a t1 t0 b X1

27. Area Strain = and Strain Rate = In 2D - Original area at t0 - New area at t1 and

28. In 3D * Diagonal terms of eij are responsible for dilatation In incompressible flow, ( is the velocity) • Thus (for incompressible flows), • in 2D, areas are preserved • in 3D, volumes are preserved

29. A A X2 t0 t1 B B O O X1 (iii) Shear Strain – Angular Deformation

30. Shear Strain Rate Rate of decrease of the angle formed by 2 mutually perpendicular lines on an element Iff small Average Strain Rate The off diagonal terms of eij are responsible for angular strain.

31. (iv) Rotation A A B t0 t1 B O O

32. Rotation due to due to Average Rotation Rate (due to superposition of 2 motions)

33. Summary • Relative motion near a point is caused by • This tensor can be decomposed into a symmetric and an anti-symmetric part. • (a) Symmetric • * : Dilation of a fluid volume • * : Angular straining or shear straining • (b) Anti-symmetric • * : 0 • * : Rotation of an element