1. Lecture 3 Kinematics Part I
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. Streamlines v Steady flow y x
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 referenceX2 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. Material Derivative (Substantial or Particle)
Consider ;
As particle moves distance d in time dt
-- (1)
If increments are associated with following a specific particle whose velocity components are such that
-- (2)
Substitute (2) (1) and dt
-- (3)
Local rate of change at a point
Advective change past

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

8. Pathlines, Streaklines & Streamlines
nozzle
a b c d e
a b c d
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. Calculation of streamlines and pathlinesds c
No flux
By definition: (i)
Pathline

11. Example 1: Stagnation point flow

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

14. (b) Calculate pathlines

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

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

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

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

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

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

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

25. It is flattened and stretched
Consider the flow x1 x3
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. In 2D - Original area at t0 - New area at t1 and
Area Strain = and Strain Rate =
and

28. In incompressible flow, ( is the velocity)
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. (iii) Shear Strain – Angular DeformationB O X2 X1 t1 t0

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