Lecture 3 Kinematics Part I CEE 262A HYDRODYNAMICS Lecture 3 Kinematics Part I

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

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.

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

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.

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

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

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

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.

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

Example 1: Stagnation point flow

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

(b) Calculate pathlines

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.

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

(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

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

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.

(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

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

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

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

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

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

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

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

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

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

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.

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

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

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