Fluid Kinematics

• Continuum hypothesis:

– fluid is made up of fluid particles;

– each particle contains numerous molecules;

– infinitesimal particles of a fluid are tightly packed together

• Thus, motion of a fluid is described in terms of fluid particles rather than individual molecules.

• This motion can be described in terms of the velocity and acceleration of the fluid particles

• At a given instant of time, description of any fluid property may be given as a function of fluid location

• Representation of fluid parameters as function of spatial coordinates is termed a field representation of the flow

• Fluid parameters are functions of position ant time. For example, temperature in the room is completely specified by temperature field

Velocity Field

, , ,T T x y z t

Velocity Field

Velocity of a particle

Velocity magnitude

Velocity field

AA

d

dt

rV

2 2 2V u v w V

u x y z t v x y z t w x y z t ˆ ˆ ˆV , , , i , , , j , , , k

Particle location in terms of its position vector

x y z tV V , , ,

Eulerian and Lagrangian Flow Description

There are two approaches in analyzing fluid mechanics problem

Eulerian method uses field concept

Lagrangian method involves following individual particle moving through the flow

Lagrangian information can be derived from the Eulerian data – and vice versa

Most fluid mechanics considerations involve the Eulerian method.

Eulerian and Lagrangian descriptions of temperature of a flowing fluid

One-, Two, and Three-Dimensional Flows.Steady and Unsteady Flows

• Steady flow – the velocity at a given point in space does not vary with time, otherwise, flow is unsteady

• In general, fluid flow is three-dimensional and unsteady

• In many situations, flow can be simplified to steady, two- or one-dimensional flow in order to make solution easier without loss of accuracy

Flow visualization of the complex three-dimensional flow past a model airfoil

Streamlines, Srteaklines and Pathlines

• Streamlines, streaklines, and pathlines are used for flow visualization

• Streamline is used in analytical work while the streakline and pathline are used in experimental work

• Streamline is a line, that is everywhere tangent to the velocity field

• Streamlines are obtained by integrating differential equation of streamline. For two-dimensional flow dy/dx = v/u

• If flow is steady, streamlines are fixed lines in space

• Streakline consist of all particles in a flow that have previously passed through the common point.

• Pathline is the line traced out by a given particle as it flows from one point to another

• For steady flow streamlines, streaklines, and pathlines are the same

Example 4.3

Solution (a) Streamline is given by solution of

Integration gives

At t = 0, C = u0v0/ω , and equation of streamline is

At t = π/2ω , C = 0, and equation of streamline is

These two streamlines are not the same because flow is unsteady

At the origin

0

0 0sin

vdy v

dx u u t y v

0 0 0 0cosu v t y v v x C

0

0

cos 1u y

xv

0

0

sinu y

xv

0

0 0

at 0

at 2

v t

u v t

V j

V i j

0 00

0 1 0

(b) Pathline is obtained from velocity field

sin and

Integration gives

and s

dx y dyu t v

dt v dt

y v t C x u

12

0

0

in

For the particle that was at the origin

at time 0, the pathline is

0 and

For the particle that was at the origin

at time 2 , the pathli

Ct C

v

t

x y v t

t

0 0

ne is

and 2 2

x u t y v t

(c) Discuss the shape of the streakline that passes through the origin

Acceleration Field

• For Eulerian description one describes the acceleration field as a function of position and time

• Acceleration is the time rate of change of velocity of a given particle

• For unsteady flow the velocity at a given point in space (occupied by different particles) may vary with time, giving rise to a portion of the fluid acceleration

• In addition, a fluid particle may experience an acceleration because its velocity changes as it flows from one point to another in space

Acceleration Field

A A A A A A At x t y t z t V V r , V , ,

or D

u v wt x y z Dt

V V V V V

a a

Velocity and position of particle A at time t

Material Derivative

Operator

is termed the material derivative or substantial derivative

In vector notation:

Material derivative of any variable is the rate at which that variable changes with time for a given particle (as seen by one moving along with the fluid – Lagrangian description. Material derivative is also called comoving derivative)

For example, the time rate of change of temperature of a fluid particle (particle A) as it moves through the temperature field T = T(x,y,z,t) is given by

Du v w

Dt t x y z

D

Dt t

V

DT T T T T Tu v w T

Dt t x y z t

V

Material Derivative. Unsteady Effects

Portion of material derivative represented by time derivative is termed the local derivative

Local derivative is the result of the unsteadiness of the flow

Uniform, unsteady flow in a constant diameter pipe

Material Derivative. Convective Effects

Portion of the material derivative represented by the spatial derivative is termed the convective derivative

Convective derivative is a result of the spatial variation of the flow

Steady state operation of a water heater

Material Derivative. Convective Effects

Portion of the material derivative represented by the spatial derivative is termed the convective derivative

Convective derivative is a result of the spatial variation of the flow

Uniform, steady flow in a variable area pipe

• System is a collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings

• Control volume is a volume in space (geometric entity, independent of mass) through which fluid may flow

Control Volume and System Representation

Typical control volumes: (a) fixed control volume, (b) fixed or moving control volume, (c) deforming control volume

• Both, control volume and system concepts can be used to describe fluid flow

• Governing laws of fluid motion are stated in terms of fluid systems, not control volume

• To shift from one representation to the other Reynolds transport theorem is used

Control Volume and System Representation

Reynolds Transport Theorem

Physical laws are stated in terms of physical parameters (velocity, acceleration, mass, temperature, momentum etc.)

Let B represent fluid parameter and b represent amount of that parameter per unit mass. Then

Parameter B is termed an extensive property, and the parameter b is termed an intensive property

Amount of extensive property that system possesses at a given instant is

Time rate of change of extensive property of a system

B mb

sys sysB bdV

syssysd bdVdB

dt dt

Reynolds Transport Theorem. Derivation

Simplified version of the Reynolds transport theorem for fixed control volume with one inlet and one outlet having uniform properties (density, velocity, and the parameter b) across the inlet and outlet with the velocity normal to sections (1) and (2) is

Control volume and system for flow through a variable area pipe

sys cv2 2 2 2 1 1 1 1

DB BA V b AV b

Dt t

Reynolds Transport Theorem. General Form

General form for of the Reynolds transport theorem for a fixed, nondeforming control volume is given by (details)

Control volume and system for flow through an arbitrary, fixed control volume

sys

cv cs

DBbdV b dA

Dt t

ˆV n

Physical Interpretation

Possible velocity configurations on portions of the control surface: (a) inflow, (b) no flow across the surface, (c) outflow

sys

cv cs

DBbdV b dA

Dt t

ˆV n

Reynolds Transport Theorem. Moving CV

Example of a moving control volume

Reynolds Transport Theorem. Moving CV

sys

cv cs

DBbdV b dA

Dt t

ˆW n

cv V W V

Selection of a Control Volume

END OF CHAPTER

Supplementary slides

Outflow across a typical portion of the control surface

Outflow across a typical portion of the control surface

cosB b V b V t A

out

0 0

coslim lim cost t

bV t Ab VB bV A

t t

out outout outcs cs

cosB dB bV A out

out csB b dA ˆV n

back

Inflow across a typical portion of the control surface

in inin cs cs

B bV dA b dA ˆcos V n