Historically the First Fluid Flow Solution ….

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Second Class of Simple Flows

Mach View of NS Equations

• Reference velocity : Velocity of sound in a selected fluid.

• Velocity sound, c:

For an ideal gas: RTc

For a real gas:

v

p

Tz

Tz

Tz

Tz

n nzRTc

Property Inertial

property Elastic

B

c

For an incompressible liquids:

Average bulk modulus for water is 2 X109 N/m2.

In Mach’s view it is not possible to differentiate compressible and incompressible flows with Ma<0.2.

Reduces the resolution of flow solutions….

The Ideal Simple Flow

Reversible flow : No vorticity generation between x & y,

2222 0 xyxy MMMM

An isentropic flow….

It was not till Daniel Bernoulli published his celebrated treatise in the eighteenth century that the science of fluid Flow really began to get off the ground – G.I. Taylor

0221 222222 xyxyxy MMMMMM

Bernoulli’s View of Ideal Flow

vvp

gzv

t

v

2

2

A flow field is a collection of stream lines.

Derivation of Bernoulli equation

• The Bernoulli equation can be obtained as a scalar product of the Euler differential equation and a differential displacement vector.

• For steady flow cases, the differential distance along the particle path is identical with a distance along a streamline.

• Thus, the multiplication of Euler equation with the differential displacement , gives:

vvxdp

gzv

xdt

vxd

2

2

vvp

gzv

t

v

2

2

Consider Euler equation:

dtvvvp

gzv

xddtt

vv

2

2

The terms in above equation must be rearranged as follows.

The first term starts with a scalar product of two vectors, eliminating the Kronecker delta and utilizing the Einstein summation convention results in:

dt

t

vdt

t

vvdt

t

vvdt

t

vv iiii

i

2

2

1

2

1

dxt

vdt

t

vvdt

t

v

2

2

1

02

2

dp

gzv

ddxt

v

Integrating above Equation results in:

Constant2

2

dp

gzv

ddxt

v

Constant 2B

2

E E

B

E

B

dpgz

vddx

t

v

For rearrangement of the second and third term, use the definition of infinitesimal change in scalar variable.

dQQxd

Integrating above equation from a begin point B to an end point E:

Along a Stream line : From beginning to the end

which is the Bernoulli equation.

For an unsteady, incompressible flow, the integration of above equation delivers:

Constant 2

2

p

gzv

Constant2B

2

E E

B

E

B

dpgz

vddx

t

v

EEE

BBB

E

B

pgzv

pgzv

dxt

v

22

22

For a steady, incompressible flow, the integration of above equation delivers:

EEE

BBB pgz

vpgz

v

22

22

The Greatest Natural Flow

Constant2B

2

EE

B

dpgz

vddx

t

v

Constant2B

2

E E

B

E

B

dpgz

vddx

t

v

Barotropic Atmosphere

• The term barotropic is derived from the Greek baro, relating to pressure, and tropic, changing in a specific manner:

• that is, in such a way that surfaces of constant pressure are coincident with surfaces of constant temperature or density.

• A barotropic atmosphere is one in which the density depends only on the pressure, ρ = ρ(p).

• The isobaric surfaces are also surfaces of constant density.

• For an ideal gas, the isobaric surfaces will also be isothermal if the atmosphere is barotropic.

Along a isobar :

0constant

p

T

The geostrophic wind is independent of height in a barotropic atmosphere.

Geostrophic Wind Patterns

Geostrophic wind is defined a horizontal velocity field,

jviuv gggˆˆ

pkvg ˆ

Prediction of Simple Wind Patterns

• The knowledge of the pressure distribution at any time determines the geostrophic wind.

• This is true for large-scale motions away from the equator.

• The geostrophic wind becomes the total horizontal velocity to within 10–15% in mid latitudes.

Ancient Fluid Dynamics

Stokesian Kingdom Eulerian Kingdom

An experimental investigation of the circumstances which determine whether the motion of water in

parallel channels shall be direct or sinuous and of the law of resistance in parallel channels

Honor of Osborne Reynolds

• Consider the Navier-Stokes equations with constant density it their dimensional form:

i

j

j

i

jii

j

ij

i

x

v

x

v

xx

pg

x

vv

t

v

0 v

Define dimensionless variables as:

U

vv

*

L

xx

*2

*

U

pp

•Here U, L are assumed to be a velocity and length characteristic of the problem being studied. •In the case of flow past a body, L might be a body diameter and U the flow speed at infinity.

L

Utt *

Non-dimensionalization of NS EquationIn low Mach number region with constant viscosity:

i

j

j

i

jii

j

ij

i

x

v

x

v

xx

pg

x

vv

t

v

1

*vUv

*xLx

*2 pUp

U

Ltt

*

Non-dimensional form of NS Equation

*

*

*

*

**

*

2*

**

*

*

i

j

j

i

jii

j

ij

i

x

v

x

v

xULx

p

U

Lg

x

vv

t

v

*

*

*

*

**

*

2*

**

*

*ˆ

i

j

j

i

jij

ij

i

x

v

x

v

xULx

p

U

Lgk

x

vv

t

v

Importance of Reynolds Number

• The Reynolds number Re is the only dimensionless parameter which is always important in the equations of fluid motion.

• The Reynolds number is very small compared to unity, Re<< 1.

• Since Re = UL/, the smallness of Re can be achieved by considering

• extremely small length scales, or

• by dealing with a highly viscous liquid, or

• by treating flows of very small velocity, so-called creeping flows.

The Creeping Flows

• The choice Re << 1 is an very interesting and important assumption.

• It is relevant to many practical problems, especially in a world where fluid devices are shrinking in size.

• A particularly interesting application is to the swimming of micro-organisms.

• This assumption, unveils a special dynamical regime which is usually referred to as Stokes flow.

• To honor George Stokes, who initiated investigations into this class of fluid problems.

• We shall also refer to this general area of fluid dynamics as the Stokesian realm.

• This is of extreme contrast to the theories of ideal inviscid flow, which might be termed the Eulerian realm.