ABC Type of Flows……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Mathematical Models for Simple Fluid Flows

Curl of a vector field:

Curl of a vector field:

Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" you get when going along a path, such as a circle.Curl is simply circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point)Curl is a vector field with magnitude equal to the maximum "circulation" at each point and oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of curl is the limiting value of circulation per unit area.

VAd

dsnVVcurl

Ad

ˆ.lim

0

The Curl of Velocity Field

ky

u

x

vj

x

w

z

ui

z

v

y

w

wvu

zyx

kji

V ˆˆˆ

z

rzrr

z iu

rr

ru

ri

r

u

z

ui

z

uu

rV ˆ11ˆˆ1

i

r

ruu

ri

u

r

ru

ri

uu

r

V

rrr

ˆsin

11ˆ1ˆsin

sin

1

Define the vorticity vector as being the curl of the velocity

kjiV zyxˆˆˆ

vorticity vector in cylindrical co-ordinates:

vorticity vector in spherical co-ordinates:

Translation & Rotation of A Fluid Element

xdx

A

B

C

D

xd

Nature of Rotation Vector

kx

v

y

uj

z

u

x

wi

y

w

z

vR ˆ

2

1ˆ2

1ˆ2

1

The existence of the Rotation vector is a characteristic of Real flows that in general undergoes a rotational motion.

Irrotational Flow Field

• Flows with vorticity are said to be rotational flows.

• Flows without vorticity are said to be irrotational flows.

• If the velocity is exactly equal to gradient of a scalar, the flow filed is obviously irrotational.

0ˆˆˆ kjiV zyx

0ˆˆˆ kji zyx

• If an application calls for an irrotational flow, the problem is completely solved by finding a scalar, .

Irrotational Solenoidal Flow Field

• Irrotational flow 0 V

• Irrotational, Solenoidal flow

0 V

• Solenoidal flow

0

02

2

2

2

2

2

yyx

02 • Laplacian flow

Complex Lamellar Flow

• A complex-lamellar velocity field is defined as a flow field whose streamlines are intersect orthogonally to vortex lines.

0 VV

Complex-lamellar flows are characterized by the fact that there exists a family of surfaces orthogonal to the stream lines.Irrotational flow field is a subset of complex-lamellar flows.This is a special class of flow fields, where streamlines are steady in a accelerating flow.

Creation of Uniform Flow Through A Diverging Duct

0 VV

BELTRAMI FLOW FIELDS

• A three-dimensional vector field V is said Beltrami if

0 VV

0 & V

• A Beltrami flow field is characterized by the fact that the vorticity vector is collinear with velocity through out the flow field.

• Beltrami fields are known as force-free fields.

• The ”surface” of the sun (called the photosphere) shows Beltrami behaviour.

• Reconstruction the three-dimensional field above the photosphere is possible using Beltrami model.

0V

Classification of Approximate Nature of Beltrami Field

• From a mathematical viewpoint, above equation can be roughly rewritten in the form

0V

0 Vx

where λ(x) is a scalar function which varies in general with position

Potential fields λ ≡ 0 is an irrotational field, also known as potential field.Linear Beltrami fields: they are characterized by a constant λ. Non-linear Beltrami fields: they corresponds to a variable function λ(.)

Mathematically Understood Flows

The Natural Genius&

The Art of Generating Lift

Hydrodynamics of Prey & Predators

The Art of C-Start

The Art of Complex Swimming

The Shocking News

• 1 million to 10 million years they might be able to make a plane that would fly ?!?!?!

• People had dreamed of flying for many years. • The United States Army was trying to develop an airplane

in 1903, but the plane wouldn't fly. • The New York Times wrote that maybe in

1 million to 10 million years they might be able to make a plane that would fly.

• Only eight days later two men were successful in flying the first manned plane.

• Controlled, powered flight had seemed impossible until Orville Wright took off on the 17th December 1903.

• They were Wilbur Wright and his younger brother, Orville.

A Narrow Gap Between Possibility & Impossibility

• The would-be aeronauts of the nineteenth century closely studied the flight of birds and began building flying machines patterned after avian structures.

• Their birdlike craft failed miserably. • They quickly realized that in reality they knew nothing about

the lift and drag forces acting on surfaces cutting through the atmosphere.

• To fly, man first had to understand the flow of air over aircraft surfaces.

• This meant that he had to build instrumented laboratories in which wings, fuselages, and control surfaces could be tested under controlled conditions.

• Thus it is not surprising that the first wind tunnel was built a full 30 years before the Wrights' success at Kitty Hawk.

• A science called Aerodynamics leading to Gas dynamics & • A wonderful instrument called aerofoil was created.

Development of an Ultimate Fluid machine

19th Century Inventions

H F Phillips Otto Lilienthal

History of Airfoil Development

Fascinating Vortex Phenomena : Kutta-Joukowski Theorem

The Joukowsky transformation is a very useful way to generate interesting airfoil shapes.

However the range of shapes that can be generated is limited by range available for the parameters that define the transformation.