Scalar-Vector Interaction in Animating but non-alive Fields ……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Special Vector Calculus for Fluid Flow Field

Material Derivatives

• A fluid element, often called a material element.

• Fluid elements are small blobs of fluid that always contain the same material.

• They are deformed as they move but they are not broken up.

• The temporal and spatial change of the flow/fluid quantities is described most appropriately by the substantial or material derivative.

• Generally, the substantial derivative of a flow quantity , which may be a scalar, a vector or a tensor valued function, is given by:

dTdtt

TDT

Vddtt

VVD

ddtt

D

Understanding of Material Derivative of A Scalar Field

• The operator D represents the substantial or material change of the quantity T(t:x,y,z).

• The first term on the right hand side of above equation represents the local or temporal change of the quantity T(t:x,y,z) with respect to a fixed position vector x.

• The operator d symbolizes the spatial or convective change of the same quantity with respect to a fixed instant of time.

• The convective change of T(t:x,y,z) may be expressed as:

dTdtt

TDT

dzz

Tdy

y

Tdx

x

TdT

TxddT

Understanding of Material Derivative of A Vector Field

V as the gradient of the vector field which is a second order tensor.

Vddtt

VVD

VxdVd

z

w

z

v

z

u

y

w

y

v

y

ux

w

x

v

x

u

V

Rate of Change of Material Derivative of A Vector Field

• Dividing above equation by dt yields the acceleration vector.

Vddtt

VVD

dt

Vd

t

V

dt

VD

Vdt

xd

t

V

dt

VD

The differential dt may symbolically be replaced by Dt indicating the material character of the derivatives.

VVt

V

Dt

VD

VVt

Va

Material or substantial acceleration

Component of Material Acceleration

VVt

Va

z

uw

y

uv

x

uu

t

uax

z

vw

y

vv

x

vu

t

vay

z

ww

y

wv

x

wu

t

waz

z

w

z

v

z

u

y

w

y

v

y

ux

w

x

v

x

u

wvut

Va

Visualization of Material Acceleration

Visualization of Material Acceleration

Variety of Pumps due to Various Components of Material Acceleration

Turbo-Machines & GEOMETRIES

Variety of Compressors due to Various Components of Material Acceleration

Local Accounting of Accelerating vector field

• An essential responsibility of an engineer is to develop a measure to account for an accelerating flow field.

• A simplest but comprehensive measure of accounting is essential.

• What is it?

• Can we Define some measure?

• If A is thought to be a flux vector, a net flux out of the volume may be expressed as

The is named as Divergence of a vector field:

Divergence is an operator that measures the magnitude of a vector field's source or sink at a given point.

v

dsnAMeasureLocal

v

ˆ.lim

0

dsnAfluxnet ˆ.

A

.

Divergence of Velocity in Various Coordinate Systems

In different coordinate systems:

• Cartesian :

• Cylindrical:

• Spherical:

x

w

y

v

x

uV

z

zr

z

uu

rr

uV

1

.

sin

sin

1

sin

11 2

2

u

r

u

rr

ur

rV r

Divergence Rules

Some “divergence rules”: VVV

...

BABA

...

AcAc

..

What is the Divergence Effect in Fluid Dynamics?

• Think of a vector field as a velocity field for a moving fluid.

• The divergence measures the expansion or contraction of the fluid.

• A vector field with constant positive or negative value of divergence.