Method to Use Conservations Laws in Fluid Flows……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Mathematics of Reynolds Transport Theorem

Deformation of a differential volume at differentinstant of time

The Field quantity

• The field quantity f(x,t) may be a zeroth, first or second order tensor valued function.

• Namely, as mass, concentration, velocity vector, and stress tensor.

• A fluid element of given initial volume (dV0) mya change its volume and/or change it surface (ds0) with a given time, while moving through the flow field.

• This is due to various experiences by the element namely, dilatation, compression and deformation.

• Let us consider the same fluid particles at any time and therefore, it is called the material volume.

The volume integral of the quantity f(x,t):

This is a function of time only.

The integration must be carried out over the varying volume V(t).

The material change of the quantity F(t) is expressed as:

Since the shape of the volume V(t) changes with time, the differentiation and integration cannot be interchanged.

tV

dVtxftF ,

tV

dVtxfDt

D

Dt

tDF,

This analogy permits the transformation of the time dependent volume V(t) into the fixed volume V0 at time t = 0 by using the Jacobian transformation function:

odVJtdV

With this operation it is possible to interchange the sequence of differentiation and integration:

The chain differentiation of the expression within the parenthesis results in

0

0,V

JdVtxfDt

D

Dt

tDF

0

0

,

V

dVDt

JtxfD

Dt

tDF

0

0,,

V

dVDt

DJtxf

Dt

txDfJ

Dt

tDF

tV

dVtxfDt

D

Dt

tDF,

Introducing the material derivative of the Jacobian function and simplifying based on derivatives of transformation functions

JvJvt

J

Dt

DJ ..

0

0.,,

V

JdVvtxfDt

txDf

Dt

tDF

This equation permits the back transformation of the fixed volume integral into the time dependent volume integral.

tV

dVvtxfDt

txDf

Dt

tDF

.,,

tV

dVvtxftxfvt

txf

Dt

tDF

.,,.,

The chain rule applied to the second and third term yields:

tV

dVvtxft

txf

Dt

tDF

,.,

The second volume integral in above equation can be converted into a surface integral by applying the Gauss' divergence theorem:

tstV

dsnvtxfdVt

txf

Dt

tDFˆ.,

,

dVvtxfdVt

txf

Dt

tDF

tVtV

,.

,

This Equation is valid for any system boundary with time the dependent volume V(t) and surface s(t) at any time.Also valid at the time t = t0 , where the volume V = VC and the surface s = sC assume fixed values. We call VC and sC the control volume and control surface.These control surfaces can be inlets or exits.

tstV

dsnvtxfdVt

txf

Dt

tDFˆ.,

,

tststV inletexit

dsnvtxfdsnvtxfdVt

txf

Dt

tDFˆ.,ˆ.,

,

m

j ts

n

i tstV jinletiexit

dsnvtxfdsnvtxfdVt

txf

Dt

tDF

11,,

ˆ.,ˆ.,,

Thermodynamic form of RTT

n

iiinlet

n

iiexit

CMCV

FFdt

tdF

dt

tdF

1,

1,

V

CV dVtxfF ,

inletinlet A

in

A

inlet AdvfAdvfF ..

exitexit A

exit

A

exit AdvfAdvfF ..

Steady State Steady Flow Thermodynamic Model

n

iiinlet

n

iiexit

CV

FFdt

tdF

1,

1,

V

CV dVxfF

inletinlet A

in

A

inlet AdvfAdvfF ..

exitexit A

exit

A

exit AdvfAdvfF ..

Uniform State Uniform Flow Thermodynamic Model

n

iiinlet

n

iiexit

CMCV

FFdt

tdF

dt

tdF

1,

1,

V

CV dVtfF

inletinlet A

in

A

inlet AdvfAdvfF ..

exitexit A

exit

A

exit AdvfAdvfF ..

Mass Flow Balance in Stationary Frame of Reference

• The conservation law of mass requires that the mass contained in a material volume V=V(t), must be constant:

tV

dVm

Consequently, above equation requires that the substantial changes of the above mass must disappear:

Mass contained in a material volume

0

tV

dVDt

D

Dt

Dm

Using the Reynolds transport theorem , the conservation of mass, results in:

0.

tVtV

dVvt

dVDt

D

This integral is zero for any size and shape of material volume. Implies that the integrand in the bracket must vanish identically. The continuity equation for unsteady and compressible flow is written as:

0.

vt

This Equation is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:

0

i

i

x

v

t

0

3

3

2

2

1

1

x

v

x

v

x

v

t

Continuity Equation in Cylindrical Polar Coordinates

• Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates.

• In this system coordinates for a point P are r, and z.

The velocity components in these directions respectively are vr ,v and vz. Transformation between the Cartesian and the polar systems is provided by the relations,

22 yxr

x

y1tan

The gradient operator is given by,

zz

rrr

ˆˆ1ˆ

As a consequence the conservation of mass equation becomes,

0

11

z

vv

rr

vr

rtzr

Continuity Equation in Cylindrical Polar CoordinatesSpherical polar coordinates are a system of curvilinear coordinates

that are natural for describing atmospheric flows.Define to be the azimuthal angle in the x-y -plane from the x-axis with 0 < 2 . to be the zenith angle and colatitude, with 0 < r to be distance (radius) from a point to the origin.

The spherical coordinates (r,,) are related to the Cartesian coordinates (x,y,z) by

222 zyxr

x

y1tan

r

z1cos

or

cos

sinsin

sincos

z

ry

rx

The gradient is

ˆ1ˆsin

1ˆ

rrrr

As a consequence the conservation of mass equation becomes,

0

sin

1sin

sin

11 2

2

v

r

v

rr

vr

rtr