Tamed Effect of Normal Stress in VFF…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Negligible Bulk Viscosity Model for Momentum Equations

Philosophy of Science

• The goal which physical science has set itself is the simplest and most economical abstract expression of facts.

• The human mind, with its limited powers, attempts to mirror in itself the rich life of the world, of which it itself is only a small part…….

• In reality, the law always contains less than the fact itself.• A Law does not reproduce the fact as a whole but only in

that aspect of it which is important for us, the rest being intentionally or from necessity omitted.

Thus spake : Ernst Mach

• In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations.

• It is highly an economical reason to think that the fastness of a flying machine is described in terms of velocity (km/hr) !!!!

Bulk Viscosity Coefficient

• The second viscosity coefficient is still a controversial quantity.

• Truly saying, may not even be a thermodynamic property, since it is found to be frequency-dependent.

• The disputed term, divv, is almost always so very small that it is entirely proper simply to ignore the effect of altogether.

• Collect more discussions on This topic and submit as an assignment: Date of submission: 22nd September 2014.

The Navier-Stokes Equations : Incompressible Flow

• The desired momentum equation for a general linear (newtonian) viscous fluid is now obtained by substituting the stress relations, into Newton's law.

• The result is the famous equation of motion which bears the names of Navier (1823) and Stokes (1845).

• In scalar form, we obtain

z

u

x

w

zx

v

y

u

yv

x

u

xx

pg

Dt

Dux

2

y

w

z

v

zv

y

v

yy

u

x

v

xy

pg

Dt

Dvy

2

vz

w

zy

w

z

v

yz

u

x

w

xz

pg

Dt

Dwz

2

These are the Navier-Stokes equations, fundamental to the subject of viscousfluid flow. Considerable economy is achieved by rewriting them as a single vector equation, using the indicia1 notation:

vx

v

x

v

xpg

Dt

vDij

i

j

j

i

j

Incompressible Flow

• If the fluid is assumed to be of constant density, divv vanishes due to the continuity equation.

• The vexing coefficient disappears from Newton's law.

• NS Equations are not greatly simplified, though, if the first viscosity is allowed to vary with temperature and pressure.

i

j

j

i

j x

v

x

v

xpg

Dt

vD

vpgDt

vD

2 vp

gDt

vD

2

• This leads to assumption of is constant, many terms vanish.

• A much simpler Navier-Stokes equation for constant viscosity is

Incompressible NS Equations in Cylindrical Coordinate system

222

2

2

2

22

2

2

211

r

v

rr

v

r

v

z

vv

rr

v

r

pg

r

v

z

vv

v

r

v

r

vv

t

v

rrrrrx

rz

rrr

r

Navier- Stokes equation in r-direction:

222

2

2

2

22

2

211

r

v

rr

v

r

v

z

vv

rr

v

r

pg

r

vv

z

vv

v

r

v

r

vv

t

v

ry

rzr

Navier- Stokes equation in -direction:

Navier- Stokes equation in z-direction:

rr

v

z

vv

rr

v

z

pg

z

vv

v

r

v

r

vv

t

v

zzzzy

zz

zzr

z

2

2

2

2

22

2 11

Made Easy

• Incompressibility is an excellent point of departure in the theory of incompressible viscous flow.

• It is essential to remember that it assumes constant viscosity.

• For non-isothermal flows, it may be a rather poor approximation.

• This approximation is highly objectionable, particularly for liquids, whose viscosity is often highly temperature-dependent.

• For gases, whose viscosity is only moderately temperature-dependent, this is a good approximation

• This fails only when compressibility becomes important, i.e., when v 0.

The Navier-Stokes equations, though fundamental and rigorous, are nonlinear,nonunique, complex, and difficult to solve.

Special Case: Incompressible Euler Equation of Motion

• For the special case of steady inviscid flow (no viscosity),

p

gvvt

v

This equation is called Euler equation of motion. Its index notation is:

ii

j

ij

i

x

pg

x

vv

t

v

1

Replacing the convective by the following vector identity

vvvv

vv

2

Identify that the convective acceleration is expressed in terms of the gradient of the kinetic energy

22

2vvv

The second term which is a vector product of the velocity and the vorticity vector ×v.

If the flow field under investigation allows us to assume a zero vorticity within certain flow regions, then it is possible to assign a potential to the velocity field that significantly simplifies the equation system. This assumption is permissible for the flow region outside the boundary layer.

Simplification of Body Forces

• First rearrange the gravitational acceleration vector by introducing a scalar surface potential z.

• The gradient of z has the same direction as the unit vector in – x3 direction.

• Furthermore, it has only one component that points in the negative x3-direction.

• As a result, we may write .

zggkg ˆ

Thus, the Euler equation of motion assumes the following form:

p

gvvt

v

p

zgvvvv

t

v

2

vvp

gzv

t

v

2

2

Above equation shows that despite the inviscid flow assumption, it contains vorticities that are inherent in viscous flows but special in inviscid flows.The vortices cause additional entropy production in inviscid flows. This can be better explained using the first law of thermodynamics.