dr. r. nagarajan professor dept of chemical engineering iit madras

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 4 - Lecture 15

Momentum Transport: Steady Laminar Flow

PDEs governing steady velocity & pressure fields:

(Navier-Stokes)

and

(Mass Conservation)

“No-slip” condition at stationary solid boundaries:

at fixed solid boundaries

1p v div

v.grad v grad grad v g

0div v

0v

STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

Special cases:

Fully-developed steady axial flow in a straight duct of

constant, circular cross-section (Poiseuille)

2D steady flow at high Re-number past a thin flat plate

aligned with stream (Prandtl, Blasius)

STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

Cylindrical polar-coordinate system for the analysis of viscous flow in a straightcircular duct of constant cross section

Wall coordinate: r = constant = aw (duct radius)

Fully developed => sufficiently far downstream of

duct inlet that fluid velocity field is no longer a

function of axial coordinate z

From symmetry, absence of swirl:

0 ( 0 )wv everywhere not just at r and r a


Conservation of mass ( = constant):

vz independent of z implies:

PDEs required to find vz( r), p(r,z)

Provided by radial & axial components of linear-

momentum conservation (N-S) equations:

1 10r zrv v v

r r r z

0 ( , 0 )r wv everywhere not just at r and a


Pressure is a function of z alone, and if

p = p(z) and vz= vz( r), then:

i.e., a function of z alone (LHS) equals a function of r alone

(RHS)

Possible only if LHS & RHS equal the same constant, say C1

0p

r

1 10 z

z

vpv r g

z r r r

1 1 zz

vpg v r

z r r r


Hence:

New pressure variable P defined such that:

and

, hence P varies linearly with z as:

1

1z

dpg C

dz

1

1 zdvdv r C

r dr dr

zP p g z

)P(z+ z) P(z C z


( / ) /1C dP dz

Integrating the 2nd order ODE for vz twice:

Since vz is finite when r = 0, C3 = 0

Since vz = 0 when

Hence, shape of velocity profile is parabolic:

21

2 3.ln4z

C rv C C r

v

22

1 . 14

wz

w

C a rv r

v a


22 1, / 4w wr a C C a v

Since is a negative constant- i.e., non-

hydrostatic pressure drops linearly along duct:

and


221

( ) . . . 14

w

zw

aP rv r

z a

10,w zr a and v C

1

1.

PC

z

Total Flow Rate:

Sum of all contributions through annular rings

each of area

Substituting for vz(r) & integrating yields:

0

2wa

zm v r r dr


2zv r dr

2 r dr

(Hagen – Poiseuille Law– relates axial pressure drop to

mass flow rate)

Basis for “capillary-tube flowmeter” for fluids of

known Newtonian viscosity

Conversely, to experimentally determine fluid

viscosity

4

.8

wa Pm

v z


Total Flow Rate:

Average velocity, U, is defined by:

Then:

i.e., maximum (centerline) velocity is twice the average

value, hence:

2wU a m

2

2

/1 1. 0 . .2 8

wz

w

maP

U va z

2

( ) 2 1zw

rv r U

a


Wall friction coefficient (non-dimensional):

w wall shear stress

Cf dimensionless coef (also called f Fanning friction factor)

Direct method of calculation:

and

212

wfC

U

z rrz

v v

r z

2

| 2 (1w

w

w rz r aw

r a

d rU

dr a



Hence:

equivalent to:

Holds for all Newtonian fluids

Flows stable only up to Re ≈ 2100

8w

w

U

d

16 / RefC


Re /wUd v


Experimental and theoretical friction coefficients for incompressible Newtonianfluid flow in straight smooth-walled circular duct of constant cross section



Same result can be obtained from overall linear-

momentum balance on macroscopic control volume A

z:

Axial force balance (for fully-developed flow where

axial velocity is constant with z):

Solving for w and introducing definition of P:

Net outflow rate ofNet force on fluid

axial momentum

0 | | 2z z z z w wpA pA g A z a z

.2w

w

a P

z



Configuration and notation: steady flow of an incompressible Newtonian fluidIn a straight circular duct of constant cross section



Above Re = 2100, experimentally-measured friction

coefficients much higher than laminar-flow predictions

Order of magnitude for Re > 20000

Due to transition to turbulence within duct

Causes Newtonian fluid to behave as if non-Newtonian

Augments transport of axial momentum to duct wall


In fully-developed turbulent regime (Blasius):

Cf varies as Re-1/4 for duct with smooth walls

Cf sensitive to roughness of inner wall, nearly

independent of Re


7/4,w p vary as U

2,w p vary as U

Wall friction coefficient (non-dimensional):Effective eddy momentum diffusivity

Can be estimated from time-averaged velocity profile

& Cf measurements

Hence, heat & mass transfer coefficients may be

estimated (by analogy)

For fully-turbulent flow, perimeter-average skin friction

& pressure drop can be estimated even for non-

circular ducts by defining an “effective diameter”:


/t tv

where P wetted perimeter

Not a valid approximation for laminar duct flow

4,eff

Ad

P


Circular jet discharging into a quiescent fluid

Sufficiently far from jet orifice, a fully-turbulent round

jet has all properties of a laminar round jet, but ,

intrinsic kinematic viscosity of fluid

jet axial-momentum flow rate Constant across any plane perpendicular to jet axis

STEADY TURBULENT FLOWS: JETS

1

2

0.0161t

Jv

tv v

J

Laminar round jet of incompressible Newtonian fluid: Far-

Field

Schlichting BL approximation

PDE’s governing mass & axial momentum conservation

in r, , z coordinates admit exact solutions by method of

“combination of variables”, i.e., dependent variables

are uniquely determined by the single independent variable:

STEADY TURBULENT FLOWS: DISCHARGING JETS

11 1/2

/ /.z r

J Ju v and v v

vz vz

1/21/2 /3, .

16

J rr z

v z

Streamline pattern and axial velocity profiles in the far-field of a laminar (Newtonian) or fully turbulent unconfined rounded jet (adapted from Schlichting (1968))


Total mass-flow rate past any station z far from jet mouth

yielding

i.e., mass flow in the jet increases with downstream distance

By entraining ambient fluid while being decelerated (by

radial diffusion of initial axial momentum)

223

18 4

u

0

( ) ( , ). 2zm z v r z r dr

8m

vz


Near-field behavior:

z/dj ≤ 10

Detailed nozzle shape important

“potential core”: within, jet profiles unaltered by

peripheral & downstream momentum diffusion processes

Swirling jets:

Tangential swirl affects momentum diffusion &

entrainment rates

Predicting flow structure huge challenge for any

turbulence model

TURBULENT JET MIXING

Additional parameters:

Initially non-uniform density, viscous dissipation,

chemical heat release, presence of a dispersed phase,

etc.

Add complexity; far-field behavior can be simplified

TURBULENT JET MIXING

dr. r. nagarajan professor dept of chemical engineering iit madras

Documents

straight duct of constant

duct inlet

duct radiusfully

steady axial flow

poiseuille2d steady

mass flow ratebasis

steady laminar flow

analysis of viscous