Transcript Momentum Transport: Steady Laminar Flow

Advanced Transport Phenomena
Module 4 - Lecture 15
Momentum Transport: Steady Laminar Flow
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
STEADY LAMINAR FLOW OF INCOMPRESSIBLE
NEWTONIAN FLUID
PDEs governing steady velocity & pressure fields:
v.grad v  
and
1

grad p  v div  grad v   g (Navier-Stokes)
div v  0
(Mass Conservation)
“No-slip” condition at stationary solid boundaries:
v0
at fixed solid boundaries
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)
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Cylindrical polar-coordinate system for the analysis of viscous flow in a straight
circular duct of constant cross section
FLOW IN A STRAIGHT DUCT OF CIRCULAR
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:
v  0 everywhere (not just at r  0 and r  aw )
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Conservation of mass (  = constant):
1 
1 

 v    vz   0
 rvr  
r r
r 
z
vz independent of z implies:
vr  0 (everywhere, not just at r  0 and aw )
PDEs required to find vz( r), p(r,z)
 Provided by radial & axial components of linear-
momentum conservation (N-S) equations:
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
p
0
r
 1   vz  
1 p
0
 v
r
   gz
 z
 r r  r  
 Pressure is a function of z alone, and if
 p = p(z) and vz= vz( r), then:
 1   vz
1 p
 gz  v 
r
 z
 r r  r



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
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Hence:
1 dp
 g z  C1
 dz
1 d  dvz
v
r
r dr  dr

  C1

New pressure variable P defined such that:
P  p   gz z
and
C1  (dP / dz) /  , hence P varies linearly with z as:
P(z+z)  P(z)  C  z
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Integrating the 2nd order ODE for vz twice:
C1r 2
vz 
 C2  C3 .ln r
4v
 Since vz is finite when r = 0, C3 = 0
 Since vz = 0 when
r  aw , C2  C1aw2 / 4v
 Hence, shape of velocity profile is parabolic:
2

 r  
Ca
vz  r   
. 1    
4v   aw  


2
1 w
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Since r  aw and vz  0, C1 is a negative constant- i.e., nonhydrostatic pressure drops linearly along duct:
1  P 
 C1  .
  z 
and
2

 r  
1  P  a w
vz ( r )  . 
 . . 1    
4  z     aw  


2
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Total Flow Rate:
 Sum of all contributions vz 2 r dr through annular rings
each of area 2 r dr

aw
m    vz  r 2 r dr
0
Substituting for vz(r) & integrating yields:
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION

m
 aw4  P 
. 

8 v  z 
(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
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Total Flow Rate:
 Average velocity, U, is defined by:
U  a
2
w

m
Then:


m
/

2

 1
a
1

P


w
U   2   .vz  0   . 
.

 aw
2
8  z  
i.e., maximum (centerline) velocity is twice the average
value, hence:
  r 2 
vz (r )  2U 1    
  aw  
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Wall friction coefficient (non-dimensional):
tw  wall shear stress
Cf  dimensionless coef (also called f  Fanning friction
tw
factor)
Cf 
1
U 2
Direct method of calculation: 2
 vz vr 
t rz   



r

z


and
2


 r  
d



t w  t rz |r aw   
2U (1    
dr 
aw   


  r aw
 
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Wall friction coefficient (non-dimensional):
Hence:
8U
tw 
dw
equivalent to:
C f  16 / Re
Holds for all Newtonian fluids
Re  Udw / v
Flows stable only up to Re ≈ 2100
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Wall friction coefficient (non-dimensional):
Experimental and theoretical friction coefficients for incompressible Newtonian
fluid flow in straight smooth-walled circular duct of constant cross section
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Wall friction coefficient (non-dimensional):
 Same result can be obtained from overall linear-
momentum balance on macroscopic control volume A
  z:
 Net outflow rate of 

   Net force on fluid 
axial momentum 
 Axial force balance (for fully-developed flow where
axial velocity is constant with z):
0   pA |z   pA |z z   g z Az  t w 2 aw z
 Solving for tw and introducing definition of P:
aw  P 
t w  .

2  z 
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
Wall friction coefficient (non-dimensional):
Configuration and notation: steady flow of an incompressible Newtonian fluid
In a straight circular duct of constant cross section
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Wall friction coefficient (non-dimensional):
 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
wall
transport of axial momentum to duct
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 In fully-developed turbulent regime (Blasius):
Cf varies as Re-1/4 for duct with smooth walls
t w ,  p vary as U 7/4
Cf sensitive to roughness of inner wall, nearly
independent of Re
t w ,  p vary as U 2
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
 Wall friction coefficient (non-dimensional):
 Effective eddy momentum diffusivity vt  t / 

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 noncircular ducts by defining an “effective diameter”:
FLOW IN A STRAIGHT DUCT OF CIRCULAR
CROSS-SECTION
deff
4A

,
P
where P  wetted perimeter
 Not a valid approximation for laminar duct flow
STEADY TURBULENT FLOWS: JETS
 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 vt  v,
intrinsic kinematic viscosity of fluid
J
vt  0.0161 

1
2
 J  jet axial-momentum flow rate

Constant across any plane perpendicular to jet axis
STEADY TURBULENT FLOWS: DISCHARGING
JETS
 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
1
 J /   
 J /   

 and v  vr . 
u  vz 
 vz

 vz 


are uniquely determined by the single independent variable:
1/2
1/2
 3  J /  r
  r, z   
.

v
z
 16 
1
1/2
STEADY TURBULENT FLOWS: DISCHARGING
JETS
Streamline pattern and axial velocity profiles in the far-field of a laminar
(Newtonian) or fully turbulent unconfined rounded jet (adapted from Schlichting
(1968))
STEADY TURBULENT FLOWS: DISCHARGING
JETS
2 2
3   
u
1  
8 
4 
Total mass-flow rate past any station z far from jet mouth

m( z )    vz (r , z ).  2 r dr 
yielding
0
m

 8 vz
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)
TURBULENT JET MIXING
 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