Transcript Momentum Transport

Advanced Transport Phenomena
Module 4 - Lecture 14
Momentum Transport: Flow over a Solid Wall
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Applications:
 Design of automobiles
 Design of aircraft, etc.
 Property of interest:
 Momentum exchange between surface & surrounding
fluid
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Associated net force

“drag” in streamwise direction

‘lift” in direction perpendicular to motion

Obtained by solving relevant conservation equations,
subject to relevant boundary conditions, or

By experiments on full-scale or small-scale models
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Momentum exchange between the moving fluid and a representative
segment of a solid surface (confining wall or immersed body)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 X  approach stream direction
 x  distance along surface
 n  distance normal to surface
 p (x,0)  local pressure
 vx (x,n)  velocity field
 tnx (x,0) = tw(x)  local wall shear stress
 Associated momentum exchange:
 Force on fluid
 Equal & opposite force on solid
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Solid surface motionless => vn (x,0) = 0
 vx (x,0) ≠ 0 => nonzero “slip” velocity
 However, experimentally: local tangential velocity of
fluid = that of solid, i.e., 0, under continuum conditions
 Wall shear stress depends on local fluid-deformation rate:
 vx 
t w ( x)   w 


n

 n 0
 Can
be determined if local normal gradient of
tangential fluid velocity can be measured
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Dimensionless local momentum transfer coefficients:
 Pressure coefficient:
and
p ( x,0)  p
C p ( x) 
1
 U 2
2
 Skin-friction coefficient
C f ( x) 
t w  x
1
U 2
2
 Measured or predicted
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Alternative definition of skin-friction coefficient:
 In terms of properties at the edge of momentum transfer
boundary layer
C f ( x) 
t w ( x)
1
 e vx2,e  x 
2
 For an incompressible fluid (   e ), in the absence of
gravitational
yields:
body-force
effects,
Bernoulli’s
equation
1
1 2
2
p0  p  U  pe ( x)   vx,e ,
2
2
 Reflects negligibility of viscous dissipation far from
surface
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
This equation implies that:
v x ,e ( x )
U
 1  C p 
1/2
and hence:
c f ( x) 
C f ( x)
1  C p ( x)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Experimentally determined angular dependence of the skin-frictionand pressure- coefficients around a circular cylinder in a cross-flow
at Re= 1.7x105
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Total (net) drag force D’ per unit length of cylinder:
 Reference Force:
1
'
Dref  U 2 Aproj
,
2
'
where projected area of cylinder per unit length
A 
'
proj cylinder
and
 dw
 CD cylinder  1
2
 Calculated from Cp, Cf data
D'
U 2 d w
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 By projecting pressure & shear forces in direction of
approach flow:


dw
D '  2    p   cos   t w   sin   d 
2
0

and

 CD cylinder   C p   cos   C f ( ) sin  d ,
0
(polar angle expressed in radians)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 cos  term  from (locally normal) pressure force (“form”
drag)
 sin  term  from (locally tangential) aerodynamic shear
force (“friction” drag)
 Thus, drag coefficient may be split into:
 CD cylinder  CD  form  CD  friction
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Experimental values for the overall drag coefficient (dimensionless
total drag) for a cylinder (in cross-flow), over the Reynolds’ number
ranger 101  Re  106
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Experimental values for the overall drag coefficient (dimensionless
total drag) for a sphere over the Reynolds’ number range 101  Re  106
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Asymptotic theories: Re >> 1, Re << 1
 Re >> 1 case is of greatest engineering interest
 e.g., flow past flat plate at zero incidence
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Momentum Diffusion Boundary Layer Theory: Laminar
Flow Past Flat Plate at Zero Incidence
 1904: L Prandtl

large but finite Reynolds number

vn and vx vanish at solid surface

Thin transition layer near surface across which vx
abruptly drops to zero
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Momentum Diffusion Boundary Layer Theory: Laminar
Flow Past Flat Plate at Zero Incidence

Inside this “boundary layer”, velocity gradients
vx / n large enough to make momentum diffusion
important (though  is small)

Exterior: inviscid region
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Division of flow field at Re1/2 >>1 into an inviscid “outer” region and a thin
tangential momentum diffusion boundary layer (BL)(after L. Prandtl).
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Momentum Diffusion Boundary Layer Theory: Laminar
Flow Past Flat Plate at Zero Incidence
 1904: L Prandtl

For Re >> 1, within the BL:
 vn
<< vx
 Momentum
diffusion important, but only in normal
direction (tnx >> txx)
 Pressure
at any streamwise location x is nearly
constant– i.e., p ≈ pe(x)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Momentum Diffusion Boundary Layer Theory: Laminar
Flow Past Flat Plate at Zero Incidence

BL equations therefore simplified, solutions to match
inner behavior of external inviscid flow
 e.g.,
2D steady flow of incompressible constant-
property Newtonian fluid past a semi-infinite flat
plate at zero incidence (Blasius, 1908)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
D'
CD cylinder 

C

 D cylinder CD  form1 C2D  friction
2
U d w
Newtonian incompressible fluid flow past a flat plate; configuration, nomenclature, and coordinate system
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Laminar BL on a flat plate:
 For thin flat plate, pressure constant everywhere => no
need for y-momentum equation
 2 scalar PDE’s governing vx ≡ u(x,y), vy ≡ v(x,y)
u v

0
x y
u
u
 2u
u v
v 2
x
y
y
(mass),
( x  momentum),
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Subject to boundary conditions:
u  , y   U ,
u  x,    U ,
u ( x,0)  0,
v( x,0)  0,
 Solved by Blasius using “combination of variables”
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Laminar BL on a flat plate:
 Blasius’ solution:
and
 1 y  Ux 1/2 
u
 fct1 
.     fct1  


U
2
x
v




1/2
v  Ux 
 
U v 
 1 y  Ux 1/2 
 fct2 
.
 fct2  
 2 x  v  


FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Blasius derived & numerically solved nonlinear ODE
governing

u( )
f    
d ,
U
0
and constructed tangential fluid-velocity profiles
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
Laminar BL on a flat plate:
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Laminar BL on a flat plate:
  ( x) = local BL thickness = y-location at which u/U =
0.99  occurs at   5
 Therefore:
 Ux 
  5x  
 v 
1/2
(grows as square root of distance x from LE of plate)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Laminar BL on a flat plate:
 Wall shear stress:
 Ux 
t w ( x)  0.332 U .  
 v 
1/2
2
 Local dimensionless skin-friction coefficient cf given
by:
tw
 Ux 
 c f ( x)  0.664  
1
 v 
U 2
2
1/2
 Total friction drag coefficient:
D(both sides )
1.328
cf 

1/2
1
2
Ux
/
v


U Aw
2
(for plate of finite length L, set x = L)
FLOW OVER A SOLID WALL: SURFACE
MOMENTUM-TRANSFER COEFFICIENTS
 Laminar BL on a flat plate:
 Effect of “blowing” or “suction” through porous solid wall:

cf values are modified

Blowing can reduce skin-friction drag
c f ( x / L,Re,...)  c f ( x / L,Re,...)  . F (blowing ),
0
where (cf)0  no-blowing momentum-transfer coefficient, and
F(blowing) function of dimensionless variable
 w vw
(  w vw  m
''
w=
eue c f ,0
local mass injection rate)
CONSERVATION EQUATION GOVERNING
VELOCITY AND PRESURE FIELDS
 Navier-Stokes (linear momentum conservation) law:
2
 v



   v.grad v   grad p  div  2 Def v    div v  I    g
3
 t



 Nonlinear vector PDE
 Equivalent to 3 independent, scalar 2nd order PDEs
 Includes “Stokes’ Postulate”: bulk viscosity
neglected
 Total mass conservation (“continuity”):

 v.grad     div v
t
 can be
CONSERVATION EQUATION GOVERNING
VELOCITY AND PRESURE FIELDS
 Conservation equations provide 4 PDEs for 5 fields: v (3
scalar fields), p, 
 Hence, necessary to specify an EOS for closure
 Unless  is constant (incompressible flow;
div v  0)
 “Caloric” EOS: h as a function of T, p

In addition to usual p (  , T ),  as a function of local
state variables
 Turbulent flows:
 Conservation equations are time-averaged
  replaced by
  t
TYPICAL BOUNDARY CONDITOINS
By applying a “pillbox” control volume to straddle a
moving interface, we can write:
Gn  normal component of mass flux
t  tangential plane
Mass balance:
Gn   0
Momentum Balance:
Gn  vn     p   t nn 
Tangential linear momentum:
Gn  vt   t nt 
TYPICAL BOUNDARY CONDITOINS
 These conservation equations allow:
 Discontinuity in normal component of velocity,
 Discontinuity in pressure across interface,
 Discontinuity in tangential velocity (“slip”) across
interface
 Thus, the “classical” boundary conditions:
vn   0, vt   0,  p   0, t nt   0
are only sometimes true.
TYPICAL INITIAL CONDITOINS
 State of independent field variables at t = 0
 Start-up of a chemical reactor, separator, etc.
 The present, if we want to predict future (e.g., weather,
climate)
 Governing conservation equations are first-order in time
 Invariant wrt shift in origin (zero point) chosen for time
 Principal of “local” action in time (determinacy)
 Future cannot influence present!
 Only applies in time-domain, not space
SOLUTION METHODS
 Coupled PDEs + bc’s + ic’s need not always be solved to
extract valuable information
 e.g., similitude analysis
 Only relatively simple fluid-dynamic problems need to be
solved to interpret instrument readings
 e.g., flowmeters
 Mathematical solutions have become possible with advent
of powerful digital computers
 Computational fluid mechanics, CFD
 Discretizing by finite-difference, finite-element methods
SOLUTION METHODS
 Modularization:
 In sub-regions, explicit results may be possible in
terms of well-known special functions

e.g., Bessel functions, Legendre polynomials
 Numerical:
 Reduce problem to solution of one lor more nonlinear
ODEs
 Then solve numerically
SOLUTION METHODS
“Road map” of common methods of solution to problems in transport (convection
/diffusion ) theory
SOLUTION METHODS
 Results should be independent of method chosen
 But effort should be minimized!
 Idealizations of complex problems serve a purpose
 Capture concepts
 Bring out qualitative features
 Sanity check on more complex predictions
Thank You