Momentum Transport
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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 101 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 101 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