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 v x (x,n) velocity field t nx (x,0) = t w (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 => v n (x,0) = 0 v x (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: t
w
w
v x
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:
p
1 2
U
2
p
and Skin-friction coefficient
C f
t
w
1 2
U
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
1 2 t
w v
2 For an incompressible fluid (
e
), in the absence of gravitational body-force effects, Bernoulli’s equation yields:
p
0
p
1 2
U
2
e
1 2
v
2 , Reflects negligibility of viscous dissipation far from surface
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS This equation implies that:
v U
C p
1/2 and hence:
c f
C f
1
C p
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Experimentally determined angular dependence of the skin-friction and pressure- coefficients around a circular cylinder in a cross-flow at Re= 1.7x10
5
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Total (net) drag force D’ per unit length of cylinder: Reference Force: '
D ref
1 2
U A
'
proj
, 2 where projected area of cylinder per unit length and
proj cylinder
cylinder
d w D
' 1 2 2
U d w
Calculated from C p , C f data
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS By projecting pressure & shear forces in direction of approach flow: and
D
' 2 0
cylinder p
0
C p cos w
sin
cos
C f
d w
2
d
, (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:
D cylinder
D form
D 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 6
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 6
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 v n and v x vanish at solid surface Thin transition layer near surface across which v x 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
v x
/
n
“boundary layer”, velocity gradients 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 Re 1/2 >>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: v n << v x Momentum diffusion important, but only in normal direction ( t n x >> t xx ) Pressure at any streamwise location x is nearly constant – i.e., p ≈ p e (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 cylinder
D
D
form
1 2
D
' 2
U d w friction
Newtonian incompressible fluid flow past a flat plate; configuration, nome nclature, 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 v x ≡ u(x,y), v y ≡ v(x,y)
u
x
v
y
0 (
mass
),
u
u
x
v
u
y
v
2
u
y
2 (
x
momentum
),
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Subject to boundary conditions:
u
,
,
y
U
,
U
,
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:
u U
fct
1 1 2
y
.
x Ux
1/2
v
fct
1 and
U v
1/2
fct
2 1 2
y
.
x Ux
1/2
v fct
2
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Blasius derived & numerically solved nonlinear ODE governing
f
0
u U d
, 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: = local BL thickness = y-location at which u/U = 0.99 occurs at 5 Therefore: 5
x
U x 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: t
w
0.332
U
2 .
Ux v
1/2 Local dimensionless skin-friction coefficient c f by: 1/2 1 2 t
w U
2
c f
0.664
Ux v
given Total friction drag coefficient:
c f
1 2
U A w
) 2 (for plate of finite length L, set x = L) 1.328
1/2
FLOW OVER A SOLID WALL: SURFACE MOMENTUM-TRANSFER COEFFICIENTS Laminar BL on a flat plate: Effect of “blowing” or “suction” through porous solid wall: c f values are modified Blowing can reduce skin-friction drag
c f c f
0 .
), where (c f ) 0 no-blowing momentum-transfer coefficient, and F(blowing)
v w w
u c v w w
m
''
w e e f
,0 )
CONSERVATION EQUATION GOVERNING VELOCITY AND PRESURE FIELDS Navier-Stokes (linear momentum conservation) law:
v
t
grad
p
div
2
Def v
2 3
div
g
Nonlinear vector PDE Equivalent to 3 independent, scalar 2 nd order PDEs neglected Total mass conservation (“continuity”):
t
div
v
CONSERVATION EQUATION GOVERNING VELOCITY AND PRESURE FIELDS Conservation equations provide 4 PDEs for 5 fields:
v
scalar fields), p, (3 Hence, necessary to specify an EOS for closure
div
v
0 “Caloric” EOS: h as a function of T, p 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: G n normal component of mass flux t tangential plane Mass balance: 0 Momentum Balance:
G v n n
Tangential linear momentum:
n
t nt
t
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: 0, 0, 0, 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