chapter 9: external* incompressible viscous flows *(unbounded)
TRANSCRIPT
CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS
*(unbounded)
CHAPTER 9: EXTERNAL INCOMPRESSIBLE VISCOUS FLOWS
adverse pressure gradient
leads to separationdifficult to use theory
thicker
•Re = Ux/; Re = Uc/; …•laminar and turbulent boundary layers• displaced inviscid outer flow• adverse pressure gradient and separation
= separated bdy layer
Boundary Layer Provides Missing Link Between Theory and Practice
Boundary layer, , where viscous stresses (i.e. velocity gradient) are important we’ll define as where u(x,y) = 0 to 0.99 U above boundary.
In August of 1904 Ludwig Prandtl, a 29-year old professor presenteda remarkable paper at the 3rd International Mathematical Congress in Heidelberg. Although initially largely ignored, by the 1920s and 1930s the powerful ideas of that paper helped create modern fluid dynamics out of ancient hydraulics and 19th-century hydrodynamics.
(only 8 pages long, but arguably one of the most important fluid-dynamics papers ever written)
• Prandtl assumed no slip condition• Prandtl assumed thin boundary layer region where shear forces are
important because of large velocity gradient• Prandtl assumed inviscid external flow• Prandtl assumed boundary so thin that within it p/y 0; v << u
and /x << /y • Prandtl outer flow drives boundary layer boundary layer can greatly
effect outer outer “inviscid” flow if separates
Theodore von Karman
- 1904 PrandtlFluid Motion with Very Small Friction2-D boundary layer equations
- 1908 BlasiusThe Boundary Layers in Fluids with Little FrictionSolution for laminar, 0-pressure gradient flow
- 1921 von Karman Integral form of boundary layer equations
- 1924 Sir Horace LambHydrodynamics ~ one paragraph on bdy layers
- 1932 Sir Horace LambHydrodynamics ~ entire section on bdy layers
BOUNDARY LAYER HISTORY
INTERNAL EXTERNAL
FULLY DEVELOPED?
CAN BE NEVER
WAKE? NEVER USUALLY - PLATE IS EXCEPTION
THEORY
LAMINAR
PIPES, DUCTS,.. FLAT PLATE & ZERO PRESSURE GRADIENT
GROWING BOUNDARY
LAYER?
NOT WHEN
FULLY
DEVELOPED
ALWAYS
ADVERSE PRESSURE
GRADIENT
PIPE/DUCT=N0
DIFFUSER=YES
PLATE=MAYBE
BODIES=USUALLY
TURBULENT
EXPERIMENT
PIPE (EXAMPLE)
u(r)/Uc/l = (y/R)1/n
PLATE (EXAMPLE)
u(y)/Uo = (y/)1/n
Note – throughout figures theboundary layer thickness*,,
is greatly exaggerated!(disturbance layer*)
Airline industry had to develop flat face rivets.
Re = 20,000
Angle of attack = 6o
Symmetric Airfoil
16% thick
Flat Plate (no pressure gradient)~ what is velocity profile?~ wall shear stress/drag?~ displacement of free stream?~ laminar vs turbulent flow?
Immersed Bodies~ wall shear stress/drag?~ lift?~ minimize wake
breath
FLAT PLATE – ZERO PRESSURE GRADIENT
Laminar Flow/x ~ 5.0/Rex
1/2
THEORY
Turbulent Flow Rextransition > 500,000u(y)/U = (y/)1/7
/x ~ 0.382/Rex1/5
EXPERIMENTAL
“At these Rex numbersbdy layers so thin that displacement effect onouter inviscid layer is small”
No simple theory for Re < 1000;(can’t assume
is thin)
ReL = 10,000 Visualization is by air bubbles see that boundary+ layer, , is thin and that outer free stream is displaced, *, very little.
FLAT PLATE – ZERO PRESSURE GRADIENT
+ Disturbance Thickness, (x) (pg 412); boundary layer thickness, (x) (pg 415)
outside (x), U is constant so P is constant
u(x,y) is not constant, (x) is thin so assume P inside (x) is impressed from the outside
ReL = 10,000 Visualization is by air bubbles see that boundary+ layer, , is thin and that outer free stream is displaced, *, very little.
FLAT PLATE – ZERO PRESSURE GRADIENT
+ Disturbance Thickness, (x) (pg 412); boundary layer thickness, (x) (pg 415)
Rex = Ux/Assume Rextransition ~ 500,000
x
ReL = Ux/
L
SIMPLIFYING ASSUMPTIONS OFTEN MADE FOR ENGINERING ANALYSIS OF BOUNDARY LAYER FLOWS
Development of laminar boundary layer (0.01% salt water, free stream velocity 0.6 cm/s, thickness
of the plate 0.5 mm, hydrogen bubble method).
** * * *
Rex 1000
breath
FLAT PLATE – ZERO PRESSURE GRADIENT: (x)
BOUNDARY OR DISTURBANCE LAYER
(x) *
BOUNDARY OR DISTURBANCE LAYER
Definition: u(x,) = 0.99 of U=U=Ue
(within 1 % of U)
Boundary+ Layer Thickness (x)
+Disturbance
is at y location where u(x,y) = 0.99 U
Because the change in u in the boundary layer takes place asymptotically, there is some indefiniteness in determining exactly.
NOTE: boundary layer is much thicker in turbulent flow.
Blasius showed theoretically for laminar flow that /x = 5/(Rex)1/2
(Rex = Ux/) x1/2
Experimentally found*for turbulent flow that
x4/5
NOTE: velocity gradient at wall (w = du/dy) is significantly greater.
At same x: U/L > U /T At same x: wL < wT
Note, boundary layer is not a streamline!
From theory (Blasius 1908, student of Prandtl):
= 5x/(Rex1/2) = 5x/(U/[x])1/2 = 51/2x1/2/U1/2
d/dx = 5 (/U)1/2 (½) x-1/2 = 2.5/(Rex)1/2
V/U = dy/dxstreamline = 0.84/(Rex1/2)
dy/dxstreamline d/dx so not streamline
Behavior of a fluid particle traveling along a streamline through a boundary layer along a flat plate.
U = U = Ue
ASIDE ~ LAMINAR TO TURBULENT TRANSITION
LAMINAR TO TURBULENT TRANSITION
Emmons spot ~ Rex = 200,000Spots grow approximately linearly downstream at downstream
speed that is a fraction of the free stream velocity.
NOTE: Turbulence is not initiated at Retr all along the width of the plate
Emmons spot, Rex = 400,000 smoke in wind tunnel
Transition not fixed but usually around Rex ~ 500,000 (2x105-3x106, MYO)
For air at standard conditions and U = 30 m/s, xtr ~ 0.24 m
x=0
Turbulent boundary layer is thicker and grows faster.
Nevertheless: Treat transition as it happens all along Rex = 500,000
Experimentally transition occurs around Rex ~ 5 x 105
Water moving around 4 m/s past a ship, transitions after about 0.14 m from the bow,
representing only about 0.1 % of total length of 143 m long ship.
breath
FLAT PLATE – ZERO PRESSURE GRADIENT: *(x)
DISPLACEMENT THICKNESS
*(x)
DISPLACEMENT THICKNESS
Definition: * = 0 (1 – u/U)dy
Displacement Thickness *(x)
* is displacement of outer streamlines due to boundary layer
By definition, no flow passes through streamline, so mass through 0 to h at x = 0is the same as through 0 to h + * at x = L.
x = L
Displacement thickness *
Uh = 0h+* udy = 0h+* (U + u - U)dy
Uh = 0h+* Udy + 0h+* (u - U)dy
Uh = U(h + *) + 0h+*(u - U)dy
-U* = 0h+*(u -U)dy
* = 0h+*(– u/U + 1)dy 0 (1 – u/U)dy
* 0 (1 – u/U)dy 0
(1 – u/U)dy
displacement of outerstreamlines due to (x)
function of x!
breath
Definition: * = 0 (1 – u/U)dy
U*w = 0 (U – u)dyw 0 (U – u)dyw
the deficit in mass flux through area w due to the presence of the boundary layer.
=mass flux passing through an area [*w] in the absence
of a boundary layer
Displacement Thickness *
Definition: * = 0 (1 – u/U)dy
U* = 0 (U – u)dy 0 (U – u[y])dy
Displacement Thickness *
* problem
Displacement Thickness, *, Problem
Suppose given velocity profile:u = 0 from y = 0 to u = Ue for y > .
Show that * =
=
Displacement Thickness * Suppose u = 0 from y = 0 to and u = Ue for y > .
Show that * = * = 0 (1 – u/Ue)dy = 0 (1 – 0/Ue)dy + (1 – Ue/Ue)dy
* = 0 dy =
=
* = Distance that an equivalent inviscid flow is displaced from a solid boundary as a consequence of
slow moving fluid in the boundary layer.
*(x) ~ 1/3 (x)
Blasius* = 1.721x/(Rex)1/2
Laminar flow on flat plate in uniform free stream
Blasius = 5x/(Rex1/2)
Displacement Thickness *
Definition: * = 0 (1 – u/Ue)dy From the point of view of the flow outside the boundary layer, * can be interpreted as the distance that the presence of the boundary layer appears to “displace” the flow outward (hence its name). To the external flow, this streamline displacement also looks like a slight thickening of the body shape.
and *are greatly magnified
*(x)(x)
breath
*(x) EXAMPLE
2-D DUCT
PROBLEM: Find Ue as a function of U, D and *
Ue(x)
Ue(x) = ?
Continuity equation (per W):
{UD = -D2D/2 udy + -D2D/2 Uedy - -D2D/2 Uedy}
UD = -D2D/2 Uedy – {-D2D/2 (Ue-u)dy}
u(y)U f(y) Ue(x)
Continuity equation (per W):
UD = -D2D/2 Uedy – {-D2D/2 (Ue-u)dy}
UD =
-D2D/2 Uedy–{(-D/2) (-D/2+) (Ue-u)dy+(D/2- ) D/2 (Ue-u)dy}
(used approximation that outside boundary layer u =Ue)(note that Ue = function of x but not y)
INLET OF DUCT
Continuity equation (per W):
UD = -D2D/2 Uedy – {(-D/2) (-D/2+) (Ue-u)dy + (D/2- ) D/2 (Ue-u)dy}
UD = -D2D/2 Uedy – Ue{(-D/2) (-D/2+) (1-u/Ue)dy}
+ Ue{(D/2- ) D/2 (1-u/Ue)dy}
INLET OF DUCT*
*
* = 0 (1 –u/Ue)dy 0 (1 – u/Ue)dy
INLET OF DUCT*
*
Continuity equation (per W):
UD = -D2D/2 Uedy–Ue{(-D/2) (-D/2+) (1-u/Ue)dy+(D/2- ) D/2 (1-u/Ue)dy}
-{ * + *}
UD= UeD – 2Ue* = Ue[D – 2*]
Ue(x) = UD/[D – 2*(x)] QED
from Continuity Equation
UD = Ue [D – 2*]
We see that the free stream velocity in the duct is given by the effective decrease in the cross
sectional area due to the growth of the boundary layers, and this decrease in area is measured by the displacement thickness!!!
FLAT PLATE – ZERO PRESSURE GRADIENT: (x)
MOMENTUM THICKNESS
* (x)Momentum Thickness
MOMENTUM THICKNESS
Definition:
= 0 u/U(1 – u/U)dy
Momentum Thickness
(x)
How define drag due to boundary layer?
Momentum Thickness
Definition: = 0 u/Ue(1 – u/Ue)dy
Ue2
w = 0 u(Ue – u)dyw
The momentum thickness, , is defined as the thickness of a layer of fluid, with velocity Ue, for which the momentum flux is equal to the
deficit of momentum flux through the boundary layer. (Mass flux through = 0 udyw)
u
U[(u)([U-u])]xdy
[Flux of momentum deficit]
UU w
= 0.99U (within 1 %) * = 0(1 – u/Ue)dy
0(1 – u/Ue)dy =0 u/Ue(1 – u/Ue)dy
0 u/Ue(1 – u/Ue)dy* & Easier to calculate form data
and more physical significancebut “can be” dependent on
Summary:
breath
Relate to drag
Control volume analysis of drag force on a flat plate due to boundary shear
Relate to Drag
Relate to Drag
PRESSURE IS UNIFORM - NO NET PRESSURE FORCE!- NO DU/DX!
1. From (0,0) to (0,h): V•n = -Uo = -Ue; V = Ue
2. From (0,h) to (L,): streamline, V•n = 0 (no shear)3. From (L, ) to (L,0): V•n = u(x=L,y) 4. From (L, 0) to (0,0): V•n = 0 (shear)
Fx = -Don CV = (d/dt) udVol + u(V•n)dA
D
0(NO BODY FORCES) (STEADY)
Fx = -D = u(V•n)dA
u(V•n)dy(w) = 0
h Uo(-Uo)wdy + 0 u(L,y) u(L,y)wdy
1. From (0,0) to (0,h): V•n = -Uo = -Ue; V = Ue
2. From (0,h) to (L,): no shear, V•n = 03. From (L, ) to (L,0): V•n = u(x=L,y); V = u(x,y) 4. From (L, 0) to (0,0): V•n = 0
-D = - Uo2wh + 0 u(L,y)2wdy
Not so useful because don’t know h as a function of (x)
-D = - Uo2wh + 0 u(x,y)2wdy
Get rid of h by using conservation of mass.
(V•n)dA= 0 0Uowdy + 0u(x,y)wdy = 0
Uoh = 0u(x,y)dy
-D = - Uow 0u(x,y)dy + 0 u(x,y)2wdy
h
-D = - Uow 0(x)u(y)dy + 0
(x) u(x,y)2wdy
D = w [0(x)Uou(x,y)dy - u(x,y)2dy]
D = w [0(x) u(x,y) [Uo - u(x,y)]dy
D w [0 u(x,y) [Uo - u(x,y)]dy
Uo2w [0
[u(x,y)/Uo][1o – [u(x,y)/Uo]]dy
D(x) = Uo2w (x)
(First derived by Von Karman in 1921)
breath
Relate to wall shear stress
D(x) = Uo2w (x)
D(x) = 0x wwdx = Uo
2 w (x)
w(x) = d/dx {Uo2 (x)}
No pressure gradient; dU/dx = 0
w(x)/ = Uo2 d/dx {(x)}
The shear stress at the wall is proportional to the rate of change of momentum thickness with distance.
(Cf =w(x)/[(1/2) Uo2)])
Book ~ derivation includes pressure gradient casew(x)/ = d/dx { Uo(x)
2 (x)} *Uo(x)dUo(x)/dx
NOTE
Class ~ derivation for constant pressure casew(x)/ = Uo
2 d/dx {(x)}
w(x)/ = Uo2 d/dx {(x)}
w(x)/ = d/dx { Uo(x) 2 (x)} *Uo(x)dUo(x)/dx
book
me
Uo = constant; dP/dx = 0
Uo constantdP/dx 0
breath
Estimating (x), *(x), (x), Drag
For laminar / turbulent / and pressure gradient flows
w(x)/ = Uo2 d/dx {(x)}
w(x)/ = d/dx { Ue(x) 2 (x)} *Ue(x)dUe(x)/dx
book
me
w(x)/ = Uo2 d/dx {(x)}
No pressure gradient; dU/dx = 0
w(x)/ = Uo2 d/dx {0 u/Ue(1 – u/Ue)dy
u(x)/U is assumed to be similar for all x and is normally specified as a function of y/(x)
Defining = y/
w(x)/=Uo2d/dx{ 01 u/Ue(1– u/Ue)d}
w(x)/=Uo2d/dx{01 u/Ue(1– u/Ue)d}
Dimensionless velocity profile for a laminar boundary layer:comparison with experiments by Liepmann, NACA Rept. 890,1943. Adapted from F.M. White, Viscous Flow, McGraw-Hill, 1991
y/
turbulent bdy layer velocity profileU = 10 m/s; x1=10cm
00.020.040.060.080.10.120.140.16
600 650 700 750 800 850 900 950 1000 1050
u (cm/sec)
y (cm
)
turbulent bdy layer velocity profileU = 10 m/s; x1=100cm
0
0.2
0.4
0.6
0.8
1
600 650 700 750 800 850 900 950 1000 1050
u (cm/sec)
y (cm
) 0
0.2
0.4
0.6
0.8
1
1.2
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
Turbulent bdy layer* ~ u/Ue = (y/)1/7; Ue = 1000 cm/s
y/(x)
u(x)/U
* du/dy|wall =
Blasius developed an exact solution (but numerical integration was necessary) for laminar flow with no pressure variation.
Blasius could theoretically predict boundary layer thickness (x), velocity profile u(x,y)/U vs y/, and wall shear stress w(x).
Von Karman and Poulhausen derived momentum integral equation (approximation) which can be
used for both laminar (with and without pressure gradient) and
turbulent flow
Von Karman and Polhausen method (MOMENTUM INTEGRAL EQ. Section 9-4)
devised a simplified method by satisfying only the boundary
conditions of the boundary layer flow rather than satisfying Prandtl’s
differential equations for each andevery particle within the boundary layer.
breath
QUESTIONS
? In laminar flow along a plate, (x), (x), *(x) andw(x):(a) Continually decreases(b) Continually increases(c) Stays the same
? In turbulent flow along a plate, (x), (x), *(x) andw(x):(a) Continually decreases(b) Continually increases(c) Stays the same
?At transition from laminar to turbulent flow, (x), (x), *(x) andw(x):(a) Abruptly decreases(b) Abruptly increases(c) Stays the same
forced
natural
6:1 ellipsoid
wall
Turbulent
Laminar
TurbulentLaminar
What is wrong with this figure?
What is wrong with this figure?
Are Antarctic Icebergs Towable Arctic News Record – Summer 1984; 36
Cf = 0.074/Re1/5 (Fox:Cf = 0.0594/Re1/5); Area = 1 km long x 0.5 km wide
sea water = 1030 kg/m3; sea water = 1.5 x 10-6 m2/secPower available = 10 kW; Maximum speed = ?
P = UD104 = U Cf½ U2A = U [0.074 1/5/(U1/5L1/5)]( ½ U2A)
104 = U14/5 [0.074(1.5x10-6)1/5/10001/5] x [½ (1030)(1000)(500)]
U14/5 = 0.03054
U = 0.288 m/s
Rex ~ 3x108
80% of fresh water found in world – Antarctic ice
Biggest problem not meltingbut is = ?
THE END
extra “stuff”
breath
Derivation
Blasius developed an exact solution (but numerical integration was necessary) for laminar flow with no pressure variation.
Blasius could theoretically predict boundary layer thickness (x), velocity profile u(x,y)/U vs y/, and wall shear stress w(x).
Von Karman and Poulhausen derived momentum integral equation (approximation) which can be
used for both laminar (with and without pressure gradient) and
turbulent flow
MOMENTUM INTEGRAL EQUATIONdP/dx is not a constant!
Deriving: MOMENTUM INTEGRAL EQ
so can calculate (x), w.
u(x,y)
Surface Mass Flux Through Side ab
w
Surface Mass Flux Through Side cd
Surface Mass Flux Through Side bc
Surface Mass Flux Through Side bc
Assumption : (1) steady (3) no body forces
Apply x-component of momentum eq. to differential control volume abcd
u
mf represents x-component of momentum fluxFsx will be composed of shear force on boundaryand pressure forces on other sides of c.v.
X-momentumFlux =
cvuVdA
Surface Momentum Flux Through Side ab
uw
X-momentumFlux =
cvuVdA
Surface Momentum Flux Through Side cd
u
U=Ue=U
X-momentumFlux =
cvuVdA
Surface Momentum Flux Through Side bc
u
a-b c-d
c-d b-c
X-Momentum Flux Through Control Surface
IN SUMMARYX-Momentum Equation
X-Force on Control Surface
w is unit width into pagep(x)
Surface x-Force On Side bc
Note that p f(y)inside bdy layer w
w is unit width into pagep(x+dx)
Surface x-Force On Side cd
w
Note that since the velocity gradient goes to zero at the top of the boundary layer, then viscous shears go to zero there (bc).
p + ½ (dp/dx)dx
Surface x-Force On Side bc
Why w and not w (bc)?
pressure
Surface x-Force On Side bc
p + ½ dp/dx along bc
In x-direction: [p + ½ (dp/dx)] wd
-(w + ½ dw/dx]xdx)wdx
Surface x-Force On Side ad
Fx = Fab + Fcd + Fbc + Fad
Fx = pw -(p + [dp/dx]x dx) w( + d) + (p + ½ [dp/dx]xdx)wd - (w + ½ (dw/dx)xdx)wdx
Fx = pw -(p w + p wd + [dp/dx]x dx) w + [dp/dx]x dx w d) + (p wd + ½ [dp/dx]xdxwd) - (w + ½ (dw/dx)xdx)wdx
d <<
=
dw << w
p(x)
** ++
#
#
=
ad = 0
ab -cd bc
U
Divide by wdx
dp/dx = -UdU/dx for inviscid flow outside bdy layer
= from 0 to of dy
-(dp/dx) = -(0dy) (-UdU/dx) = (dU/dx)(0
Udy)
dp/dx = -UdU/dx for inviscid flow outside bdy layer
= from 0 to of dy
p + ½ U2 + gz = constant outside boundary layer (“inviscid”)
dp/dx + UdU/dx = 0 outside bdy layer (dz = 0)
Integration by parts
Multiply by U2/U2 Multiply by U/U
QED
FLAT PLATE – ZERO PRESSURE GRADIENTLAMINAR FLOW
Blasius Theoretical Solution
for u(x,y), (x), *(x), (x)Background
Blasius* (1908) developed an exact solution for laminar flow over a flat plate with no pressure variation. Blasius could
theoretically predict: (x), *(x), (x),
velocity profile u(x,y)/U vs y/,and wall shear stress w(x).
*(first graduate student of Prandtl)
Blasius* (1908) developed an exact solution but required numerical integration,
not good if there is a pressure gradient and not good for turbulent flow
hence we develop the momentum integral equation.
Newton’s Law: F = mau(u/x) + v(u/y) = -(1/) p/x + (2u/x2 + 2u/y2)u(v/x) + v(v/y) = -(1/) p/y + (2v/x2 + 2v/y2)
Cons. of Massu/x + v/y = 0
Boundary Conditions: u = v = 0 at y = 0
Ue outside boundary layer
2-D Governing Equations
Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow
v << u; d/dx << d/dy Conservation of Mass
u/x + v/y = 0
Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat
plate more tractable:
Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow
v << u; d/dx << d/dyConservation of Y-Momentum
u(v/x) + v(v/y) = -(1/) p/y + (2v/x2 + 2v/y2)
p/y 0
Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat
plate more tractable:
Pressure can vary only along boundary layer not through it!
Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow
v << u; d/dx << d/dyConservation of X-Momentum
u(u/x) + v(u/y) = -(1/) p/x + (2u/x2 + 2u/y2)p/y 0
u(u/x) + v(u/y) = -(1/) dp/dx + (2u/x2 + 2u/y2)
u(u/x) + v(u/y) UdU/dx + (1/)(/y)
Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat
plate more tractable:
B.Eq.
Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow
u(u/x) + v(u/y) = (2u/y2)u/x + v/y = 0u = v = 0 at y = 0
u = Ue at y = (outside boundary layer)
With Prandtl’s simplifying assumptions plus nopressure gradient for laminar flow, left the 2-D mass and momentum
equations over flat plate “ripe” for Blasius:
BOUNDARY LAYER EQUATIONS
Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow
By an ingenious coordinate transformation of the boundary layer equation Blasius showed that the dimensionless velocity distribution, u(x,y)/Ue,
was only a function of one composite dimensionless variable, [y/x][Uex/]1/2.
u(x,y)/Ue = f([y/x][Uex/]1/2) = f() = y/
Dimensional analysis = u(x,y)/Ue = f (Uex/, y/x)
Dimensionless velocity profile for a laminar boundary layer:comparison with experiments by Liepmann, NACA Rept. 890,1943. Adapted from F.M. White, Viscous Flow, McGraw-Hill, 1991
y/(x)
From Blasius solution it is found that:
= 5 (x/U)1/2 or /x = 5(Rex)-1/2
* = (x/U)1/2 or */x = 1.72(Rex)-1/2
= (x/U)1/2 or /x = 0.664(Rex)-1/2
w = 0.332 U3/2 (/x)
u(x,y)/Ue =1 at y(Ue/[x])1/2 = u(x,y)/Ue = 0.99 at y(Ue/[x])1/2 5.0
u(x,y)