2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary layer Equations Contents:
– Boundary Layer Equations;
– Boundary Layer Separation;
– Effect of londitudinal pressure gradient on boundary layer evolution
– Blasius Solution
– Integral parameters: Displacement thickness and momentum thickness
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Laminar Thin Boundary Layer Equations (<<x) over flat plate
Steady flow, constant and . Streamlines slightly divergent0 yp
dxdpxp e
2
2
2
21
y
u
x
u
x
p
y
uv
x
uu
2D Navier-Stokes Equations along x direction:
Compared with 2
2
y
u
dxdpe
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Laminar Thin Boundary Layer Equations (<<x) over flat plate
2
21
y
u
dx
dp
y
uv
x
uu e
Laminar thin boundary layer equations (<<x) for flat plates
pe external pressure, can be calculated with Bernoulli’s Equation as there are no viscous effects outside the Boundary Layer
Note 1. The plate is considered flat if is lower then the local curvature radius
Note 2. At the separation point, the BD grows a lot and is no longer thin
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
z
wu
y
vu
x
uu
y
u
dx
dp
y
uv
x
uu e
2
21
Turbulent Thin Boundary Layer Equations (<<x) over flat plate
2D Thin Turbulent Boundary Layer Equation (<<x) to flat plates:
Resulting from Reynolds Tensions (note the w term)
0 0
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary Layer Separation Boundary Layer Separation: reversal of the flow by
the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects
mfm: BL / Separation / Flow over edges and blunt bodies
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary Layer Separation Boundary layer separation: reversal of the flow by the
action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary Layer Separation Bidimensional (2D) Thin Boundary Layer (<<x)
Equations to flat plates:
2
21
y
u
dx
dp
y
uv
x
uu e
Close to the wall (y=0) u=v=0 :
dx
dp
y
u e
y1
0
2
2
Similar results to turbulent boundary layer - close to the wall there is laminar/linear sub-layer region.
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary Layer Separation Outside Boundary layer: 0
2
2
y
u
The external pressure gradient can be:o dpe/dx=0 <–> U0 constant (Paralell outer streamlines):
o dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):
o dpe/dx<0 <–> U0 increases (Convergent outer streamlines):
Close to the wall (y=0) u=v=0 :
dx
dp
y
u e
y1
0
2
2
Same sign
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Zero pressure gradient:dpe/dx=0 <–> U0 constant (Paralell outer streamlines):
y
u
Inflection point at the wall
No separation of boundary layer
02
2
yy
u
00
2
2
yy
u
Boundary Layer Separation
Curvature of velocity profile is constant
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Favourable pressure gradient:dpe/dx<0 <–> U0 increases (Convergent outer streamlines):
02
2
yy
u y
00
2
2
yy
u
Curvature of velocity profile remains constant
No boundary layer separation
Boundary Layer Separation
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Adverse pressure gradient:dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):
02
2
yy
u
00
2
2
yy
u Curvature of velocity profile can change
Boundary layer Separation can occur
y
P.I.
Boundary Layer Separation
Separated Boundary Layer
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Sum of viscous forces:2
2
y
u
Become zero with velocity
Can not cause by itself the fluid stagnation (and the separation of Boundary Layer)
Boundary Layer Separation
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Effect of longitudinal pressure gradient:
0dx
dpe (Convergent outer streamlines)
0dx
dpe (Divergent outer streamlines)
Viscous effects retarded Viscous effects reinforced
Fuller velocity profiles
Less full velocity profiles
...11
dx
dp
ux
u e
Decreases BL growth Increases BL growths
Boundary Layer Separation
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Effect of longitudinal pressure gradient:
Fuller velocity profiles
Less full velocity profiles
...11
dx
dp
ux
u e
Decreases BL growth Increases BL growths
Fuller velocity profiles – more resistant to adverse pressure gradients
Turbulent flows (fuller profiles)- more resistant to adverse pressure gradients
Boundary Layer Separation
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Boundary Layer Sepaation
Longitudinal and intense adverse pressure gradient does not cause separation
=> there’s not viscous forces
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Bidimensional (2D) Thin Boundary Layer (<<x) Equations to flat plates:
2
2
y
u
y
uv
x
uu
0
y
v
x
u
Boundary Condition: y=0 u=v=0y=∞ u=U
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius hypothesis: with fU
u
The introdution of η corresponds to recognize that the nondimension velocity profile is stabilized.
nx
Ay
A and n are unknowns
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Remark: eyx
A
y n
x
ny
x
nA
x n
1
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Procedure:
oUsing current function:y
u
x
v
o Remark:
yyd
d
A
xUf
A
xu
nn
F
dfA
xU
n
o Replace u/U=f(η) e at the boundary layer equation, choose n such that the resulting equation does not depend on x and A in order to simplify the equation.
.
xv
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
o FUy
u
results:
Fx
nU
x
u
o
o Fx
UA
y
un
o Fx
UA
y
un
2
2
2
2
o FxnFnxA
U
xv nn
11
FA
xU
n
From:
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
We will obtain:
02
12
FFA
UnxF
n
2
2
y
u
y
uv
x
uu
00, xu
00, xv
Uxu ,
Boundary Conditions:
o Making n=1/2 and the equation comes:UA
02 FFFx
Uy
with
00 FU 00 F
00 FF 00 F
UFU 1F
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Graphical Solution:
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001
x
Uy
FU
u F
0
0,4
0,8
1,2
0 2 4 6 8 10
x
Uy
FU
u
F
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Solution:
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001
x
Uy
FU
u F
0Fx
UU
x
FUx Re
664,00
2
0
0
y
y
u
oShear stress at the wall
2
0
2
1U
fc
o Friction coefficent
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Solution:
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001
x
Uy
FU
u F
dxDL
o 0
o Drag
L
D
LU
DC
Re
328,1
21 2
o Drag Coefficent
UL
L Re
02
1F
L
UU
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Solution :
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Uyu 99,0
o Boundary layer thickness
0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001
x
Uy
FU
u F
xUxx Re
55
η=5
%8,10
5
0
F
F
o Shear stress at y=
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Displacement thickness:
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0
* 1dyuU
Ud
0
1dyuU
Ud
0
udyUU d
U
0
dyuU
Ideal Fluid flow rate
Real Flow rate
Déficit of flow rate due to velocity reduction at BD
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Displacement thickness :
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0
1dyuU
Ud
0
1dyuU
Ud
0
udyUU d
Ideal Fluid flow rate
Real Flow rate
Déficit of flow rate due to velocity reduction at BD
dUq
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Displacement thickness :
0
1dyuU
Ud
0
1dyuU
Ud
0
1udy
Ud
Initial deviation of BD
δ
Deviation of outer streamlines
Section where the streamline become part of boundary layer
δdq/U LC
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Blasius Solution for displacement thickness:
x
d
x Re
72,1
δ dq/U LC
Ux
x Recom
344,0dou
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Momentum thickness:
0
2
1udyuU
Um
0
2
1udyuU
Um
dU mdUdyu
2
0
2
mUudyUdyu
2
00
2
0
2
0
2 dyuudyUU m
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Momentum flow rate through a section of BD:
mdqm UUUdyuqx
222
0
2
Momentum flow rate of uniform
profile
UU
Reduction due to deficit of flow
rate
dUU
Reduction due to deficit
momentum flow rate at BD
mUU
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Longitudinal momentum balance between the leading edge and a cross section at x:
xxqmxqm xx
qqD
0
dU 2 mU d 2
mU 2
δ d-dLC
x
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Blasius Solution to momentum thickness:
x
m
x Re
664,0
Ux
x Rewith
133.0mor
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Laminar Boundary Layer Equations Contents:
– Thin Boundary Layer Equations with Zero Pressure Gradient;
– Boundary Layer Separation;
– Effect of longitudinal pressure gradient on the evolution of Boundary Layer
– Blasius Solution
– Local Reynolds Number and Global Reynolds Number
– Integral Parameters: displacement thickness and momentum thickness
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Recommended study elements:– Sabersky – Fluid Flow: 8.3, 8.4
– White – Fluid Mechanics: 7.4 (sem método de Thwaites)
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
Exercise
L=2m
U=2m/s 0dxdpe
Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. (=1,2 kg/m3, =1,810-5 Pa.s) with U=2 m/s. Zero pressure gradient over the flat plate. Transition to turbulent at Rex=106.
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
a) Find boundary layer thickness at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge
Exercise
mU
x cxc 5,72
2,1108,110Re
56
Find xc:
xx Re
5
Laminar Boundary layer at x1 and x2 – We can apply Blasius
Solution
=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
L=2m
U=2m/s 0dxdpe
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
xx Re
5
Laminar Boundary layer at x1 and x2 – We can apply Blasius
Solution
mx 75,01 5
51 10105,1
75,02Re
x m0119,010
575,051
mx 5,12 52 102Re x
m0168,02
a) Find boundary layer thickness at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
b) Check that it is a thin boundary layer.
=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
L=2m
U=2m/s 0dxdpe
A: Thin Blayer if /x<<1: 0159,075,0
0119,0
1
x
0112,05,1
0168,0
2
x
Why/x at 2 is lower than /x at 1?
y=(x)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.
=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
L=2m
U=2m/s 0dxdpe
Streamline
x2=1,5mx1=0,75m
y1=?
A: We have the same flow rate between the streamline and the plate at both cross sections
Flow rate through a cross section of BD: dUq
Flow rate through section 2: 22 dUq
Flow rate through section 1: 11 yU 1dU 1
1
11
00
1
yy
udyudyudyq
1221 ddy
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.
=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
L=2m
U=2m/s 0dxdpe
Linha de corrente
x2=1,5mx1=0,75m
y1=?
A: We have the same flow rate between the streamline and the plate at both cross sections
1221 ddy Laminar BD: 344,0d
0,0168m 0,0058m 0,0041m y1=0,0151m
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
e) Find the force per unit leght between sections S1 and S2.
L=2m
U=2m/s 0dxdpe=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
xmx UD 2,0
A: There are no other forces applied except that imposed by the resistance (Drag) of plate:
The applied force between the leading edge and the cross section at x is:
Laminar BD: 133,0m
Drag force to section 2: D0,2=0,0107N/mmm 00223,0133,0 22 mm 00158,0133,0 11 Drag force to section 1: D0,1=0,0076N/m
Drag force between 1 and 2: D1,2=D0,2-D0,1=0,0031N/m
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L ), the boundary layer would eventually separate?
L=2m
U=2m/s 0dxdpe=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.
False: The BD will separate only with adverse pressure gradient. The drag forces will decrease with the velocity
over the plate. The drga forces are not able to stop the fluid flow.