1. 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
Displacement thickness and momentum thickness
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

2. Laminar Thin Boundary Layer Equations (d<<x) over flat plate
Steady flow, constant r and m.
Streamlines slightly divergent
2D Navier-Stokes Equations along x direction:
Compared with
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

3. Laminar Thin Boundary Layer Equations (d<<x) over flat plate
Laminar thin boundary layer equations (d<<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 d 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

4. Turbulent Thin Boundary Layer Equations (d<<x) over flat plate
2D Thin Turbulent Boundary Layer Equation (d<<x) to flat plates: 0
Resulting from Reynolds Tensions (note the w term)
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

5. 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

6. 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

7. Boundary Layer Separation
Bidimensional (2D) Thin Boundary Layer (d<<x) Equations to flat plates:
Close to the wall (y=0) u=v=0 :
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

8. Boundary Layer Separation
Outside Boundary layer:
Close to the wall (y=0) u=v=0 :
Same sign
The external pressure gradient can be:
dpe/dx=0 <–> U0 constant (Paralell outer streamlines):
dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):
dpe/dx<0 <–> U0 increases (Convergent outer streamlines):
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

9. Boundary Layer Separation
Zero pressure gradient:
dpe/dx=0 <–> U0 constant (Paralell outer streamlines): y u
Curvature of velocity profile is constant
No separation of boundary layer
Inflection point at the wall
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

10. Boundary Layer Separation
Favourable pressure gradient:
dpe/dx<0 <–> U0 increases (Convergent outer streamlines): y
No boundary layer separation
Curvature of velocity profile remains constant
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

11. Boundary Layer Separation
Adverse pressure gradient:
dpe/dx>0 <–> U0 decreases (Divergent outer streamlines): y
Boundary layer Separation can occur
Separated Boundary Layer
Curvature of velocity profile can change
P.I.
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

12. Boundary Layer Separation
Sum of viscous forces:
Become zero with velocity
Can not cause by itself the fluid stagnation (and the separation of Boundary Layer)
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

13. Boundary Layer Separation
Effect of longitudinal pressure gradient:
(Convergent outer streamlines)
(Divergent outer streamlines)
Viscous effects retarded
Viscous effects reinforced
Fuller velocity profiles
Less full velocity profiles
Decreases BL growth
Increases BL growths
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

14. Boundary Layer Separation
Effect of longitudinal pressure gradient:
Less full velocity profiles
Fuller velocity profiles
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
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

15. Boundary Layer Sepaation
Longitudinal and intense adverse pressure gradient does not cause separation
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

16. 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 (d<<x) Equations to flat plates:
Blasius Solution to Laminar Boundary Layer Equation over a flat plate
Boundary Condition: y= u=v=0
y=∞
u=U
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

17. 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 hypothesis: with
The introdution of η corresponds to recognize that the nondimension velocity profile is stabilized.
A and n are unknowns
Remark: e
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

18. 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:
Using current function:
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. .
Remark:
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.
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

19. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
From:
results:
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

20. 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:
Making n=1/2 and the equation comes:
with
Boundary Conditions:
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

21. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Graphical Solution:
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

22. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Shear stress at the wall
Friction coefficent
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

23. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Solution:
Drag
Drag Coefficent
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

24. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Boundary layer thickness
η=5
Shear stress at y=
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

25. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Displacement thickness:
Déficit of flow rate due to velocity reduction at BD
Real Flow rate
Ideal Fluid flow rate U 
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

26. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Displacement thickness :
Déficit of flow rate due to velocity reduction at BD
Real Flow rate
Ideal Fluid flow rate
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

27. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Displacement thickness :
Deviation of outer streamlines
Initial deviation of BD δd
q/U LC δ
Section where the streamline become part of boundary layer
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

28. 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:
com ou d δ
q/U LC
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

29. 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:
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

30. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Momentum flow rate through a section of BD:
Momentum flow rate of uniform profile
Reduction due to deficit of flow rate
Reduction due to deficit momentum flow rate at BD
Reduction due to deficit momentum flow rate at BD
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

31. 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:
Longitudinal momentum balance between the leading edge and a cross section at δ d
-d LC x
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

32. 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:
with or
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

33. 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
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

34. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
Recommended study elements:
Sabersky – Fluid Flow: 8.3, 8.4
White – Fluid Mechanics: 7.4 (sem método de Thwaites)
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

35. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. (=1,2 kg/m3, =1,810-5 Pa.s) with U=2 m/s. Zero pressure gradient over the flat plate. Transition to turbulent at Rex=106.
=1,2 kg/m3,
=1,810-5 Pa.s
(Rex)c =106.
L=2m
U=2m/s
Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. Zero pressure gradient over the flat plate. Transition to turbulent occurs for
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

36. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
=1,2 kg/m3,
=1,810-5 Pa.s
(Rex)c =106.
L=2m
U=2m/s
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
Find xc:
Boundary layer thickness at S1
Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

37. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
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
Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

38. Exercise =1,2 kg/m3, U=2m/s =1,810-5 Pa.s (Rex)c =106. y=(x) L=2m
b) Check that it is a thin boundary layer.
A: Thin Blayer if /x<<1:
Why/x at 2 is lower than /x at 1?
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

39. Exercise =1,2 kg/m3, U=2m/s =1,810-5 Pa.s (Rex)c =106.  y1=?
Streamline
y1=? 
x1=0,75m
x2=1,5m
L=2m
d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.
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:
We have the same flow rate between the streamline na
Flow rate through a cross section of CL:
Flow rate through section 2:
Flow rate through section 1:
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

40. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
=1,2 kg/m3,
=1,810-5 Pa.s
(Rex)c =106.
U=2m/s
Linha de corrente
y1=? 
x1=0,75m
x2=1,5m
L=2m
d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.
A: We have the same flow rate between the streamline and the plate at both cross sections
Laminar BD:
y1=0,0151m
0,0168m
0,0058m
0,0041m
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

41. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
=1,2 kg/m3,
=1,810-5 Pa.s
(Rex)c =106.
L=2m
U=2m/s
e) Find the force per unit leght between sections S1 and S2.
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:
Find the force per unit leght between sections
The applied force between the leading edge and the cross section at
Laminar BD:
Drag force to section 2: D0,2=0,0107N/m
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

42. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Exercise
=1,2 kg/m3,
=1,810-5 Pa.s
(Rex)c =106.
L=2m
U=2m/s
f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L ), the boundary layer would eventually separate?
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.
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.
2004
Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST