Seminar class presentation at IIT Kharagpur, Department Of Aerospace Engineering

1. A STUDY ON VISCOUS FLOW
(With A Special Focus On Boundary
Layer And Its Effects)
COMPLETED BY
RAJIBUL ALAM
M. Tech. (1st year)
Roll No-14AE60R03

2. Viscosity is an aspect of friction. Whenever a fluid moves over a solid surface or over an
another fluid , the mutual frictional resistance arising in the contact surfaces is known as
viscosity and such a flow is known as viscous flow. The viscous shear stress is given by
Ƭ=µ.
𝜕𝑢
𝜕𝑥
| 𝑦=0
where, Ƭ=shear stress
µ=coefficient of viscosity
For air at standard room temperature ,
µ =1.7894*10−5 kg/ms
FUNDAMENTAL EQUATION:
Newton’s second law gives 𝐹𝑥=m. 𝑎 𝑥 ------------- (1)
where 𝐹𝑥is the force in X direction, m is the mass and 𝑎 𝑥 is the acceleration in X
direction . Now consider a fluid element as shown in figure.
Here shear stresses on three faces are shown.

4. Now considering forces on the fluid element in X direction only, we have
𝑭 𝑿= ( p- ( p +
𝝏𝒑
𝝏𝒙
.dx ) ) .dy. dz + ( ( Ƭ 𝒙𝒙 +
𝝏
𝝏𝒙
Ƭ 𝒙𝒙 .dx ) - Ƭ 𝒙𝒙 ) .dy . dz + ( ( Ƭ 𝒚𝒙+
𝝏
𝝏𝒚
Ƭ 𝒚𝒙 .dy ) –
Ƭ 𝒚𝒙 ) ) .dx .dz + ( ( Ƭ 𝒛𝒙 +
𝝏
𝝏𝒛
Ƭ 𝒛𝒙.dz ) - Ƭ 𝒛𝒙 ) ) .dx .dy
On simplification which gives,
𝑭 𝑿= ( -
𝝏𝑷
𝝏𝒙
+
𝝏
𝝏𝒙
Ƭ 𝒙𝒙 +
𝝏
𝝏𝒀
Ƭ 𝒚𝒙 +
𝝏
𝝏𝒁
Ƭ 𝒛𝒙 ) .dx .dy. dz -------------------(2)
again mass m = ρ.dx.dy.dz where ρ is the density of the fluid
also 𝒂 𝑿=
𝑫𝒖
𝑫𝒕
=
𝝏𝒖
𝝏𝒕
+ V.𝜵𝒖 =
𝝏𝒖
𝝏𝒕
+ ( u
𝝏𝒖
𝝏𝒙
+ v
𝝏𝒖
𝝏𝒚
+ w
𝝏𝒖
𝝏𝒛
)
Putting these values in equation (1) we have
-
𝝏𝒑
𝝏𝒙
+
𝝏
𝝏𝒙
Ƭ 𝒙𝒙 +
𝝏
𝝏𝒚
Ƭ 𝒚𝒙 +
𝝏
𝝏𝒛
Ƭ 𝒛𝒙 = ρ.
𝝏𝒖
𝝏𝒕
+ 𝝆𝒖.
𝝏𝒖
𝝏𝒙
+ 𝝆𝒗.
𝝏𝒖
𝝏𝒚
+ ρw.
𝝏𝒖
𝝏𝒛
This is the Navier-Stokes equation in X direction for a viscous flow . Similarly equations
For other directions can also be found.

5. Types of viscous flows :
Viscous flows can be differentiated into two types viz Laminar
and Turbulent flow.
Laminar Flows : Laminar flows are those in which streamlines are smooth and regular and
a fluid particle moves smoothly along a streamline.
Turbulent Flows : Turbulent flows are those in which streamlines break up and fluid
particles move randomly in a zigzag fashion.

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7. Boundary Layer :
When a solid body moves in viscous fluid or vice versa , the fluid
layer adjacent to the solid body sticks to it . As we move perpendicular to the fluid
motion from the solid body, the velocities of fluid layers keep on increasing till it
reaches free stream fluid velocity . Thus a velocity gradient exists in the direction
perpendicular to the fluid motion. This region where a velocity gradient exists in the
direction perpendicular to the fluid motion is known as boundary layer.

8. Types Of Boundary Layer :
a) Blasius Boundary Layer : It is the boundary layer attached to a flat plate
held in oncoming unidirectional flow.
b) Stokes Boundary Layer : It is a thin shear layer develops on an oscillatory
body in a viscous liquid.
In such a case when a fluid rotates, viscous forces are balanced by Coriolis
effect.
Coriolis effect is the deflection of a body when viewed in a rotating frame
of reference.
BLASIUS B.L. STOKES B.L.
S

9. c) Laminar Boundary Layer: When a laminar flow takes place over a bounding
surface , the associated boundary layer is known as Laminar Boundary Layer.
d) Turbulent Boundary Layer: When a turbulent flow takes place over a bounding
surface the associated boundary layer is known as Turbulent Boundary Layer.
Boundary Layer Properties:
1)Velocity Boundary Layer Thickness (∂): Boundary Layer thickness at a point on
the solid surface is the height from that point to a point where velocity is 99% of the
free stream velocity . Boundary layer thickness at different points may be different.
2)Thermal Boundary Layer Thickness ( 𝝏 𝑻 ): Thermal boundary layer thickness at
point on the solid surface is height from that point to a point where the fluid
temperature is equal to the free stream flow temperature.

10. For full presentation click
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special.html

11. 3) Displacement Thickness ( ∂* ) : It is the height proportional to the missing mass flow rate
due to the presence of boundary layer . It also gives the displacement through which a
streamline drifts due to the presence of boundary layer.
Let u and ρ be the instantaneous velocity and density of a fluid inside a boundary layer . Let
free stream values be 𝑢 𝑒 𝑎𝑛𝑑 ρ 𝑒 .
Then missing mass flow due to the presence of boundary layer
𝟎
𝝏
𝝆 𝒆. 𝒖 𝒆 . ⅆ𝒚 - 𝟎
𝝏
𝝆. 𝒖. ⅆ𝒚 ------------(3)
Now if ∂* be the momentum thickness then, equation (3)= 𝝆 𝒆.𝒖 𝒆.∂*
This gives
∂* = 𝟎
𝝏
( 1- 𝝆.𝒖
𝝆 𝒆..𝒖 𝒆 ). ⅆ𝒚

12. Boundary layer equations: For a steady two dimensional flow continuity and momentum
equations are given by,
𝝏
𝝏𝒙
(ρ.u ) +
𝝏
𝝏𝒙
( ρ.v )=0 ------------------(4)
u.
𝝏𝒖
𝝏𝒙
+ v.
𝝏𝒖
𝝏𝒚
= -
𝟏
ρ
.
𝝏𝒑
𝝏𝒙
+ µ .
𝝏 𝟐 𝒖
𝝏𝒚 𝟐 ---------(5)
SOME RESULTS:
(A) LAMINAR B.L. : For an incompressible flow over a flat plate
∂=
𝟓.𝟎 𝒙
√𝑹 𝒆𝒙
∂*=
𝟏.𝟕𝟐𝒙
√𝑹 𝒆𝒙
For compressible flow
∂=
𝒇 𝟏( 𝑴∞)
√𝑹 𝒆𝒙
∂*=
𝒇 𝟐( 𝑴∞)
√𝑹 𝒆𝒙
(B) TURBULENT B.L. : For an incompressible flow over a flat plate
∂=
.𝟑𝟕 𝒙
( 𝑅 𝑒𝑥 )ˆ.2

13. Aerodynamic Boundary Layer : It was first proposed by Ludwig Prandtl in a paper presented
on August 12,1904 at the third International Congress Of Mathematics in Heidelberg,Germany.
It simplifies equations of a flow around an airfoil by dividing the flow field into two areas viz
flow adjacent to the airfoil i.e. inside the boundary layer where viscosity is dominant and
outside the boundary layer where the flow can be assumed to be inviscid.
Viscous flow solution: This differentiation of flow leads to a simplified solution to viscous
flow problems .
INVISCID FLOW
VISCOUS
FLOW
IN
(a) First for the flow outside the boundary layer inviscid solution is carried out and
ρ 𝑒,𝑢 𝑒 and 𝑇𝑒 are found at the outer extreme of the boundary layer.

14. (b) Using the above values the boundary layer equations (4) and (5) are solved and then
momentum thickness ∂* is found. Once momentum thickness is found effective body shape i.e.
body + boundary layer is determined. Now within this region viscous calculations are carried
out.
Effects of types of boundary layer on shear stress : Velocity profile of a laminar boundary
layer is completely different from that of a turbulent boundary layer.
Incase of a laminar boundary layer, velocity gradually reduces from free stream value at the
outer layer to zero at the wall while incase of a turbulent boundary layer velocity is almost
uniform near to the solid surface and suddenly becomes zero as evident from the following
graph.

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16. Pressure distribution in a boundary layer:
Pressure through a boundary layer in a direction perpendicular to the surface is constant.
Thus in figure though pressure increases in the direction of the flow, but it remains constant in
a given section, viz 1-1, 2-2 etc.
This is an important phenomenon. This is why pressure calculated from inviscid calculation
gives correct value for real life surface pressure.

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18. Explanation:
For full presentation click
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u.𝝏𝒖/𝝏𝒙 + v. 𝝏𝒖/𝝏𝒚 = - 𝟏/ρ . ( 𝝏𝒑/𝝏𝒙 ) + µ . ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 )
Now at y=0, v=0 and
𝜕𝑢
𝜕𝑥
=0 .
hence,
𝟏/ρ . ( 𝝏𝒑/𝝏𝒙 ) = µ . ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 )
however boundary layer suction is provided then equation becomes
v.𝝏𝒖/𝝏𝒚 + 𝟏/ρ . ( 𝝏𝒑/𝝏𝒙 ) = µ . ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 )
𝟏/ρ . ( 𝝏𝒑/𝝏𝒙 ) = µ . ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 )
Now if 𝝏𝒑/𝝏𝒙 =0 ,then ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 ) =0 at the wall i.e.
𝜕𝑢
𝜕𝑦
is at a maximum there
and falls away steadily.

19. If however
𝜕𝑝
𝜕𝑥
> 0 i.e. if there is an adverse pressure gradient ( 𝝏^𝟐 𝒖)/( 𝝏.y^𝟐 ) > 0 i.e.
𝝏𝒖
𝝏𝒚
is at
a minimum there. Hence 𝝏𝒖/(𝝏𝒚 ) first increases and then decreases with y. At this point flow is
at the verge of separation.
If pressure gradient further increases velocity profile gets distorted and ( 𝝏𝒖/(𝝏𝒚 ) )y=0
becomes zero first and flow is at the verge of separation now. On further increase of pressure
gradient( 𝝏𝒖/(𝝏𝒚 ) )y=0 becomes negative and flow reversal happens.

20. WAKE REGION
PRESSURE REDUCES SHARPLY
Wake due to separation

21. How does lift reduce due to separation :
It is now clear that due to separation , pressure
downstream an airfoil on the upper surface reduces sharply which increases drag drastically.
But at the same time any decrease in pressure on the upper surface should increase the lift.But
why does the lift decrease due to separation?

22. For full presentation click
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23. IMPORTANCE OF INVISCID FLOW: Though the concept of perfect inviscid fluid is absurd, yet
in many cases viscosity associated is negligible. In such cases inviscid flow calculation gives
almost accurate results.
Moreover in case of viscous flow over an aerofoil the flow can be divided into two regime as
discussed earlier, viz viscous and inviscid which makes the life easier.
Most importantly , some recent research by NASA is hinting that even as complex problem
as flow separation till date to be thought of completely viscous dominated phenomenon may
be in reality be an inviscid dominated flow which requires only a rotational flow. For example
some inviscid flow field numerical solutions for flow over a circular cylinder when vorticity is
introduced by means of a curved shock wave , accurately predicting the separated flow on the
rear side of the cylinder.

24. Boundary layer and shock wave interaction :
Boundary layer does not mix
with a shock wave, rather when
a viscous supersonic flow takes
place over a flat plate shock
wave impinges on the boundary
layer. Due to strong adverse
pressure gradient across a
shock wave the boundary layer
gets separated . Since high
pressure behind the shock feeds
upstream through the subsonic
portion of the boundary layer , hence
separation takes place ahead of the theoretical inviscid flow impingement point of the shock
wave . In turn separated boundary layer deflects the oncoming flow into itself , thus creating a
second oblique shock wave known as leading edge shock wave. Separated boundary layer
subsequently reattaches the surface at some downstream position. Here the flow is again
turned into itself creating a third shock wave known as reattachment shock wave. In between
the leading edge shock and reattachment shock supersonic flow is turned away from itself
through some expansion fans.
The scale and severity of the interaction depends on the type of boundary layer. A laminar
boundary layer more readily separates than a turbulent boundary layer .

25. Boundary layer-shock wave interaction in a nozzle : We know for pressure ratio higher than
some critical value normal shock wave stands right inside the nozzle as shown in the adjacent
figure.
However the study of Craig A. Hunter presented in a paper called “Experimental
Investigation Of Separated Nozzle Flows “ shows that normal shock wave interacts with
boundary layer inside the nozzle to cause local separation of the flow and again reattachment
of the flow. This causes the formation of Lamda type shock wave as shown in the adjacent
figure.
LAMDA TYPE
SHOCK WAVE

26. REFERENCES :
A) Fundamentals Of Aerodynamics
--by John D Anderson Jr
B) Principle Of Flight
--by John D Anderson Jr
C) Edinburgh University Publication On Boundary Layer
D) Internet