1. JHADA SIRHA GOVERNMENT ENGINEERING COLLEGE
JAGDALPUR
DEPARTMENT OF MECHANICAL ENGINEERING
COURSE NAME: FLUID MACHINES
COURSE CODE: C037513(037)
B.TECH. , 5th SEM.
COURSE CO-ORDINATOR:
GULAB VERMA
ASSISTANT PROFESSOR
DEPARTMENT OF MECHANICAL ENGG.
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
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2. Boundary Layer
When a viscous fluid flow over solid surface (such as a
plate) , the fluid layer adjacent to the surface attain the
velocity of surface.
In other words, the relative velocity between the plate and
adjacent fluid layer is zero.
This phenomenon has been established through
experimental observations and is known as the no-slip
condition.
The layer that sticks to the surface slows the adjacent fluid
layer because of viscous forces between the fluid layers,
which slows the next layer, and so on.
Consequently, a region with a velocity gradient (viscous
region) is set up in the fluid in a direction normal solid
surface.
The flow region adjacent to the solid surface in which the
velocity gradients (and thus the viscous effects) are
significant is called the boundary layer.
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
2
3. Boundary Layer Thickness (δ)
“The boundary layer is a thin region adjacent to the solid surface where the velocity of fluid varies from zero
to free stream velocity in the direction normal to the solid surface.”
Therefore boundary layer thickness is defined as:
“The normal distance from the solid surface in which the velocity reaches 99 percent of the velocity of the free
stream (u = 0.99U͚ ). It is denoted by the symbol δ.”
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
3
4. Characteristics of Boundary Layer
The stages of the formation of the boundary layer and its characteristics over a flat plate are shown in the
figure below:
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
4
5. Near the leading edge of a flat plate, the boundary layer is wholly laminar. For a laminar boundary layer, the
velocity distribution is parabolic.
The thickness of the boundary layer (δ) increases with distance from the leading edge x. Then laminar boundary
layer becomes unstable and breaks into the turbulent boundary layer.
For a turbulent boundary layer, if the boundary is smooth, the roughness projections are covered by a very thin
layer next to the wall where viscous effects are dominant, called the laminar sublayer. Because of its very low
small thickness variation of velocity is assumed to be linear.
Next to the viscous sublayer is the buffer layer, in which turbulent effects are becoming significant, but the flow is
still dominated by viscous effects. Above that is the turbulent layer in the remaining part of the flow in which
turbulent effects dominate over viscous effects.
For a turbulent boundary layer, the velocity distribution is given by Log law or Prandtl's one-seventh power law.
The critical the Reynold’s number for the transition from the laminar boundary layer to turbulent boundary layer is
Re =
.
= 5 × 10 .
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
5
6. Displacement Thickness ( ∗
The displacement thickness represents the vertical distance that a solid boundary must be displaced
upward so that the ideal flow has the same mass flow rate as the real flow due to the development of the
boundary layer.
Expression of displacement thickness 𝛅∗ = ∫ 𝟏 −
𝐮
𝐔
𝐝𝐲
𝛅
𝟎
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
6
7. Momentum Thickness (θ
The momentum thickness represents the vertical distance that a solid boundary must be displaced upward
so that the ideal flow has the same momentum as the real flow due to the development of the boundary
layer.
Expression of momentum thickness 𝛉 = ∫
𝐮
𝐔
𝟏 −
𝐮
𝐔
𝐝𝐲
𝛅
𝟎
Energy Thickness ( ∗∗
The momentum thickness represents the vertical distance that a solid boundary must be displaced upward
so that the ideal flow has the kinetic energy as the real flow due to the development of the boundary layer.
Expression of Energy thickness 𝛅∗∗ = ∫
𝐮
𝐔
{𝟏 − (
𝐮
𝐔
)𝟐}𝐝𝐲
𝛅
𝟎
11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
7
8. 11/20/2022
GULAB VERMA, ASSISTANT PROF., DEPT. OF MECH. ENGG.,
GEC-JDP
8
Von Karman Momentum Integral Equation
τ
ρU
=
dθ
dx
Von Karman suggested a method based on the momentum equation by the use of which the growth of a boundary layer
along a flat plate, the wall shear stress, and the drag force could be determined (when the velocity distribution in the
boundary layer is known). The method can be used for both laminar and turbulent boundary layers.
Local co-efficient of drag (𝐂𝐃
∗
) =
𝛕𝟎
𝟏
𝟐
𝛒𝐔𝟐
Average co-efficient of drag (𝐂𝐃) =
𝐅𝐃
𝟏
𝟐
𝛒𝐀𝐔𝟐
Starting from the beginning of the plate, the method
can be used for both laminar and turbulent boundary
layers.
