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University of Tikrit Chapter Thirteen Second Grade
Chapter
13
BOUNDARY LAYER THEORY
13.1. Introduction 13.1. INTRODUCTION A.
13.2. Boundary layer definitions The concept of boundary layer was first introduced
and characteristics- by L. Prandtl in 1904 and since then it has been applied
boundary layer thickness to several fluid flow problems.
(δ)—displacement thickness
(δ*)—momentum thickness When a real fluid (viscous fluid) flows past a
S.
(q)— energy thickness (δ ). stationary solid boundary, a layer of fluid which comes
13.3. Momentum equation e in contact with the boundary surface, adheres to it (on
for boundary layer by account of viscosity) and condition of no slip occurs
Vonkarman. (The no-slip condition implies that the velocity of fluid at
13.4. Laminar boundary layer. a solid boundary must be same as that of boundary itself).
13.5. Turbulent boundary layer. Thus the layer of fluid which cannot slip away from the
13.6. Total drag due to laminar boundary surface undergoes retardation; this retarded
and turbulent layers. layer further causes retardation for the adjacent layers of
13.7. Boundary layer separation the fluid, thereby developing a small region in the
and its control immediate vicinity of the boundary surface in which the
Highlights velocity of the flowing fluid increases rapidly from zero
Objective Type Questions at the boundary surface and approaches the velocity of
Theoretical Questions main stream. The layer adjacent to the boundary is
Unsolved Examples. known as boundary layer. Boundary layer is formed
whenever there is relative motion between the boundary
and the fluid. Since τ=µ ∂u , the fluid exerts a
0
∂
y
Burhan y=0
shear stress on the boundary and boundary exerts an
equal and opposite force on fluid known as the shear
resistance.
According to boundary layer theory the extensive fluid
medium around bodies moving in fluids can be divided
into following two regions:
(i) A thin layer adjoining the boundary is called the
Dr. boundary layer where the viscous shear takes
place.
(ii) A region outside the boundary layer where the flow
behaviour is quite like that of an ideal fluid and
the potential flow theory is applicable.
725
Dr. Burhan S. Abdulrazzaq BOUNDARY LAYER THEORY
University of Tikrit Chapter Thirteen Second Grade
726 Fluid Mechanics
13.2. BOUNDARY LAYER DEFINITIONS AND CHARACTERISTICS
Consider the boundary layer formed on a flat plate kept parallel to flow of fluid of velocity
U (Fig. 13.1) (Though the growth of a boundary layer depends upon the body shape, flow over
a flat plate aligned in the direction of flow is considered, since most of the flow surfaces can be
approximated to a flat plate and for simplicity).
— The edge facing the direction of flow is called leading edge.
— The rear edge is called the trailing edge.
— 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, as
more and more fluid is slowed down by the viscous boundary, becomes unstable and breaks
into turbulent boundary layer over a transition region. A.
Laminar Transi- Turbulent boundary layer
boundary layer tion
U
0.99 U
U u~logy
U S.
0.99 U
Y u~y Laminarsublayer
X Parabolic
0
O x Trailing edge
Leading edge Smoothflat plate
Fig. 13.1. Boundary layer on a flat plate.
For a turbulent boundary layer, if the boundary is smooth, the roughness projections are covered
by a very thin layer which remains laminar, called laminar sublayer. The velocity distribution in the
turbulent boundary layer is given by Log law or Prandtl’s one-seventh power law.
The characteristics of a boundary layer may be summarised as follows:
(i) δ (thickness of boundary layer) increases as distance from leading edge x increases.
(ii) δ decreases as U increases.
(iii) δ increases as kinematic viscosity (ν) increases.
U
(iv) τ ≈µ ; hence τ decreases as x increases. However, when boundary layer becomes
0 0
δ
Burhan
turbulent, it shows a sudden increase and then decreases with increasing x.
(v) When U increases in the downward direction, boundary layer growth is reduced.
(vi) When U decreases in the downward direction, flow near the boundary is further retarded,
boundary layer growth is faster and is susceptible to separation.
(vii) The various characteristics of the boundary layer on flat plate (e.g variation of δ, τ or force
Ux 0 UL
F) are governed by inertial and viscous forces; hence they are functions of either ν or ν .
Dr.
Ux 5
(viii) If 5 10 ... boundary layer is laminar (velocity distribution is parabolic).
v <×
Ux 5
If 5 10 ... boundary layer is turbulent on that portion (velocity distribution follows
v >×
Log law or a power law).
Dr. Burhan S. Abdulrazzaq BOUNDARY LAYER THEORY
University of Tikrit Chapter Thirteen Second Grade
Chapter 13 : Boundary Layer Theory 727
(ix) Critical value of Ux at which boundary layer changes from laminar to turbulent depends on:
v
— turbulence in ambient flow,
— surface roughness,
— pressure gradient,
— plate curvature, and
— temperature difference between fluid and boundary.
(x) Though the velocity distribution would be a parabolic curve in the laminar sub-layer zone,
but in view of the very small thickness we can reasonably assume that velocity distribution
is linear and so the velocity gradient can be considered constant.
13.2.1. Boundary Layer Thickness (δ) A.
The velocity within the boundary layer increases from zero at the boundary surface to the
velocity of the main stream asymptotically. Therefore the thickness of the boundary layer is
arbitrarily defined as that distance from the boundary in which the velocity reaches 99 per cent of the
velocity of the free stream (u = 0.99U). It is denoted by the symbol δ. This definition however gives
an approximate value of the boundary layer thickness and hence δ is generally termed as nominal
thickness of the boundary layer. S.
The boundary layer thickness for greater accuracy is defined in terms of certain mathematical
expressions which are the measure of the boundary layer on the flow. The commonly adopted
definitions of the boundary layer thickness are:
1. Displacement thickness (δ*)
2. Momentum thickness (q)
3. Energy thickness (δ ).
e
13.2.2. Displacement Thickness (δ*)
The displacement thickness can be defined as follows:
“It is the distance, measured perpendicular to the boundary, by which the main/free stream is
displaced on account of formation of boundary layer.”
Or
“It is an additional “wall thickness” that would have to be added to compensate for the reduction
in flow rate on account of boundary layer formation”.
The displacement thickness is denoted by δ*.
Burhan
Let fluid of density ρ flow past a stationary plate with velocity U as shown in the Fig. 13.2.
Consider an elementary strip of thickness dy at a distance y from the plate.
U
Boundary layer Velocity
dy u=0.99U distribution
U
Dr.
y
Stationary plate
Fig. 13.2. Displacement thickness.
Dr. Burhan S. Abdulrazzaq BOUNDARY LAYER THEORY
University of Tikrit Chapter Thirteen Second Grade
728 Fluid Mechanics
Assuming unit width, the mass flow per second through the elementary strip
= ρudy
Mass flow per second through the elementary strip (unit width) if the plate were not there
= ρ U·dy ...(ii)
Reduction of mass flow rate through the elementary strip
= ρ (U – u) dy
[The difference (U – u) is called velocity of defect].
Total reduction of mass flow rate due to introduction of plate
δ
= ∫ρ (U –)u dy ...(iii)
0
(if the fluid is incompressible) A.
Let the plate is displaced by a distance δ* and velocity of flow for the distance δ* is equal to
the main/free stream velocity (i.e. U). Then, loss of the mass of the fluid/sec. flowing through the
distance δ*
= ρUδ* ...(iv)
Equating eqns. (iii) and (iv), we get: S.
* δ
ρUδ = ∫ρ (U –)u dy
0
δ u
or, *
δ = 1– dy ...(13.1)
∫
U
0
13.2.3. Momentum Thickness (q)
Momentum thickness is defined as the distance through which the total loss of momentum per
second be equal to if it were passing a stationary plate. It is denoted by q.
It may also be defined as the distance, measured perpendicular to the boundary of the solid
body, by which the boundary should be displaced to compensate for reduction in momentum of the
flowing fluid on account of boundary layer formation.
Refer to fig. 13.2. Mass of flow per second through the elementary strip = ρudy
2
Momentum/sec. of this fluid inside the boundary layer = pudy × u = ρu dy
Momentum/sec. of the same mass of fluid before entering boundary layer = ρuUdy
2
Burhan
Loss of momentum/sec. = ρuUdy – ρu dy = ρu (U – u) dy
∴ Total loss of momentum/sec.
δ
= ∫ρu (U –)u dy ...(i)
0
Let, q = Distance by which plate is displaced when the fluid is flowing with a constant velocity U.
Then loss of momentum/sec. of fluid flowing through distance q with a velocity U
= ρqU2 ...(ii)
Dr.
Equating eqns. (i) and (ii), we have:
2 δ
ρqU = ∫ρu (U –)u dy
0
Dr. Burhan S. Abdulrazzaq BOUNDARY LAYER THEORY
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