THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation: what are we doing today? ]

In a flow, the boundary layer represents a (relatively) thin
layer of fluid that is nearest the solid boundary

The behaviour of the boundary layer is very different from that
of the fluid above it

The boundary layer concept, first discovered by Prandtl, is
very powerful as it allows us to divide large Reynolds
number flows into viscous and inviscid portions
Who Cares!?

 our knowledge of boundary layers has been the driving force behind radical
improvements in aircraft performance, automobile aerodynamics, naval
hydrodynamics, geophysical modeling, and general bluff body flows to name a
few !!
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation ]

The thickness of a boundary layer can be influenced by the
fluid viscosity, surface smoothness, and the rate at which
the flow travels over that surface
flat plate
bluff body aerofoil
pipe flow

We see in the clips how there exists a real division in how
the flow behaves within the boundary layer and beyond it

If the Reynolds number is large enough, we can treat the
flow as divided into viscous in the boundary layer and
inviscid beyond it
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation ]


Much can be learned of the boundary layer concept through
examination of flow over a flat plate
It is important to note in particular, how the Reynolds
number is formulated, the changes in velocity gradient, and
the location of regime transitions
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation ]


Here the Reynolds number is expressed as Re=Ux/n as the closest thing to
a characteristic length is the distance, x, from the plate leading edge
Inspection of the figure reveals that the boundary layer starts as thin laminar
layer, then over distance becomes unstable, goes through a transitional
stage and then goes turbulent
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation ]




Another characteristic distinction within the B/L is its non-zero vorticity
Observe the behaviour of a fluid particle in the flow (one within) and one
above the boundary layer
The fluid particles outside the B/L do not rotate as they flow along
(irrotational, or w/o vorticity)
Inside the B/L, particles rotate and distort under the heavy velocity
gradients present there
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ physical interpretation ]

Mixing in a laminar B/L is really limited to the molecular level

Once the fluid particles reach the turbulent zone mixing
ensues that affects parcels of fluid of various scales  this
is characteristic of turbulent flow
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: d ]

Knowing the B/L thickness is essential in defining its region of influence in a flow

One method for delineating the B/L thickness is to consider the height above the
plate at which the velocity of the boundary layer flow, u, is equivalent to some
fraction of the free stream velocity. This is typically the location where u=0.99U

This is to say that the thickness of the boundary layer (normal to the plate) can be
estimated to be the vertical location where the boundary layer flow velocity is 99%
that of the free stream flow
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: d* the displacement thickness ]

If we consider a uniform velocity profile (left) and the boundary
layer velocity profile (right), we can attempt to adjust the uniform
profile until its flow rate matches that of the boundary layer profile

The height of this vertical displacement we refer to as the boundary
layer “displacement thickness, d*”
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: d* the displacement thickness ]
 we can express this as
 here, b is the plate width
- (1)
 so we write
- (2)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: d* the displacement thickness ]
- (2)

“The displacement thickness represents the amount of thickness a body
must increase by, such that the fictitious uniform inviscid flow, has the
same mass flowrate as the actual viscous flow”
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: q the momentum thickness ]


Now, let us imagine that we are considering the flow of momentum over the
plate, the velocity deficit in the non-uniform profile will also produce a deficit
in momentum flow
The plate can be displaced by a suitable height to allow an inviscid flow the
same momentum throughput as the viscous flow, i.e.,
- (3)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: q the momentum thickness ]
 which we rearrange for
- (4)
 or
- (5)

The momentum thickness is a convenient formulation used
when considering the drag on an object
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: example 1 ]
GIVEN:
 boundary layer
displacement thickness for
the given flow:
d* = 0.0070 x0.5
 assume incompressible
flow (reasonable in this
velocity range)
REQD:
Acquire an expression for the velocity, U, as a function of x for the
flow within the duct but outside the boundary layer
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: example 1 (cont’d) ]
SOLU:
 boundary layer
displacement thickness for
the given flow:
d* = 0.0070 x0.5
1. Based on mass conservation we know we can write that the flow rate at
section (1) must be very close to that in section (2)
- (E1)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: example 1
(cont’d) ]
SOLU:
 boundary layer
displacement thickness for
the given flow:
d* = 0.0070 x0.5
2. And, through our definition of displacement thickness, the flowrate at (2)
must be the same as that of a uniform flow with a velocity U, through a duct
whose walls have been moved inward by d*
- (E2)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ boundary layer thickness: example 1
(cont’d) ]
SOLU:
 boundary layer
displacement thickness for
the given flow:
d* = 0.0070 x0.5
V
3. Now we sub our expression for d* into (E2)
- (E2)
10.25
10.20
10.15
10.10
10.05
10.00
V
0
1
2
3
x
 and rearrange to get
- (ans)
 we observe what
happens to U as we
progress further down the
duct  what is the reason
for this?
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]


One of the powerful aspects of B/L theory is its simple facilitation of drag
force computation
The momentum integral equation method discussed here provides a simpler
alternative to assessing the drag through solving the governing differential
equations

We consider a CV fixed around a uniform flow over a flat plate
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag ]

We make some simple assumptions about the CV: that the pressure
throughout is constant, that the incoming velocity profile is uniform, and that
the exiting profile varies from zero at the CV bottom to free stream speed at
the top of the CV

The fluid adjacent to the plate makes up the bottom bounding surface in the
CV, and the top control surface follows a streamline that need only coincide
with the B/L at section (2)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

We now apply the x-component of the momentum equation
to the steady flow through the CV
- (6)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

For a plate of width b we can write
- (7)

D is the drag that the plate exerts on the fluid, we also note
the there is no net force due to the uniform pressures on the
CV, further there is no flow through the top (streamline) or
bottom (plate) boundaries, only through (1) and (2)
- (8)
 or
- (9)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

We do not know the height, h straight out, but we can make
use of mass conservation here
- (10)

Flow through 1 equals flow through 2
 which we can re-write as
- (11)

Now we combine expressions (9) and (11), and we yield
- (12)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]
- (12)

(12) gives us the drag in terms of the momentum flux deficit
across the CV outlet

Examination of (12) confirms some relationships
 for an inviscid flow, the drag over the plate would be zero (U would be equal to u)
 we also see how drag on a flat plate is a balance between shear drag and a
decrease in the momentum of the fluid (= an increase in the momentum thickness),
the thickening of the boundary layer is required to overcome the viscous shear
stress on the plate
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

If we combine (12) with (5) we can write (13)
- (12)
- (5)
- (13)

(13) was first expressed by Von Karman (1921)

The shear stress distribution can be had by differentiating
both sides of (13) wrt x
- (14)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

Recalling (7), we can now express
as (15)
- (7)
- (15)

Now we combine (15) and (14) to arrive at (16)
- (14)
- (16)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag
]

Here (16) empowers us to make fairly useful boundary layer
computations like drag and shear stress, with relatively
crude assumptions
- (16)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1 ]
GIVEN:
 flow of a laminar,
incompressible fluid past a
flat plate at y=0
 the boundary layer is
approximated as
 where
REQD:
Determine the shear stress for this profile and compare with the
Blasius solution
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1
(cont’d) ]
SOLU:
1. We recall our recently derived expression for shear stress
- (E1)
2. We recall for laminar flow, shear stress can be expressed
as
- (E2)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1
(cont’d) ]
SOLU:
 or, for this profile
- (E3)
3. And we know our expression for momentum thickness
- (E4)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1
(cont’d) ]
SOLU:
 where this becomes
- (E5)
4. Now we combine (E1), (E3) and (E5), and obtain the
differential equation for d
- (E6)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1
(cont’d) ]
SOLU:
 separating variables we have
- (E7)
5. We can integrate (E7) from x=0 (where d=0 to some
arbitrary location, x, where the thickness = d), we get
- (E8)
87-351 Fluid Mechanics
THE BOUNDARY LAYER: AN INTRODUCTION
[ momentum integral equation and drag: example 1
(cont’d) ]
SOLU:
 which we can rearrange for
- (E9)
 we now compare with the much
more labour intensive result for
Blasius’ solution
- (E10)
 this compares fairly well
considering the effort saved in
achieving the result
6. Finally, we combine (E1), (E5), and (E9) to express the shear stress
**this result is within about 13% of Blasius’ result
- (ans)
87-351 Fluid Mechanics