In 1921 Theodore von Karman introduced the idea of the momentum integral analysis.
This approach is a simple bulk integral analysis of a boundary layer over a flat plate that
provides more insight and quantitative information about the physics of boundary layer flows.
For example, we will be able to predict the drag acting on a plate, that can be a first
approximation of very slender bodies like a really thin wing of an airplane or like
the sides of a truck trailer.
Let's consider a flat plate immersed in a fluid moving parallel to the wall.
This will generate a boundary layer on the plate and a specific profile of velocity inside
the boundary layer.
We define a control volume consisting of the plate, an inlet at the leading edge, a station
a distance L downstream of the leading edge, and a streamline at a distance h from the
plate that meets the boundary layer at station L. The velocity profile of the boundary layer
is assumed to be a known quantity, this makes the analysis assumptions hold for both laminar
and turbulent boundary layers.
The integral form of the conservation of mass for constant density flow tells us that the
volume flux through the domain must be conserved.
In our control volume only the two vertical boundaries at location 0 and location L contribute
to this balance, since there cannot be flow across a wall or a streamline.
We can define the height H_L as the sum of H_not plus a Delta star term.
Substituting this expression into the second integral and rearranging the terms we obtain
an expression for the Delta star term.
Taking the limit of H_L going to infinity we obtain the formal expression for the displacement
thickness.
The displacement thickness represents the distance through which streamlines outside
of the boundary layer are displaced by the layer.
It is a measure of blockage of the external flow due to the formation of boundary layers.
The same definition of the displacement thickness is used for turbulent boundary layers, where
the velocity is taken in time-averaged sense.
Through experimental measurements of the velocity profile in the boundary layer it is hard to
identify the exact location where the velocity reaches the 99% of the freestream velocity
and estimate the boundary layer thickness.
However, the displacement thickness can be easily deduced from velocity measurements
taken across the boundary layer.
Let's now use the integral approach to analyze the moment conservation of momentum along
the X direction.
Like the previous case only the two vertical boundaries participate in the X momentum calculations.
The variation of momentum due to the boundary layer is equal to the drag acting on the plate.
From the relation that connects H_L, H_not and the displacement thickness we can derive
an expression for H_not.
Substituting this expression into the momentum equation and taking the limit of H_L to infinity
we obtain this final relation.
The integral term is defined as the momentum thickness that represents the loss of momentum
due to the presence of the boundary layer as compared to an equivalent inviscid flow.
The definition of momentum thickness holds for any incompressible boundary layer, laminar
or turbulent.
For the boundary layer over a flat plate, the momentum thickness is equivalent to the
non-dimensional drag.
While for a generic shape the momentum thickness is not .
We can now analyze how the different terms behave in the boundary layer along the Y direction.
The first term U over V_infinity changes and grows gradually from zero to one.
The displacement thickness presents a mirrored trend compared to the previous term, going
from one to zero.
The momentum thickness instead initially grows and then goes back to zero.
As you can see the displacement thickness is always larger than the momentum thickness.
The ratio of these two terms is typically referred to as the shape factor.
This parameter can be used to identify if the boundary layer is laminar turbulent or
if we have separation.
The main point of the integral analysis of the boundary layer is to estimate the friction
and drag acting on the plate.
Let’s first recall the expression for the momentum thickness and introduce the coefficient
of drag that represents a non-dimensional value of drag.
From these two expressions we can easily get a relation for the drag coefficient based
on the momentum thickness.
The drag can be also calculated as the integral of the shear stress acting on the plate.
The friction coefficient term is a non- dimensional way to represent the shear stress.
From these relations we can see that the friction coefficient is directly related to the drag
through a simple derivation.
The final result gives us that the friction coefficient is equal to twice the derivative
along the X direction of the momentum thickness.
Our first assumption for the analysis is that the velocity profile in the boundary layer
is known either from experimental results or numerical simulations.
But what if you don't know the velocity profile?
Well in that case you can guess it!
The profile must satisfy the boundary conditions that are: the non-slip condition on the wall
and a smooth transition to the free stream at the end of the boundary layer.
Based on these Information we can see that a 2nd order polynomial can provide a suitable
velocity profile.
Using this profile in the expressions for the momentum thickness and the displacement
thickness we obtain these simple relations for displacement and momentum thickness.
The next step is to use these relations to obtain an expression for the drag coefficient.
In order to do so, we can first get an expression for the shear stress based on this 2nd order
polynomial profile and then substitute it into the expression for the friction coefficient.
Doing so we can get approximate relations to estimate the boundary layer thickness,
displacement and momentum thickness at specific locations over the plate.
The Reynolds number with the X subscript simply means that the characteristic length used
for the estimation of the Reynolds number is the distance between the leading edge of
the plate and the position analyzed, so it varies along the plate.
Now as last step we can obtain the final expression for the coefficient of drag from the friction
coefficient expression.
Here is the beauty of this analysis, all these expressions are based on a guessed velocity
profile and relations obtained using simple integral considerations and still they provide
results within 10% of the exact laminar boundary layer solution.
In addition to that we can also perform some dimensional analyses and gather additional
insight on the boundary layer.
A thin boundary layer presents a thickness that is extremely smaller compared to the
plate or body size.
Based on the result of the integral analysis we obtain that the Reynolds number must be
much larger than one.
Let’s now analyze the deviation of the streamline due to the boundary layer.
Combining the expression for the displacement thickness and the assumption of large Reynolds
number, we have that in a thin boundary layer the velocity parallel to the wall is much
larger than the normal component.
That quadratic velocity profile gives us an additional information, differentiating it
respect to x and to y we can find that the variation along x of the velocity component
is much smaller than its variation normal to the wall.
The same holds also for the normal component of the velocity.