The boundary layer equations provide us a tool to analyze the viscous effects contained into the
boundary layer and estimate the friction and the drag acting on a surface. In this lesson, we are
going to analyze different methodologies that have been used to extend the range of applicability
of the boundary layer theory. The different approaches will give us a set of equations
that we can use to study different cases such: non constant mean flow,
non-zero pressure gradients, corrections for low and high Reynolds numbers,
solutions for geometries different than flat plate, and three-dimensional flows.
We have several methods to study so let’s get started!
We can extend the range of applicability of the boundary layer equations
to the case of non-constant mean flow, meaning that the free-stream velocity can vary with
X and time. We can then recast the equations into this form,
where the terms in the brackets takes into account the pressure gradient.
This expression of the boundary layer is valid for both laminar and turbulent boundary layers.
Starting from these equations Karman came up with a form of the momentum integral equation
multiplying the continuity equation by the difference of the X component of the velocity and
the free-stream velocity, and then subtracting it from the momentum equation. In the integral form
obtained we can recognize the expressions of the momentum thickness and the displacement thickness.
The term V_W is the normal velocity to the wall that takes into account injection or suction of
air through the wall. Substituting the momentum thickness and displacement thickness terms
we can rewrite this equation in this form and connect it to the skin friction coefficient.
Under the assumption of steady flows and non-porous walls
we can reduce the expression to this form where H is the shape factor.
This form of the Karman integral relation is used in the analysis of turbulent boundary layers.
Now, starting from this equation we can introduce the correlation method of Thwaites
to develop a solution, based on correlations, that works for non-zero pressure gradients
and up to the separation point. The idea is to multiply the Karman relation by a factor
and defining a correlation parameter Lambda to develop a new form of the equation. We can see
that the equation can be written in this form, where we can identify some correlation
functions: one is the shear correlation in one is the shape factor correlation.
Wait wait wait…. Ok… let's step back for a moment and understand what we mean with “correlations”.
Back in those days numerical techniques were not available or extremely popular and scientists
could rely mostly on experimental data to understand physical phenomena. In this regard many
of them started trying to understand how different variables can correlate with specific behaviors
to understand what parameters affect a specific kind of physics. Correlations were found after
intense studies of extremely large amount of experimental results, to identify specific
patterns that can lead to mathematical models that we can use to describe the flow.
Going back to the analysis, we can rewrite the equations in terms of this correlation functions.
Thwaites found that for a large set of experimental results the F function could
be described by a simple linear fit. Substituting this expression into the ODE we can finally solve
it and obtain an estimate of the moment thickness then we can use to calculate friction and drag.
In this particular expression the constant C it's typically set to 0 to avoid
Theta to be infinite when the location X0 is a stagnation point. The final quadratic relation
for the momentum thickness give us an accurate prediction of the of data within 5%
for favorable or mild adverse pressure gradients and within 15%
near separation points. We can also estimate the shear stress and the displacement thickness
once we know the momentum thickness using the following expressions, where we can use a curve
fit for the shear function and a polynomial form for the shape factor function.
The next method that we're going to analyze is the Falkner-Skan equation.
This is a similarity solution approach for flows over wedge shape geometries.
They generalized the Blasius solution for variable free-stream velocity and then they found out that,
if the velocity as a power law distribution, a similarity solution can be obtained.
The non-dimensional coordinate ETA is different compared to the one used in the Blasius solution.
Combining these relations and following steps similar to the ones done in the Blasius solution
we can obtain this single ODE that describes the boundary layer over wedge shapes,
as long as the boundary conditions are satisfied. Wall as streamline. Non-slip condition at the wall
and freestream velocity at infinity are the boundary conditions. In the expression
we have a beta term that appears. This term represents a measure of the pressure gradient
in the flow field and is connected to the angle of the geometry. With this in mind,
we can have special cases such as a flat plate and a vertical plate.
The Falkner-Skan equation includes the effect of the pressure gradient in the solution.
The gradient can be calculated using Bernoulli's equation
from the free-stream velocity. The solution of the equation is similar to the Blasius boundary layer
except that a numerical integration method is required to obtain the solution.
The equation relies on the parameters beta and m so let’s analyze the physical meaning of M.
If M is less than zero the study case presents an adverse pressure gradient
meaning that the pressure increases along the flow stream direction. The pressure change along the
surface acts as an obstacle slowing down the fluid or even reverting the fluid direction and creating
separation. If M is larger than 0 we have instead a favorable pressure gradient, hence the pressure
decrease along the flow stream direction. That helps the fluid to stay in contact with the wall.
Different combinations of beta and m can represent
different configurations such as expansion corners and wedges.
Here we can see the predicted velocity profiles for different combinations of M and BETA terms.
As you can see the profile shape can vary significantly, and for the red profile we
reach flow separation. The Falkner-Skan equation gives us a good prediction that takes in account
pressure gradients and it's valid up to the separation point for different geometries.
The boundary layer equations that we've seen so far deal with two-dimensional flows
but we can also extend our analysis to three-dimensional boundary layers.
Consider a plate flat plate aligned with the XZ plane we can write the 3D boundary layer equations
where U_Infinity and W_Infinity are known freestream velocity components.
Three dimensional flows can present cross flows that affect the velocity distribution
in the boundary layer. Cross flows are typically referred as secondary flows since their magnitude
is generally smaller than the intensity of velocity in the main flow direction.
Examples of secondary floors in boundary layers can be aircrafts with swept back wings
or cross flows on turbo machinery blades or propellers. If we look at the distribution
of the velocity along the mainstream direction, we can identify our typical boundary layer profile.
If we then include the components of velocity
acting on the plane normal to it, we can see that there are non-zero for secondary floors,
and we can have a unidirectional skewing or bidirectional skewing. In the example
presented here we have a duct with an s-bend that generates a secondary motion in the fluid.
The analyses introduced up to here are based on the assumption of thin boundary layer
and relatively large Reynolds number with a limited range of applicability. The boundary
layer problem has a singular perturbation nature, so, if we introduce the non-dimensional form of
the boundary layer momentum equation and take the limit of Reynolds to infinity,
you can see that the term on the right hand side vanishes. This create a singularity where
the equation cannot satisfy anymore the non-slip condition at the wall.
However, even for large Reynolds number we have no sleep at the wall,
hence the boundary layer theory is not physical for extremely large Reynolds numbers. Friedrichs
approached the problem to correct this limitation and successively Van Dyke introduced an asymptotic
methodology to provide a complete solution. Following Friedrichs approach, the boundary layer
momentum equation can be roughly approximated by this ODE and it's respective boundary conditions.
The equation can provide us two different solutions for two distinct regions. One for
the inner layer dominated by the viscous effects where the no-slip condition must be satisfied. The
second is the outer layer where we are approaching the free stream velocity and the viscous effects
are limited. The two solutions are valid on the only in their respective regions. Van Dyke
Introduced the matching principle, the idea behind it, is too blend the solutions of the two regions
to create a composite function that represents the entire region. Shown here is the first order
analysis of the physics model for the inner layer and the outer layer. We can see in the plot that
the are disconnected. Blending those 2 functions we can obtain a description of the entire profile.
Van Dyke further extended and generalized this first order procedure into a matching approach
for inner and outer expansions of any order. This methodology has been used to correct leading edge
effects at low Reynolds number and trailing edge effects. As an example, reported here is a second
order asymptotic correction for the flat plate boundary layer. The final result for the drag
coefficient is a sum of the Blasius solution and the second order correction term. This let
us extend the Blasius solution to the range of Reynolds number between one and one thousand.
The matching principle is widely used also in the analysis of turbulent boundary layers.
In this lecture we have seen different methodologies
that have been used in the past to extend the range of applicability
of the boundary layer theory to predict friction and drag for a wider set of phenomena.
Effects like the presence of pressure gradients, three dimensional flows, low and
high Reynolds numbers as well as non-constant inflow conditions can in this way be analyzed.