Take a look at a concept sport car racing on a track. The flow adjusts to the presence of the
body surrounding it. If you take a closer look at the flow, near the body surface,
you could notice that the flow looks somehow different. Can you see it ? let's get closer….
even closer. Can you see it now ? That is a boundary layer,
a really thin region where viscous effects are important.
Boundary layers are essential physical features of bounded flows,
that they provide insight on physical mechanism, which control phenomena like drag or separation
and must be accounted for any fluid-dynamic based design, like aircrafts, cars or ships.
The concept of the boundary layer was first introduced by Ludwig Prandtl in 1904.
He observed that the viscous effects are contained within the thin boundary layer and
that the fluid in region, outside the boundary layer, behaves as if inviscid. This concept
let us obtain analytical solutions of a wide range of flow problems.
Let's now analyze a viscous flow over a flat plate. The flow passing over the flat plate
generates a boundary layer starting from the leading edge of the plate.
As we move along the plate we can see that the boundary layer thickness grows. Looking more in
detail, we can also notice that the boundary layer is first smooth and then bursts into
eddies and swirls. (4a) This can be seen on a free surface flow where the surface is first smooth and
then presents several vortical structures. These two regions define the laminar and
the turbulent boundary layer. In between we find the transitional zone where the laminar boundary
layer starts becoming turbulent. Now let's find out how a boundary layer forms.
Let's imagine we have a flat plate that's moving in a fluid with a certain velocity.
We can describe the fluid at different locations from the flat plate using some particles. As the
flat plate moves, it will pull the fluid that is in direct contact with it and this will affect
the particles near the plate. Further away the fluid will not react to the plate motion. Hence,
the different particles shown here will move with a different velocity depending on the
distance from the moving wall. The particle that is directly touching the plate surface will move
with the same velocity of the plate. This is like having a hand immerged in water and then suddenly
pulling it out. Part of the water will follow your hand and stay on it during the motion.
Similarly, the fluid will follow and stay attached to the plate that is moving. Now,
if we move with the plate, it will look stationary and the fluid will move around the plate.
In this case the particle touching the wall won’t move. This is due to the non- slip condition that
states that a particle on a boundary will not move relatively to it. As we move away from the wall,
the particles will have a finite velocity that increases, as the particle is farther
away from the wall, up to a maximum that is the freestream velocity.
We can now define some useful parameters for the description of a boundary layer.
As we mentioned before, the boundary layer grows from the leading edge of the plate.
Here we can see the boundary layer edge that represents the position
where the flow velocity is equal to the 99% of the free steam velocity.
The distance between the plate and the boundary layer edge is defined as boundary layer thickness
and is commonly represented by the Greek letter delta. Now you can see that we have two different
regions: an inner one, that is a viscous zone that defines the boundary layer, and the outer
one that is an inviscid zone where we can assume that the fluid is inviscid.
Let's define now the reasoning underlying the boundary layer theory.
The boundary layer is assumed to be thin. We can analyze the characteristic timescales of the
boundary layer to define when this layer is thin. The first timescale is connected to convection
and represents the time a particle will take to move over the plate. The second time scale
is connected to the viscous diffusion that represents how the viscous effects diffuse
normal to the wall. Namely, the time for the momentum defect to propagate normal to the wall.
For a boundary layer to be thin, the convection timescale must be much
larger than the viscous diffusion time scale. This means that taking the ratio of the
convection timescale over the viscous diffusion time scale this result should be larger than
one. The terms presented here under the square root are nothing but the Reynolds number. Hence,
we found that that the Reynolds number must be much larger than one for the boundary layer to
be thin. A Reynolds number larger than one means that the convective effect are more significant
than the viscous effects. In our analysis we will assume that a boundary layer is thin for
Reynolds numbers larger than one thousand. The boundary layer is laminar for the range of
Reynolds number values that span from a thousand to a million. This range is only for guidance,
since different conditions such as surface roughness and free-stream conditions can
trigger the transition to turbulence at lower or higher Reynolds numbers.
A boundary layer is not always present around a body. For example there are no boundary layers
in the separated regions. In this picture we can see a small region where the flow
separates due to the sharp curvature of the body and then later reattaches to the body.
Flow separation occurs when the main flow stream detaches from the body. Once the flow
stream is no more in contact with the surface it cannot generate anymore a boundary layer.
Our goal in analyzing the boundary layer is to develop a solution for the velocity distribution
within the boundary layer for given geometries. Knowing the velocity distribution, we can
calculate the shear stress at the wall and thus the skin friction drag force acting on the body.