In this lesson, we will look at aspects of turbulent wall-bounded flows.
Wall-bounded flows are situations where fluid flows through or around a solid body.
A few examples are flow through pipes or tubes, flow over a car, in fact even the flow
over you when you are cycling or swimming. Because of viscosity, a very thin layer close to the body,
called the boundary layer, is formed within which viscous forces play a crucial role.
All boundary layers start of in an orderly manner I.e. laminar and eventually transition
to attain a turbulent nature. Based on our understanding of the laminar boundary layers,
we will now develop a framework to understand the behavior of turbulent boundary layers
formed due to air flow over a flat plate.
Let us consider the 2D dimensional version of the RANS equations, written in cartesian coordinates.
Because the boundary layer is a very thin region, its thickness is very small compared
to the length of the plate. Furthermore, we can assume the variation of properties in the
streamwise direction is negligible compared to the variation in the wall normal direction. Also,
since we are assuming 2D, all the z-components and their derivatives can be neglected.
As a result, the RANS equations reduce to those shown here.
For external flows like this case, since most of the viscous effects are restricted to the boundary
layer, we can assume an inviscid region outside the boundary layer. Consequently, the Bernoulli's
equation is satisfied here and at the edge of the boundary layer, we have the following relation.
Substituting this into the momentum equation, we obtain the equation shown here.
These equations can be solved by applying the no-slip boundary condition on the flat-plate and
mean flow matching condition at the edge of the boundary layer. Recollect the x-momentum equation
for the laminar boundary layer. The only difference between the two equations
is the presence of the Reynolds stress term in the turbulent boundary layer equations.
From the integral forms of conservation of mass and momentum equations, we know the displacement
thickness and momentum thickness of the boundary layer. The Karman momentum integral equation
which is obtained by manipulating the continuity and the momentum equations is shown here.
This equation, written in terms of displacement and momentum thickness,
also holds for the turbulent boundary layers, with one difference. The stress term has contributions
from the turbulent velocity fluctuations in addition to the laminar stresses.
Now that we know the governing equations for the turbulent boundary layer,
we can use numerical techniques to solve and obtain a solution for a given problem.
We will not discuss those methods in this course. However, we will slightly switch gears
and directly discuss the behavior of the turbulent boundary layers. Prandtl's mixing
length hypothesis provides an algebraic estimate for calculating the eddy viscosity. Based on this
theory, it is possible to classify the turbulent boundary layer into 3 different zones.
The inner Layer or the near-wall region where the viscous forces play a dominant role.
This region covers about ~15% of the total boundary layer thickness !
It can be further divided into two zones.
Laminar Sublayer -- This is the zone that is closest to the wall. It is so named because of
the highly viscous nature of the flow in this region causing the flow to be laminar.
Buffer zone -- This region is a transition region where the viscous forces from the laminar sublayer
still play a role but the turbulent nature of the flow starts to kick in
Outer Layer -- The region beyond the Near-wall is called the outer-layer
where only the turbulence effects dominate and the influence of viscosity is negligible.
Overlap Layer -- This is the transition region between the inner and the outer layers
through which the effect of the wall slowly fades away.
In order to describe the flow in these different layers, Prandtl and Karman reasoned that in the
Inner layer, the freestream properties does not really impact the flow characteristics.
Instead, the mean velocity in the boundary layer depends on properties such as
wall shear stress, density, viscosity and wall normal distance. Based on dimensional analysis,
we can obtain two dimensionless variables that have the following form.
Here, v* has the units of velocity and is called the wall-friction velocity.
Conversely, in the outer layer, the velocity distribution is dictated
by the freestream pressure gradient, wall shear stress and boundary layer thickness, and not on
viscous forces. Once again based on dimensional analysis, we obtain the following relations.
This relation is sometimes also referred to as the velocity-defect law, where V-u
is the reduction in the velocity due to the influence of the wall.
In the overlap region, F_inner and F_outer functions seen in the above two relations,
are got by imposing a velocity match. After matching, we can show that the these two
functions are in fact logarithmic. Based on this, we can rewrite the velocity profiles
for the two layers for the overlap region as shown here. These are equivalent relations.
This logarithmic velocity profile distribution is generally referred to as the law of the wall.
Because of the logarithmic variation, the overlap region is also referred to as the log-law region.
In these equations, k and B are universal constants and are approximately 0.41 and 5
respectively. The value of A is primarily defined by the mean flow pressure gradient, among other
variables. If we plot the relation for the inner layer with the experimental data, we notice
that in the range of y+, i.e the non-dimensional wall normal distance, extending from 30 to 350,
there is an excellent correlation between the log profile and the experiments. The upper range of y+
increases with increasing Reynolds numbers. For example, for atmospheric boundary layer flows,
the log law region can extend to the order of 1000's of y+.
In the region very close to the wall i.e. the viscous sublayer, where y+ is below 5,
the velocity profile is linear.
Between y+ values of 5 and 30 i.e. the buffer region, the velocity profile neither follows
the linear nature of the viscous sublayer nor the logarithmic nature of the overlap region.
It adopts a smooth profile that transitions from the linear to the logarithm character.
Spalding in 1961 proposed a composite blend of the law of the wall,
which is valid for the entire wall region, including the buffer layer.
Beyond y+ of 350, the logarithm profile is no longer valid. This is b because the
outer layer at this location is significantly influenced by the free stream pressure gradient.
Coles observed from multiple experiments
that the deviation of the velocity profile from the log distribution takes the shape of a wake.
He proposed a modification to the log law to include a wake function.
This wake function is determined by a curve fit, a common form of which is shown here.
This modified log-law profile is valid all through the overlap and the outer layer region.
i.e. for y+>30. For, region below y+=30, the wake function can be neglected and
the Spalding formula can be used to calculate the velocity distribution.
By integrating the Coles' wall-wake formulation, we can obtain boundary layer parameters
such as the displacement and momentum thicknesses and the skin-friction coefficient.
These equations need to be solved in order to estimate the boundary layer parameters.
However, based on certain experimental data, correlations can be written that can circumvent
the equation solving. If we assume mean flow equilibrium i.e. zero pressure gradient,
the velocity profile can be described by a value of pi=0.45. At the edge of the boundary layer, we
can obtain the following relation for the Cf and Reynolds number based on boundary layer thickness.
By curve fitting, we obtain a power-law approximation for the skin-friction coefficient.
Based on pipe flows, Prandtl suggested the use of one-seventh power law correlation
for the velocity to determine the momentum thickness.
But we know that Cf is related to the momentum thickness by the following relation.BY
substituting and solving, we obtain the relations for boundary layer thickness
and skin-friction coefficient in terms of Reynolds number based on axial length
along the flat plate. Prandtl originally proposed another set of relations given
here. These relations are quite often used in literature but are not very accurate.
By now, you probably have a good idea regarding the behavior of turbulent boundary layers.
A question you might be having is, how do they compare with the laminar boundary layers?
Recall the relations for a laminar boundary layer over a flat-plate. Let us
consider an example problem to get a sense of how different he two boundary layer profiles are.
Assume that air at 20c is flowing over a smooth flat plate of length 1m at a velocity of 10m/s.
The Reynolds number of this flow is approximately 650,000. The critical
Reynolds number for a smooth flat plate is 10^5 and the transitional Reynolds number is 3*10^6.
Based on this range, the current problem is in the transitional regime. So, it is a fair to compare
the solutions from both the boundary layer theories. At the trailing edge of the plate,
the boundary layer thickness assuming a laminar profile is approximately 6.04 mm, whereas
the turbulent profile is 23.59 mm i..e about 4 times that of the laminar profile.
The skin-friction coefficient values also show a similar trend.
A representative profile for the axial velocity distribution as it varies with the wall normal
distribution, for both the boundary layer types is shown here. It is easy to see
that the turbulent profile is more "fuller" compared to the laminar profiles. This is
because of the enhanced momentum exchange for the turbulent boundary layers, that brings in high
velocity fluid from the outer regions towards the wall. With that let us wrap up this lesson.