
English: 
Hello welcome to my talk, All about Fluids.
this talk is on turbulent boundary layer, in which I will show how
the standard turbulent boundary layer is derived and some details within the
turbulent boundary layer will be discussed. This development is very
useful for practical purpose, as well as for providing the basis for developing
the enhanced or improved turbulent boundary layers for different
applications.
Why we need to study the turbulent boundary layer? In principle the full

English: 
Hello welcome to my talk, All about Fluids.
This talk is on turbulent boundary layer, in which I will show how
the standard turbulent boundary layer is derived and some details within the
turbulent boundary layer will be discussed. This development is very
useful for practical purpose, as well as for providing the basis for developing
the enhanced or improved turbulent boundary layers for different applications
Why we need to study the turbulent boundary layer? In principle the full

English: 
solution to the Navier-Stokes equation would be only to use the direct numerical
simulation (DNS), which would resolve all eddies from the largest to the Kolmogorov
length scales, a requirement for this direct solution is the number of the
grid, which in principle is a proportional to power of 9/4
of the Reynolds number, and this can lead to a huge number of grids if
the Reynolds number is large. As deduced by Spalart in 2000 DNS would
be ready for practical applications in around 2080 if the advancing speed

English: 
solution to the Navier-Stokes equation would be only to use the direct numerical
simulation (DNS), which would resolve all eddies from the largest to the Kolmogorov
length scales, a requirement for this direct solution solution is the number of the
grid, which in principle is a proportional to power of 9/4
of the Reynolds number, and this can lead to a huge number of grids if
the Reynolds number is large. As deduced by Spalart in 2000 DNS would
be ready for practical application in around 2080 if the advancing speed

English: 
of computer technologies can be maintained. Take an aeroplane or a car as an example
the grid need for RANS is 10^7,  for LES  is 10^11.5,
and for DNS is 10^16. Currently we are pretty much in 10^9.
However, in 2016 he and his a co-author revised the claim and stated: by now
we are not confident of this for the 21st century, or even that it will ever happen.
We know for the viscous flow, an

English: 
of computer technologies can be maintained. Take an aeroplane or a car as an example
the grid need for RANS is 10^7, for LES is 10^11.5,
and for DNS is 10^16. Currently we are pretty much in 10^9.
However, in 2016 he and his co-author revised the claim and stated: by now
we are not confident of this for the 21st century, or even that it will ever happen.
We know for the viscous flow, an

English: 
important aspect is to calculated the skin friction drag accurately, which is a
challenge so far for numerical modeling. The most popular method, the RANS modeling
which has a least requirement on grid, would have issues with the grid
generation in the boundary layer since for a good modelling, a requirement would
be that there must be points within the wall unit y+ equalling to five or less.
This requirement would lead to a very large number of grids for RANS
modeling. Another issue is in the boundary layer the complicated processes

English: 
important aspect is to calculated the skin friction drag accurately, which is a
challenge so far for numerical modeling. The most popular method, the RANS modeling
which has a least requirement on grid,  would have issues with the grid
generation in the boundary layer since for a good modelling, a requirement would
be that there must be points within the wall unit y+ equalling to five or less.
This requirement would lead to a very large number of grids for RANS
modeling. Another issue is in the boundary layer the complicated processes

English: 
are involved in the transition from laminar to turbulent, from low Reynolds
number - high Reynolds number. Relatively the confidence of the turbulence
modeling is high only for the flows of high Reynolds number.
In reality, the modeling of the near wall flow would be made often use the wall
functions, such as in the commercial software and in the open source. This
avoids the resolution of the boundary layer directly. In this regard, the
understanding of the boundary layer would be very important
Turbulent flow can be simply regarded as the superposition of the highly

English: 
are involved in the transition from laminar to turbulent, from low Reynolds
number - high Reynolds number. Relatively the confidence of the turbulence
modeling is high only for the flows of high Reynolds number.
In reality, the modeling of the near wall flow would be made often use the wall
functions, such as in the commercial software and in the open source. This
avoids the resolution of the boundary layer directly. In this regard, the
understanding of the boundary layer would be very important
Turbulent flow can be simply regarded as the superposition of the highly

English: 
irregular and oscillatory flows upon a smoothed flow, more specifically, the
superposition of the fluctuating flows on the mean flow
however an accurate definition on turbulent  flow would be
difficult, probably because it is too difficult to use the few words to
describe the extremely complicated phenomena from case to case.
Generally turbulent flow happens when the Reynolds numbers are large enough,
for instance, in a horizontal circular pipe, the critical Reynolds number is
2300,  while in the boundary layer on a flat plate it is 500,000.

English: 
irregular and oscillatory flows upon a smoothed flow, more specifically, the
superposition of the fluctuating flows on the mean flow.
However, an accurate definition on turbulent flow would be
difficult, probably because it is too difficult to use the few words to
describe the extremely complicated phenomena from case to case.
Generally turbulent flow happens when the Reynolds numbers are large enough,
for instance, in a horizontal circular pipe, the critical Reynolds number is
2300, while in the boundary layer on a flat plate it is 500,000.

English: 
this can be seen from the free online
book, 'Fluid mechanics, Turbulent flow and turbulence modelling' by L.Davidson
if we look at the physics of the Reynolds
number, we can see the Reynolds number is actually the ratio of the fluid
inertia force over the viscous force. In this regard, it can be understood that
the turbulent flow occurs when its inertia force dominates the viscous
force. There is a similar statement in Wikipedia on how the flow turbulence is
cause: turbulence is caused by excessive kinetic energy in parts of a fluid flow

English: 
this can be seen from the free online
book, 'Fluid mechanics, Turbulent flow and turbulence modelling' by L.Davidson
if we look at the physics of the Reynolds
number, we can see the Reynolds number is actually the ratio of the fluid
inertia force over the viscous force. In this regard, it can be understood that
the turbulent flow occurs when its inertia force dominates the viscous
force. There is a similar statement in Wikipedia on how the flow turbulence is
cause: turbulence is caused by excessive kinetic energy in parts of a fluid flow

English: 
which overcomes the damping effect of the fluid's viscosity.
However, how the turbulent flow starts is really unclear and still challenging too.
having difficulties in defining the flow turbulence, there are some common
characteristics for the turbulent flows,  these include: irregularity,  this is
very obvious when we compare the turbulent flow to a laminar flow, the
fluctuating part is always unsteady and the chaotic; Diffusivity, diffusivity
would be significantly increased in turbulent flows and this would increase
the resistance and heat transfer too. Hence in some practical applications

English: 
which overcomes the damping effect of the fluid viscosity.
However, how the turbulent flow starts is really unclear and still challenging too.
having difficulties in defining the flow turbulence, there are some common
characteristics for the turbulent flows, this include: irregularity, this is
a very obvious when we compare the turbulent flow to a laminar flow, the
fluctuating part is always unsteady and the chaotic; Diffusivity, diffusivity
would be significantly increased in turbulent flows and this would increase
the resistance and heat transfer too. Hence in some practical application

English: 
turbulence is enhanced to increase the diffusivity in the flow; Turbulent flows
always happen in large Reynolds numbers for instance, in pipe flow it is 2300 and
in the boundary layer flow it is 500,000; Turbulent flows are always
three-dimensional, however in the 2D setups, the mean flow may be
2-dimensional, but the fluctuating flows are always three-dimensional;
Dissipation. the turbulence would increase the energy dissipation. this is different
from those in laminar flows, in which the dissipation happens in the molecular

English: 
turbulence is enhanced to increase the diffusivity in the flow; Turbulent flows
always happen in large Reynolds numbers for instance, in pipe flow it is 2300 and
in the boundary layer flow it is 500,000; Turbulent flows are always
three-dimensional, however in the 2D setups, the mean flow may be
2-dimensional, but the fluctuating flows are always three-dimensional;
Dissipation. the turbulence would increase the energy dissipation. this is different
from those in laminar flow, in which the dissipation happens in the molecular

English: 
scale. In turbulent flow the energy dissipation happens at the smallest
Kolmogorov scale, the smallest eddies, which would (be) still significantly larger than
the molecular scale. Continuum: the smallest eddies are much
larger than the molecular scales. Hence the turbulent flow can be regarded as a continuum
unlike the laminar flow, in turbulent flow, the flow can be separated into two
parts: the mean flow, the average flow, which is very similar to the flow in a
laminar flow; and the fluctuating part of the flow, which is the unsteady part
superimposing on the mean flow. Mathematically the turbulent flow

English: 
scale. In turbulent flow the energy dissipation happens at the smallest
Kolmogorov scale, the smallest Eddie's which would still significantly larger than
the molecular scale. Continuum: the smallest Eddie's are much
larger than the molecular scales. Hence the turbulent flow can be regarded as a continuum
unlike the laminar flow, in turbulent flow, the flow can be separated into two
parts: the mean flow, the average flow, which is very similar to the flow in a
laminar flow; and the fluctuating part of the flow, which is the unsteady part
superimposing on the mean flow. Mathematically the turbulent flow

English: 
velocity ui can be expressed as this,
the mean part, the capital Ui and the fluctuating part u'_i. Here i equals
to 1, 2, 3 for three different spatial directions x, y and z.
if we take it the capital U as the mean flow, and then, we can easily
obtain the average of the fluctuating part of the flow velocity is zero.
Similarly we can make the operations on the turbulent pressure, which can be
regarded as a combination of the mean pressure, the capital P, and the

English: 
velocity ui can be expressed as this
the mean part, the capital Ui and the fluctuating part u'_i. Here i equals
to 1, 2, 3 for three different spatial directions: x, y and z.
if we take the capital U as the mean flow, and then, we can easily
obtain the average of the fluctuating part of the flow velocity is zero.
Similarly we can make the operations on the turbulent pressure, which can be
regarded as a combination of the mean pressure, the capital P, and the

English: 
fluctuating part p', so we have this and this.
To make the problem simpler, we consider the incompressible flows. hence the
continuity equation can be expressed simply as this in Einstein summation.
it is equivalent to this. Here the double subscript i means the repeat from 1
to 3, given as this. So if we average the expression
as this, we have this, and then average each term of the left-hand side
of the equation. Since this term can be expressed as this, the average of the

English: 
fluctuating part p', so we have this and this
To make the problem simpler, we consider the incompressible flows. hence the
continuity equation can be expressed simply as this in Einstein summation.
it is equivalent to this. Here the double subscript i means the repeat from 1
to 3, given as this. So if we average the expression
as this, we have this, and then average each term of the left-hand side
of the equation. Since this term can be expressed as this, the average of the

English: 
mean velocity is same as this, and this can be replaced by this mean
velocity and mean of this term, so this will be zero
Hence we have the continuity equation for the mean flow velocity as this
Accordingly we would have the continuity equation for the fluctuating
velocity as this.
Having said the irregularity and chaotic motions in turbulent flow,
it is accepted that the turbulent flows are still governed by the
Navier-Stokes equation.  For a simplification, in this talk
incompressible flows are assumed, thus the Navier-Stokes equation can be

English: 
mean velocity is same as this, and this can be replaced by this mean
velocity and mean of this term, so this will be zero
Hence we have the continuity equation for the mean flow velocity as this
Accordingly, we would have the continuity equation for the fluctuating velocity as this.
Having said the irregularity and chaotic motions in turbulent flow,
it is accepted that the turbulent flows are still governed by the
Navier-Stokes equation.  For a simplification, in this talk
incompressible flows are assumed, thus the Navier-Stokes equation can be

English: 
given as this, in Einstein summation. So we take the average of the equation
we have the expression as this, so for the linear terms, the averaging
operations would be very straightforward as these, on the velocity
time rate, and on the pressure gradient and  the
term for the viscous force, while the body force fi is normally
well described, for instance, the gravitational force, it would there be
same for both turbulent and laminar flows.
The significant difficulties are coming from the nonlinear term in the
momentum equation, the convective term as this. We substitute the mean and

English: 
given as this, in Einstein summation. So we take the average of the equation
we have the expression as this, so for the linear terms, the averaging
operations would be very straightforward as this on the velocity
time rate, and on the pressure gradient and the
term for the viscous force, while the body force fi is normally
well described, for instance, the gravitational force, it would there be
same for both turbulent and laminar flows
The significant difficulties are coming from the nonlinear term in the
momentum equation, the convective term as this. We substitute the mean and

English: 
fluctuating velocities into the expression, and the averaging of the
terms as this. Since the mean values Ui and Uj are
constants for the averaging operation, therefore, we can take them out, so we
have the expression as this. Now the gradient of the fluctuating velocity
with xj, its average would be zero. and obviously, this term would be zero as well.
so all these two terms will vanish.
therefore we have the expression for this nonlinear term and we can see
there is an additional term because of the flow turbulence

English: 
fluctuating velocities into the expression, and the averaging of the
terms as this. Since the mean values Ui and Uj are
constants for the averaging operation, therefore, we can take them out, so we
have the expression as this. Now the gradient of the fluctuating velocity
with xj, its average would be zero. and obviously, this term would be zero as well.
so all these two terms will vanish.
Therefore, we have the expression for this nonlinear term and we can see
there is an additional term because of the flow turbulence

English: 
So we can write this additional term in this form, and if we apply the
continuity equation for incompressible flows as this, and this term will
vanish. Therefore we have the expression as this.  we write this back to the
momentum equation, we have the expression for the mean velocity as this. here is
the additional term. Now if we define a new stress tensor for the turbulent flow,
Tij as this.
this term is named as Reynolds stress, so use the newly defined stress
tensor Tij, the momentum equation for the mean flow velocity Ui would be given

English: 
So we can write this additional term in this form, and if we apply the
continuity equation for incompressible flows as this, and this term will
vanish. Therefore, we have the expression as this.  we write this back to the
momentum equation, we have the expression for the mean velocity as this. here is
the additional term. Now if we define a new stress tensor for the turbulent flow, Tij as this.
this term is named as Reynolds stress, so use the newly defined stress
tensor Tij, the momentum equation for the mean flow velocity Ui would be given

English: 
as this. this is the same as the momentum equation for the laminar flows,
in this slide, a discussion would be made for the Reynolds stress
Given the turbulent stress expression as this:  this term represents
the fluid viscous stress, and this is the Reynolds stress.
From the expression we can see Reynolds stress is a symmetric tensor, with
six unknowns involving in Reynolds stress. Reynolds stress is a very
difficult and challenging term and, generally there is no rational method to
determine the Reynolds stress in analytical or numerical solutions for

English: 
as this. this is the same as the momentum equation for the laminar flows,
in this slide, a discussion would be made for the Reynolds stress
Given the turbulent stress expression as this:  this term represents
the fluid viscous stress, and this is the Reynolds stress.
From the expression we can see Reynolds stress is a symmetric tensor, with
six unknowns involving in Reynolds stress. Reynolds stress is a very
difficult and challenging term and, generally there is no rational method to
determine the Reynolds stress in analytical or numerical solutions for

English: 
Navier-Stokes equation. For the flows of high Reynolds number, the
Reynolds stresses are generally more important than the real viscous stress,
since the viscous stress is proportional to the viscosity MU, it is
inversely proportional to Reynolds number. Due to the Reynolds stress, the level of
total stress in turbulent flow is increased, which implies the friction
drag on a body immersed in the turbulent flow will be larger than that in the
laminar flow. A physical explanation for this is the
turbulent momentum convection would cause the velocity profile to become

English: 
Navier-Stokes equation. For the flows of high Reynolds number, the
Reynolds stresses are generally more important than the real viscous stress,
since the viscous stress is proportional to the viscosity MU, it is
inversely proportional to Reynolds number, due to the Reynolds stress, the level of
total stress in turbulent flow is increased, which implies the friction
drag on a body immersed in the turbulent flow will be larger than that in the
nominal flow. A physical explanation for this is the
turbulent momentum convection would cause the velocity profile to become

English: 
more uniform and therefore it will result in a larger velocity gradient
near the boundary. Here in the plot, we can see the turbulent and nominal
boundary layer over a flat plate with a free stream velocity U0 equalling to
55.56 m/s, so here we can see the large gradient for
the turbulent boundary layer, therefore a large is skin-drag in the turbulent boundary layer.
For a simplification of the problem, the turbulent the boundary layer on a
flat plate is discussed here. The problem analysis would show the mean velocity u
in the boundary layer, given as this. Here the normal u is used for the mean

English: 
more uniform and therefore it will result in a larger velocity gradient
near the boundary. Here in the plot, we can see the turbulent and laminar
boundary layer over a flat plate with a free stream velocity U0 equalling to
55.56 m/s,  so here we can see the large gradient for
the turbulent boundary layer, therefore a large skin-drag is in the turbulent
boundary layer.
For a simplification of the problem, the turbulent boundary layer on a
flat plate is discussed here. The problem analysis would show the mean velocity u
in the boundary layer, given as this. Here the normal u is used for the mean

English: 
velocity in the boundary layer, rather than the capital U. so we can see the
velocity distribution would be a function of the parameters of
boundary layer thickness Delta, the distance from the boundary, y; the fluid
density, Rho; the fluid viscosity, NU, and  the free stream velocity U0.
However, in the analysis of the turbulent boundary layer, the free stream velocity
U0 is normally replaced by a so-called the friction velocity u_tau, which is
calculated as this, given (calculated) by the shear stress on the boundary.

English: 
velocity in the boundary layer, rather than the capital U. so we can see the
velocity distribution would be a function of the parameters of
boundary layer thickness Delta, the distance from the boundary, y; the fluid
density, Rho; the fluid the viscosity, NU, and the free stream velocity U0.
However, in the analysis of the turbulent boundary layer, the free stream velocity
U0 is normally replaced by a so called the friction velocity u_tau, which is
calculated as this, given (calculated) by the shear stress on the boundary.

English: 
This replacement can be justified since for a given U0 and a fixed x,
there will be a unique value for shear stress on the wall TAU_0, thus the unique
friction velocity, u_tau, so after the replacement we have the general
expression for the velocity distribution as this.
We can make the dimensional analysis to this expression and that would lead to an
expression as this: the non-dimensional velocity distribution u with regard to
u_tau is a given by a function of two non-dimensional parameters.
the first parameter is the Reynolds number based on the friction velocity u_tau and
the distance from the boundary y and the second non dimensional

English: 
This replacement can be justified since for a given U0 and a fixed x,
there will be a unique value for shear stress on the wall TAU_0, thus the unique
friction velocity, u_tau, so after the replacement we have the general
expression for the velocity distribution as this.
We can make the dimensional analysis to this expression and that would lead to an
expression as this: the non-dimensional velocity distribution u with regard to
u_tau is given by a function of two non-dimensional parameters.
the first parameter is the Reynolds number based on the friction velocity u_tau and
the distance from the boundary y and the second non-dimensional

English: 
parameter is y divided by the boundary layer thickness Delta
As the main parameter for the boundary layer, we can assume that the
non-dimensional velocity within the boundary layer, it's a function of the Reynolds
number u_tau*y divided by NU, while the non-dimensional parameter y
divided by Delta would be used before completing the formulation of the
boundary layer. As such we have the expression for this non-dimensional
velocity u as the function f1, so if we apply the boundary condition,
we have on the boundary where y=0 and no slip boundary

English: 
parameter is y divided by the boundary layer thickness Delta
As the main parameter for the boundary layer, we can assume that the
non-dimensional velocity within the boundary layer, it's a function of the Reynolds
number u_tau*y divided by NU, while the non-dimensional parameter y
divided by Delta would be used before completing the formulation of the
boundary layer. As such we have the expression for this non-dimensional
velocity u as the function f1, so if we apply the boundary condition,
we have on the boundary where y=0 and no slip boundary

English: 
condition would lead to f1(0)=0, and also we can calculate the shear
stress on the boundary tau_0, given by this and that we can calculate the
differentiation of the velocity with regard to y as this, and then we
have the expression as this. This equation would the lead to an expression
as this: f1'(0)=1, here the prime denotes the derivative
with regard to its own argument of the function.
So if we put all this together, we can have an approximation for the function
f1 as this.  This expression is valid for the viscous sublayer where the Reynolds

English: 
condition would lead to f1(0)=0, and also we can calculate the shear
stress on the boundary tau_0, given by this and then we can calculate the
differentiation of the velocity with regard to y as this, and then we
have the expression as this. This equation would lead to an expression
as this: f1'(0)=1, here the prime denotes the derivative
with regard to its own argument of the function.
So if we put all this together, we can have an approximation for the function
f1 as this.  This expression is valid for the viscous sublayer where the Reynolds

English: 
number based on the friction velocity and y is smaller than 5.
outside the viscous sublayer there may be a certain region within the
boundary layer, where the non-dimensional parameter y/DELTA would be
important. Based on this deduction, we can assume
a velocity defect law as this expression, so this velocity defect law can be
understood: at the certain region outside the viscous sublayer, the momentum
flux may be balanced by both viscous stress and the Reynolds stress.

English: 
number based on the friction velocity and y is smaller than 5.
outside the viscous sublayer there may be a certain region within the
boundary layer, where the non-dimensional parameter y/DELTA would be
important. Based on this detection, we can assume
a velocity defect law as this expression, so this velocity defect law can be
understood: at the certain region outside the viscous sublayer, the momentum
flux may be balanced by both viscous stress and the Reynolds stress.

English: 
thus the velocity defect U0-u is used in the
expression. the velocity defect law means that the
defector velocity in the certain region within the boundary would be a balanced
result by both the shear stress on the boundary, thus the friction velocity, u_tau
and the non-dimensional boundary parameter y/DELTA.
In seeking the solution for the boundary layer, we can assume in certain region
within the boundary layer, these two functions f1 and f2 are both
valid for approximating the boundary layer, so we could formulate an equation as this.

English: 
thus the velocity defect U0-u is used in the
expression. the velocity defect law means that the
defect velocity in the certain region within the boundary would be a balanced
result by both the shear stress on the boundary, thus the friction velocity, u_tau
and the non-dimensional boundary
parameter y/DELTA.
In seeking the solution for the boundary layer, we can assume in certain region
within the boundary layer, these two functions f1 and f2 are both
valid for approximating the boundary layer, so we could formulate an equation as this.

English: 
This assumption has been proven very useful for solving the boundary
layer problem. It is generally assumed that the Reynolds
number based on the friction velocity and the boundary layer
thickness Delta is significantly larger than a unit, given by this.
Now we take the differentiation of the equation with regard to y and multiply y
on both sides, we have the expression as this.
From this equation it can be seen that in both sides of the equation,
they are functions of the non-dimensional parameter u_tau*y divided by NU and y

English: 
This assumption has been proven very useful for solving the boundary
layer problem. It is generally assumed that the Reynolds
number based on the friction velocity and the boundary layer
thickness Delta is significantly larger than  a unit, given by this.
Now we take the differentiation of the equation with regard to y and multiply y
on both sides, we have the expression as this.
From this equation it can be seen that in both sides of the equation,
they are functions of the non-dimensional parameter u_tau*y divided by nu and y

English: 
divided by Delta, respectively. As such the equation holds correctly
if and only if they both are a constant, A, given as these equations.
Now we can integrate the first equation, so we have the function f1,
and integrate the second equation, we have the function f2. both C1
and C2 are the constants of integrations.
and using the experimental data, we can decide the constants in these
expressions. Here we have A=2.5, C1 is 5.1 and
C2 is 2.35. and KAPPA equals to 0.41,

English: 
divided by Delta, respectively. As such the equation holds correctly
if and only if they both are a constant, A, given as these equations.
Now we can integrate the first equation, so we have the function f1,
and integrate the second equation, we have the function f2. both C1
and C2 are the constants of integrations.
and using the experimental data, we can decide the constants in these
expressions. Here we have A=2.5, C1 is 5.1 and
C2 is 2.35. and KAPPA equals to 0.41,

English: 
roughly equals to the inverse of for the constant A, KAPPA is the von Karman
constant used in fluid dynamics. Therefore, we have the expressions for
the boundary layer and these equations can be used in different regions within
the boundary layer, so in this figure we can see the standard turbulent
boundary layer formulation gives very a good agreement with the experimental
data. According to Newman's book, Marine Hydrodynamics, there are more data
available, and the experimental data include the boundary layer profile for
flat plate, as well as the tubes and channels, all these boundary layers are same.

English: 
roughly equals to the inverse of for the constant A, KAPPA is the von Karman
constant used in fluid dynamics. Therefore we have the expressions for
the boundary layer and these equations can be used in different regions within
the boundary layer, so in this figure we can see the standard turbulent
boundary layer formulation gives very a good agreement with the experimental
data. According to Newman's book, 'Marine  Hydrodynamics', there are more data
available, and the experimental data include the boundary layer profile for
flat plate, as well as the tubes and channels, all these boundary layers are same.

English: 
A simple and very popular boundary layer profile
is the 1/7 power law which was proposed by Prandtl and correlated by Blasieus using
the experimental data. If you want to know more how this 1/7th power law is derived,
please read the book, ëAerodynamics for Engineering Studentsí.
The 1/7th power law is given as this, and this 1/7th power law gives
a very good agreement with the experimental data for the Reynolds number Rx smaller

English: 
A simple and very popular boundary layer profile
is the 1/7 power law which was proposed by Prandtl and correlated by Blasieus using
the experimental data. If you want to know more how this 1/7th power law is derived,
please read the book, Aerodynamics for Engineering Students
The 1/7th power law is given as this, and this 1/7th power law gives
a very good agreement with the experimental data for the Reynolds number Rx smaller

English: 
than 10^7.
However there is an obvious drawback of the formula, since it would lead an
infinite sheer stress on the boundary. A direct calculation for the
sheer stress on the boundary, given by this, here when y=0 on the
boundary, it is infinite. So alternatively we have to use an
approximation, which was derived from the standard boundary layer we talked in the
previous slides, the logarithmic boundary layer, it is given as this, tau_0
If we draw the 1/7th power law against the experimental data for the

English: 
than 10^7.
However there is an obvious drawback of the formula, since it would lead an
infinite sheer stress on the boundary. A direct calculation for the
sheer stress on the boundary, given by this, here when y=0 on the
boundary, it is infinite. So alternatively we have to use an
approximation, which was derived from the standard boundary layer we talked in the
previous slides, the logarithmic boundary layer, it is given as this, tau_0
If we draw the 1/7th power law against the experimental data for the

English: 
turbulent boundary layer,  we can see for different Reynolds number from 100,000 to
10^7, we can see this 1/7th power law gives a very good approximation
to all these turbulent boundary layer profiles
and in this plot we also draw out the laminar boundary layer, the Blasieus
solution and the parabolic approximation. So all these data for this drawing are from
the book, Fluid Mechanics, by F.M. White
Based on the 1/7 power law of the turbulent boundary layer, we could have
the following approximations: the turbulent boundary layer thickness is

English: 
turbulent boundary layer, we can see for different Reynolds number from 100,000 to
10^7, we can see this 1/7th power law gives a very good approximation
to all these turbulent boundary layer profiles
and in this plot we also draw out the laminar boundary layer, the Blasieus
solution and the parabolic approximation. So all these data for this drawing are from
the book, ëFluid Mechanicsí, by F.M. White.
Based on the 1/7 power law of the turbulent boundary layer, we could have
the following approximations: the turbulent boundary layer thickness is

English: 
given by Delta, equals to 0.373 x Rx power
of -1/5,
here the Reynolds number Rx, given as this. so from this expression we can compare
the boundary layer thickness to the laminar boundary thickness.
it can be seen: laminar boundary layer thickness is the proportional to x
square root, and this is the laminar  boundary layer thickness expression,
the turbulent boundary layer thickness is proportional to x power of 4/5,
therefore turbulent boundary layer thickness grows much faster than the
laminar boundary layer thickness. So in this approach, we assume the free stream
velocity U0=55.56m/s,  this is a
laminar boundary layer thickness and this is turbulent boundary layer thickness.

English: 
given by Delta, equals to 0.373 x Rx power of -1/5,
here the Reynolds number Rx, given as this. so from this expression we can compare
the boundary layer thickness to the laminar boundary thickness.
it can be seen: laminar boundary layer thickness is the proportional to x
square root, and this is the laminar boundary layer thickness expression,
the turbulent boundary layer thickness is proportional to x power of 4/5,
Therefore, turbulent boundary layer thickness grows much faster than the
laminar boundary layer thickness. So in this approach, we assume the free stream
velocity U0=55.56m/s, this is a
laminar boundary layer thickness and this is turbulent boundary layer thickness.

English: 
so we can see the turbulent boundary layer thickness grows much
faster than the laminar boundary layer thickness. And the turbulent
displacement thickness given by Delta star, and the momentum thickness
THETA as this. And this is one important parameter,
the shear stress on the boundary, tau_0 given by this. This is very
important because when we need to calculate the friction velocity, u_tau,
we must calculate tau_0 first. So this is the formula we may use for
calculating the shear stress on boundary.

English: 
so we can see the turbulent boundary layer thickness grows much
faster than the laminar boundary layer thickness. And the turbulent
displacement thickness given by Delta star, and the momentum thickness
THETA as this. And this is one important parameter,
the shear stress on the boundary, tau_0 given by this. This is very
important because when we need to calculate the friction velocity, u_tau,
we must calculate tau_0 first. So this is the formula we may use for
calculating the shear stress on boundary.

English: 
We have talked about the basis of the turbulent boundary layer and this
could provide the standard wall functions for CFD applications, such as in the
commercial software ANSYS Fluent and the open source OpenFOAM.
In ANSYS Fluent version 18.0, the available wall functions include:
Standard wall functions;  scalable wall functions; non- equilibrium wall functions;
Menter-Lechner method, and also has an option for users to define their wall functions.
in OpenFOAM, the available wall functions include: epsilon wall function;

English: 
We have talked about the basis of the turbulent the boundary layer and this
could provide the standard wall functions for CFD applications, such as in the
commercial software ANSYS Fluent and the open source OpenFOAM.
In ANSYS Fluent version 18.0, the available wall functions include:
Standard wall functions; scalable wall functions; non-equilibrium wall functions;
Menter-Lechner method, and also has an option for users to define their wall functions.
in OpenFOAM, the available wall functions include: epsilon wall function;

English: 
k q R wall functions; omega wall functions; f wall functions;
nut wall functions and v2 wall functions. and for all these wall functions
there may also have the corresponding wall functions of low-
Reynolds number.  If you want to know more information about the wall functions,
please read the paper: A Thorough Description of How Wall Functions are
Implemented in the OpenFOAM, this paper is available online and you can download it

English: 
k, q, R wall functions; omega wall functions; f wall functions;
nut wall functions and v2 wall functions. and for all these wall functions
there may also have the corresponding wall functions of low-
Reynolds number. If you want to know more information about the wall functions,
please read the paper: A Thorough Description of How Wall Functions are
Implemented in the OpenFOAM, this paper is available online and you can download it.
