
English: 
Hello welcome to my talk, All about Fluids.
This talk is the first part of the talk on laminar boundary layer, with a focus on solving
the 2D boundary layer equation, specifically on solving the simple 2D
boundary layer equation on a flat plate. Also, it might be the simplest viscous
boundary layer, it would be very useful for
understanding the physics of the laminar boundary layer, as well as to provide the
basis for analyzing the more complicated boundary layer.
Why the viscous boundary layer is so important?  In principle, the full solution

English: 
 
Hello welcome to my talk, All about Fluids,
this talk is the first part of the talk on laminar boundary layer, with a focus on solving
the 2D boundary layer equation, specifically on solving the simple 2D
boundary layer equation on a flat plate. Also, it might be the simplest viscous
boundary layer, it would be very useful for
understanding the physics of the laminar boundary layer, as well as to provide the
basis for analyzing the more complicated boundary layer.
Why the viscous boundary layer is so important?  In principle, the full solution

English: 
to the Navier-Stokes equations would be use the direct numerical simulation (DNS)
which resolve all eddies from the largest to the Kolmogorov length scale.
a requirement for this direct solution is the huge number of fluid grid which
is proportional to the 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 reckoned by Spalart in 2000, DNS would be ready for practical applications in
around 2080 if the advancing speed of the computer technology can be
maintained. However, in 2016,

English: 
to the Navier-Stokes equations would be use the direct numerical simulation (DNS)
which resolve all eddies from the largest to the Kolmogorov length scale.
a requirement for this direct solution is the huge number of fluid grid which
is proportional to the power of 9/4 of the Reynolds number,
and this can lead to a huge numbers of grids if the Reynolds number is large
As reckoned by Spalart in 2000, DNS would be ready for practical applications in
around 2080 if the advancing speed of the computer technology can be
maintained. However in 2016,

English: 
he and his co-author revised that claim, and stated: by now we are not
confident of this for the 21st century or even that it will ever happen.
The next possible method would be the large eddy simulation (LES).
LES solves all the large eddies and the modeling is only for the small
eddies, which could be easier to be modelled and more isotopic in
physics. A difficulty in LES is the eddies are very small in the boundary layer, and if
one uses LES to resolve most of the stress-bearing, the grid spacing and the
time steps would be close to the requirement of the DNS modeling.

English: 
he and his co-author revised that claim, and stated: by now we are not
confident of this for the 21st century or even that it will ever happen.
The next possible method would be the large eddy simulation (LES).
LES solves all the large eddies and the modeling is only for the small
eddies, which could be easier to be modelled and more isotopic in
physics. A difficulty in LES is the eddies are very small in the boundary layer, and if
one uses LES to resolve most of the stress-bearing, the grid spacing and the
time steps would be close to the requirement of the DNS modeling.

English: 
The popular mitigation is use the hybrid method, such as, DES, detached eddy
simulation or hybrid RANS/LES, both taking advantages of the LES and RANS.
for the problem. However critical issues are to be solved
the most popular method currently is the RANS modelling and challenge is
when the boundary layer is resolved and the irrelevant the skin drag is calculated.
the issues are with the grid generation in boundary layer since the requirement would
be there will be some point within the wall unit, y+ equalling to five or less,

English: 
The popular mitigation is use the hybrid method, such as, DES, detached eddy
simulation or hybrid RANS/LES, both taking advantages of the LES and RANS.
for the problem. However critical issues are to be solved
the most popular method currently is the RANS modelling and challenge is
when the boundary layer is resolved and the irrelevant the skin drag is calculated.
the issues are with the grid generation in boundary layer since the requirement would
be there will be some point within the wall unit, y+ equalling to five or less,

English: 
This requirement would lead to very large number of grids for RANS
modeling. Another issue in the boundary layer is the complicated process are involved
from laminar to transition and to turbulent, thus the suitable turbulence
models for all these processes must be used. Currently the confidence for the
turbulence modeling is relatively 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 function, rather than resolve the boundary layer directly.
in this regard the understanding of the boundary layer would be very important

English: 
This requirement would lead to very large number of grids for RANS
modeling. Another issue in the boundary layer is the complicated process are involved
from laminar to transition and to turbulent, thus the suitable turbulence
models for all theses processes must be used. Currently the confidence for the
turbulence modeling is relatively 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 function, rather than resolve the boundary layer directly.
in this regard the understanding of the boundary layer would be very important

English: 
let's start from the example of Hagen- Poiseuille flow, the laminar flow in a long
horizontal pipe under the pressure
gradient delta p over delta x, if we assume
the flow is very slow and its corresponding Reynolds number is less than
2300, the flow would be laminar. As such, the Navier-Stokes equation can be much
simplified as this. Applying the no-slip boundary condition
on the pipe wall, we can obtain the flow velocity profile, given by this
It can be seen in the low Reynolds number, the viscous effect would be important in
the whole region of the flow, and flow is rotational. And this case can be

English: 
let's start from the example of Hagen- Poiseuille flow, the laminar flow in a long
horizontal pipe under the pressure gradient delta p over delta x, if we assume
the flow is very slow and its corresponding Reynolds number is less than
2300, the flow would be laminar. As such, the Navier-Stokes equation can be much
simplified as this. Applying the no-slip boundary condition
on the pipe wall, we can obtain the flow velocity profile, given by this
It can be seen in the low Reynolds number, the viscous effect would be important in
the whole region of the flow, and flow is rotational. And this case can be

English: 
regarded as the flow boundary layer is expanded from the wall to the center of
the pipe. If the Reynolds number is very large, for instance, 10 million, then the
flow becomes fully turbulent and the velocity profile can be seen as this
The large portion of the flow in the middle of the pipe can be taken as irrotational and
the thin regions near the walls are rotational, in which the viscous
effects are significant, thus these thin layers are called as the viscous
boundary layer. In solving such a flow in a pipe, we can start by assuming all flow is
irrotational, and using the irrotational flow for calculating the

English: 
regarded as the flow boundary layer is
expanded from the wall to the center of
the pipe. If the Reynolds number is very large, for instance, 10 million, then the
flow becomes fully turbulent and the velocity profile can be seen as this
The large portion of the flow in the middle of the pipe can be taken as irrotational and
the thin regions near the walls are rotational,  in which the viscous
effect are significant,  thus these thin layers are called as the viscous
boundary layer. In solving such a flow in a pipe, we can start by assuming all flow is
irrotational, and using the irrotational flow for calculating the

English: 
boundary layer. If the boundary layers are thick, then we need to iterate the
solutions of irrotational flows and the viscous boundary layer.
This method was proposed by Prandtl in 1904 to solve the d'Alembert paradox by
including the fluid drag.
In this talk for a simplification to the boundary layer problem
we will examine the uniform flow passing a flat plate. We assume the width of the plate
is large. thus the flow developed would be
considered as 2D. if the plate is long enough, we can see
the boundary layer development starting from the laminar boundary layer and after

English: 
boundary layer. If the boundary layers are thick, then we need to iterate the
solutions of irrotational flows and the
viscous boundary layer.
This method was proposed by Prandtl in 1904 to solve the d'Alembert paradox by
including the fluid drag.
In this talk for a simplification to the boundary layer problem
will will examine the uniform flow passing a flat plate. We assumed the width of the plate
is large. thus the flow developed would be
considered as 2D. if the plate is long enough, we can see
the boundary layer development starting from the laminar boundary layer and after

English: 
a certain length, the local Reynolds number would be large enough, for instance,
500,000, the boundary layer starts the transition
from laminar to turbulent.  Principally, the transition phenomena to turbulence is
not instant and the full turbulent boundary layer would be only achieved
after the transition zone.
In this talk, our attention would be on the  study of
the laminar boundary layer. We assume here the length of the plate L0
would be limited, and at the boundary layer over it would be all laminar.
Our interest would be the boundary layer thickness and the distribution of the

English: 
a certain length, the local Reynolds number would be large enough, for instance,
500,000, the boundary layer starts the transition
from laminar to turbulent.  Principally, the transition phenomena to turbulence is
not instant and the full turbulent boundary layer would be only achieved
after the transition zone.
In this talk, our attention would be on the study of
the laminar boundary layer. We assume here the length of the plate L0
would be limited, and at the boundary layer over it would be all laminar.
Our interest would be the boundary layer thickness and the distribution of the

English: 
flow velocity in the boundary layer u(x,y). Based on the flow condition we can
define the Reynolds number R as this,  U is
the uniform flow velocity, L0 is the
length of the plate, and NU is the kinematic viscosity of the fluid
To examine the boundary layer, the corresponding simplified the Navier-Stokes equation
must be solve. To achieve the goal, we start to examine the non-dimensional
Navier-Stokes equation by using the non-dimensional parameters, the space
variables x*, y* and z*, the non-dimensional  velocities u*
v* and w*, the non-dimensional pressure p*, and

English: 
flow velocity in the boundary layer u(x, y). Based on the flow condition we can
define the Reynolds number R as this, U is the uniform flow velocity, L0 is the
length of the plate, and NU is the kinematic viscosity of the fluid
To examine the boundary layer, the corresponding simplified the Navier-Stokes equation
must be solve. To achieve the goal, we start to examine the non-dimensional
Navier-Stokes equation by using the non-dimensional parameters, the space
variables x*, y* and z*, the non-dimensional velocities u*
v* and w*, the non-dimensional pressure p*, and

English: 
the non-dimensional time t*
substitute these non-dimensional parameters into the NS equation, the Navier-Stokes
equation can be changed into the non-dimensional Navier-Stokes equation
It can be seen that the continuity
equation has the same form as that in the Navier-Stokes equation
but the equation for the momentum component, we have two new
non-dimensional parameters, Reynolds number R, which is related to the fluid
viscous force, and the Froude number, Fr, which is derived and based on the
fluid gravitational force. and this fluid number is important for studying the

English: 
the non-dimensional time t*.
substitute these non-dimensional parameters into the NS equation, the Navier-Stokes
equation can be changed into the non-dimensional Navier-Stokes equation
It can be seen that the continuity
equation has the same form as that in the Navier-Stokes equation
but the equation for the momentum component, we have two new
non-dimensional parameters, Reynolds number R, which is related to the fluid
viscous force, and the Froude number, Fr, which is derived and based on the
fluid gravitational force. and this fluid number is important for studying the

English: 
flow with free surface, for instance, the wave-structure interaction. However here
we focus on the flow around a fully submerged body, the flat plate.
Thus the effect of the gravitational force we are not be included
To obtain the analytical solution for the boundary layer equation
simple laminar flows must be considered. Here we assume the flat plate has a
larger width, therefore the flow can be treated as 2 dimensional.
Generally speaking, there are no easy solutions or in most cases there are no

English: 
flow with free surface, for instance,  the wave-structure interaction. However here
we focus on the flow around a fully submerged body, the flat plate.
Thus the effect of the gravitational force we are not be included
To obtain the analytical solution for the boundary layer equation
simple laminar flows must be considered. Here we assume the flat plate has a
larger width, therefore the flow can be treated as 2 dimensional.
Generally speaking, there are no easy solutions or in most cases there are no

English: 
analytical solutions for 3d cases. For 2D boundary layer
the dimensional navier-stokes equation can be expressed as this. In this talk we are
examine the very simple case of 2d boundary layer, which is a flow past a
flat plate as seen in the plot, where DELTA is the boundary layer thickness
which would be one of our interests. Following the research result from the
Schlichting and Gersten's book, Boundary Layer theory, the boundary layer
thickness would be proportional to the
square root of the kinematical viscosity
coefficient NU or the ratio of the boundary layer thickness to the

English: 
analytical solutions for 3d cases. For 2D boundary layer
the dimensional Navier-Stokes equation can be expressed as this. In this talk we are
examine the very simple case of 2d boundary layer, which is a flow past a
flat plate as seen in the plot, where DELTA is the boundary layer thickness
which would be one of our interests. Following the research result from the
Schlichting and Gersten's book, ‘Boundary Layer Theory’, the boundary layer
thickness would be proportional to the square root of the kinematical viscosity
coefficient NU or the ratio of the boundary layer thickness to the

English: 
plate length would be proportional to the reverse of the square root of the Reynolds
number. Thus the boundary layer thickness would be significantly smaller
than the length of the plate. Our other interest would be the velocity
distribution u(x, y).
In analysis of the boundary layer, the boundary layer thickness DELTA is
assumed to be very small compared to the plate length L0, as such, the non-
dimensional boundary layer thickness DELTA star, defined as the ratio of the
boundary layer thickness DELTA divided by the plate length
L0, it would be much smaller than a unit. In the magnitude analysis of the non-

English: 
plate length would be proportional to the reverse of the square root of the Reynolds
number. Thus the boundary layer thickness would be significantly smaller
than the length of the plate. Our other interest would be the velocity
distribution u(x,y).
In analysis of the boundary layer, the boundary layer thickness DELTA is
assumed to be very small compared to the
plate length L0, as such, the non-
dimensional boundary layer thickness DELTA star, defined as the ratio of the
boundary layer thickness DELTA divided 
by the plate length
L0, it would be much smaller than a unit. In the magnitude analysis of the non

English: 
dimensional parameters in the non-dimensional Navier-Stokes equation, we
have the normal value which are comparable to a unit
including x*, u* and t*, we have the small values which are in a
magnitude of non-dimensional boundary layer thickness, DELTA_star, including y*
and v*, and the very small values which are
comparable to the square of the DELTA_star, including the reverse of the Reynolds number.
Based on the magnitude analysis for the non-dimensional parameters, we can check

English: 
dimensional parameters in the non-dimensional navier-stokes equation, we
have the normal value which are comparable to a unit
including x*, u* and t*, we have the small values which are in a
magnitude of non-dimensional boundary layer thickness, DELTA_star, including y*
and v*, and the very small values which are
comparable to the square of the DELTA_star, including the reverse of the Reynolds number.
Based on the magnitude analysis for the non-dimensional parameters, we can check

English: 
and therefore simplified the non-dimensional navier-stokes equation
as Prandtl did more than 100 years ago in 1904, for instance, both terms in
the continuity equation have magnitude of  1 as this, therefore both terms
should be kept as shown in the Prandtl  boundary layer equation as this,
for the moment equation in x direction, the corresponding magnitudes of
each term are given as this, it can be seen there is only one term of small
value as this, it's a combination of this small term and the unit. As such we
can drop this term, we have the momentum equation for the

English: 
and therefore simplified the non-dimensional Navier-Stokes equation
as Prandtl did more than 100 years ago in 1904, for instance, both terms in
the continuity equation have a magnitude of 1 as this, therefore both terms
should be kept as shown in the Prandtl boundary layer equation as this,
for the moment equation in x direction, the corresponding magnitudes of
each term are given as this, it can be seen there is only one term of small
value as this, it's a combination of this small term and the unit. As such we
can drop this term, we have the momentum equation for the

English: 
boundary layer as this, for the momentum equation in y
direction, the corresponding magnitude of each term are given as this.
it can be seen all terms are small if we time all these terms together, therefore we
can drop all these terms, but keep the term of the pressure gradient only, which
would be an important equation in the boundary layer analysis
and all of this magnitude analysis can be found in the book of Schlichting and
Gersten. After the dimensional analysis, the
dimensional boundary layer equation can be re-written back to the dimensional

English: 
boundary layer as this, for the momentum equation in y
direction, the corresponding magnitude of each term are given as this.
it can be seen all terms are small if we time all these terms together, therefore we
can drop all these terms, but keep the term of the pressure gradient only, which
would be an important equation in the boundary layer analysis
and all of these magnitude analysis can be found in the book of Schlichting and
Gersten. After the dimensional analysis, the
dimensional boundary layer equation can be re-written back to the dimensional

English: 
boundary layer equation as this, and our target is to solve the boundary layer equation
to simplify the problem more we consider a steady uniform flow passing a flat
plate, so we have this equation.  Based on the boundary layer equation, we can see
the gradient of the pressure in y-direction is zero within the boundary
layer, this equation means an important conclusion: the pressure will not
change across the thin boundary layer from the outside of the boundary
layer to the solid boundary in y direction as seen in this plot. Now we
can consider the flow outside of the boundary layer, it can be assumed as

English: 
boundary layer equation as this, and our target is to solve the boundary layer equation
to simplify the problem more, we consider a steady uniform flow passing a flat
plate, so we have this equation.  Based on the boundary layer equation, we can see
the gradient of the pressure in y-direction is zero within the boundary
layer, this equation means an important conclusion: the pressure will not
change across the thin boundary layer from the outside of the boundary
layer to the solid boundary in y direction as seen in this plot. Now we
can consider the flow outside of the boundary layer, it can be assumed as

English: 
irrotational, since the fluid viscous effect can be ignored for the flow
outside of the boundary layer, thus the Bernoulli equation is valid, and
the pressure can be calculated as this, then we can calculate the pressure
gradient in x-direction as this, so this equation is correct when y is larger
than the boundary layer thickness Delta, and for the uniform flow outside of
the boundary layer, the gradient of the velocity with regard to x would be 0, so
we have the pressure gradient with regard to x is zero
and this should be valid for y larger than the boundary layer thickness

English: 
irrotational, since the fluid viscous effect can be ignored for the flow
outside of the boundary layer, thus the Bernoulli equation is valid, and
the pressure can be calculated as this, then we can calculate the pressure
gradient in x-direction as this, so this equation is correct when y is larger
than the boundary layer thickness Delta, and for the uniform flow outside of
the boundary layer, the gradient of the velocity with regard to x would be 0, so
we have the pressure gradient with regard to x is zero
and this should be valid for y larger than the boundary layer thickness

English: 
and within the boundary layer, since the pressure can carry over
from the outside to the solid wall, given by this equation, and then the steady
boundary equation for a flat plate can be finally simplified as this.
to solve the think boundary layer, we can construct a new parameter, ETA using both x
and y as this,  the choice of this expression would be based on both the
good reasoning and the experience. For the reasoning, please refer to the
Newman's book, 'Marine Hydrodynamics', here I can say this choice of the new

English: 
and within the boundary layer, since the pressure can carry over
from the outside to the solid wall, given by this equation, and then the steady
boundary equation for a flat plate can be finally simplified as this.
to solve the think boundary layer, we can construct a new parameter, ETA using both x
and y as this, the choice of this expression would be based on both the
good reasoning and the experience. For the reasoning, please refer to the
Newman's book, 'Marine Hydrodynamics', here I can say this choice of the new

English: 
parameter works perfectly for solving the boundary layer equation and from
the continuity equation of an incompressible flow, we can obtain a
stream function for the flow and the stream function Psi is given as this,
here f(ETA) is the so-called non-dimensional stream function
based on the stream function we can calculate the velocity component u, given
as this, and v is calculated as this, here the two terms: the first term
to represent the explicit of the available x in the stream function, Psi
and this term is implicit of x because ETA is
the function of x, so we have this, u and v, and we calculate this,

English: 
parameter works perfectly for solving the boundary layer equation and from
the continuity equation of an incompressible flow, we can obtain a
stream function for the flow and the stream function Psi is given as this,
here f(ETA) is the so-called non-dimensional stream function
based on the stream function we can calculate the velocity component u, given
as this, and v is calculated as this, here the two terms: the first term
to represent the explicit of the available x in the stream function, Psi
and this term is implicit of x because ETA is
the function of x, so we have this, u and v, and we calculate this,

English: 
the velocity gradient with x and y, and we put together so the continuity
equation is satisfied automatically, because we choose the stream function
and the parameter ETA.
now we go back to the momentum equation for the boundary layer on a flat plate
as this, and substitute the velocity components and the mathematical
manipulation would lead to an equation as this, the Blasieus equation, which was
derived by Blasieus in 1908 under the supervision of Prandtl.  and the
corresponding boundary conditions are given as this. numerical solution to

English: 
the velocity gradient with x and y, and we put together so the continuity
equation is satisfied automatically, because we choose the stream function
and the parameter ETA.
now we go back to the momentum equation for the boundary layer on a flat plate
as this, and substitute the velocity components and the mathematical
manipulation would lead to an equation as this, the Blasieus equation, which was
derived by Blasieus in 1908 under the supervision of Prandtl.  and the
corresponding boundary conditions are given as this. numerical solution to

English: 
the Blasieus equation will give the solution as this in the plot, and
in the plot, experimental data are also used for
comparison, so it can be seen the Blasieus solution is very accurate for the
laminar boundary layer. In the book, 'Boundary Layer Theory', you can find the
more data and all data are very close to the Blasieus solution, and from the
solution, we can work out when the velocity u equals 0.99*U,
the uniform flow velocity, and we get the number, ETA=3.5,
Thus we can calculate the boundary thickness Delta as this, here Rx defined as this

English: 
the Blasieus equation will give the solution as this in the plot, and
in the plot, experimental data are also used for
comparison, so it can be seen the Blasieus
solution is very accurate for the
laminar boundary layer. In the book, 'Boundary Layer Theory', you can find the
more data and all data are very close to
the Blasieus solution, and from the
solution, we can work out when the velocity u equals 0.99*U,
the uniform flow velocity, and we get the number, ETA=3.5,
Thus we can calculate the boundary thickness Delta as this, here Rx defined as this

English: 
the local Reynolds number, based on the distance x
in this slide, we will calculate some useful thickness for the laminar
boundary layer, the first thickness is called the displacement thickness,
DELTA_star, which is given as this, the physical meaning of the
displacement sickness is due to the boundary layer the flow rate is reduced
which is equivalent to the body wall sickened by Delta_star. this concept is
very useful in the method of the coupled irrotational and viscous
flow, in which the irrrotational flow boundary would be replaced by the

English: 
the local Reynolds number, based on the distance x
in this slide, we will calculate some useful thickness for the laminar
boundary layer, the first thickness is called the displacement thickness,
DELTA_star, which is given as this, the physical meaning of the
displacement sickness is due to the boundary layer the flow rate is reduced
which is equivalent to the body wall sickened by Delta_star. this concept is
very useful in the method of the coupled irrotational and viscous
flow, in which the irrrotational flow boundary would be replaced by the

English: 
displacement thickness, rather than the real body boundary. so for
accommodating the effect of the viscous boundary layer.  this displacement
thickness is about 1/3 of the conventional boundary thickness, given
as this, Delta. And the second thickness 
is called momentum thickness, given as
this. The momentum thickness is defined in relation to the moment of flow rate
within the boundary layer, this rate is smaller than the rate if the
viscous boundary layer is not existed. And the momentum thickness
is about 0.13 times of the conventional boundary layer thickness.
and in some text the book a kinetic

English: 
displacement thickness, rather than the real body boundary. so for
accommodating the effect of the viscous boundary layer.  this displacement
thickness is about 1/3 of the conventional boundary thickness, given
as this, Delta. And the second thickness is called momentum thickness, given as
this. The momentum thickness is defined in relation to the moment of flow rate
within the boundary layer, this rate is smaller than the rate if the
viscous boundary layer is not existed. And the momentum thickness
is about 0.13 times of the conventional boundary layer thickness.
and in some text the book a kinetic

English: 
energy thickness DELTA**is given as this, which is about 0.2
times of the actual boundary layer thickness, Relatively this
thickness is not as popular as the first two.
based on the Blasieus solution, we can now calculate the shear stress TAU_xy on the
flat plate, based on the rate of stream on the surface of the flat plate
which would be calculated as this, here Rx is the local Reynolds number if the plate
has a length L0 and the width B, then the total drag D acting on the plate can

English: 
energy thickness DELTA**is given as this, which is about 0.2
times of the actual boundary layer thickness, Relatively this
thickness is not as popular as the first two.
based on the Blasieus solution, we can now calculate the shear stress TAU_xy on the
flat plate, based on the rate of stream on the surface of the flat plate
which would be calculated as this, here Rx is the local Reynolds number if the plate
has a length L0 and the width B, then the total drag D acting on the plate can

English: 
be integrated from the above shear stress as this, here this drag is only
for one side of the plate, and if both sides of the plate are immersed in the
fluid, then the drag should be doubled.
the corresponding drag coefficient Cf is calculated as this.  now it can be seen the frictional drag
coefficient Cf is proportional to the inverse of the
square root of the Reynolds number R.

English: 
be integrated from the above shear stress as this, here this drag is only
for one side of the plate, and if both sides of the plate are immersed in the
fluid, then the drag should be doubled.
the corresponding drag coefficient Cf is  calculated as this.  now it can be seen the frictional drag
coefficient Cf is proportional to the inverse of the
square root of the Reynolds number R.
