
English: 
Hello welcome to my talk, All about Fluids
this talk is the second part of the talk on laminar boundary layer. in this talk I
will show you how we can extend the analysis method for the flow past the
flat plate to a more general case of steady flow past the 2d surface, such as
an aerofoil. this would include the establishment and the solution of the
boundary layer equation. In addition, some
simple approximations are introduced and
compared with the accurate Blasieus solution for the flow past a flat plate
unlike the flow past a flat plate, in which the pressure gradient in the x

English: 
Hello welcome to my talk, All about Fluids.
This talk is the second part of the talk on laminar boundary layer. in this talk I
will show you how we can extend the analysis method for the flow past the
flat plate to a more general case of steady flow past the 2d surface, such as
an aerofoil. this would include the establishment and the solution of the
boundary layer equation. In addition, some simple approximations are introduced and
compared with the accurate Blasieus solution for the flow past a flat plate.
Unlike the flow past a flat plate, in which the pressure gradient in the x

English: 
direction would be 0, the flow past the general 2d surface is subject to the
different pressure gradient, such as the flow shown the figure in a similar
manner of the flow past a flat plate, the boundary layer for the flow past a 2d
surface develops started from the laminar flow Region 1 to a transition process
Region 2 and to the fully turbulent boundary layer, Region 6. the difference
is that in the flow past the general 2d surface, due to the existence of the
pressure gradient in the x direction, the flow might be separate and reverse
point 4 and Region 5. here we must notice the difference between the

English: 
direction would be 0, the flow past the general 2d surface is the subject to the
different pressure gradient, such as the flow showing the figure in a similar
manner of the flow past a flat plate, the boundary layer for the flow past a 2d
surface develops started from the laminar flow Region 1 to a generation process
Region 2 and to the fully turbulent boundary layer, Region 6. the difference
is that in the flow past the general 2d surface, due to the existence of the
pressure gradient in the x direction, the flow might be separate and reverse
point 4 and Region 5. here we must notice the difference between the

English: 
uniform flow velocity U0 and the flow velocity outside of the boundary
layer the capital U, which is not a constant here.
due to the existence of the pressure gradient, the laminar boundary
layer equation would be much more complicated than that for the flat
plate, however there are still many similarities between their flows past
the flat plate and the past the general 2D case. If we take the local
coordinate x, y, the tangential and normal coordinates on the surface in the
general 2D cases.
generally speaking, the boundary layer equation is still valid if the
coordinates are given by the local coordinate system, where x is the

English: 
uniform flow velocity U0 and the flow velocity outside of the boundary
layer the capital U, which is not a constant here.
due to the existence of with the pressure gradient, the laminar boundary
layer equation would be much more complicated than that for the flat
plate, however there are still many similarities between their flows past
the flat plate and the past the general 2D case. If we take the local
coordinate x, y, the tangential and normal coordinates on the surface in the
general 2D cases
generally speaking, the boundary layer equation is still valid if the
coordinates are given by the local coordinate system, where x is the

English: 
tangential coordinates on the 2D surface, and y is the normal coordinate from the
local surface. and the corresponding boundary
conditions for the boundary layer equation are given by these here
we must notice the velocity outside the boundary layer capital U is the
function of x, but independent of y. if the radius of the curvature of
the surface is significantly larger than the boundary layer thickness.
looking at the last equation, it is a very important equation in the study of
the boundary layer, which implies that the pressure outside of the boundary
layer could carry over to the solid wall in y direction without a change.

English: 
tangential coordinates on the 2D surface, and y is the normal coordinate from the
local surface. and the corresponding boundary
conditions for the boundary layer equation are given by these here.
we must notice the velocity outside the boundary layer capital U is the
function of x, but independent of y, if the radius of the curvature of
the surface is significantly larger than the boundary layer thickness.
looking at the last equation, it is a very important equation in the study of
the boundary layer, which implies that the pressure outside of the boundary
layer could carry over to the solid wall in y direction without a change.

English: 
now we can examine the pressure outside
the boundary layer, where the flow can be taken as an irrotational flow, and thus
the Bernoulli's equation is valid and the pressure is given as this.
so the pressure gradient with regard to x is given by this, this should
be principally correct for the flow outside the boundary,
however for the thin layer, the pressure does not change in the normal
direction, hence this pressure gradient in x direction would be valid for the
flow in the boundary layer. As such the boundary layer equations can be written
as this, and next we are going to solve the equations in this talk.

English: 
now we can examine the pressure outside
the boundary layer, where the flow can be taken as an irrotational flow, and thus
the Bernoulli's equation is valid and the pressure is given as this
so the pressure gradient with regard to x is given by this, this should
be principally collect for the flow outside the boundary
however for the thin layer, the pressure does not change in the normal
direction, hence this pressure gradient in x direction would be valid for the
flow in the boundary layer. As such the boundary layer equations can be written
as this,  and next we are going to solve the equations in this talk.

English: 
we can first construct a coordinate transformation as ETA given by this,
here s is a transformed coordinate, corresponding to x,
apply this transform for a flat plate, where capital U would be a constant and
then the transformation would be exactly same as that one we used for the flat
plate, and we can calculate the derivative of ETA with regard to x
as this. Here capital U is a function of x, so we
have the term as this. For the derivative of ETA with regard

English: 
we can first construct a coordinate transformation as ETA given by this
here s is a transformed coordinate, corresponding to x
apply this transform for a flat plate, where capital U would be a constant and
then the transformation would be exactly same as that one we used for the flat
plate, and we can calculate the derivative of ETA with regard to x
as this. Here capital U is a function of x, so we
have the time as this. For the derivative of ETA with regard

English: 
to y is a much simpler. if we get the derivative with regard
to x, it would be given by this, and its derivative with regard to y is zero,
that means s is independent of y.
From the equation of an incompressible flow
we can have a stream function defined as Psi here, given by this and f is a
function of both s and ETA, the transformed stream function,
so based on this stream function, we can calculate the velocity component, u
given by this, and we can get a simple result as this. capital U times f',

English: 
to y is a much simpler. if we get the derivative with regard
to x, it would be given by this, and its derivative with regard to y is zero,
that means s is independent of y.
From the equation of an incompressible flow
we can have a stream function defined as
Psi here, given by this and f is a
function of both s and ETA, the transformed stream function
so based on this stream function, we can calculate the velocity component, u
given by this, and we can get a simple result as this. capital U times f'

English: 
here f' indicates the f derivative with regard to ETA, so from this
expression, we can see f' is the ratio of u over U,
and for the velocity component v, given by this, and it's much more complicated
than u.
further calculations of velocity derivative of u with regard to x is 
given as this. here the transformed stream function f
is the function of both s and ETA, so we have the complicated expression

English: 
here f' indicates the f derivative with regard to ETA, so from this
expression, we can see f' is the ratio of u/U,
and for the velocity component v, given by this, and it's much more complicated
than u.
further calculations of velocity derivative of u with regard to x is a
given as this. here the transformed stream function f
is the function of both s and ETA, so we have the complicated expression

English: 
for the derivative of u with regard to x, for the derivative of v with regard to y
is given as this. Here ETA and f would be a function of y, so
we have the expression as this. If we put these together, we have the
continuity equation as this for an incompressible flow, this means the
stream function Psi with the transformed function f
and the transformed coordinate, s, and ETA are so chosen
that they could satisfy the continuity equation automatically.
for applying the momentum equation in x direction as this, we need more derivatives

English: 
for the derivative of u with regard to x, for the derivative of v with regard to y
is given as this. Here ETA and f would be a function of y, so
we have the expression as this. If we put these together, we have the
continuity equation as this for an incompressible flow, this means the
stream function Psi with the transformed  function f
and the transformed coordinate, s, and ETA are so chosen
that they could satisfy the continuity equation automatically.
for applying the momentum equation in x
direction as this, we need more derivatives

English: 
as this, the derivative of u with regard to y,
given as this, and the second-order derivation of u
with regard to y given by this. Also we need to transform the derivative
of u with regard to x to the derivative of u with regard to s,
so substitute all the components into the momentum equation in x direction
we can do this following the derivation in the book, 'Aerodynamics For Engineers',
we can obtain the equation as this, here we can define the parameter BETA as this,

English: 
as this, the derivative of u with regard to y,
given as this, and the second-order derivation of u
with regard to y given by this. Also we need to transform the derivative
of u with regard to x to the derivative of u with regard to s,
so substitute all the components into the momentum equation in x direction
we can do this following the derivation in the book, 'Aerodynamics For Engineers',
we can obtain the equation as this, here we can define the parameter BETA as this,

English: 
if the boundary is thin, BETA
can be a constant for a given s or x.
we can also assume the s derivative of f
and f' are 0, that is, f and f' would be independent of s. In other word
we could have a similarity solution for the boundary layer if BETA is given.
for different s, the boundary layer shape would be similar.
As such we can have with the famous Falkner-Skan equation, given as this. it is
a third order ordinary differential equation and the corresponding boundary

English: 
if the boundary is thin, BETA
can be a constant for a given s or x.
we can also assume the s derivative of f
and f' are 0, that is, f and f' would be independent of s. In other word,
we could have a similarity solution for the boundary layer if BETA is given.
for different s, the boundary layer shape would be similar.
As such we can have with the famous Falkner-Skan equation, given as this. it is
a third order ordinary differential equation and the corresponding boundary

English: 
conditions for the Falkner-Skan equation are given as this.
on the solid boundary, ETA equals zero, the transformed function f would
be 0. This would guarantee the stream function Psi on the solid
boundary would be 0, which is a constant and represents a stream line on the
solid boundary and the differentiation of f with regard to ETA,
the f' is 0 on the solid boundary, which is required by the no-slip
boundary condition. On the outside of the boundary, f' would be unit since
outside the boundary u equals to capital U.

English: 
conditions for the Falkner-Skan equation
are given as this.
on the solid boundary, ETA equals zero, the transformed function f would
be 0. This would guarantee the stream function Psi on the solid
boundary would be 0, which is a constant and represents a stream line on the
solid boundary and the differentiation of f with regard to ETA,
the f' is 0 on the solid boundary, which is required by the no-slip
boundary condition. On the outside of the boundary, f' would be unit since
outside the boundary u equals to capital U

English: 
generally there is no analytical solution to the Falkner-Skan equation,
however we can apply a numerical method to solve the Falkner-Skan equation. for
different BETA, the boundary layer would be different, but we can obtain the
similarity solution, which means for a given BETA, velocity distribution against
ETA would be same for different s or x.
the solution of the Falkner-Skan equation is given in this figure fro different
BETA, and the from this solution, we can have two limits: the upper limit
BETA=2.0, if you BETA is larger than 2.0, it would

English: 
generally there is no analytical solution to the Falkner-Skan equation,
however we can apply a numerical method to solve the Falkner-Skan equation. for
different BETA, the boundary layer would be different, but we can obtain the
similarity solution, which means for a given BETA, velocity distribution against
ETA would be same for different s or x.
the solution of the Falkner-Skan equation is given in this figure fro different
BETA, and the from this solution, we can have two limits: the upper limit
BETA=2.0, if you BETA is larger than 2.0, it would

English: 
generate a velocity within the boundary layer larger than U, which
would violate the boundary layer definition. The lower limit BETA equals to
-0.1988, if BETA is less then -0.1988
it would generate a reverse flow within the boundary layer.
so the lower limit is a good indication for the flow separation.
for the case of BETA equalling to 0, which corresponds to the flow past a flat
plate, since the flow velocity outside the boundary layer would be a constant
equalling to the incoming flow U0.
now we can calculate the shear stress on the solid wall for different BETA.

English: 
generate a velocity within the boundary layer larger than U, which
would violate the boundary layer definition. The lower limit BETA equals to
-0.1988, if BETA is less then -0.1988
it would generate a reverse flow within the boundary layer.
so the lower limit is a good indication for the flow separation.
for the case of BETA equalling to 0, which corresponds to the flow past a flat
plate, since the flow velocity outside the boundary layer would be a constant
equalling to the incoming flow U0.
now we can calculate the shear stress on the solid wall for different BETA.

English: 
TAU_xy on the solid wall is calculated as this, and it is given by
the second order derivation of the transformed stream function f with
regard to ETA. For a different BETA, f'' on
the solid boundary is given in this figure. it should be noted the f''
on the solid wall would be independent of s, that's because the
similarity solution for a given BETA. consider the case of BETA=0, the
corresponding f'' is 0.4696, so this corresponding shear stress TAU_xy
equalling to this given in a form of local Reynolds number Rx, defined in

English: 
TAU_xy on the solid wall is calculated as this, and it is given by
the second order derivation of the transformed stream function f with
regard to ETA. For a different BETA, 'f' on
the solid boundary is given in this figure. it should be noted the 'f'
on the solid wall would be independent of s, that's because the
similarity solution for a given BETA. consider the case of BETA=0, the
corresponding 'f' is 0.4696, so this corresponding shear stress TAU_xy
equalling to this given in a form of local Reynolds number Rx, defined in

English: 
as this. So this is exactly same as we have obtained
for the flow past a flat plate
in this slide, an example is a given for the boundary layer,on a thin aerofoil,
NACA65 - 006, which is a thin airfoil of thickness ratio of 6 percent, and at
an angle of attack of 0 degree,  the pressure coefficients are given, seen in
the figure. we can now examine the development of
the laminar boundary layer over the thin aerofoil. Generally BETA would be a

English: 
as this. So this is exactly same as we have obtained
for the flow past a flat plate.
in this slide, an example is given for the boundary layer, on a thin aerofoil,
NACA65-006, which is a thin airfoil of thickness ratio of 6 percent, and at
an angle of attack of 0 degree, the pressure coefficients are given, seen in
the figure. we can now examine the development of
the laminar boundary layer over the thin aerofoil. Generally BETA would be a

English: 
good parameter to examine whether the laminar boundary layer would separate
as shown in one of the previous slides, BETA would be limited
between -0.1988 and 2.0.
BETA is equal to or smaller than -0.1988, we can think the flow separates.
BETA is given based on the definition as this,
and based on the definition of the pressure coefficient,
we can deduce this formulation for BETA, here x_hat is a non-dimensional x,
normalised by the aerofoil chord, c. So based on the coefficient Cp, and we

English: 
good parameter to examine whether the laminar boundary layer would separate
as shown in one of the previous slides, BETA would be limited
between -0.1988 and 2.0.
BETA is equal to or smaller than -0.1988, we can think the flow separates.
BETA is given based on the definition as this
and based on the definition of the pressure coefficient,
we can deduce this formulation for BETA, here x_hat is a non-dimensional x,
normalised by the aerofoil chord, c. So based on the coefficient Cp, and we

English: 
can get beta on the different position along the chord, so from the
curve, we can see the aerofoil has a favorable pressure gradient for the
first half of the chord, where the BETA is larger than zero, and it's also seen
there will be a separation at 60% of the chord, where BETA reaches -0.1988
this means the aerofoil would have a separation,
even in the angle of attack of zero, if the boundary layer is considered all
laminar. This may be true in reality, maintaining for laminar flow boundary
layer on the aerofoil would be difficult, although it may have a smaller skin drag

English: 
can get beta on the different position along the chord, so from the
curve, we can see the aerofoil has a favorable pressure gradient for the
first half of the chord, where the BETA is larger than zero, and it's also seen
there will be a separation at 60% of the chord, where BETA reaches -0.1988
this means the aerofoil would have a separation,
even in the angle of attack of zero, if the boundary layer is considered all
laminar. This may be true in reality, maintaining for laminar flow boundary
layer on the aerofoil would be difficult, although it may have a smaller skin drag.

English: 
for the laminar boundary layer than that for the turbulent boundary layer,
but the laminar boundary layer can easily separate, which is a case we
should avoid for many practical applications.
in fact the flow passing an aerofoil of a flight condition would be normally
turbulent, due to the large flow velocity and the corresponding
boundary layer thickness and thus the displacement thickness would be much
thicker and it would help to delay or avoid the flow separation in some practical cases.
We may intentionally trigger a turbulent flow so to delay or avoid the
flow separation on the aerofoil, for instance, using vortex generator near

English: 
for the laminar boundary layer than that for the turbulent boundary layer,
but the laminar boundary layer can easily separate, which is a case we
should avoid for many practical applications.
in fact the flow passing an aerofoil of a flight condition would be normally
turbulent, due to the large flow velocity and the corresponding
boundary layer thickness and thus the displacement thickness would be much
thicker and it would help to delay or avoid the flow separation in some practical cases.
We may intentionally trigger a turbulent flow so to delay or avoid the
flow separation on the aerofoil, for instance, using vortex generator near

English: 
the wing leading edge, we can find more information in the book 'Aerodynamics for
Engineers'
To avoid the solving the boundary layer equation, the approximation methods have
been developed, for instance, Pohlhausen has developed an approximation for the
2D laminar boundary layer, using the fourth-order polynomial function, which can be
written as this. it can be seen this expression
satisfied the no-slip boundary condition since u=0 if y=0
however we have four coefficients: a1, a2, a3, and a4
to be decided. Following Pohlhausen's suggestion, this four coefficients can be

English: 
the wing leading edge, we can find more information in the book 'Aerodynamics for Engineers'
To avoid solving the boundary layer equation, the approximation methods have
been developed, for instance, Pohlhausen has developed an approximation for the
2D laminar boundary layer, using the fourth-order polynomial function, which can be
written as this. it can be seen this expression
satisfies the no-slip boundary condition since u=0 if y=0
however we have four coefficients: a1, a2, a3, and a4
to be decided. Following Pohlhausen's suggestion, this four coefficients can be

English: 
decided using the relevant boundary
conditions: the first boundary condition is the inner boundary at y=0
on the solid surface, where u and v are both 0.
because of the no-penetration and the no-slip boundary conditions
in addition, the pressure does not a change on y direction in the laminar boundary
layer, so we have boundary condition on y=0 as this.
on the outer boundary, we have the boundary condition as this, for the first
condition it is obvious, because (in) the outer boundary, the velocity u is same as the

English: 
decided using the relevant boundary
conditions: the first boundary condition is the inner boundary at y=0
on the solid surface, where u and v are both 0.
because of the no-penetration and the no-slip boundary conditions
in addition, the pressure does not a change on y direction in the laminar boundary
layer, so we have boundary condition on y=0 as this
on the outer boundary, we have the boundary condition as this, for the first
condition it is obvious, because the outer boundary, the velocity u is same as the

English: 
capital U. the first-order of u with regard
to y is zero, that's because u is to approach capital U asymptotically.
and the second order differentiation u with regard to y, is because
outer boundary viscous term must vanish, so this would be zero
now we have four boundary conditions for solving four coefficients,
by defining a non-dimensional parameter, the capital LAMBDA as this, it is a shape
function, so these boundary conditions would lead to the coefficients: a1, a2, a3 and
a4, given as this, and if we plot out the profile of the boundary layer, we

English: 
capital U. the first-order of u with regard
to y is zero, that's because u is to approach capital U asymptotically.
and the second order differentiation u with regard to y, is because
outer boundary viscous term must vanish, so this would be zero.
now we have four boundary conditions for solving four coefficients,
by defining a non-dimensional parameter, the capital LAMBDA as this, it is a shape
function, so these boundary conditions would lead to the coefficients: a1, a2, a3 and
a4, given as this, and if we plot out the profile of the boundary layer, we

English: 
can see this plot, and similar to the limits, for BETA between -0.1988
and 2.0,  we have limits for the non-dimensional LAMBDA: it would be
between -12.0 and 12.0, so the lower limit LAMBDA equals to -12.0, if
LAMBDA is smaller than -12.0, it would generate a backflow within the
boundary, that is, the flow separation we can see here.
if LAMBDA is larger than the upper limit, 12.0, it would generate a larger
velocity within the boundary than U, as we see in this case, LAMBDA= 30, so

English: 
can see this plot, and similar to the limits, for BETA between -0.1988
and 2.0,  we have an limits for the non-dimensional LAMBDA: it would be
between -12.0 and 12.0, so the lower limit LAMBDA equals to -12.0, if
LAMBDA is a smaller than -12.0, it would generate a backflow within the
boundary, that is, the flow separation we can see here
if LAMBDA is larger than the upper limit, 12.0, it would generate a larger
velocity within the boundary than U, as we see in this case, LAMBDA= 30, so

English: 
we can see this larger velocity than the unit of the ratio of u/U.
so this part would violate the boundary layer definition, thus LMABDA must be
smaller than 12.0.
Now in this slide, we will compare some other approximations to the Blasieus'
solution. In this case the flow is considered past a flat plate, therefore
LAMBDA is 0.  so we have Pohlhausen's approximation for the flow past a flat
plate as this,  and we have also cubic approximation which is given by this,
this approximation would give a slightly better for shear stress on the

English: 
we can see this larger velocity than the unit of the ratio of u/U.
so this part would violate the boundary layer definition, thus LMABDA must be
smaller than 12.0.
Now in this slide, we will compare some other approximations to the Blasieus'
solution. In this case the flow is considered past a flat plate, therefore
LAMBDA is 0.  so we have Pohlhausen's approximation for the flow past a flat
plate as this,  and we have also cubic approximation which is given by this
this approximation would give a slightly better for shear stress on the

English: 
flat plate, this formulation is from the book 'Aerodynamics for Engineers', and
we have also a parabolic approximation, which would give a better overall
profile for the boundary layer given by this. this formula is the from the book
'Aerodynamics for Engineering Students', so if we plot all the approximations
against the exact Blasieus' solution, we can see the parabolic resolution, the
black line is a closest result to the Blasieus result, while the fourth order
polynominal and the cubic approximation are not good as the
parabolic approximation. however some more comments must be made

English: 
flat plate, this formulation is from the book 'Aerodynamics for Engineers', and
we have also a parabolic approximation, which would give a better overall
profile for the boundary layer given by this. this formula is the from the book
'Aerodynamics for Engineering Students', so if we plot all the approximations
against the exact Blasieus' solution, we can see the parabolic resolution, the
black line is the closest result to the Blasieus result, while the fourth order
polynominal and the cubic approximation are not good as the
parabolic approximation. however some more comments must be made

English: 
here. In all these approximations,  only Pohlhausen's fourth-order polynomial
function can include the effect of the shape function LAMBDA, and its derivation
has a more physical background, while the parabolic approximation seems
to give the closest result to the Blasieus' solution, but the cubic approximation
could give a closer shear stress on the solid boundary than other two shear
stress calculations.

English: 
here. In all these approximations,  only Pohlhausen's fourth-order polynomial
function can include the effect of the shape function LAMBDA, and its derivation
has a more physical background, while the parabolic approximation seems
to give the closest result to the Blasieus' solution, but the cubic approximation
could give a closer shear stress on the solid boundary than other two shear
stress calculations.
