
English: 
Hello, welcome to my talk, on the history of
aerodynamics, Part 3, the Age of Fruition in the Theoretical
Aerodynamics. This period covers almost
all 20th century. As I have talked in Part 2 on this topic,
up to 1906, all the necessary principles for
aerodynamics were in place. However,
how to use these principles to solve the practical problems
is a different issue. In the first half of the 20th century,
researchers worked on this series
and solved many practical problems
and some of those methods are still
being used

English: 
Hello, welcome to my talk, on the history of
aerodynamics, Part 3, the Age of Fruition in the Theoretical
Aerodynamics. This period covers almost
all 20th century. As I have talked in Part 2 on this topic,
up to 1906, all the necessary principles for
aerodynamics were in place.
However, how to use these principles to solve the practical problems
is a different issue.
In the first half of the 20th century,
researchers worked on these theories, and solved many practical problems,

English: 
today the second half of the century
is mainly the computer age computers
have been used to solve very complicated
problems today if we use
cfd to solve an aerodynamic problem
we may not think about the kuta
zhukovsky syrian or coda condition
in our numeric modeling only
in the panel method this may be severe
required
but the war of wider importance
we use the potential flow theory to
solve the
aerodynamics problem especially
the problem with the lifting bodies
let us have a look at the achievement
at the very early 20th century

English: 
and some of those methods are still being used today.
The second half of the century is mainly the computer age: computers
have been used to solve very complicated problems. Today if we use
CFD to solve an aerodynamic problem, we may not think about the Kutta-
Joukovski Theorem or Kutta condition in our numeric modeling, only
in the panel method, this may be still required.
But they were  of vital importance when use the potential flow theory to
solve the aerodynamics problem, especially
the problem with the lifting bodies.
Let us have a look at the achievements at the very early 20th century.

English: 
in 1903 the wright brothers
made the first successful heavier
than air flies and the less than a year
in 2004 the maid
closed the circled flight and
in the same year pronter developed
the boundary layers theory which is
sedia in use
even in the modern cfd modeling
and in 1906 kuta
yukovsky syrian was established
up to then all the necessary principles
were in place after that
it is a period how to use and
improve the theory especially before the
computer age

English: 
In 1903, the Wright brothers made their first successful heavier-
than-air flight, and less than a year, in 1904 (not 2004), they made
a closed-circuit flight. And in the same year,  Prandtl developed
the boundary layer theory which is still in use
even in the modern CFD modeling. and in 1906 Kutta-
Joukowski theorem was established. Up to then, all the necessary principles
were in place. After that it is a period how to use and
improve the theory especially before the computer age.

English: 
Here some key achievements in the modern aerodynamics are listed:
- In 1909 Louis Bleriot flies across the English channel from
Calais France to Dover in England.
- In 1910 Joukowski proposed the Joukowski transformation
and the Joukowski airfoil;
- In 1915 Prandtl established the lifting line theory; and in the same year, NACA was established.
- In 1922, Munk developed the thin airfoil theory
for calculating the lift and the moment acting on the airfoil.
- In 1925 Prandtl's mixing length theory was developed; and in 1928

English: 
here some key achievements
in the modern aerodynamics are listed
in 1909 louis various
flies across the english china from
karai france
to dover in england in 1910
yukovsky proposed the yukovsky
transformation
and the yokosuki air foil
in 1915 planter
established lifting nice theory
and in the same year naka was
established in 1922
monk developed the thin airfoil theory
for calculating the lift and the moment
acting on their fire in 1925
pronter's mixing lens series
was developed and in 1928

English: 
Prandtl-Glauert compressibility correction was established;
- In 1933, Theodorsen's general potential flow for arbitrary airfoils using the conformal
mapping method; - in 1947, the first supersonic flight was
made by (with) BEll X-1;
In early 1960s researchers
studied the panel methods for calculating the airfoils,
wings and aeroplanes;
- in 1970s turbulence  models, such as the k-EPSILON,
k-OMEGA models for CFD have been developed and accepted.
-In 1990s, large eddy simulation has been used for studying the
turbulent flow;

English: 
pronter cloud compressibility
collection was established in 1933
cr dozens the general potential flow for
abichi airfoils using the conformal
mapping method in 1947
the first supersonic flight was
made by bear x1
in early 1960s researchers
studied the panel methods for
calculating the air foil
wings and aeroplanes in 1970s
populous models such as the k epsilon
k omega models for cfd
have been developed and accepted
in 1990s large ad simulation
has been used for studying the
turbulence flow
and in 1997

English: 
hybrid leds was proposed by sparat
the yokowski transformation
is given by this here
a is the ladies of the reference
soccer see here the lead soccer
so to create uh zukowski
airfoil and modified and shifted circle
that the prime is calculated at this
so this would be given in this
blue circle here are
of its the distance between the
center of the original circle the red
one
and that of the modified blue circle
so this parameter is used for controlled
zhukovsky airfoil thickness
and the data is the shifting angle here

English: 
and in 1997 hybrid-LES was proposed by Spalart.
The Joukowski transformation is given by this,
Here a is the radius of the reference circle, see here the red circle.
so to create a Joukowski airfoil, a modified and shifted circle,
ZETA' is calculated at this, so this would be given in this
blue circle, here r_off is the distance between the
center of the original circle the red one
and that of the modified blue circle, so this parameter is used for controlling
the Joukowski airfoil thickness; and DELTA is the shifting angle, here

English: 
see the figure, this parameter is used for controlling the camber line.
This plot shows the original circle in green and the modified circle in blue,
and the Joukowski airfoil in red. From this Joukowski airfoil, we
can see this airfoil has a cusp at
the trailing edge, this might apply the limit for the Joukowski aerofoil.
For solving the problem, Karman-Trefftz made a transformation in 1918,
which could generate the much broader aerofoils for practical applications.
The idea behind the Joukowski transform is that there is an analytical solution

English: 
see the figure this parameter is used
for controlling the camber 9
this plot showing the original soccer
in green and the modified circle in blue
and the zhukovsky airfoil
in red from this zhukov ski airfoil we
can see
this airfoil has cops at
the cheating age this might play
the limit for the ryokovsky aerofoil
for solving the problem common cafes
made the transform in 1918
which could generate the much broader
aerofoil for practical applications
the idea behind the joke of sk transform
is that there is a analytical solution

English: 
for the flow around the soccer with a
certain circulation
this can be expressed as the uniform
flow plus
as circulatory flow with a circulation
capital so the combined flow
would be the circle with a circulation
gamma and we can see the staggering the
points
on the circle are not symmetric anymore
due to the existence of
the circular flow so here the
stagnant point and here so
such a type of flow we have the
mathematical
or analytic solution for this type of
row
so if we apply the zhukovsky transform
or chord conform use this formula
and we can have the flow past an
aerofoil
the yukovsky aerofoil and we can see
from this plot the stagnant point at the
leading edge

English: 
for the flow around the circle with a certain circulation,
this can be expressed as the uniform flow plus
a circulatory flow, with a circulation, capital GAMMA, so the combined flow
would be the circle with a circulation GAMMA, and we can see the stagnant
points on the circle are not symmetric anymore,
due to the existence of the circulatory flow, so here the
stagnant points are here, so such a type of flow we have the
mathematical or analytic solution for this type of
flow, so if we apply the Joukowski transform
or called conformal mapping using this formula, and we can have the flow past an
aerofoil, the Joukowski aerofoil and we can see
from this plot, the stagnant point at the leading edge

English: 
and the stagnant point at the chilling
age
and the flow would be separate from the
chilling age
this is in continence with the cuta
condition
the pronter lifting 9 series or
the manchester plantar wing theory is a
mathematical model
for protecting lift distribution of
a 3d wing based on
the geometry see here so this
left knife series states that the lift
of each wing segment the local
left per unit span does not
correspond simply to the 2d analysis
protect but also is strongly affected by
the neighboring
wing section so we see this bomb the
vertex
and the distribution along the span of

English: 
and the stagnant point at the trailing edge,
and the flow would be separate from the trailing edge,
this is in accordance with the Kutta condition.
The Prandtl lifting line theory or
the Lanchester-Prandtl wing theory is a mathematical model
for predicting lift distribution on a 3D wing, based on
its geometry, see here. so this lifting line theory states that the lift
over each wing segment, the local lift per unit span, does not
correspond simply to the 2D analysis predict, but also is strongly affected by
the neighboring wing section, so we see this bound
vertex and the distribution along the span of the wing.

English: 
the wing the left deny theory is based
on the concept
of circulation and the kuta
yukovsky theorem it produces the left
distribution around the span wise based
on the distribution of the bond vertex
so the overall lift on the wing is given
by this integration from tip 1 to
tip 2.
seeing alpha theory is a simple
theory of airfoil in incompressible
invasive flows it was developed
by max monk in 1922
and the father defined by hermann
groet and others
the cld idealizes the flow along
an aerofoil as a two-dimensional flow
around the scene aerofoil more
specifically

English: 
The lifting line theory is based on the concept
of circulation and the Kutta-Joukowski theorem, it produces the lift
distribution around the spanwise, based on the distribution of the bound vertex,
so the overall lift on the wing is given by this integration from tip1 to tip2.
Thin airfoil theory is a simple theory of airfoil in incompressible,
inviscid flows. It was developed by Max Munk in 1922,
and further refined by Hermann Glauert and others.
The theory idealizes the flow along an aerofoil as a two-dimensional flow
around a thin aerofoil, more specifically,

English: 
the aerofoil is represented by the camber line,
and the airfoil has a zero thickness.
Since the study is
in the incompressible and inviscid flow,
thus the airfoil friction is not included.
By applying the Kutta condition, we can obtain the vorticity
distribution, see the drawing here, thus we can derive the lift and
moment acting on the aerofoil.
The thin airfoil theory was very important,
because it provides the sound theoretical bases for the following properties of
the airfoil: - the center of the pressure and the
aerodynamic center are coincident and lie
at quarter of the chord behind the leading edge;

English: 
the aerofoil is represented by the
cambodi
and the airfoil has zero
sickness since the study is
in the incompressible and the invasive
flow
thus the airflow reflection is not
included
by applying the cuda condition we can
obtain the vorticity
distribution see the drawing here
thus we can derive the lift and
moment acting on the aerofoil
the thing air force was very important
because it provides the sound theoretic
bases for the following properties of
their therefore
the center of the pressure and the
aerodynamic
center are constant and lie
at the quarter of the court behind the
leading age

English: 
- the theoretical lift coefficient for thin aerofoil
is given as this simple equation, here ALPHA is the angle of attack (in
radians) and C_l0 would be zero for the
symmetric airfoil. Take an example of the thin airfoil, NACA0009,
the prediction used the thin airfoil theory
is in black line, and the dot and dashed line is from the experiment
data: this for the lift coefficient and this
is for the moment coefficient. So we can see
when the angle of attack is small, say less than 12 degrees,
the thin airfoil theory predicts the lift
and the moment very well, see the comparison here.

English: 
the theoretical lift coefficient for
seeing elephant
is given as this simple equation
here alpha it's the angle of attack in
radians and the cl0 would be zero for
the
symmetric airfoil take an example
of the scene airfoil naka009
the prediction used the thin air for
seri
is in black nine and the dot
and dash deny is from the experiment
data
this for the left coefficient and this
is for the moment the coefficient so we
can see
when the angle of attack is small
say less than 12 degrees
the same airfoil theory predicts the
lift
and the moment very well see the
comparison here

English: 
in 1925 pronter
studied the molecular shares just in
fluid
see the figure here so he could
determine the shear stress t x y
in terms of the mean flow velocity
u y and the fluid turbulence
viscosity mu t given
in the form that this and based on the
prontous proposal the mixing lens theory
the fluid turbulence viscosity is
given as this and here
air mix is the mixing lens
it needs to be specified in the actual
simulation
so this in a way would limit the
application
of the planters mixing lens theory
for this approximation model it is they
are

English: 
In 1925, Prandtl studied the molecular shear stress in
fluid, see the figure here, so he could
determine the shear stress t_xy (correction: TAU_xy) in terms of the mean flow velocity
U(y) and the fluid turbulence viscosity MU_t given
in the form as this,
and based on the Prandtl's proposal, the mixing length theory,
the fluid turbulence viscosity is given as this, and here
l_mix is the mixing length, it needs to be specified in the actual simulation.
so this in a way would limit the
application of the Prandtl's mixing length theory.
For this approximation model, it is still
quite useful:

English: 
quite useful for providing a tool
for solving some specific turbulence
flows
if the mixing lens air mix can be
specified
this model can be used for enhancing the
basic understanding of
the turbulent flow and provide the
basics
for the laser development encounters one
equation turbulence model
this model has found the applications in
different
areas including atmospheric science
oceanography and sternal structure
in 1920s researchers have worked
on their high-speed flows in 1925
the nine united supersonic aerodynamics
was developed and in 1928

English: 
- for providing a tool for solving some specific turbulence
flows, if the mixing length l_mix can be specified;
- this model can be used for enhancing the
basic understanding of the turbulent flow, and provide the
basics for the later development in Prandtl's one-
equation turbulence model;
- this model has found the applications in
different areas, including atmospheric science,
oceanography and stellar structure.
In 1920s researchers have worked on the high-speed flows. In 1925
the linearised supersonic aerodynamics was developed, and in 1928

English: 
Prandtl-Glauert made the correction for the compressibility; and in 1940
the application of the supersonic flow theory had
been made; and in 1947 the faster then speed-of-sound flight
Bell X-1 was made.
In the development, an important
advancement was the Prandtl-Glauert transformation
or  Prandtl-Glauert correction was established, based on the linearized
equation with the compressible, inviscid flows.
And the correction was made for the pressure coefficient
Cp, given by this, here Cp is the pressure coefficient
with the compressibility; and Cp0 is the pressure coefficient for the

English: 
planta grodd made the collection for the
compressibility and in 1940
the application of the supersonic flow
theory had
been made and in 1947
the faster then speed of sound flight
bear x1 was made
in the development an important
advancement
was that plantar glowed transform
or plantar growth collection
was established based on the linearized
equation
with the compressible inviscid
flow and that the collection
was made for the pressure coefficient
cp given by this
ercp is the pressure coefficient
with the compressibility and the cp0
is the pressure coefficient for the

English: 
incompressible flow; and M0 is the free  stream Mach number.
Obviously we can see the Prandtl-Glauert correction has a
singularity when M0 = 1.0,
see this figure, when M0 = 1.0, Cp would be infinite.
if we look at this picture, we can see the sonic boom around the airplane,
when the airplane passes the sound barrier,
that is, the Mach number is 1.0, so the sonic boom may (be) linked with
the singularity of the pressure coefficient
at Mach number M0=1.0.

English: 
incompressible flow
and m0 is there for this stream
mach number obviously we can see
the bronzer growth collection has the
singularity when m0
equals to 1.0
see this figure when m0 equals
1 cp would be infinite
if we look at this picture we can see
the sonic boom around the airplane
when the airplane passes the sound
barrier
that is the mach number is 1.0
so the sonic boom my linked with
the singularity of the pleasure
coefficient
at mach number m0 1.0

English: 
Theodore Theodorsen was the Norwegian-American aerodynamicist,
who was famous for his contributions in the theoretical aerodynamics, as well
as his studies for turbulence.
In 1933, he developed the
general potential theory for arbitrary wing sections, in which he built the
method for transforming the arbitrary airfoil into circle, such that
he could extend the thin aerofoil theory and the Joukowski transformation,
both were limited to the specific cases, but the general potential theory can be
used for any aerofoil and even other different bodies as well.
So in summary given by

English: 
ceo ceo dawson was the
norwegian american aerodynamicist
who was famous for his contribution
in the theological aerodynamics as well
as his studies for turbulence
in 1933 he developed the
general potential theory for apache
wing section in which he builds the
method for transforming the rbc
airfoil into soccer such that
he could extend the singh aerofoil cad
and the zhoukovskiy transform
both were limited to the specific cases
but the january potential theory can be
used for any
aerofoil and even other different the
body
as well so in summary given by

English: 
Theodorsen, the transformation or the conformal mapping to transform the arbitrary
airfoil in the ZETA-plane into a circle
in the z1- plane here. There are three steps in the transformation:
the first step is transform the arbitrary airfoil
into a circle-like closed curve, in the z'-plane here,
and use the further transformation, this circle-like curve can be
transformed into full circle in the z-plane; and the last step
is the shifting made from z-plane to z1-plane by shifting
a distance c1 here. And from this transformation, we can see

English: 
cr dawson the transform or the
conformal mapping to transform the rpg
airfoil
in the data plan into a circle
in the z1 plane here
there are three steps in the transform
the first step is transform the rpg
airfoil
into a circle like close the curve
in the zeta prime plane here
and i use the further transform
this circle-like curve can be
transformed into four soccer in the
zeta plane and the last step
is the shifting made from data plane to
the one plane by shifting
a distance c1 here
and from this transformation we can see

English: 
the flow direction to the aerofoil here is ALPHA, but the flow direction
for the circle in the z1-plane is ALPHA+BETA. So all these
transforms are quite complicated, but it is workable, even without a computer.
This method has been also used by
Eppler for designing his airfoil, see the reference 2.
With the digital computer available for computation,
the panel codes were developed from early 1960s. The advanced panel codes, such as
PAN AIR, developed by Boeing, was first introduced in the late 1970s

English: 
the flow direction to the aerofoil here
is alpha but the flow direction
for the circle in the zeta one plane
is alpha plus beta so all these
transforms are quite complicated
but it is workable even without the
computer
this method has been also used by
apple for designing his airfoil
see the reference 2.
with the digital computer available
for computation the panel
code was developed from early 1960s
the advanced panel codes such as
panerai derived by boeing was
first introduced in the late 1970s

English: 
and such panel method has been very popular,
even with their great advancement in CFD,
panel codes are still used for preliminary aerodynamic analysis, because of
its faster turnaround time and the reliable
predictions for some parameters.
There are some basic assumptions for the
panel methods which are basically based on the
potential flow in which the flow is assumed to be irrotational and inviscid.
and for such flows, Kutta condition must be satisfied on the lifting surfaces.
For the incompressible potential flows,

English: 
and such panel method has been very
popular
even with their grade advancement
in cfd panel code
as they are used for preliminary
aerodynamic analysis because of
its faster turnaround time and the
reliable
prediction for some parameters
there are some basic assumptions for the
panel method which are basically based
on the
potential flow in which the flow
is assumed to be irrotational
and invasive and for saggy flows
coda condition must be satisfied
on the lefting surfaces
for the incompressible potential flow

English: 
the normal laplace equation would be
used given as this
and for the compressible flows with
the assumption of the small disturbance
on the main flow around x axis
so the dynamic equation for the
potential function
phi for the subsonic flow given at
this and for the supersonic flow
the dynamic equation for the potential
flow
is given as this also
the dynamic equation for the
compressible flow has a different
dynamic equation for the potential
function
use some specific coordinates
transform we can still write
the dynamic equation for the potential
flow in
subsonic and supersonic to be a form
at the conventional laplace equation

English: 
the normal Laplace equation would be used, given as this.
and for the compressible flows, with the assumption of the small disturbance
on the main flow along x-axis, so the dynamic equation for the
potential function PHI for the subsonic flow given as this,
and for the supersonic flows, the dynamic equation for the potential
flow is given as this.
Although the dynamic equation for the compressible flow has a different
dynamic equation for the potential function,
use some specific coordinates transformation, we can still write
the dynamic equation for the potential flow in
subsonic and in supersonic to be a form as the conventional Laplace equation,

English: 
hence all this flow can be served
using the panel method
details can be found in the relevant
text books or in my
talks later
powerful computers in 1970s
were made for the implementations
and the validations of the proposed
providence model
possible adaz provided
the opportunities for tuning and
enhancing the turbulence model
therefore in 1970s the topless mothers
have been well established and
were accepted this
including the famous two equation
turbulence models

English: 
hence all these flows can be solved using the panel methods.
Details can be found in the relevant textbooks or in my talks later.
Powerful computers in 1970s were made for the implementations
and the validations of the proposed turbulence models possible,
as thus provided the opportunities for tuning and
enhancing the turbulence models, therefore in 1970s the turbulence models
have been well established and well accepted.
including the famous two equation turbulence models.
k-EPSILON model:

English: 
ke bistro moda the fourth dakira mother
was formulated at imperial college
in late 1960s and
tuned and enhanced in 1970s
the transport equations for both
turbulent the kinetic energy
k and the dissipation rate epsilon
are both directly derived from the
nano's
average the navier-stokes equation
and since then the k epsilon model has
again the widest bread use
since 1970s
and the enhanced model were available
including the
denomination group ing and the
realizable model
and many other variants
the first the ke bis remodel was
proposed
by komoglov in 1942 and
independently by seven in nineteen

English: 
- the first k-EPSILON model was formulated at Imperial College
in late 1960s; and tuned and enhanced in 1970s.
- the transport equations for both turbulent kinetic energy
k and the dissipation rate EPSILON are both directly derived from the
Reynolds averaged Navier-Stokes equation.
- and since then the k-EPSILON model has gained the wider spread use since 1970s;
- and enhanced models were available,
including the Renomination group (RNG) and
Realizable model and many other variants.
The first k-EPSILON (correction: k-OMEGA) model was proposed
by Komoglov in 1942 and independently by Saffman in 1970;

English: 
- Wilcox kept improving the Saffman's
k-OEMGA model since 1970s;
- the  k-OMEGA model is getting popular
since the Menter's SST  k-OMEGA model
in 1994. This SST model is to merge
the Wilcox  k-OMEGA model (which is good for modeling the near wall
region) and the k-EPSILON model (which is good
for the free stream in the outer part of the boundary layer),
and in the SST model, the formulation for the eddy viscosity has been revised.
And from the examples in research and in industry applications, this SST model
is one of the most popular models.

English: 
seventy
real cox keep improved the seventh
k omega model since 1970s
the k omega mod are getting popular
since
the mantas sst k omega model
in 1994
this sst model is to merge
the rear k omega model
which is good for modeling the near wall
region
and the k epsilon model which is good
for the free stream in the old part of
the bundle layer
and in the sst model the formulation for
the added viscosity has been devised
and from the example in research and
in industry applications this sst model
is one of the most popular model

English: 
Soon after the various turbulence models, the research activities in this field
have realized the limits of the turbulence
models. Hence a lot of research had been shifted to large eddy simulation, (LES)
in earlier 1990s and later
to the hybrid RANS/LES modeling, according to the reference [1].
It is well known in LES, large energy
carrying eddies would be resolved numerically, and only the small eddies
are needed to be modelled. Generally speaking,
these small eddies would be more isotropic, thus it would
be easier to be modelled, at least in principle.

English: 
so after the various turbulence models
the research activities in this field
has realized the limits of the
turbulence
models hence a lot of
research had been shifted to large ad
simulation air yes in earlier
1990s and later
to the hybrid lens less
modeling according to the reference
one it is well known
in air yes large energy
carrying ids would be reserved
numerically and only the smileys
are needed to be murdered generally
speaking
this small eddies would be more
isotropic thus it would
be easier to be murdered at least in
principle

English: 
- the LES model was first proposed by Smargorinsky for atmospheric
research in 1963, and
its first engineering application in 1970s
- As the late 1970s a research group at Stanford University took the lead
in the development and application of LES.
however, LES
applies a severe demand on computations, in many cases well beyond the general
practical applications. For instance, the computation of LES
would be proportional to the Reynolds number
power of 1.8 to 2.0, so for the practical application in high
Reynolds number, for instance, for a car or an airplane,

English: 
the less mother was first proposed
by smart grensky for a mesophytic
research
in 1963 and
its first engineering application in
1970s
as the late 1970s a research group
at stanford university took the lead
in the development and application of
air years however air yes
applies a severe demand on computations
in many cases where beyond the general
practical applications
for instance the computation of air yes
would be proportional to the nano's
number
for 1.8 to two point
so for the platinic application in high
nearness number
for instance for a car or airplane

English: 
the l year's mesh would be 10 power of
11.5 and the time steps
would be 10 power of 6.7
country the large computation
of cfd would be roughly temporal of
9. thus the pure
less modeling are still very limited
to the simple flows at low reynolds
number
and the primary used as the research
tool
for studying the details of the
turbulent flow
to ease the created the mind in air yes
sabara proposed a hybrid
dance less model
he landed the hybrid mode as

English: 
the LES mesh would be 10^11.5 and the time steps
would be 10^6.7.
Currently the large computation
of CFD would be roughly 10^9.
Thus the pure LES modelings are still very limited to the simple flows at low Reynolds
number, and primarily used as the research
tool for studying the details of the
turbulent flows.
To ease the grid demand in LES, Spalart proposed a hybrid
RANS-LES model, he named the hybrid model as

English: 
detached at the simulation model
des model and this
hybrid method becomes very popular
approach especially in the research
community the hybrid
lens and the less model
so the original ds was proposed
for the boundary layer would be cheated
by friends and the legion of the massive
separation are cheated with their years
therefore
the reduced computation can be
significant
the mesh now is 10 power of
8 compared to the requirement
in air yes for 11.5
the time steps is 10 power 4
compared to tempo of 6.7

English: 
detached eddy simulation model (DES model),
and this hybrid method becomes a very popular approach, especially in the research
community.
the hybrid RANS-LES model:
so the original DES was proposed for the boundary layer would be treated
by RANS and the regions of the massive separation are treated with LES.
therefore, the reduced computation can be
significant: the mesh now is 10^8,
compared to the requirement in LES: 10^11.5;
the time steps is 10^4, compared to 10^6.7.

English: 
and now many different hybrid lens
less model has been proposed
and being tested one
important issue is on how we can better
implement the hybrid lens less mode
and some some challenging issue
in the hybrid lens areas model
using hybrid lens aes mode
sometimes can be very tricky
generally leds could have some
very similar mathematic equation as
those
in range thus the numeric
implementation would be same as
lens model according reference
3 and even easier

English: 
and now many different hybrid RANS-LES models have been proposed
and being tested. One important issue is on how we can better
implement the hybrid RANS-LES model, and solve some challenging issues
in the hybrid RANS-LES model.
using hybrid RANS-LES model
sometimes can be very tricky. Generally, LES could have some
very similar mathematic equations as those
in RANS, thus the numerical implementation would be same as
RANS model, according reference [3], and even easier
according the reference [4].

English: 
according the reference 4.
in some cases people would like to
perform a conventional
leds on a course grid
for the lower bounded flows
especially in the attached flow region
this approach would be very erroneous
since this type of the leds
would be inferior to even simple
conventional
lens modeling so when we
use the leds we need to pay
a lot of attention on the grid
you

English: 
In some cases, people would like to
perform a conventional LES on a coarse-grid
for the wall bounded flows, especially in the attached flow region.
this approach would be very erroneous since this type of the LES
would be inferior to even simple conventional
RANS modeling.
So when we use LES, we need to pay
a lot of attention on the grid.
