
English: 
Hello welcome to my talk, All about Fluids. This talk is on the basic
equations for the turbulence modeling, Part 1, Reynolds averaged
Navier-Stokes equation. In this talk I will introduce the method for averaging
in the fluid dynamics, namely time averaging for the continuous functions
or the ensemble averaging for the sampled system, and then I will show how
the Reynolds averaged Navier-Stokes equation is derived. This is the
fundamental equation for turbulent flows, and it is the basis for establishing the

English: 
Hello welcome to my talk, All about Fluids. This talk is on the basic
equations for the turbulence modeling, Part 1, Reynolds averaged
navier-stokes equation. In this talk I will introduce the method for averaging
in the fluid dynamics, namely time averaging for the continuous functions
or the ensemble averaging for the sampled system, and then I will show how
the Reynolds averaged navier-stokes equation is derived. This is the
fundamental equation for turbulent flows, and it is the basis for establishing the

English: 
turbulent transport equations for Reynolds stress components; turbulent
kinetic energy and the dissipation rate.
For fluid dynamics, we may have two different flows: steady flow and
unsteady flows. here we're talking about turbulent flows,
Principally, turbulent flows are inherently unsteady, for some flows the
mean flow is steady, with small turbulence superposed on the main mean flow, see the
figure here. The red line is the mean flow velocity,
and the fluctuation of the velocity is the turbulent velocity.

English: 
turbulent transport equations for Reynolds stress components; turbulent
kinetic energy and the dissipation rate.
For fluid dynamics, we may have two different flows: steady flow and
unsteady flows. here we're talking about turbulent flows,
Principally, turbulent flows are inherently unsteady, for some flows the
mean flow is steady, with small turbulence superposed on the main mean flow, see the
figure here. The red line is the mean flow velocity,
and the fluctuation of the velocity is the turbulent velocity.

English: 
so the averaged velocity of a steady flow is given as this, here T is the
entire time period of the problem, as such the averaged variables would
be time-independent. It should be noted here T is a time period, long enough for
the steady averaged variable, not necessary to be infinite as shown in
some textbooks. As such, the averaged variables would
be time-independent.
For the unsteady flows, as seen in this figure,
we could have an always time-dependent averaged variable superposed on (with) the

English: 
so the averaged velocity of a steady flow is given as this, here T is the
entire time period of the problem, as such the averaged variables would
be time-independent. It should be noted here T is a time period, long enough for
the steady averaged variable, not necessary to be infinite as shown in
some textbooks. As such, the averaged variables would be time-independent.
For the unsteady flows, as seen in this figure,
we could have an always time-dependent averaged variable superposed on (with) the

English: 
small fluctuating variable, so the averaging is made as given in this formula,
so here T is the sectional time period, a time period longer than most of
the turbulence time scales if it is not all. but it would be significantly shorter
than the entire time period of the problem, therefore the part time-averaged
velocity is still time-dependent.
For the unsteady flows, it can be the
unsteady flow driven by the unsteady boundary conditions or simply the
spontaneous unsteady flows, such as the vortex shedding after the
cylinder in this video (video from Wikimedia Commons).

English: 
small fluctuating variable, so the averaging is made as given in this formula,
so here T is the sectional time period, a time period longer than most of
the turbulence time scales if it is not all. but it would be significantly shorter
than the entire time period of the problem, therefore the part time-averaged
velocity is still time-dependent.
For the unsteady flows, it can be the
unsteady flow driven by the unsteady boundary conditions or simply the
spontaneous unsteady flows, such as the vortex shedding after the
cylinder in this video (video from Wikimedia Commons).

English: 
the unsteadiness of the flow after the cylinder is caused as the vortex shedding
although the incoming flow and the boundary conditions are steady.
the spatial averaging is a volume averaging of the physical parameter at
the geometrical center of a small volume, given by this
see here, the center point xn is at the geometrical center
Or we may have the fluid momentum centre of the small fluid element,
defined as this, here the averaged velocity is calculated at the

English: 
the unsteadiness of the flow after the cylinder is caused as the vortex shedding
although the incoming flow and the boundary conditions are steady.
the spatial averaging is a volume averaging of the physical parameter at
the geometrical center of a small volume, given by this
see here, the center point xn is at the geometrical center
Or we may have the fluid momentum centre of the small fluid element,
defined as this, here the averaged velocity is calculated at the

English: 
momentum center of the fluid element DELTA_V
nowadays in deriving the Reynolds averaged Navier-Stokes equation, we
don't use the spatial averaging as shown in here. However, in CFD, such as the
method of finite volume, we may use the spatial averaging and we normally take
the physical parameters at certain point within the fluid element for
representing the physical parameters
Ensemble averaging was given by Reynolds in his derivation in 1895.

English: 
momentum center of the fluid element DELTA_V
nowadays in deriving the Reynolds averaged navier-stokes equation, we
don't use the spatial averaging as shown in here. However, in CFD, such as the
method of finite volume, we may use the spatial averaging and we normally take
the physical parameters at certain point within the fluid element for
representing the physical parameters
Ensemble averaging was given by Reynolds in his derivation in 1895.

English: 
his averaging scheme is given as this, according (to) his own word, u_bar here
is the velocity of x-direction at each instantaneous center of gravity of
the fluid of a small volume DELTA_V, however this expression is not for
today's understanding for the problem, since SIGMA RHO has no physical meaning here
and Reynolds also gave the
component momentum at the center of gravity, written as this
and then we can easily derive the velocity expression: the mean velocity
plus the fluctuation velocity.
nowadays the ensemble averaging for the

English: 
his averaging scheme is given as this, according (to) his own word, u-bar here
is the velocity of x-direction at each instantaneous center of gravity of
the fluid of a small volume DELTA_V, however this expression is not for
today's understanding for the problem, since SIGMA RHO has no physical meaning here
and Reynolds also gave the
component momentum at the center of gravity, written as this
and then we can easily derive the velocity expression: the mean velocity
plus the fluctuation velocity.
Nowadays the ensemble averaging for the

English: 
timed samples is given as this, here we assume we have M samples within the
short period T here, so this averaging of the velocity is a constant for the
time period T. and again the velocity can be
expressed as this, so here we suppose in a sampled system, we have M sample point
within the time period,
thus the averaged velocity is still time-dependent, as we have seen in
many textbooks and papers, such as, the U-RANS
Unsteady RANS, T-RANS (Transient RANS).
Following the Wilcox's book,

English: 
timed samples is given as this, here we assume we have M samples within the
short period T here, so this averaging of the velocity is a constant for the
time period T. and again the velocity can be
expressed as this, so here we suppose in a sampled system, we have M sample point
within the time period,
thus the averaged velocity is still time-dependent, as we have seen in
many textbooks and papers, such as, the U-RANS
Unsteady RANS, T-RANS (Transient RANS).
Following the Wilcox's book,

English: 
in the talk I'll use the capital U and the capital P for the mean velocity and pressure,
and u', p' would be the fluctuating velocity and pressure
so all the averaging methods mentioned in the previous slides are linear operators
in a simple averaging, we have an expression as this
it should be noted so that the averaging value, U would be a constant for the
averaging operation, but it would be both time- and spatial- dependent
therefore the averaging of the averaged velocity here is given as this

English: 
in the talk I'll use the capital U and the capital P for the mean velocity and pressure,
and u', p' would be the fluctuating velocity and pressure
so all the averaging methods mentioned in the previous slides are linear operators
in a simple averaging, we have an expression as this
it should be noted so that the averaging value, U would be a constant for the
averaging operation, but it would be both time- and spatial- dependent,
Therefore, the averaging of the averaged velocity here is given as this

English: 
so the averaging of the averaged velocity would be the averaged velocity itself.
the averaging of the fluctuating variable would be zero
Take the velocity u as an example, and take an average of this expression we
have this and it can be written as this, so we have the expression as this.
from this, we can deduce the averaged fluctuating velocity would be zero.
here we must mention the averaging operation and the conventional
mathematical operations: +, - and the differentiations, integrations and
the summations are all linear operations, thus the order of the operations can be

English: 
so the averaging of the averaged velocity would be the averaged velocity itself.
the averaging of the fluctuating variable  would be zero
Take the velocity u as an example, and take an average of this expression we
have this and it can be written as this, so we have the expression as this.
from this, we can deduce the averaged fluctuating velocity would be zero.
here we must mention the averaging operation and the conventional
mathematical operations: +, - and the differentiations, integrations and
the summations are all linear operations, thus the order of the operations can be

English: 
swapped, because of the linear operators.
In here we look at the averaging of the
fluctuating velocity derivation with regard to time, this is the expression
and averaging of this expression, we have
expression at this.
so since this part,
we can swap the order of the differentiation and the average ,
so this is the expression for the averaging of the velocity derivation with regard
to time, therefore, we have the expression for this, that means the derivation
of the fluctuation velocity with regard to time, its average is zero.
and this actually can be can be easily obtained because we can change the order of the

English: 
swapped, because of the linear operators.
In here we look at the averaging of the
fluctuating velocity derivation with regard to time, this is the expression
and averaging of this expression, we have
expression at this.
so since this part, we can swap the order of the differentiation and the average,
so this is the expression for the averaging of the velocity derivation with regard
to time, therefore, we have the expression for this, that means the derivation
of the fluctuation velocity with regard to time, its average is zero.
and this actually can be can be easily obtained because we can change the order of the

English: 
averaging with the derivation, and this can be easily derived.
for the incompressible flows, the conventional expression of the
continuity equation is given as this, in details the differentiation form of the
continuity equation is written as this, here u v and w are the velocity components.
and this is equivalent to the expression
as this, using the indices.
so in Einstein summation convention,
this continuity equation can be written as this simple form

English: 
averaging with the derivation, and this can be easily derived.
for the incompressible flows, the conventional expression of the
continuity equation is given as this, in details the differentiation form of the
continuity equation is written as this, here u, v and w are the velocity components.
and this is equivalent to the expression as this, using the indices.
so in Einstein summation convention,
this continuity equation can be written as this simple form

English: 
so if we take the average of the left hand side, we have this and we
can exchange of the operations: the averaging operation and the derivation,
we have this and this,
so we have the continuity equation for
the mean flow velocity as this. now use the averaged and the
fluctuating velocities, ui = Ui+u'_i, so the continuity
equation would be in this form. Considering the continuity equation for
the mean velocity, so we can have the continuity equation for the fluctuating
velocity as this,
therefore we can see for the incompressible flows,

English: 
so if we take the average of the left hand side, we have this and we
can exchange of the operations: the averaging operation and the derivation,
we have this and this, so we have the continuity equation for
the mean flow velocity as this. now use the averaged and the
fluctuating velocities, ui = Ui+u'_i, so the continuity
equation would be in this form. Considering the continuity equation for
the mean velocity, so we can have the continuity equation for the fluctuating
velocity as this,
Therefore, we can see for the incompressible flows,

English: 
the continuity equation for the instantaneous, the average and the fluctuating
velocities have the same form, seen in the circles.
the Navier-Stokes equation for incompressible flows can be written as
this in Einstein's summation convention.
Here Ui, Uj are the velocity components, i, j equal to 1 to 3; fi is the body
force per volume; p is the fluid pressure; NU is the kinematic viscosity

English: 
the continuity equation for the instantaneous, the average and the fluctuating
velocities have the same form, seen in the circles.
the navier-stokes equation for incompressible flows can be written as
this in Einstein's summation convention.
Here Ui, Uj are the velocity components, i, j equal to 1 to 3; fi is the body
force per volume;  p is the fluid pressure; NU is the kinematic viscosity

English: 
of the Newtonian fluid; ui*uj is the tensor of the double velocity vectors.
use the averaged and the fluctuating velocities and pressure, we have the
Navier-Stokes equation as this. Take the average of both sides of the
equation, we have the expression as this: this term can be written in this
and we can swap the order of the average and the plus (+), so we have the equation as this.
Now in this slide we are examine each term in the averaged Navier-Stokes
equation as this, termed 1 to 5.

English: 
of the Newtonian fluid; ui*uj is the tensor of the double velocity vectors.
use the averaged and the fluctuating velocities and pressure, we have the
navier-stokes equation as this. Take the average of both sides of the
equation, we have the expression as this: this term can be written in this
and we can swap the order of the average and the plus (+), so we have the equation
as this
now in this slide we are examine each term in the averaged navier-stokes
equation as this, termed 1 to 5.

English: 
so Term 1, we have this, and we can separate this into this
because of the average of the fluctuating velocity is zero
we have the expression as this and this.
for the second term, we can separate
each term into this expression, now considering the mean velocity,
the capital Ui and capital Uj would be constants for the averaging operation,
thus we can take the mean velocities, capital Ui, capital Uj out of the

English: 
so Term 1, we have this, and we can separate this into this,
because of the average of the fluctuating velocity is zero
we have the expression as this and this.
for the second term, we can separate
each term into this expression, now considering the mean velocity,
the capital Ui and capital Uj would be constants for the averaging operation,
thus we can take the mean velocities, capital Ui, capital Uj out of the

English: 
average operation, here and here, here as well. Now we can see the average of the
single fluctuating velocity u'_j and u'_i  are both zero.
so we have the final expression for the second term as this.
so from this mathematical manipulation, we can see the averaging of a single
fluctuating velocity is zero, but the average of the double fluctuating
velocities is not zero, And then in turbulent flows, this term would be
very important.
the 3rd term is for body force,
in most cases the body force is given, so it is not relevant to the

English: 
average operation, here and here, here as well. Now we can see the average of the
single fluctuating velocity u'_j and u'_i are both zero.
so we have the final expression for the second term as this.
so from this mathematical manipulation, we can see the averaging of a single
fluctuating velocity is zero, but the average of the double fluctuating
velocities is not zero, And then in turbulent flows, this term would be very important.
the 3rd term is for body force,
in most cases the body force is given, so it is not relevant to the

English: 
fluid turbulence, thus its average is itself, so the expression is this.
For the fourth term, the pressure gradient, we can separate the expression as this.
and obviously for the pressure fluctuation, its average is zero, thus we have the
expression (for the) for the fourth term.
Similarly, in the 5th term, we have the
expression as this, and we can separate the expression into two terms. now we can
see this term would be 0, because it is for the averaging of the fluctuation
u'_i, therefore, the 5th term is given by this.

English: 
fluid turbulence, thus its average is itself, so the expression is this.
For the fourth term, the pressure gradient, we can separate the expression as this.
and obviously for the pressure fluctuation, its average is zero, thus we have the
expression (for the) for the fourth term.
similarly in the 5th term, we have the
expression as this, and we can separate the expression into two terms. now we can
see this term would be 0, because it is for the averaging of the fluctuation
u'_i, therefore, the 5th term is given by this.

English: 
now put all these terms together, we have the Reynolds averaged
Navier-Stokes equation as this. This equation is for the averaged
velocities and pressure, and a significant difference for the
equation is that this Reynolds averaged Navier-Stokes equation has an
additional term when compare it to the original Navier-Stokes equation here.
And this term is all about the flow turbulence, and causes that
tremendous difficulties for solving the fluid dynamic equation.
Turbulence modeling is a way to try to find a solution for this term.

English: 
now put all these terms together, we have the Reynolds averaged
navier-stokes equation as this. This equation is for the averaged
velocities and pressure, and a significant difference for the
equation is that this Reynolds averaged navier-stokes equation has an
additional term when compare it to the original navier-stokes equation here.
And this term is all about the flow turbulence, and causes that
tremendous difficulties for solving the fluid dynamic equation.
Turbulence modeling is a way to try to find a solution for this term.
this equation can

English: 
this equation can be written in this form, so this is the conventional Reynolds averaged
Navier-Stokes equation, with the Reynolds stress tensor defined as TAU_ij
equals minus average of the double velocities u'_i and u'_j,
this equation can be derived from the conventional symmetrical strain rate
Sij, and the mathematical operation on this viscous stress tensor will have this
expression and this.
and the incompressible flow, this
term would be zero, thus this term would be same as this term.
Use the newly proposed asymmetrical viscous stress tensor in the reference below,

English: 
be written in this form, so this is the conventional Reynolds averaged
navier-stokes equation, with the Reynolds stress tensor defined as TAU_ij
equals minus average of the double velocities u'_i and u'_j,
this equation can be derived from the conventional symmetrical strain rate
Sij, and the mathematical operation on this viscous stress tensor will have this
expression and this.
and the incompressible flow, this
term would be zero, thus this term would be same as this term.
Use the newly proposed  asymmetrical viscous stress tensor in the reference below,

English: 
The Reynolds averaged Navier-Stokes equation can be given in this form, here
Aij is the asymmetric strain-rate tensor, defined as this.
now we have the conventional Navier-Stokes equation, in a form as this,
here the mean and the fluctuating velocities and pressures are used.
and from the previous slides, we have also the Reynolds averaged
Navier-Stokes equation given in this form.
so if we subtract the Navier-Stokes equation
with the Reynolds averaged Navier-Stokes equation,
we can obtain the Navier-Stokes equation for the fluctuating velocities and pressures as this.

English: 
The Reynolds averaged navier-stokes equation can be given in this form, here
Aij is the asymmetric strain-rate tensor, defined as this.
now we have the conventional navier-stokes equation, in a form as this,
here the mean and the fluctuating velocities and pressures are used.
and from the previous slides, we have also the Reynolds averaged
navier-stokes equation given in this form.
so if we subtract the Navier-Stokes equation
with the Reynolds averaged navier-stokes equation,
we can obtain the navier-stokes equation for the fluctuating velocities and
pressures as this.

English: 
if we use the continuity equation for the
incompressible flow this and this, and we can write this navier-stokes equation
for the fluctuating velocity and pressure into this expression.
now we can see for the turbulent flow, the mean variables and the fluctuating variables
are actually linked together. mathematically for the mean flow the
Reynolds stress tensor, this term from the flow turbulence, would be an
important influencing factor which would change the flow pattern of the mean flow.
and in the navier-stokes equation for the fluctuating velocities
and pressure, we can see the mean flow velocities Uj and Ui are engaged,

English: 
if we use the continuity equation for the
incompressible flow this and this, and we can write this Navier-Stokes equation
for the fluctuating velocity and pressure into this expression.
now we can see for the turbulent flow, the mean variables and the fluctuating variables
are actually linked together. mathematically for the mean flow the
Reynolds stress tensor, this term from the flow turbulence, would be an
important influencing factor which would change the flow pattern of the mean flow.
and in the Navier-Stokes equation for the fluctuating velocities
and pressure, we can see the mean flow velocities Uj and Ui are engaged,

English: 
which can be regarded as the generator of the flow turbulence or from the point
of view of energy, it can be understood that the engagement of the mean
velocities here is a way transforming energy from the mean flow into the
turbulent flow.

English: 
which can be regarded as the generator of the flow turbulence or from the point
of view of energy, it can be understood that the engagement of the mean
velocities here is a way transforming energy from the mean flow into the turbulent flow.
