
English: 
Hello welcome to my talk, All about Fluids.
this talk is on the turbulence modelling, basic equations, Part II, Turbulent
transport equation. In this talk the detailed derivations of the turbulent
transport equations are presented, including the transport equation for the
Reynolds stress components, turbulent kinetic energy and the dissipation rate,
In addition, the Boussinesq's assumption is presented, with a short discussion with
the assumption. I must admit that in this talk there are

English: 
Hello welcome to my talk, All about Fluids.
this talk is on the turbulence modelling, basic equations, Part II, Turbulent
transport equation. In this talk the detailed derivations of the turbulent
transport equations are presented, including the transport equation for the
Reynolds stress components, turbulent kinetic energy and the dissipation rate,
In addition, the Boussinesq's assumption is presented, with a short discussion with
the assumption. I must admit that in this talk there are

English: 
a lot of mathematics and the relevant  equations. I will try to present these
equations in a more understandable way, and
to make the equation correct, I derived all the equation myself and
then checked and double-checked all the equations against some well-known
books and published papers.  and thus hope all these would be useful for you.
in this slide, a Navier-Stokes operator is defined based on the conventional
navier-stokes equation, we consider the incompressible flows here.
this navier-stokes operator is of vital importance in the derivations of the
turbulent transport equation. As we have seen in many textbooks,

English: 
a lot of mathematics and the relevant  equations. I will try to present these
equations in a more understandable way, and
to make the equation correct, I derived all the equation myself and
then checked and double-checked all the equations against some well-known
books and published papers.  and thus hope all these would be useful for you.
in this slide, a Navier-Stokes operator is defined based on the conventional
navier-stokes equation, we consider the incompressible flows here.
this navier-stokes operator is of vital importance in the derivations of the
turbulent transport equation. As we have seen in many textbooks,

English: 
the Navier-Stokes operator, the fancy capital L, is defined as this. Basically this
moves the terms of the right hand side together with the left hand side terms,
Here ui is the velocity component i. For the navier-stokes equation, this can
be understandable, since this is the moment equation for the velocity
component ui. However when we define the
navier-stokes operator, it is an operator, with ui as the main variable, but it is
also with some other components, here fi and xi.

English: 
the Navier-Stokes operator, the fancy capital L, is defined as this. Basically this
moves the terms of the right hand side together with the left hand side terms,
Here ui is the velocity component i. For the navier-stokes equation, this can
be understandable, since this is the moment equation for the velocity
component ui. However when we define the
navier-stokes operator, it is an operator, with ui as the main variable, but it is
also with some other components, here fi and xi.

English: 
thus the author believes a better expression for the navier-stokes operator
should be given as this, here qi is the generalized the variable of the
component i, and it can be either ui, fi or xi, and in this talk I will use this
expression, but I must say here for this expression it would lead (to) the exactly
same result as those in the traditional navier-stokes operator, but I think this
is with a better physics. As such, for the component j, we have the N-S operator
as this. so based on the navier-stokes equation

English: 
thus the author believes a better expression for the navier-stokes operator
should be given as this, here qi is the generalized the variable of the
component i, and it can be either ui, fi or xi, and in this talk I will use this
expression, but I must say here for this expression it would lead (to) the exactly
same result as those in the traditional navier-stokes operator, but I think this
is with a better physics. As such, for the component j, we have the N-S operator
as this. so based on the navier-stokes equation

English: 
we could have both the equations for L(qi) and L(qj) are zero.
thus, we can construct an equation as this, and take the average of the
equation we have with the equation we can use to derive the transport
equation for Reynolds stress components.
now we can substitute the navier-stokes operator into this expression, we have
this, and we can exchange the order of the averaging operations and the
pluses or minuses.
so we have the expression as this for the first part of the expression, and the

English: 
we could have both the equations for L(qi) and L(qj) are zero.
thus, we can construct an equation as this, and take the average of the
equation we have with the equation we can use to derive the transport
equation for Reynolds stress components.
now we can substitute the navier-stokes operator into this expression, we have
this, and we can exchange the order of the averaging operations and the
pluses or minuses
so we have the expression as this for the first part of the expression, and the

English: 
second part of the expression. and then we can put all these
corresponding terms together, the first terms together into this and
the second terms this, so we have this.
the 3rd terms for the body force we have the expression as this;  the 4th term
with the pressure gradient, so we have this; and the last term with the fluid
viscous terms, we have this. And we can term these terms into 1 to 5, and the
in the next few slides we will do more mathematical derivations for each of these terms

English: 
second part of the expression. and then we can put all these
corresponding terms together, the first terms together into this and
the second terms this, so we have this.
the 3rd terms for the body force we have the expression as this;  the 4th term
with the pressure gradient, so we have this; and the last term with the fluid
viscous terms, we have this. And we can term these terms into 1 to 5, and the
in the next few slides we will do more mathematical derivations for each of these terms

English: 
for the first term, we can separate the expression by
timing the relevant terms together as this, Now since the averaged
velocity capital Ui and Uj would be constants
regarding (to) the averaging operation, thus they can be taken out of the
averaging operation, so we have this. put these two terms together we have
this term. Obviously the averaging on the single
fluctuating velocities u'_i and u'_j would be 0, therefore, the
final expression for the for first term is given as this.

English: 
for the first term, we can separate the expression by
timing the relevant terms together as this, Now since the averaged
velocity capital Ui and Uj would be constants
regarding (to) the averaging operation, thus they can be taken out of the
averaging operation, so we have this. put these two terms together we have
this term. Obviously the averaging on the single
fluctuating velocities u'_i and u'_j would be 0, therefore, the
final expression for the for first term is given as this.

English: 
the second item has an expression as this, now we time the relevant terms
together, so we have the expression as this. now we can take the mean velocity
capital Ui, capital Uk, capital Uj out of the averaging operation,
we have the expression as this, in here the continuity equation for the mean
velocity capital U and for the fluctuating velocity
u' are also used.

English: 
the second item has an expression as this, now we time the relevant terms
together, so we have the expression as this. now we can take the mean velocity
capital Ui, capital Uk, capital Uj out of the averaging operation,
we have the expression as this, in here the continuity equation for the mean
velocity capital U and for the fluctuating velocity
u' are also used.

English: 
look at the expression so the first two terms would be 0, therefore, we have the
expression as this, we also put this and this together, we have the term as this.
so the final expression for the 2nd term is given as this.
for the 3rd term, the body force would be generally given
and specified, thus it is not relevant to the flow turbulence.
for instance, the gravitational force of the fluid is the one of the body
force, which is proportional to the fluid volume or mass,

English: 
look at the expression so the first two terms would be 0, therefore, we have the
expression as this, we also put this and this together, we have the term as this.
so the final expression for the 2nd term is given as this.
for the 3rd term, the body force would be generally given
and specified, thus it is not relevant to the flow turbulence.
for instance, the gravitational force of the fluid is the one of the body
force, which is proportional to the fluid volume or mass,

English: 
therefore it is independent of the fluid velocity. As such we can take the
body force out of averaging operation, so we have the expression as this.
look at the averaging of the single fluctuating velocity, so we have the
expression for the 3rd term as zero.
the 4th term is for the pressure gradient,
timing the relevant terms together, and also take out of the averaged
pressure out of the averaging operation, the capital P here and here.
so from this expression we can see this two terms would be 0.

English: 
therefore it is independent of the fluid velocity. As such we can take the
body force out of averaging operation, so we have the expression as this.
look at the averaging of the single fluctuating velocity, so we have the
expression for the 3rd term as zero.
the 4th term is for the pressure gradient,
timing the relevant terms together, and also take out of the averaged
pressure out of the averaging operation, the capital P here and here.
so from this expression we can see this two terms would be 0.

English: 
hence, we have the expression as this. Now we can do more mathematical
manipulations for this expression, we have the final expression for the 4th term
as this. For the fifth term, this is relevant to
the fluid viscosity. we can separate all these terms. so we
have the expression as this. and we can take the mean velocity out of the
averaging operation, we have the expression as this and this, here again
the averaging of the single fluctuating velocity
ui_j and u'_i would be zero, thus we have the expression as this,

English: 
hence, we have the expression as this. Now we can do more mathematical
manipulations for this expression, we have the final expression for the 4th term
as this. For the fifth term, this is relevant to
the fluid viscosity. we can separate all these terms. so we
have the expression as this. and we can take the mean velocity out of the
averaging operation, we have the expression as this and this, here again
the averaging of the single fluctuating velocity
ui_j and u'_i would be zero, thus we have the expression as this,

English: 
and we can do a little bit more the derivation for this term, we have this,
since we can see this term comes to here, and this comes to here, so we have the
expression as this. And put these two terms together we have the expression as
this, so the final expression for the 5th term is given as this.
in this slide we put all these five terms together, so we have the transport
equation for the Reynolds stress components, so the equation given as this
complicated equation. and we can write this into a

English: 
and we can do a little bot more the derivation for this term, we have this,
since we can see this term comes to here, and this comes to here, so we have the
expression as this. And put these two terms together we have the expression as
this, so the final expression for the 5th term is given as this.
in this slide we put all these five terms together, so we have the transport
equation for the Reynolds stress components, so the equation given as this
complicated equation. and we can write this into a

English: 
conventional transport equation for the Reynolds stress components, we have
this. and the two terms on the left hand side are these, and all other terms we
put on the right hand side. So if we use the Reynolds stress
tensor TAU_ij equals to minus average of the fluctuating velocities ui_i and
u'_j, and then we can write the transport pressure for Reynolds
stress tensor as this.
The transport equation for the Reynolds stress tensor is an equation we
derived for solving the Reynolds stress tensor. For the Reynolds stress tensor,

English: 
conventional transport equation for the Reynolds stress components, we have
this. and the two terms on the left hand side are these, and all other terms we
put on the right hand side. So if we use the Reynolds stress
tensor TAU_ij equals to minus average of
the fluctuating velocities ui_i and
u'_j, and then we can write the transport pressure for Reynolds
stress tensor as this.
The transport equation for the Reynolds stress tensor is an equation we
derived for solving the Reynolds stress tensor. For the Reynolds stress tensor,

English: 
it is a symmetric tensor, thus it has six unknowns in the Reynolds stress
tensor. As we have shown in the previous slides, the transport equation for
Reynolds stress tensor component would be as this.
here the pressure-strain term, capital PI_ij given by this, so for this term we have
six new unknowns. and for the diffusion term Cijk, we
introduce 10 unknowns, and for the dissipation term, EPSILON_ij, we

English: 
it is a symmetric tensor, thus it has six unknowns in the Reynolds stress
tensor.As we have shown in the previous
slides, the transport equation for
Reynolds stress tensor component would be as this.
here the pressure-strain term, capital PI_ij given by this, so for this term we have
six new unknowns. and for the diffusion term Cijk, we
introduce 10 unknowns, and for the
dissipation term, EPSILON_ij, we

English: 
introduce six unknowns. So therefore, to solve the Reynolds
stress component, we formulated the transport equation, but we have also
introduced 22 new unknowns, and unfortunately, we have no more physical
laws for building more equations for these unknowns. More details can be found
in the Wilcox's book, 'Turbulence Modelling for CFD'.
For the turbulent flow, we can define the turbulence kinetic energy, k, as this.
and its relation with the Reynolds stress tensor component TAU_ii as this,
therefore if we set the indices i= j, in the transport equation

English: 
introduce six unknowns. So therefore to solve the Reynolds
stress component, we formulated the transport equation, but we have also
introduced 22 new unknowns, and unfortunately, we have no more physical
laws for building more equations for these unknowns. More details can be found
in the Wilcox's book,  'Turbulence Modelling for CFD'.
For the turbulent flow, we can define the turbulence kinetic energy, k, as this.
and its relation with the Reynolds stress tensor component TAU_ii as this,
therefore if we set the indices i= j, in the transport equation

English: 
for the Reynolds stress component, we can obtain the transport equation for the
kinetic energy k as this.
to solve the energy transport equation
above as we see in the conventional one- and two-equation turbulence models,
we must model some terms on the right hand side of the equation here,  so the
dissipation rate must be solved with the additional equation in the two-equation
turbulence models, and we'll see the details in the next slides in this talk.
TAU_ij must be linked with the turbulent kinetic energy, as well as with
the mean flow, for instance, using the Boussinesq's eddy viscosity assumption,

English: 
for the Reynolds stress component, we can obtain the transport equation for the
kinetic energy k as this.
to solve the energy transport equation
above as we see in the conventional one- and two-equation turbulence models,
we must model some terms on the right hand side of the equation here,  so the
dissipation rate must be solved with the additional equation in the two-equation
turbulence models, and we'll see the details in the next slides in this talk.
TAU_ij must be linked with the turbulent kinetic energy, as well as with
the mean flow, for instance, using the Boussinesq's eddy viscosity assumption,

English: 
given as this, and the term in the red ellipse must be approximated and a most used
approximation in the turbulence modeling is the simple expression, given as this.
so put these together, the transport
equation for the kinetic energy k in the standard k-Epsilon model is given as this.
to understand the transport equation, we can write the equation in this form,
following the reference by L Davidson, the online book,
'Fluid mechanics, turbulent flows and turbulence modeling', we generally
have 4 different regions in the transport equation  for the turbulent

English: 
given as this, and the term in the red ellipse must be approximated and a most used
approximation in the turbulence modeling is the simple expression, given as this.
so put these together, the transport
equation for the kinetic energy k in the standard k-Epsilon model is given as this.
to understand the transport equation, we can write the equation in this form,
following the reference by L Davidson, the online book,
'Fluid mechanics, turbulent flows and turbulence modeling', we generally
have 4 different regions in the transport equation  for the turbulent

English: 
kinetic energy, termed using Roman number 1-4, and this transport
equation corresponds to the energy cascade process, as assumed by Kolmogorov
as shown in this figure. In the region I, it can be
derived from the Reynolds transport Theorem for the quantity of the
kinetic energy, including the local change of the turbulence energy, and the
connection of the turbulent energy. The region II is the production term,
meaning the product of the Reynolds stress tensor and the velocity gradient.
in this region the large energy-carrying eddies interact with the mean flow and
extract energy from the mean flow, and transfer energy to the eddies

English: 
kinetic energy, termed using Roman number 1-4, and this transport
equation corresponds to the energy cascade process, as assumed by Kolmogorov
as shown in this figure. In the region I, it can be
derived from the Reynolds transport Theorem for the quantity of the
kinetic energy, including the local change of the turbulence energy, and the
connection of the turbulent energy. The region II is the production term,
meaning the product of the Reynolds stress tensor and the velocity gradient.
in this region the large energy-carrying eddies interact with the mean flow and
extract energy from the mean flow, and transfer energy to the eddies

English: 
of slightly smaller scales. Region III is the diffusion term,
this region represents the mid-range and isotropic turbulence, the energy
transport or diffusion in this cascade process, energy flows from the large
eddies to smaller eddies. Here an unusual fact is the inclusion of the
viscous term in this diffusion term, the main reason for this combination
might be that the other two terms can be simply approximated as the eddy viscous term
as we see in the turbulence models, the item 3 in the previous slide.  Region
IV is the dissipation term, Epsilon. This term is responsible for the transformation

English: 
of slightly smaller length scales. Region III is the diffusion term,
this region represents the mid-range and isotropic turbulence, the energy
transport or diffusion in this cascade
process, energy flows from the large
eddies to smaller eddies. Here an unusual fact is the inclusion of the
viscous term in this diffusion term, the main reason for this combination
might be that the other two terms can be simply approximated as the eddy viscous term
as we see in the turbulence models, the item 3 in the previous slide.  Region
IV is the dissipation term, Epsilon. This term is responsible for the transformation

English: 
of kinetic energy from the small eddies to thermal energy, and this term itself
is positive, meaning the kinetic energy is transferred to the internal energy,
thus the turbulent energy is dissipated into heat.
now in this slide we can use the navier-Stokes operator to construct an equation
for the dissipation rate, Epsilon, defined as this. So the constructed equation is
given as this, here L(qi) is the navier-stokes operator for the component Qi,
so substitute the N-S operator into the
constructed equation, we have the expression as this, here we have five

English: 
of kinetic energy from the small eddies to thermal energy, and this term itself
is positive, meaning the kinetic energy is transferred to the internal energy,
thus the turbulent energy is dissipated into heat.
now in this slide we can use the navier-Stokes operator to construct an equation
for the dissipation rate, Epsilon, defined as this. So the constructed equation is
given as this, here L(qi) is the navier-stokes operator for the component qi,
so substitute the N-S operator into the
constructed equation, we have the expression as this, here we have five

English: 
terms on the right-hand side of the expression, and we will examine all
these terms in details in the next two slides.
for the term one we can simply time the terms out, and take the average velocity
out of the averaging operation, so we have the expression as this, and drop
the term with the averaging of the single fluctuating velocity, we have this,
and finally we have the expression as this, for the first term, here Epsilon is
defined as this.
the second term is much more complicated, we can time the
relevant terms together, and we take the terms of the average double velocities out

English: 
terms on the right-hand side of the expression, and we will examine all
these terms in details in the next two
slides.
for the term one we can simply time the terms out, and take the average velocity
out of the averaging operation, so we have the expression as this, and drop
the term with the averaging of the single fluctuating velocity, we have this,
and finally we have the expression as this, for the first term, here Epsilon is
defined as this.
the second term is much more complicated, we can time the
relevant terms together, and we take the terms of the average double velocities out

English: 
of the (averaging) operation, so we have the expression as this.
this term can be dropped because of the averaging of the single fluctuating velocity.
so using the continuity equation for the mean flow velocity
capital U and the fluctuating velocity u',
these two equations, and we can obtain the expression as this.
so further using the rule of calculus for each item in this expression, we get
two terms in this expression, so using the expression Epsilon_ij, we have the
expression of this, and further we can write the expression as this

English: 
of the (averaging) operation, so we have the expression as this.
this term can be dropped because of the averaging of the single fluctuating velocity.
so using the continuity equation for the mean flow velocity
capital U and the fluctuating velocity u',
these two equations, and we can obtain the expression as this.
so further using the rule of calculus for each item in this expression, we get
two terms in this expression, so using the expression Epsilon_ij, we have the
expression of this, and further we can write the expression as this

English: 
Here actually it is the epsilon, so we have the final expression for the second term as this
for the 3rd term, the body force is independent of the turbulence of the
flow, thus the body force can be taken out of the averaging operation,
therefore the 3rd term is zero. The 4th term can be timed out for this averaged
and the fluctuating pressure, so we have the expression as this, here the averaged
pressure is a constant, with regarding to the averaging operation, so
from this expression we can see the first term here would be 0, therefore

English: 
Here actually it is the epsilon, so we have the final expression for the second term as this
for the 3rd term, the body force is independent of the turbulence of the
flow, thus the body force can be taken out of the averaging operation,
therefore the 3rd term is zero. The 4th term can be timed out for this averaged
and the fluctuating pressure, so we have the expression as this, here the averaged
pressure is a constant, with regarding to the averaging operation, so
from this expression we can see the first term here would be 0, therefore

English: 
we have the expression for the 4th term as this, and we can exchange the
order of the derivations here, with regard to xi and with regard
to xj. if we look at this expression, we can see when we change the order of the
derivation, and we have this, and this term would be 0, therefore, we have the
final expression for the 4th term as this.
for the fifth term, in a similar way we can separate the mean velocity and the
fluctuating velocity, and take the mean velocity out of the averaging
operation, we have the expression as this, and dropped the first term,
we have the expression as this, in here we change the order of the

English: 
we have the expression for the 4th term as this, and we can exchange the
order of the derivations here, 
with regard to xi and with regard
to xj. if we look at this expression, we can see when we change the order of the
derivation, and we have this, and this term would be 0, therefore, we have the
final expression for the 4th term as this.
for the fifth term, in a similar way we can separate the mean velocity and the
fluctuating velocity, and take the mean velocity out of the averaging
operation, we have the expression as this, and dropped the first term,
we have the expression as this, in here we change the order of the

English: 
derivations with regard to xk and xj, and the further exchange the order
inside of the square brackets, we have the expression as this.
so if we consider an expression as this, so we can separate this term into
two terms as this, and the expression for
the fifth term would be this, and use
the definition for the Epsilon, we have the expression as this for the fifth term.
here we put all these terms together we have the transport equation for the
dissipation rate, Epsilon, and this can be rewritten as in this form, we put these

English: 
derivations with regard to xk and xj, and the further exchange the order
inside of the square brackets, we have the expression as this.
so if we consider an expression as this, so we can separate this term into
two terms as this, and the expression for the 5th term would be this, and use
the definition for the Epsilon, we have the expression as this for the fifth term.
here we put all these terms together we have the transport equation for the
dissipation rate, Epsilon, and this can be rewritten as in this form, we put these

English: 
two terms on the left hand side and all other terms on the right hand side.
the is the transport equation for the dissipation rate, Epsilon, and Epsilon
and Epsilon_ij are defined in this two equations.
based on the book of Wilcox, 'Turbulence modeling for CFD', we normally
have an isotropic relation for the Epsilon_ij and epsilon, this is a very
simple relation and it is generally accepted because of the fact of the
dissipations happening at the small scales, for which the turbulence flow
would be more isotropic. and an anisotropic expression is given by
this, with the correction term, here fs is the low Reynolds number factor

English: 
two terms on the left hand side and all other terms on the right hand side.
the is the transport equation for the dissipation rate, Epsilon, and Epsilon
and Epsilon_ij are defined in this two equations.
based on the book of Wilcox, 'Turbulence modeling for CFD', we normally
have an isotropic relation for the Epsilon_ij and epsilon, this is a very
simple relation and it is generally accepted because of the fact of the
dissipations happening at the small scales, for which the turbulence flow
would be more isotropic. and an anisotropic expression is given by
this, with the correction term, here fs is the low Reynolds number factor

English: 
and b_ij is the Reynolds stress anisotropy  tensor. There is a difficult
problem for solving the transport equation for the dissipation rate, we must
model all these terms in the big ellipse.
in this slide, a presentation and the discussion would be presented for the
Boussinesq's assumption. Based on Schmitt, in 1877 Boussinesq
reasoned that the navier-stokes equation would be still valid for the mean flow
in the turbulent flows, but the fluid viscosity is significantly
increased due to the flow turbulence. Thus he obtained the stresses of the flow

English: 
and b_ij is the Reynolds stress anisotropy  tensor. There is a difficult
problem for solving the transport equation for the dissipation rate, we must
model all these terms in the big ellipse.
in this slide, a presentation and the discussion would be presented for the
Boussinesq's assumption. Based on Schmitt, in 1877 Boussinesq
reasoned that the navier-stokes equation would be still valid for the mean flow
in the turbulent flows, but the fluid viscosity is significantly
increased due to the flow turbulence. Thus he obtained the stresses of the flow

English: 
given by this T_ij link the flow velocity gradient of the mean velocity.
here, NU_B is the imaginary fluid
viscosity, and it is equivalent to today's total viscosity: NU_B equals (to) NU
plus NU_t, the fluid viscosity plus turbulent eddy viscosity, as we see in
the turbulence modeling. and Boussinesq obtained the relation: k equals (to) 1.5
(times) pressure. it should be noted Boussinesq actually made this assumption
18 years earlier than Reynolds who proposed the Reynolds stress tensor in 1895.
in today's populace modeling, for instance, the most applied two-equation
turbulence model, it is always assumed that the Boussinesq's hypothesis is

English: 
given by this T_ij link the flow velocity gradient of the mean velocity.
here, NU_B is the imaginary fluid
viscosity, and it is equivalent to today's total viscosity: NU_B equals (to) NU
plus NU_t, the fluid viscosity plus turbulent eddy viscosity, as we see in
the turbulence modeling. and Boussinesq obtained the relation: k equals (to) 1.5
(times) pressure. it should be noted Boussinesq actually made this assumption
18 years earlier than Reynolds  who proposed the Reynolds stress tensor in 1895.
in today's populace modeling, for instance, the most applied two-equation
turbulence model, it is always assumed that the Boussinesq's hypothesis is

English: 
linked the Reynolds stress tensor, with an expression as this, here TAU_ij is
the Reynolds stress tensor, NU_t is the turbulent eddy viscosity, capital Ui
and capital Uj are the mean velocities, and k here is the turbulent
kinetic energy. if we set the indices, i
equalling to j, so we have the expression as this, and for the incompressible
flows, these two terms will be 0. therefore we have the expression as this
we can see TAU_ii would be equal to minus 2k, and here k is the turbulent energy
given by this.

English: 
linked the Reynolds stress tensor, with an expression as this, here TAU_ij is
the Reynolds stress tensor, NU_t is the turbulent eddy viscosity, capital Ui
and capital Uj are the mean velocities, and k here is the turbulent
kinetic energy. if we set the indices, i
equalling to j, so we have the expression as this, and for the incompressible
flows, these two terms will be 0. therefore we have the expression as this
we can see TAU_ii would be equal to minus 2k, and here k is the turbulent energy
given by this.
