Hi friends
Would anybody mind to explain me what are ‘divdiscretization’ and ‘gradPdiscretization’ in Navier-Stokes equations? How do they appear and what are their roles?
Regards,
Navier-Stokes equations
Re: Navier-Stokes equations
Hi,
These options relate to how the weak formulations of the viscous and
the pressure gradient terms are written.
In the case of "DivDiscretization" the viscous term is integrated by
parts by using the equation
-(2 div D(u),v) = (grad u, grad v) + (div u, div v) + boundary terms.
Otherwise one uses
-(2 div D(u),v) = 2 (D(u), grad v) + boundary terms.
Here (x,y) is the L2 inner product of x and y, and D(u) is the
symmetric part of grad u. If the case of "GradPDiscretization"
the pressure gradient term is not integrated by parts so that
the pressure gradient term is written as (grad p, v). Otherwise
integration by parts is applied so that
(grad p, v) = -(p, div v) + boundary terms.
These options have naturally impact on what natural boundary conditions
are obtained. The "DivDiscretization" option generally leads to prescribing
pseudo traction forces, i.e. the natural bc terms do not represent true
surface forces on the boundary.
-mika
These options relate to how the weak formulations of the viscous and
the pressure gradient terms are written.
In the case of "DivDiscretization" the viscous term is integrated by
parts by using the equation
-(2 div D(u),v) = (grad u, grad v) + (div u, div v) + boundary terms.
Otherwise one uses
-(2 div D(u),v) = 2 (D(u), grad v) + boundary terms.
Here (x,y) is the L2 inner product of x and y, and D(u) is the
symmetric part of grad u. If the case of "GradPDiscretization"
the pressure gradient term is not integrated by parts so that
the pressure gradient term is written as (grad p, v). Otherwise
integration by parts is applied so that
(grad p, v) = -(p, div v) + boundary terms.
These options have naturally impact on what natural boundary conditions
are obtained. The "DivDiscretization" option generally leads to prescribing
pseudo traction forces, i.e. the natural bc terms do not represent true
surface forces on the boundary.
-mika
-
kimsongoik
- Posts: 35
- Joined: 14 Oct 2009, 18:55
Re: Navier-Stokes equations
Dear Mika
First I want to appreciate you and the Elmer group for answering all questions kindly and completely. Second I want to ask one more question: where is it recommended to use "GradPDiscretization" and "DivDiscretization" and where not?
Regards,
First I want to appreciate you and the Elmer group for answering all questions kindly and completely. Second I want to ask one more question: where is it recommended to use "GradPDiscretization" and "DivDiscretization" and where not?
Regards,
Re: Navier-Stokes equations
Hi,
It is not easy to say what is the best way of formulating the NS system.
Here are however some possibly noteworthy points:
- If the intention is to prescribe the true surface force on a part of the
boundary, the default formulation for handling the diffusive term is the
only consistent choice. In the case of flow problems surface forces are not
usually known, so this argument may appear to be somewhat vague.
- In the case of incompressible flow using the "DivDiscretization" option
may lead to more stable discretization when the Reynolds number increases.
That is, including the term (div u,div v) has a stabilizing effect.
- I guess that using the "GradPDiscretization" principally has an impact on
how the boundary condition on the outflow region is treated. My experience
with using this option is so limited that I cannot say what is the relative
merit of this formulation. Maybe someone else has more experience on
using this strategy. Note that these formulations generally lead to
different treatment of boundary conditions on regions where the velocity is
not specified.
To sum up, I see the default formulation to be physically sound in general.
In the case of moderate Reynolds number, the "DivDiscretization" option can
have an advantage due to improved stability. Finally, considerations of
what is seen to be the "correct" way of formulating the boundary condition
on regions where the velocity is not specified may also be an argument for
preferring a certain formulation.
-mika
It is not easy to say what is the best way of formulating the NS system.
Here are however some possibly noteworthy points:
- If the intention is to prescribe the true surface force on a part of the
boundary, the default formulation for handling the diffusive term is the
only consistent choice. In the case of flow problems surface forces are not
usually known, so this argument may appear to be somewhat vague.
- In the case of incompressible flow using the "DivDiscretization" option
may lead to more stable discretization when the Reynolds number increases.
That is, including the term (div u,div v) has a stabilizing effect.
- I guess that using the "GradPDiscretization" principally has an impact on
how the boundary condition on the outflow region is treated. My experience
with using this option is so limited that I cannot say what is the relative
merit of this formulation. Maybe someone else has more experience on
using this strategy. Note that these formulations generally lead to
different treatment of boundary conditions on regions where the velocity is
not specified.
To sum up, I see the default formulation to be physically sound in general.
In the case of moderate Reynolds number, the "DivDiscretization" option can
have an advantage due to improved stability. Finally, considerations of
what is seen to be the "correct" way of formulating the boundary condition
on regions where the velocity is not specified may also be an argument for
preferring a certain formulation.
-mika
Re: Navier-Stokes equations
Hi,
a few comments on the 'grad p' discretization:
In addition to pressure not being integrated by parts, the (incompressible)
continuity equations IS integrated by parts. These changes have effect
on the boundary conditions, as opposed to the default stuff:
- pressure Dirichlet boundary conditions can be used (and pressure level
must be fixed by such a condition)
- mass flux is available as a natural boundary condition
The default settings should be the way to go in most cases, though.
Regards, Juha
a few comments on the 'grad p' discretization:
In addition to pressure not being integrated by parts, the (incompressible)
continuity equations IS integrated by parts. These changes have effect
on the boundary conditions, as opposed to the default stuff:
- pressure Dirichlet boundary conditions can be used (and pressure level
must be fixed by such a condition)
- mass flux is available as a natural boundary condition
The default settings should be the way to go in most cases, though.
Regards, Juha
-
kimsongoik
- Posts: 35
- Joined: 14 Oct 2009, 18:55
Re: Navier-Stokes equations
Hi
What are the default values for ‘divdiscretization’ and ‘gradPdiscretization’ in Navier-Stokes solver for incompressible flows? Are they "True" or "False"?
Regards,
What are the default values for ‘divdiscretization’ and ‘gradPdiscretization’ in Navier-Stokes solver for incompressible flows? Are they "True" or "False"?
Regards,
Re: Navier-Stokes equations
HI
both "False" by default,
Juha
both "False" by default,
Juha