## Re: [Libmesh-users] Navier-Stoke equation in ex13

 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: John Peterson - 2008-01-24 17:23:52 ```You mean how do you linearize it? Newton's method and iterate within each timestep. -J li pan writes: > hi Roy, > thanx for the explaination. But how did you solve ((u > * grad)u, v)_Omega? It's a square term. I heard there > are some other methods, streamline, least square FEM > ... I would like to hear your comments. > > pan > > > > --- Roy Stogner wrote: > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > I've worked with Newton type flow equation. To > > make it > > > sure, I would like to know the exact expression of > > > equation in ex13. Can you tell me? > > > > The system of equations with variables (u,p) is : > > (partial u)/(partial t) = - (u * grad)u - div(sigma) > > div(u) = 0 > > > > Where sigma is the stress tensor (normalized to have > > unit viscosity) > > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > > > Then the weak form we use in ex13 and ex18, with > > test functions (v,q) > > is: > > ((partial u)/(partial t), v)_Omega = - ((u * grad)u, > > v)_Omega > > + (sigma, grad v)_Omega + (sigma * n, v)_dOmega > > (div(u), q) = 0 > > > > In ex13 we use Dirichlet boundaries everywhere, so v > > = 0 on the > > boundary and we drop the dOmega term. Otherwise, > > you'd substitute > > into that term the natural boundary condition: > > sigma * n = 0 > > > > which is actually what David wanted in the first > > place. ;-) > > > > You know, we probably ought to have something like > > this in the > > comments heading examples 13 and 18. "The > > Navier-Stokes equations" is > > definitive enough, but the fact that we integrate > > all of sigma > > (including the pressure term) by parts isn't set in > > stone. > > --- > > Roy > > > > > ------------------------------------------------------------------------- > > This SF.net email is sponsored by: Microsoft > > Defy all challenges. Microsoft(R) Visual Studio > > 2008. > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > _______________________________________________ > > Libmesh-users mailing list > > Libmesh-users@... > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > ____________________________________________________________________________________ > Be a better friend, newshound, and > know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > ------------------------------------------------------------------------- > This SF.net email is sponsored by: Microsoft > Defy all challenges. Microsoft(R) Visual Studio 2008. > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > _______________________________________________ > Libmesh-users mailing list > Libmesh-users@... > https://lists.sourceforge.net/lists/listinfo/libmesh-users ```

 [Libmesh-users] Navier-Stoke equation in ex13 From: li pan - 2008-01-24 10:38:43 ```Dear developers, I've worked with Newton type flow equation. To make it sure, I would like to know the exact expression of equation in ex13. Can you tell me? thanx pan ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: Roy Stogner - 2008-01-24 16:23:41 ```On Thu, 24 Jan 2008, li pan wrote: > I've worked with Newton type flow equation. To make it > sure, I would like to know the exact expression of > equation in ex13. Can you tell me? The system of equations with variables (u,p) is : (partial u)/(partial t) = - (u * grad)u - div(sigma) div(u) = 0 Where sigma is the stress tensor (normalized to have unit viscosity) sigma = ((grad(u) + transpose(grad(u)))/2 - pI) Then the weak form we use in ex13 and ex18, with test functions (v,q) is: ((partial u)/(partial t), v)_Omega = - ((u * grad)u, v)_Omega + (sigma, grad v)_Omega + (sigma * n, v)_dOmega (div(u), q) = 0 In ex13 we use Dirichlet boundaries everywhere, so v = 0 on the boundary and we drop the dOmega term. Otherwise, you'd substitute into that term the natural boundary condition: sigma * n = 0 which is actually what David wanted in the first place. ;-) You know, we probably ought to have something like this in the comments heading examples 13 and 18. "The Navier-Stokes equations" is definitive enough, but the fact that we integrate all of sigma (including the pressure term) by parts isn't set in stone. --- Roy ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: John Peterson - 2008-01-24 16:44:38 ```Roy Stogner writes: > > On Thu, 24 Jan 2008, li pan wrote: > > > I've worked with Newton type flow equation. To make it > > sure, I would like to know the exact expression of > > equation in ex13. Can you tell me? > > The system of equations with variables (u,p) is : > (partial u)/(partial t) = - (u * grad)u - div(sigma) > div(u) = 0 > > Where sigma is the stress tensor (normalized to have unit viscosity) > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) I don't think it should have a 1/2. The stress tensor is typically defined as (see e.g. Panton's fluid mechanics book, http://www.cs.otago.ac.nz/postgrads/alexis/FluidMech/node10.html, http://en.wikipedia.org/wiki/Newtonian_fluid ): sigma = -pI + 2*mu*epsilon(u) where epsilon(u) = (1/2)*(grad(u) + grad(u)^t) is known as the "symmetric part" of the velocity gradient. http://mathworld.wolfram.com/SymmetricMatrix.html Now I think this is confusing because the 2 and the 1/2 always cancel, but whatever, this is how people define things. I agree that this email should be in the introductory comments of ex13, a lot of people have asked the same question. -J ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: li pan - 2008-01-24 17:15:49 ```hi Roy, thanx for the explaination. But how did you solve ((u * grad)u, v)_Omega? It's a square term. I heard there are some other methods, streamline, least square FEM ... I would like to hear your comments. pan --- Roy Stogner wrote: > > On Thu, 24 Jan 2008, li pan wrote: > > > I've worked with Newton type flow equation. To > make it > > sure, I would like to know the exact expression of > > equation in ex13. Can you tell me? > > The system of equations with variables (u,p) is : > (partial u)/(partial t) = - (u * grad)u - div(sigma) > div(u) = 0 > > Where sigma is the stress tensor (normalized to have > unit viscosity) > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > Then the weak form we use in ex13 and ex18, with > test functions (v,q) > is: > ((partial u)/(partial t), v)_Omega = - ((u * grad)u, > v)_Omega > + (sigma, grad v)_Omega + (sigma * n, v)_dOmega > (div(u), q) = 0 > > In ex13 we use Dirichlet boundaries everywhere, so v > = 0 on the > boundary and we drop the dOmega term. Otherwise, > you'd substitute > into that term the natural boundary condition: > sigma * n = 0 > > which is actually what David wanted in the first > place. ;-) > > You know, we probably ought to have something like > this in the > comments heading examples 13 and 18. "The > Navier-Stokes equations" is > definitive enough, but the fact that we integrate > all of sigma > (including the pressure term) by parts isn't set in > stone. > --- > Roy > > ------------------------------------------------------------------------- > This SF.net email is sponsored by: Microsoft > Defy all challenges. Microsoft(R) Visual Studio > 2008. > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > _______________________________________________ > Libmesh-users mailing list > Libmesh-users@... > https://lists.sourceforge.net/lists/listinfo/libmesh-users > ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: John Peterson - 2008-01-24 17:23:52 ```You mean how do you linearize it? Newton's method and iterate within each timestep. -J li pan writes: > hi Roy, > thanx for the explaination. But how did you solve ((u > * grad)u, v)_Omega? It's a square term. I heard there > are some other methods, streamline, least square FEM > ... I would like to hear your comments. > > pan > > > > --- Roy Stogner wrote: > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > I've worked with Newton type flow equation. To > > make it > > > sure, I would like to know the exact expression of > > > equation in ex13. Can you tell me? > > > > The system of equations with variables (u,p) is : > > (partial u)/(partial t) = - (u * grad)u - div(sigma) > > div(u) = 0 > > > > Where sigma is the stress tensor (normalized to have > > unit viscosity) > > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > > > Then the weak form we use in ex13 and ex18, with > > test functions (v,q) > > is: > > ((partial u)/(partial t), v)_Omega = - ((u * grad)u, > > v)_Omega > > + (sigma, grad v)_Omega + (sigma * n, v)_dOmega > > (div(u), q) = 0 > > > > In ex13 we use Dirichlet boundaries everywhere, so v > > = 0 on the > > boundary and we drop the dOmega term. Otherwise, > > you'd substitute > > into that term the natural boundary condition: > > sigma * n = 0 > > > > which is actually what David wanted in the first > > place. ;-) > > > > You know, we probably ought to have something like > > this in the > > comments heading examples 13 and 18. "The > > Navier-Stokes equations" is > > definitive enough, but the fact that we integrate > > all of sigma > > (including the pressure term) by parts isn't set in > > stone. > > --- > > Roy > > > > > ------------------------------------------------------------------------- > > This SF.net email is sponsored by: Microsoft > > Defy all challenges. Microsoft(R) Visual Studio > > 2008. > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > _______________________________________________ > > Libmesh-users mailing list > > Libmesh-users@... > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > ____________________________________________________________________________________ > Be a better friend, newshound, and > know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > ------------------------------------------------------------------------- > This SF.net email is sponsored by: Microsoft > Defy all challenges. Microsoft(R) Visual Studio 2008. > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > _______________________________________________ > Libmesh-users mailing list > Libmesh-users@... > https://lists.sourceforge.net/lists/listinfo/libmesh-users ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: li pan - 2008-01-24 17:34:18 ```thanx John, there is another point. I read the Navier-Stokes equation in wiki (http://en.wikipedia.org/wiki/Navier-Stokes_equations). There is a grad(p) term. But it doesn't appear in the equaiton of ex13. pan --- John Peterson wrote: > You mean how do you linearize it? Newton's method > and iterate > within each timestep. > -J > > > li pan writes: > > hi Roy, > > thanx for the explaination. But how did you solve > ((u > > * grad)u, v)_Omega? It's a square term. I heard > there > > are some other methods, streamline, least square > FEM > > ... I would like to hear your comments. > > > > pan > > > > > > > > --- Roy Stogner wrote: > > > > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > > > I've worked with Newton type flow equation. > To > > > make it > > > > sure, I would like to know the exact > expression of > > > > equation in ex13. Can you tell me? > > > > > > The system of equations with variables (u,p) is > : > > > (partial u)/(partial t) = - (u * grad)u - > div(sigma) > > > div(u) = 0 > > > > > > Where sigma is the stress tensor (normalized to > have > > > unit viscosity) > > > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > > > > > Then the weak form we use in ex13 and ex18, > with > > > test functions (v,q) > > > is: > > > ((partial u)/(partial t), v)_Omega = - ((u * > grad)u, > > > v)_Omega > > > + (sigma, grad v)_Omega + (sigma * n, > v)_dOmega > > > (div(u), q) = 0 > > > > > > In ex13 we use Dirichlet boundaries everywhere, > so v > > > = 0 on the > > > boundary and we drop the dOmega term. > Otherwise, > > > you'd substitute > > > into that term the natural boundary condition: > > > sigma * n = 0 > > > > > > which is actually what David wanted in the > first > > > place. ;-) > > > > > > You know, we probably ought to have something > like > > > this in the > > > comments heading examples 13 and 18. "The > > > Navier-Stokes equations" is > > > definitive enough, but the fact that we > integrate > > > all of sigma > > > (including the pressure term) by parts isn't > set in > > > stone. > > > --- > > > Roy > > > > > > > > > ------------------------------------------------------------------------- > > > This SF.net email is sponsored by: Microsoft > > > Defy all challenges. Microsoft(R) Visual Studio > > > 2008. > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > _______________________________________________ > > > Libmesh-users mailing list > > > Libmesh-users@... > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > > > > > > ____________________________________________________________________________________ > > Be a better friend, newshound, and > > know-it-all with Yahoo! Mobile. Try it now. > http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > > > > > > ------------------------------------------------------------------------- > > This SF.net email is sponsored by: Microsoft > > Defy all challenges. Microsoft(R) Visual Studio > 2008. > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > _______________________________________________ > > Libmesh-users mailing list > > Libmesh-users@... > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > ____________________________________________________________________________________ Looking for last minute shopping deals? Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: John Peterson - 2008-01-24 17:36:07 ```This is what Roy discussed. The divergence theorem has been applied to that term. -J li pan writes: > thanx John, > there is another point. I read the Navier-Stokes > equation in wiki > (http://en.wikipedia.org/wiki/Navier-Stokes_equations). > There is a grad(p) term. But it doesn't appear in the > equaiton of ex13. > > pan > > > --- John Peterson > wrote: > > > You mean how do you linearize it? Newton's method > > and iterate > > within each timestep. > > -J > > > > > > li pan writes: > > > hi Roy, > > > thanx for the explaination. But how did you solve > > ((u > > > * grad)u, v)_Omega? It's a square term. I heard > > there > > > are some other methods, streamline, least square > > FEM > > > ... I would like to hear your comments. > > > > > > pan > > > > > > > > > > > > --- Roy Stogner wrote: > > > > > > > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > > > > > I've worked with Newton type flow equation. > > To > > > > make it > > > > > sure, I would like to know the exact > > expression of > > > > > equation in ex13. Can you tell me? > > > > > > > > The system of equations with variables (u,p) is > > : > > > > (partial u)/(partial t) = - (u * grad)u - > > div(sigma) > > > > div(u) = 0 > > > > > > > > Where sigma is the stress tensor (normalized to > > have > > > > unit viscosity) > > > > sigma = ((grad(u) + transpose(grad(u)))/2 - pI) > > > > > > > > Then the weak form we use in ex13 and ex18, > > with > > > > test functions (v,q) > > > > is: > > > > ((partial u)/(partial t), v)_Omega = - ((u * > > grad)u, > > > > v)_Omega > > > > + (sigma, grad v)_Omega + (sigma * n, > > v)_dOmega > > > > (div(u), q) = 0 > > > > > > > > In ex13 we use Dirichlet boundaries everywhere, > > so v > > > > = 0 on the > > > > boundary and we drop the dOmega term. > > Otherwise, > > > > you'd substitute > > > > into that term the natural boundary condition: > > > > sigma * n = 0 > > > > > > > > which is actually what David wanted in the > > first > > > > place. ;-) > > > > > > > > You know, we probably ought to have something > > like > > > > this in the > > > > comments heading examples 13 and 18. "The > > > > Navier-Stokes equations" is > > > > definitive enough, but the fact that we > > integrate > > > > all of sigma > > > > (including the pressure term) by parts isn't > > set in > > > > stone. > > > > --- > > > > Roy > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > > This SF.net email is sponsored by: Microsoft > > > > Defy all challenges. Microsoft(R) Visual Studio > > > > 2008. > > > > > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > > _______________________________________________ > > > > Libmesh-users mailing list > > > > Libmesh-users@... > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > > > > > > > > > > > > > ____________________________________________________________________________________ > > > Be a better friend, newshound, and > > > know-it-all with Yahoo! Mobile. Try it now. > > > http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > This SF.net email is sponsored by: Microsoft > > > Defy all challenges. Microsoft(R) Visual Studio > > 2008. > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > _______________________________________________ > > > Libmesh-users mailing list > > > Libmesh-users@... > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > ____________________________________________________________________________________ > Looking for last minute shopping deals? > Find them fast with Yahoo! Search. http://tools.search.yahoo.com/newsearch/category.php?category=shopping ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: li pan - 2008-01-24 17:44:22 ```ok, I see it. thanx a lot pan --- John Peterson wrote: > This is what Roy discussed. The divergence theorem > has > been applied to that term. > > -J > > li pan writes: > > thanx John, > > there is another point. I read the Navier-Stokes > > equation in wiki > > > (http://en.wikipedia.org/wiki/Navier-Stokes_equations). > > There is a grad(p) term. But it doesn't appear in > the > > equaiton of ex13. > > > > pan > > > > > > --- John Peterson > > wrote: > > > > > You mean how do you linearize it? Newton's > method > > > and iterate > > > within each timestep. > > > -J > > > > > > > > > li pan writes: > > > > hi Roy, > > > > thanx for the explaination. But how did you > solve > > > ((u > > > > * grad)u, v)_Omega? It's a square term. I > heard > > > there > > > > are some other methods, streamline, least > square > > > FEM > > > > ... I would like to hear your comments. > > > > > > > > pan > > > > > > > > > > > > > > > > --- Roy Stogner > wrote: > > > > > > > > > > > > > > On Thu, 24 Jan 2008, li pan wrote: > > > > > > > > > > > I've worked with Newton type flow > equation. > > > To > > > > > make it > > > > > > sure, I would like to know the exact > > > expression of > > > > > > equation in ex13. Can you tell me? > > > > > > > > > > The system of equations with variables > (u,p) is > > > : > > > > > (partial u)/(partial t) = - (u * grad)u - > > > div(sigma) > > > > > div(u) = 0 > > > > > > > > > > Where sigma is the stress tensor > (normalized to > > > have > > > > > unit viscosity) > > > > > sigma = ((grad(u) + transpose(grad(u)))/2 > - pI) > > > > > > > > > > Then the weak form we use in ex13 and > ex18, > > > with > > > > > test functions (v,q) > > > > > is: > > > > > ((partial u)/(partial t), v)_Omega = - ((u > * > > > grad)u, > > > > > v)_Omega > > > > > + (sigma, grad v)_Omega + (sigma * n, > > > v)_dOmega > > > > > (div(u), q) = 0 > > > > > > > > > > In ex13 we use Dirichlet boundaries > everywhere, > > > so v > > > > > = 0 on the > > > > > boundary and we drop the dOmega term. > > > Otherwise, > > > > > you'd substitute > > > > > into that term the natural boundary > condition: > > > > > sigma * n = 0 > > > > > > > > > > which is actually what David wanted in the > > > first > > > > > place. ;-) > > > > > > > > > > You know, we probably ought to have > something > > > like > > > > > this in the > > > > > comments heading examples 13 and 18. "The > > > > > Navier-Stokes equations" is > > > > > definitive enough, but the fact that we > > > integrate > > > > > all of sigma > > > > > (including the pressure term) by parts > isn't > > > set in > > > > > stone. > > > > > --- > > > > > Roy > > > > > > > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > > > This SF.net email is sponsored by: > Microsoft > > > > > Defy all challenges. Microsoft(R) Visual > Studio > > > > > 2008. > > > > > > > > > > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > > > > _______________________________________________ > > > > > Libmesh-users mailing list > > > > > Libmesh-users@... > > > > > > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > > > > > > > > > > > > > > > > > > > > > ____________________________________________________________________________________ > > > > Be a better friend, newshound, and > > > > know-it-all with Yahoo! Mobile. Try it now. > > > > > > > http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ > > > > > > > > > > > > > > > > > > > > > ------------------------------------------------------------------------- > > > > This SF.net email is sponsored by: Microsoft > > > > Defy all challenges. Microsoft(R) Visual > Studio > > > 2008. > > > > > > > > > > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > > > > > _______________________________________________ > > > > Libmesh-users mailing list > > > > Libmesh-users@... > > > > > > > > > > https://lists.sourceforge.net/lists/listinfo/libmesh-users > > > > > > > > > > > > ____________________________________________________________________________________ > > Looking for last minute shopping deals? > > Find them fast with Yahoo! Search. > http://tools.search.yahoo.com/newsearch/category.php?category=shopping > > ------------------------------------------------------------------------- > This SF.net email is sponsored by: Microsoft > Defy all challenges. Microsoft(R) Visual Studio > 2008. > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > _______________________________________________ > Libmesh-users mailing list > Libmesh-users@... > https://lists.sourceforge.net/lists/listinfo/libmesh-users > ____________________________________________________________________________________ Never miss a thing. Make Yahoo your home page. http://www.yahoo.com/r/hs ```
 Re: [Libmesh-users] Navier-Stoke equation in ex13 From: Benjamin Kirk - 2008-01-24 18:43:52 ```> hi Roy, > thanx for the explaination. But how did you solve ((u > * grad)u, v)_Omega? It's a square term. I heard there > are some other methods, streamline, least square FEM > ... I would like to hear your comments. By square I guess you mean asymmetric? The convection term is asymmetric and causes problems when convection is stronger than diffusion, e.g. Cell Reynolds numbers (u*h/nu) exceed 2. In that case you need to do something line the streamline/upwind Petrov-Galerkin method, Galerkin-Least squares, etc... We sidestep that problem in ex13 by keeping the cell Reynolds numbers low enough that it is not an issue. -Ben ```