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Old   August 13, 2014, 10:34
Arrow Time step in LES
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M. Nabi
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Hi all,

I am writing a code for LES. This code is based on a structured curvilinear grids with strong stretching beside the walls. In order to reduce the restrictions for time step, I used Adams-Bashforth for the convection term and Crank-Nicholson for the diffusion term.

\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \right] 
=
\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_i}{\partial x_j} \right) \right]
+
\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_j}{\partial x_i} \right) \right]


The first term in the right-hand side is ok, by using Crank-Nicholson method. But the second term yielded a strong restriction in the CFL number. I tried both Adams-Bashforth and Crank-Nicheolson for the second term, but both of them needed very small dt. In the case I remove \nu_t from the second term, it works fine, but including \nu_t makes strong restrictions in time step.

And my question: How can I remove this restriction from the second term? Is there any way?

Thanks
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Old   August 13, 2014, 11:21
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Quote:
Originally Posted by mnabi View Post
Hi all,

I am writing a code for LES. This code is based on a structured curvilinear grids with strong stretching beside the walls. In order to reduce the restrictions for time step, I used Adams-Bashforth for the convection term and Crank-Nicholson for the diffusion term.

\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \right] 
=
\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_i}{\partial x_j} \right) \right]
+
\frac{\partial}{\partial x_j} \left[  \left( \nu + \nu_t \right)   \left( \frac{\partial u_j}{\partial x_i} \right) \right]


The first term in the right-hand side is ok, by using Crank-Nicholson method. But the second term yielded a strong restriction in the CFL number. I tried both Adams-Bashforth and Crank-Nicheolson for the second term, but both of them needed very small dt. In the case I remove \nu_t from the second term, it works fine, but including \nu_t makes strong restrictions in time step.

And my question: How can I remove this restriction from the second term? Is there any way?

Thanks

I don't understand your procedure...the eddy viscosity at time n+1 is unknown and the system would be not linear...
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Old   August 13, 2014, 11:26
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Originally Posted by FMDenaro View Post
I don't understand your procedure...the eddy viscosity at time n+1 is unknown and the system would be not linear...
Right, it is unknown, but I iterate all momentum equations and the eddy viscosity (using Smagorinsky) until velocity field converges. Then use pressure correction equation, and correct the velocity field.
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Old   August 13, 2014, 11:41
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the terms in the RHS require (dt/2)* for the CN integration.
However, the restriction for the time step is very important, you need a small value
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Old   August 13, 2014, 11:52
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Originally Posted by FMDenaro View Post
the terms in the RHS require (dt/2)* for the CN integration.
However, the restriction for the time step is very important, you need a small value
Right, it is dt/2 after discretization. A half part is explicit, and the other half part is implicit. But CN method for second term of the diffusion doesn't reduce the restriction of the CFL number. The time step restriction is very strong because \nu_t is few orders larger than \nu, which makes the total viscosity large. The first term has no any problem as it is unconditionally stable by using CN.

In my case, the second term reduces the dt from 2e-3 to 1e-6

I wonder either there is a way to reduce the restriction for the second term. . I saw several papers use AB2-CN method with relatively large dt, but they didn't mention about the details.

Thank you
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Old   August 13, 2014, 12:47
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the LINEAR stability analysi of the CN discretization leads to two eigenvalues from which one has stability unconditionally for one but not for the other. That means the coupling with the all terms of the equations leads to a stability region (cfl,Reh) with a strong restriction in the dt at low Reh.
Considering that you have a non linear case, the constraint on the dt is clearly strong.

Actually, I am used to work by letting the CN scheme for the molecular viscous term and using explicit second order AB for the eddy viscosity term.
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Old   August 13, 2014, 12:54
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Quote:
Originally Posted by FMDenaro View Post
the LINEAR stability analysi of the CN discretization leads to two eigenvalues from which one has stability unconditionally for one but not for the other. That means the coupling with the all terms of the equations leads to a stability region (cfl,Reh) with a strong restriction in the dt at low Reh.
Considering that you have a non linear case, the constraint on the dt is clearly strong.

Actually, I am used to work by letting the CN scheme for the molecular viscous term and using explicit second order AB for the eddy viscosity term.
Thank you for reply.
But using AB for eddy viscosity term, does it reduce the CFL restriction? Do you use also AB for the viscosity of the first term, or only for the second term?
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Old   August 13, 2014, 13:38
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First, the eddy viscosity can become locally high and the time scale needs to be small. Then, using an explicit AB scheme for the SGS term you avoid to work with a non-linear system having coefficents in the SGS terms.
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Old   August 13, 2014, 21:57
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Quote:
Originally Posted by FMDenaro View Post
First, the eddy viscosity can become locally high and the time scale needs to be small. Then, using an explicit AB scheme for the SGS term you avoid to work with a non-linear system having coefficents in the SGS terms.
But how to avoid having coefficients in the SGS term? Even if it is explicit, still the eddy viscosity can be locally high from time step n, and it is included in SGS!!! Do you mean we have to linearize it by, for instance, Newton linearization?
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Old   August 14, 2014, 03:19
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just using the explicit AB discretization requires eddy viscosity only at tn and tn-1, no coefficents are therefore present in the algebric system
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Old   August 14, 2014, 03:58
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Originally Posted by FMDenaro View Post
just using the explicit AB discretization requires eddy viscosity only at tn and tn-1, no coefficents are therefore present in the algebric system
It is right, they will be known. But they will still restrict the CFL number as the value of eddy viscosity is large. Using explicit method such as AB2, lets the CFL number dependent on the size of cell quadratic, isn't it ?
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Old   August 14, 2014, 04:14
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Originally Posted by mnabi View Post
It is right, they will be known. But they will still restrict the CFL number as the value of eddy viscosity is large. Using explicit method such as AB2, lets the CFL number dependent on the size of cell quadratic, isn't it ?

the stability constraint is due to the combination of the type of time integration along with convective, diffusive and SGS terms.
Generallly, the CFL must be quite smaller than 1
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Old   August 14, 2014, 20:38
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Quote:
Originally Posted by FMDenaro View Post
the stability constraint is due to the combination of the type of time integration along with convective, diffusive and SGS terms.
Generallly, the CFL must be quite smaller than 1
I changed the second term to explicit, but still have problems with dt.
Do you know any reference which explains the discretizations in details for LES?
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Old   August 15, 2014, 06:03
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I changed the second term to explicit, but still have problems with dt.
Do you know any reference which explains the discretizations in details for LES?
if you want, some of my papers about discretization for LES are published on IJNMF. Details are provided for projection method with AB/CN discretization
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Old   August 15, 2014, 21:57
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Originally Posted by FMDenaro View Post
if you want, some of my papers about discretization for LES are published on IJNMF. Details are provided for projection method with AB/CN discretization
Thank you very much. I will search for your papers.
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Old   August 16, 2014, 00:42
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Originally Posted by FMDenaro View Post
if you want, some of my papers about discretization for LES are published on IJNMF. Details are provided for projection method with AB/CN discretization
I found several papers you published in IJNMF. Which one do you suggest?
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