Time step in LES

mnabi · August 13, 2014, 10:34

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

FMDenaro · August 13, 2014, 11:21

Quote:

Originally Posted by mnabi

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...

mnabi · August 13, 2014, 11:26

Quote:

Originally Posted by FMDenaro

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.

FMDenaro · August 13, 2014, 11:41

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

mnabi · August 13, 2014, 11:52

Quote:

Originally Posted by FMDenaro

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

FMDenaro · August 13, 2014, 12:47

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.

mnabi · August 13, 2014, 12:54

Quote:

Originally Posted by FMDenaro

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?

FMDenaro · August 13, 2014, 13:38

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.

mnabi · August 13, 2014, 21:57

Quote:

Originally Posted by FMDenaro

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?

FMDenaro · August 14, 2014, 03:19

just using the explicit AB discretization requires eddy viscosity only at tn and tn-1, no coefficents are therefore present in the algebric system

mnabi · August 14, 2014, 03:58

Quote:

Originally Posted by FMDenaro

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 ?

FMDenaro · August 14, 2014, 04:14

Quote:

Originally Posted by mnabi

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

mnabi · August 14, 2014, 20:38

Quote:

Originally Posted by FMDenaro

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?

FMDenaro · August 15, 2014, 06:03

Quote:

Originally Posted by mnabi

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

mnabi · August 15, 2014, 21:57

Quote:

Originally Posted by FMDenaro

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.

mnabi · August 16, 2014, 00:42

Quote:

Originally Posted by FMDenaro

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?

August 13, 2014, 10:34	Time step in LES	#1
mnabi Member M. Nabi Join Date: Jun 2009 Posts: 44 Rep Power: 16	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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August 13, 2014, 11:41		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	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

August 13, 2014, 12:47		#6
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	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.

August 13, 2014, 13:38		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	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.

August 14, 2014, 03:19		#10
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	just using the explicit AB discretization requires eddy viscosity only at tn and tn-1, no coefficents are therefore present in the algebric system