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Critical DiffiusionNumber / Peclet-Number

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Old   June 4, 2020, 08:45
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Lennard Porter
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Hi everybody,
Im new to thermal transient simulations, and have come across a problem.
I searched for anwers, and have found a lot, but nothing really answering my questions.

I have a transient thermal simulation, with a stationary flowfield (only solving energy-equation, so no need to look after Courant-Number).
The flowtemperature changes in time, transfering the heat into a glass-shield. So i have heatconduction within the glass.
Is there a general (theoretical) critical number for the Diffusion-Number (in OpenFOAM)?

For as far as i know, the Courant-Number is defined as CFL=\frac{w \cdot \Delta t}{\Delta x}, and usually has to be smaller than 1, for stability/precision reasons due to loss of information being "passed through" a cell within a single timestep.

The Diffusion-Number is defined as DN=\frac{a \cdot \Delta t}{(\Delta x)^2}, with a being a material specific constant a=\frac{\lambda }{\rho \cdot c_p}.

How do I calculate a critical timestep for heattransfer (conduction) issues? Is there a critical Diffusion-Number , like there is for Peclet<2?

Thanks in advance for your help!
Best regards
Lolo

Last edited by lolo97; July 1, 2020 at 15:16.
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Old   June 5, 2020, 00:17
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The diffusion-number for heat conduction is called Fourier number. There also is a stability criterion which ends up being around 1.

But just like the CFL criterion for Courant number, it highly depends on the problem. If you really want to know what that exact number is, you need to do the stability analysis in depth (or at least state all your numerics/discretization/etc.)

And although you have a stationary velocity field, you still have an advection term, which means a similar Courant number constraint still applies.
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Old   June 5, 2020, 03:40
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Filippo Maria Denaro
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Quote:
Originally Posted by lolo97 View Post
Hi everybody,
Im new to thermal transient simulations, and have come across a problem.
I searched for anwers, and have found a lot, but nothing really answering my questions.

I have a transient thermal simulation, with a stationary flowfield (only solving energy-equation, so no need to look after Courant-Number).
The flowtemperature changes in time, transfering the heat into a glass-shield. So i have heatconduction within the glass.
Is there a general (theoretical) critical number for the Diffusion-Number (in OpenFOAM)?

For as far as i know, the Courant-Number is defined as CFL=\frac{w \cdot \Delta t}{\Delta x}, and usually has to be smaller than 1, for stability/precision reasons due to loss of information being "passed through" a cell within a single timestep.

The Diffusion-Number is defined as DN=\frac{a \cdot \Delta t}{(\Delta x)^2}, with a being a material specific constant a=\frac{\lambda }{\rho \cdot c_p}.

If I apply the same rules as for the CFL-Number (DN<=1, and x = cell-length (0,5mm)), my critical timestep is t_{critical}=\frac{(\Delta x)^2 \cdot 1}{a}=\frac{0,5 \cdot 10^{-3} m^2 \cdot 1}{4\cdot 10^{-7} \frac{m^2}{s}}=0,625s.

Is that somewhat realistic? (For heatconduction in glass)
If not so, what do I have to look out for? How do I calculate a critical timestep for heattransfer (conduction) issues? Is there a critical Diffusion-Number , like there is for Peclet<2?

Thanks in advance for your help!
Best regards
Lolo



These non-dimensional numbers can be defined for any type of numerical discretization but their role in the numerical stability depends on the discretization, from case to case. However, for a steady case you should also consider the Peclet cell number.
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Old   June 6, 2020, 05:51
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Lennard Porter
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ok allright thank you

Last edited by lolo97; July 1, 2020 at 15:02.
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conduction, courant, heattransfer, peclet number, timestep

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