Critical DiffiusionNumber / Peclet-Number

lolo97 · June 4, 2020, 08:45

Hi everybody,
I´m 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

= $\frac{\lambda }{\rho \cdot c_p}$ .

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

LuckyTran · June 5, 2020, 00:17

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.

FMDenaro · June 5, 2020, 03:40

Quote:

Originally Posted by lolo97

Hi everybody,
I´m 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

= $\frac{\lambda }{\rho \cdot c_p}$ .

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.

lolo97 · June 6, 2020, 05:51

ok allright thank you

June 4, 2020, 08:45	---------	#1
lolo97 New Member Lennard Porter Join Date: Mar 2020 Posts: 5 Rep Power: 6	Hi everybody, I´m 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 = $\frac{\lambda }{\rho \cdot c_p}$ . 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.

June 6, 2020, 05:51		#4
lolo97 New Member Lennard Porter Join Date: Mar 2020 Posts: 5 Rep Power: 6	ok allright thank you Last edited by lolo97; July 1, 2020 at 15:02.

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June 5, 2020, 00:17		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,704 Rep Power: 66	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.