Mesh& steptime independant: conduction-convection problem

Fati1 · October 27, 2018, 08:33

Discussion in CFD forum
Dear all,
I’m working on the modeling of the temperature distribution in a convection-conduction system (dynamic problem/ One dimensional/ Laminar/1order Upwind for the convective term). I have two equations (for the solid material & the fluid), the two are coupled with the convective heat transfer coefficient.
I have conducted an implicit finite volume discretization; the system converges without problem. But I’m confused when conducting the mesh/ step time independent analysis. I’m not sure if I understand what I should exactly do! I feel like my scheme is numerical-velocity dependent (delta-x/delta-x).
- When refining meshes: I go from 10 to 10000 meshes (dx=0.1 to 0.0001) (just to test if there is any problem), the values of the temperature distribution don’t move that much from the initial temperature, I feel like if I’m going further with decreasing dx, the temperature distribution are being reduced to the initial temperature (before the heat transfer)
- when taking smaller (dt), the temperature increases more rapidly than when taking higher dt.
- In between the lowest & highest values, there is an interval, where the results don’t change that much, and they are nearly-equivalent (4 to 5%) to some experimental results I compared my model to. When studying this interval, I found that the numerical-velocity is near to the physical-velocity of the problem.
So, depending on the values of (dx & dt) considered for the simulation, the transient evolution of the temperature distribution changes. If I resume these two parameters in the numerical velocity (Vn=dx/ dt) I would say that: when Vn decreases, it takes more time for the problem to converge & if it increases, the problem converges more rapidly to the steady state. So here, I have the convergence any ways, but my problem is that I’m studying how the temperature is changing during the heat transfer (through time), and I need to have correct values for the distribution during the period of the energy transfer (Evolution of the temperature), which I cannot have when changing (dx & dt) parameters.
(Mathematically speaking, it makes sense : Because the terms in the matrix increases as some are multiplied by (1/dx) when dx decreases , in the other side the values of vector decreases as some are multiplied by dx) so the solution, after inversing the matrix have like (something going to zero/something very high) it’s like the problem is not moving and the temperature distribution can’t move from the initial conditions or move very slowly!
I’m very confused!!! Is there any problem with the stability of the grid/scheme? Or I need more considerations for the choice of dx & dt, & how Can I get that mesh/ step times independent studies?
Thank you very much!

FMDenaro · October 28, 2018, 14:52

Not sure to understand your questions, I found some confusion...
I assume you are using the equation

dT/dt + u dT/dx = k d^2T/dx^2

where u is zero in the solid region and some BC.s applied along with the coupling at the interface.

This equation has to be discretized in time and space and if you use an explicit method, once chosen a step dx, you need to fulfill the stability constraint for the time step in the region (Pe_h, cfl).

The solution must reach a steady state, measured by some norm on the time derivative.

On the other hand, grid independence should be measured in some norm from grid to grid.

October 27, 2018, 08:33	Mesh & steptime dependency studies: conduction-convection problem	#1
Fati1 New Member Ano Join Date: Oct 2018 Posts: 1 Rep Power: 0	Discussion in CFD forum Dear all, I’m working on the modeling of the temperature distribution in a convection-conduction system (dynamic problem/ One dimensional/ Laminar/1order Upwind for the convective term). I have two equations (for the solid material & the fluid), the two are coupled with the convective heat transfer coefficient. I have conducted an implicit finite volume discretization; the system converges without problem. But I’m confused when conducting the mesh/ step time independent analysis. I’m not sure if I understand what I should exactly do! I feel like my scheme is numerical-velocity dependent (delta-x/delta-x). - When refining meshes: I go from 10 to 10000 meshes (dx=0.1 to 0.0001) (just to test if there is any problem), the values of the temperature distribution don’t move that much from the initial temperature, I feel like if I’m going further with decreasing dx, the temperature distribution are being reduced to the initial temperature (before the heat transfer) - when taking smaller (dt), the temperature increases more rapidly than when taking higher dt. - In between the lowest & highest values, there is an interval, where the results don’t change that much, and they are nearly-equivalent (4 to 5%) to some experimental results I compared my model to. When studying this interval, I found that the numerical-velocity is near to the physical-velocity of the problem. So, depending on the values of (dx & dt) considered for the simulation, the transient evolution of the temperature distribution changes. If I resume these two parameters in the numerical velocity (Vn=dx/ dt) I would say that: when Vn decreases, it takes more time for the problem to converge & if it increases, the problem converges more rapidly to the steady state. So here, I have the convergence any ways, but my problem is that I’m studying how the temperature is changing during the heat transfer (through time), and I need to have correct values for the distribution during the period of the energy transfer (Evolution of the temperature), which I cannot have when changing (dx & dt) parameters. (Mathematically speaking, it makes sense : Because the terms in the matrix increases as some are multiplied by (1/dx) when dx decreases , in the other side the values of vector decreases as some are multiplied by dx) so the solution, after inversing the matrix have like (something going to zero/something very high) it’s like the problem is not moving and the temperature distribution can’t move from the initial conditions or move very slowly! I’m very confused!!! Is there any problem with the stability of the grid/scheme? Or I need more considerations for the choice of dx & dt, & how Can I get that mesh/ step times independent studies? Thank you very much! Last edited by Fati1; October 28, 2018 at 08:52.

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October 28, 2018, 14:52		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	Not sure to understand your questions, I found some confusion... I assume you are using the equation dT/dt + u dT/dx = k d^2T/dx^2 where u is zero in the solid region and some BC.s applied along with the coupling at the interface. This equation has to be discretized in time and space and if you use an explicit method, once chosen a step dx, you need to fulfill the stability constraint for the time step in the region (Pe_h, cfl). The solution must reach a steady state, measured by some norm on the time derivative. On the other hand, grid independence should be measured in some norm from grid to grid.