Mesh Independence in MFIX-TFM for Bubbling Fluidized Bed Simulation

[wuming]

Hello everyone,

I’m currently working with MFIX-TFM to simulate a bubbling fluidized bed, and I’ve run into a challenge that I’m hoping the community can help me understand.

Using the tutorial case “fluid_bed_tfm_2d,” I conducted a mesh independence study where only the mesh size was varied while all other settings remained constant. According to the rule of thumb for achieving mesh independence in TFM simulations, the mesh size should be smaller than ten times the particle diameter. In my tests, I used meshes sized at 10x, 5x, and 2.5x the particle diameter.

However, I’ve encountered an issue where the computed flow fields appear inconsistent as the mesh size changes. This is different from what we typically observe in single-phase fluid simulations, where refining the mesh generally leads to a more consistent flow field. For TFM simulations, it seems decreasing the mesh size does not necessarily lead to a converging solution for the flow field, which is puzzling to me.

Fig. 10x particle diameter

Fig. 5x particle diameter

Fig. 2.5x particle diameter

My confusion lies in understanding why, for TFM simulations, refining the mesh does not yield a consistent flow field as expected. Although I have reviewed some literature over the past several days, I have yet to find papers that directly discuss this issue.

Could anyone provide an explanation for this phenomenon or point me towards relevant papers that discuss this specific challenge? Any insights would be greatly appreciated!

Thank you in advance for your help.

Best regards,

[jeff.dietiker]

What is your expectation in terms of “consistent flow field”? You should not be comparing snapshots at a give time, but quantities of interest that can be averaged over time, say the pressure drop across the bed, the bed expansion, bubble size/velocity. Fluidized bed are highly chaotic and the flow pattern cannot be compared on individual snapshots. Please see below a reference that shows the effect of grid resolution:

Helal Uddin, Charles J. Coronella, Effects of grid size on predictions of bed expansion in bubbling fluidized beds of Geldart B particles: A generalized rule for a grid-independent solution of TFM simulations, Particuology, Volume 34, October 2017, Pages 61-69, ISSN 1674-2001, Redirecting.

[wfullmer]

I disagree with the authors’ conclusion even based on their own findings. If you look at Fig. 4, you can see that the QoI isn’t even reaching the asymptotic region until a resolution (or below) that which they are saying is converged. But, regardless, Jeff is right, you have to time average a QoI, ideally for a sufficiently long period of time to make statistical uncertainty (in this case from deterministic chaos not true randomness) negligibly small. And then you can extrapolate to a “true” solution in the limit of $\Delta^* \to 0$ with Richardson extrapolation. Note that the convergence rates might depend on the QoI.

[wuming]

Dear Dr. Dietiker and Dr. Fullmer,

Thank you both for your insightful responses and discussion on the topic. I appreciate the explanation that, as the mesh is refined, the time-averaged physical quantities can tend to become grid-independent. This concept is clear to me.

However, from another perspective, CFD solvers are fundamentally solving a set of mathematical equations. As the mesh size decreases, one would expect the computational results to converge towards the true solution, which is expected to be unique (this uniqueness is one of the foundational principles for using numerical methods to solve systems of partial differential equations).

In the case of the TFM example I presented, I am puzzled by why the computational results at 2.0 seconds or other instances appear so different. Could you enlighten me on the underlying reasons for this discrepancy?

Best regards,

[wfullmer]

deterministic chaos.

here’s a self-serving ref for the KT-TFM: https://doi.org/10.1063/1.4977513
but if this is new to you, I strongly suggest looking up a textbook on the subject and at least reviewing the chapter on the Lorenz attractor

[wuming]

Dear Dr. Fullmer,

Thank you very much for your detailed response and for sharing your insights through your paper. Your explanations have significantly enhanced my understanding of numerical computations in gas-solid two-phase flows.

I would like to extend our discussion to non-uniform meshes. Specifically, I am curious about whether the conclusions we’ve discussed still hold true when using non-uniform grids with varying cell sizes. For instance, in simulating wall-to-bed heat transfer, it is often necessary to refine the mesh near the walls to a scale smaller than the particle diameter.

In achieving mesh independence in cases with non-uniform meshes, could you provide some guidance? From my literature review, besides variables such as bed height, pressure drop, bubble size, and velocity, the heat transfer coefficient near the walls appears to be a critical parameter. Would it be feasible to achieve mesh independence by time-averaging the heat transfer coefficients?

Best regards,

[wfullmer]

Celik etal (2008) JFE Editorial Policy Statement.pdf (190.2 KB)

It should work in the same way for heat transfer coefficients as other QoI’s (quantities of interest, i.e., \phi in the above ref). If the physics being simulated is chaotic (in your case it is), it should also be time averaged. Regarding non-uniform meshes, it really works the same way as uniform meshes, just that that you have to calculate the average volume size. I’ve attached ASME’s Journal of Fluids Engineering editorial policy on estimating discretization error, see Eq. (1). Note that they also provide some guidance that the refinement factor should be at least 1.3.

[wuming]

Dear Dr. Fullmer,

Thank you very much for your detailed response and the attached reference material.

I am currently going through it carefully. I will be conducting tests using MFIX-TFM and plan to share the results under this topic in a few days.

Thanks again for your assistance.

Best regards,