Lid Driven Triangular and Trapezoidal Cavity Flow: Vortical Structures for Steady Solutions and Hopf Bifurcations
Round 1
Reviewer 1 Report
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Author Response
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Reviewer 2 Report
In the manuscript “Steady solutions of the 2D lid-driven triangular and trapezoidal cavity flow”, the evolution of steady solutions of lid-driven right-angled isosceles triangular and trapezoidal cavity flows was explored by lattice Boltzmann method. For the triangular one, the flow topologies of steady solutions were captured and analyzed at wide Reynolds numbers, and the critical Reynolds number of Hopf bifurcation is obtained by investigating the perturbation decay rate. For the trapezoidal one, bowl-shaped and pyramid-shaped trapezoids were considered, and the effect of different top-line/base-line ratios on flow topologies and critical Reynolds number were also studied. This work is quite detailed. However, the purpose of this work is not clear enough. The authors need to clarify the innovation of this paper and the significance of the results. Besides, there still exists some other problems in this paper:
1. In introduction, a great number of previous works about cavity flows are listed. Do they all make sense? Similar works should be classified and serve the purpose of your idea. And it is also not clear that the difference between this article and other papers, it seems that only the geometry of cavity changed. In a word, what is the innovation and purpose of this work?
2. Why is the LBM rather then the NS equations used? Since the LBM is used to capture the rarefied nonequilibrium effect which is not analyzed in this paper.
3. Can the author give a simple explanation of Hopf bifurcation and its relationship with flow stability?
4. In section 3.1, it is recommended to put some figures rather than reference for comparison. Meanwhile, how is the position of the main vortex calculated?
5. Figure 6 and Figure 9 are not clear enough.
Comments for author File: Comments.pdf
Author Response
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Reviewer 3 Report
There is not any novelty in the submitted MS and the authors solved a simple basic problem. Also the numerical solution has been written very poorly, same as abstract section.
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Reviewer 4 Report
In this paper, the 2D lid-driven triangular and trapezoidal cavity flow is simulated by LBM. The flow topologies and critical Reynolds number of Hopf bifurcation are analyzed. The influence of geometric parameter on the flow is also investigated. I suggested that the following questions can be clarified before to be accepted.
1) The introduction section should be re-organized. It contains only two paragraphs, and the first big paragraph is only a collection of existing research. The shortage of current research and the main improvement of the present paper are not clear. It is suggested that the authors can explain what the main difference in the study of triangular and trapezoidal geometries is.
2) Equation (1) in line 119 is not the common complete expression for lattice Boltzmann evolution equation. In Eq. (3) in line 130, the dot products between ei and u should be added.
3) How do the authors deal with the tilted boundaries of the triangular and trapezoidal cavities. Is the interpolation boundary condition for curved boundaries used, or zigzag boundaries approximated by square mesh are used? The zero density gradient in Eq. (4) on these boundaries is not trivial. Will this detailed boundary geometry influence the flow characteristics?
4) Since the LBM is an unsteady method in nature, how to determine that the steady-states are reached. It seems that unsteady conditions are also studied in this work.
5) In Figure 4(b), (c), how to calculate the physical time t’? In Fig. 4 (a), do different convergence criteria result in different converged velocity value? If the flow is unsteady periodic, how to define a converged velocity value U?
6) Lines 196-197 is confusing. Is it Re>8000 or Re<8000?
7) The title of the paper should be modified. The main findings of flow structures are better be mentioned.
8) Name expressions in Ref. 28 are incorrect.
Author Response
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Round 2
Reviewer 2 Report
The author carefully revised the manuscript according to the comments of reviewers. It is recommended to accept this article.
Author Response
Thank you Sir.
Reviewer 4 Report
My questions are well addressed, and I have no more major problems.
One minor problem is that, the modified Eq. 4 is kind of more confusing, because the reader does not know the bar means to choose an average density. And to make the density gradients in both directions equal zero is impossible in most of the conditions. I suggest the authors can consider it before publishing.
Author Response
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