Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
Round 1
Reviewer 1 Report
The authors carried out studies on the statistical uncertainty in DNS of back step flow. The DNS model is not validated and details on the DNS of the back step flow is not included. Spectral element method is hp-method, they only showed the polynomial order p = 6, but no detailed information on the element size. The data analysis is shown without detailed explanation on their finding. I recommend to reject the manuscript at its current form. My detailed comments are as follow:
- For the numerical model section, the authors listed Channel flow study using a different model (pseudo spectral method ). The main focus of this study is back step flow, they used Nek5000. I don't see the reason to keep the channel flow results or discussion in the manuscript, which they already published a paper. Also within the channel flow part, there are repetitive contents, such as line 100-101 and 106-107, and ambiguous content such as line 109-110.
- BFS model study is not validated, detailed information of Nek5000 parameters are missing. What is the element size? Is de-aliasing used? what is the time-marching scheme? What is the CFL number? What is model-spin up time? PN-PN-2 formulation?
- Not clear data at different monitor points used in analysis are from different time intervals? (Line 251-253)
- Line 293-299 belongs to the numerical method section.
- Figures, the legend covers the axis in some figures
- For most of the figures, the authors simply described their finding without explanation. Also, what is the significance of their finding.
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The manuscript studies the uncertainties in statistical averaging turbulent quantities. Flow examples are channel flow and a backward-facing step. A detailed CFD study has been carried out, the results of which have been used in the averaging process.
CFD results for turbulent flows are usually very sensitive to unphysical amounts of diffusion, e.g. due to the convective terms that have not been discretized in an energy-preserving way, or due to excessive turbulent viscosity of the turbulence model. The present approach is no exception.
In order to study turbulent statistics it must be made convincingly very clear that these are not contaminated with these excessive amounts of diffusion
To start with, the equations of motion (1) and (2) have not been written in conservation form. This already is no good sign - it is certain that the convective terms are disturbing the energy budgets in the flow. Also, the spectral method inside nec5000 is not energy preserving. Hence, the first step that a reader wants to see is a convincing story that the turbulent details of the CFD results are physical. The authors refer the reader to other publications (references [1] and [8]), but that will not reassure a reader.
The delicate situation gets even worse when recirculation bubbles are present in the flow, as in the BFS case. When I read on line 574, dealing with the BFS, that "the authors conjecture that the estimator can be combined with a RaNS simulation", I must disappoint the authors. RaNS are not to be trusted with recirculating flow. That type of flow is the most difficult flow for CFD methods. To achieve some reliability a CFD method for this type of flow must be energy-preserving and the turbulence model should be very careful with turbulent diffusion: a low-dissipation model is necessary (nowadays there are several of them).
Having said this, the manuscript also should demonstrate that the approach is viable in situations where the CFD results are inaccurate. The absolute level of the results will be wrong, but maybe estimates about the averaging times will still have some value. Please also comment on this issue.
Some other comments:
= On line 156 it is mentioned that for the BFS the boundary condition are no-slip (except for in- and outflow). Does this also hold for the lateral sides, where full-slip or periodic conditions could have been used (to make life easier).
= Lines 526-529: "the results significantly diverged from the proper results and from results where only every tenth or hundredth time step is used for estimation. Therefore, our conclusion is that the frequency of monitoring data can safely be reduced". I cannot combine the phrasing "significantly diverged" with the conclusion "can safely be reduced". I would have concluded that the frequency should have been increased. Please rephrase these sentences.
= The green symbols in figures 10-13 and 15-17 are hardly visible - make them (dark)blue. Why are so many of these figures necessary? Is it not possible to make a more concise presentation of these results?
In summary: The subject of the manuscript is interesting. But in its present version a reader will not be convinced about the conclusions. A major rewriting of the text is necessary.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 3 Report
Dear authors
The paper is interesting, but it needs some improvements:
- Please see the yellow markings in the attached paper.
- The appendixes would be placed inside the main body of the paper.
- The references may be enriched.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors addressed the comments and I only have a minor comment here
It is inconsistent in 3.2.1 and in 2.2, in 2.2 they stated the PN/PN formulation was used and in 3.2.1 they suggested PN/PN-2 formulation. Some content in 3.2.1 should be moved to 2.2, such as the polynomial order.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 2 Report
In their accompanying cover letter, the authors give a long reaction on my earier comments. But I find hardly anything of this reaction in the revised version. Many readers will have similar concerns as I had, and they will be interested in the reaction of the authors. So, add more of it in the text. I will shortly discuss the most important issues.
= My first comment was on the amount of numerical diffusion. A reader would like to be convinced that these numerical artefacts are not spoiling the results. In their cover letter the authors explain that they have seriously considered this issue in the PhD thesis of the first author. This sounds great, and now I know where to look. But in the revised manuscript I only read (line 62) "A more thorough presentation of all the parameters of the BFS DNS can also be found in [9]." This is so vague. Why not write that you have studied the influence of numerical diffusion in [9] and found it to be sufficiently small. Then every reader, not just me, can feel reassured.
= My next question concerned the accuracy of solving the non-conservative form. In the cover letter the authors explain to me:
"non-conservative discretisation in NEK5000 is just as accurate as "standard" pseudospectral scheme with conservative treatment of convective terms."
Why not explain this to all readers, accompanied by a good reference?
= I had comments on the phrasing "the authors conjecture that the estimator can be combined with a RaNS simulation" (now on line 488). Some text has been added (lines 493-495), but I do not recognize the long and interesting reaction in the cover letter. Please inform the readers of your view.
In general:
A journal paper should be written with the prospective readers in mind. Therefore, realize that readers mostly will not be experts in the subject, in contrast with the authors themselves. Use language and steps in the explanation that fit to the knowledge level of these readers. In other words, be more like a school teacher. Do not be afraid to make, seemingly trivial, small steps. Readers will appreciate this.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 3 Report
I accept the author's revision; however, the pdf commented is not the best way to evaluate the final version of the paper.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
