A Constitutive Equation of Turbulence

Round 1

Reviewer 1 Report

In this article, the authors first give a fundamental/theoretical review for the constitutive equations of turbulent flows, and then, introduce the first order, nonlinear, nonlocal and fractional Difference-Quotient Turbulence Model (DQTM) as the constitutive equations of turbulent shear flows. Next, three basic examples of shear flows have been chosen to show the capability of the DQTM. Finally, the article summarizes the analogies between laminar and turbulent flow constitutive equations. The entire article is excellently written and ready for publish in present form.

From the reviewer's humble opinion, the only part that needs to be revised is the consistency of the symbols. For example, the symbol of eddy viscosity (epsilon) in section 4.4 (it is probably a typo due to word editing software).

Author Response

Dear reviewer No. 1

Please see letter to reviewer No. 1 for our answers. 

With kind regards

Peter W. Egolf

Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript reports a review of a novel modelling of turbulence developed by one of the authors in his earlier works. In this model, the turbulence closure problem is treated as the need for a constitutive equation for turbulent flows, in a manner analogous to the Newton's law of viscosity for laminar flows. Here, the Reynolds shear stress in the mean flow Navier-Stokes equation is modeled with non-local, non-linear, and fractal contributions. The Difference Quotient Turbulence Model (DQTM) arising from this approach extends the cases where eddy viscosity/size is assumed to be constant (Boussinesq) in analogy to kinetic theory of gases and eddy viscosity/size is variable (Pradtl) to more realistic cases where fractal and non-local effects are relevant. It is shown that this model gives a constitutive equation for turbulent flows, and its analogy to Newton's law of viscosity for laminar flows is discussed. It also turns out that the 'eddy viscosity' appearing in this new constitutive equation could be explained in analogy to the microscopic kinetic theory of gases which explains the Newtonian viscosity from a molecular perspective. The application of this rather simple approach is demonstrated in flows such as turbulent wake flows, axisymmetric turbulent jets, and plane turbulent Couette flows. Furthermore, the possibility of treating turbulent flow as a critical phenomenon is discussed, in analogy to magnetism. The order and control parameters in turbulence are highlighted. Lack of experimental data near criticality prevents a proper validation of the proposed 'vortisation curves'.

The study is very interesting, novel, and presented with excellent clarity. This will be a very good addition to Fluids, as a review article. A few minor comments are listed below; the authors could clarify these. I recommend the manuscript to be accepted for publication in Fluids. 

Minor comments:

1. It would be helpful if x' is defined after Eq. (1).
2. At the end of page 3: Is d << λ true for liquids in general?
3. Sec. 4.1, line 7 from the top: The subscript of u looks like l rather than i. Please check this.
4. It would be helpful if the meaning of 'fractional derivative' could be clarified after Eq. (3).
5. Page 5, line 6 from the top: It would be helpful if the physical reasoning for the replacement of the dependence of eddy viscosity on x' with dependence on u could be clarified. Furthermore, does this replacement make only the non-locality of gradients/derivatives contribute to the non-locality of the stress tensor?
6. In Prandtl's mixing-length model, if eddy viscosity is constant, then u_eddy varies inversely as the mixing length (please see the last line in page 5). Is the mixing length a constant (please see the discussion in page 6, line 3 from the top)? Please clarify.
7. Eq. 6c-d: Is linearity of the mean velocity with x₂ an assumption in this calculation (since the local gradient of mean u₁ is equated to its mean gradient over the flow domain from an arbitrary x₂ to x_2max)? Is this due to the choice of the particular form of the Heaviside function? Please clarify.
8. Perhaps, the captions of Figs. 5a and b can be combined into one Fig. 5.
9. Fig. 5b caption: do the authors mean 'Eq. (14)' instead of 'Eq. (13c)'? And, perhaps 'Fig. 6' could be replaced with 'Eq. (13c)'?
10. Table 1: In the dimensionless velocity for turbulent flow, why is this in terms of u₁ and not the mean value of u₁?
11. Table 1: In the stress parameter for laminar flows, '1/R/Re' could be replaced with '1/Re'.
12. Ref. 19: please check the year -- it seems to be published in 2006.

Author Response

Dear reviewer No. 2

Please see letter to reviewer No. 2 for our answers. 

With kind regards

Peter W. Egolf

Author Response File: Author Response.pdf