Talk:Kelvin-Helmholtz Instability and Roll-up

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    <review>message to curators</review>

    1. Consider citing Majda and Bertozzi (2002), "Vorticity and Incompressible Flow", Cambridge University Press, for an overview of the mathematical theory and computations.

    2. The sum in Equation (10) should more generally include negative wavenumbers as well as positive ones, e.g. as in Equation (8).

    3. "alternative point vortex rule" -> "alternate point vortex rule"

    4. The Author advocates using a spectrally accurate method to detect singularity formation. This requires care because spectral accuracy is typically lost at the critical time. Krasny (1986) used an alternative approach - the point vortex method together with extrapolation for several values of N. Although the point vortex method is only 1st order accurate, it apparently remains consistent at the critical time of the Moore singularity.

    5. Sakajyo -> Sakajo ?


    Reply to Reviewer A

    Thank you for comments and suggestions. I have modified the article accordingly. Details of corrections are as follows.

    1. I added this reference in "Further reading".

    2. 3. 5. Thank you for the indication. I corrected these typos in the revised article.

    4. I have rewritten the paragraph below Eq. (12) (in the revised article) in the subsection "Zero surface tension case" following the reviewer's comments. I have added a reference (R. E. Caflisch, T. Y. Hou and J. S. Lowengrub (1999) "Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering" Math. Comput. 68:1465-1496). Thank you for the important suggestions.

    Reviewer B

    1. In the section “Moore’s Curvature Singularity” define \(\Gamma_{c}\).

    2. I think \(A\ne1\) in the last paragraph of the same section should read \(A=1\) .

    3. I recommend extending the article to include the effect of viscosity and introduce the Orr-Sommerfeld equation. I believe there is no singularity when viscosity is included.

    Reply to Reviewer B

    Thank you for comments and suggestions. Details of corrections are as follows.

    1. I have defined \(\Gamma_c\) in the last part of the first paragraph in the subsection "Moore's curvature singularity".

    2. My description might be hard to understand. I added some sentences for this matter in the last paragraph of the subsection "Moore's curvature singularity".

    3. I have added a new section to describe the effect of viscosity, in which the Orr-sommerfeld equation is introduced. I have simply discussed the relation between the solution of the linear stability analysis in inviscid flow in the section "Linear stability theory" and the analytical solution of the Orr-sommerfeld equation, in the last paragraph of the subsection " Orr-Sommerfeld equation".

    I have cited some results for numerical calculations of viscous flow in the subsection "Roll-up in viscous flow".

    Reviewer C aka Alexander Velikovich

    Chihiro Matsuoka wrote a nice brief review, which is accessible to those who are not familiar with the subject and at the same time helpful to those who are. Movies showing evolving vortices are very impressive. The text is pretty good as is. My suggestions are of purely editorial nature.

    1. I would recommend adding some explanation of the relevant physics before stating in the introduction that “In order to simplify the theoretical and numerical calculations, the effects of gravity, surface tension, and the density difference between the two fluids are not considered in most cases; however, KHI can occur as long as a uniform velocity shear exists when a lighter fluid is superposed on a heavier fluid.” Most of the manifestations of the KHI observed in nature (such as white cap formation by wind blowing over gravity waves on water surface, roll-up cloud patterns in the sky, internal gravity waves in the ocean) and in demonstration laboratory experiments, such as one with inclined water tank, are made possible by stable density stratification due to gravity. This is reflected by Eq. (7), which differs from the dispersion relation for gravity/capillary waves only by the term under the square root proportional to (U_1-U_2)^2. Even when the gravity is not relevant, the density difference is helpful (in many cases necessary) to make the KHI development observable. Such observations of KHI in high-energy-density environment, where the gravity is inessential, have recently been made on the Omega laser, cf. E. C. Harding et al., PRL 103, 045005 (2009), O A Hurricane et al., Phys. Plasmas 16, 056305 (2009); Astrophysics Space Sci. 336, 139 (2011). I recommend citing these papers. I have no idea how Scholarpedia deals with the copyright issues, but these authors could probably be asked to share their radiographic images of observed gravity-free KHI development.

    After noting this, it is indeed fair to state that the basic mechanism of the KHI development does not need gravity and/or density difference. Neglecting them, as well as the surface tension and viscosity, simplifies the application of vortex theory, which is the main reason why these effects are “not considered in most cases.”

    2. It is not clear how the statement “…the curvature singularity does not appear for A=1. Although the rigorous mathematical proof does not exist, their prediction for finite Atwood numbers including A=1 is supported theoretically (Tanveer 1993) and numerically (Matsuoka and Nishihara 2006)” is related to KHI. The value A = 1 indicates that there is a single fluid, rho_2=0. Is it possible for the KHI to develop in a single fluid? Not according to Eq. (7), which for rho_2=0 reduces to the dispersion relation for stable capillary waves. I think that the author actually had in mind RTI and RMI, the subject of her paper with Nishihara cited here. Then it should be so stated to avoid confusion.

    I have no idea how much space is allowed in this article to illustrate the connection between the KHI and RTI/RMI. Some discussion obviously cannot be avoided. In this context the statement that “When rho_2 > rho_1 and U_1 =U_2 =0 , RTI occurs” is not quite accurate. It is better to say that with U_1 =U_2 =0 we are dealing with classical RTI. But as long as rho_2 > rho_1 the interface would still be distorted by combined action of the RTI and KHI, the former dominating at long wavelengths, the latter increasing the instability growth rate at short wavelengths. Interestingly, continuous, rather than localized, shear of horizontal velocity can produce an opposite effect, actually suppressing the RTI growth. This method of RTI control had been studied for mitigating hydromagnetic RMI in z-pinch implosions by U. Shumlak and N. F. Roderik, Phys. Plasmas 5, 2384 (1998), and experimentally demonstrated as a part of U. Shumlak’s ZaP Flow Z-Pinch program at the University of Washington.

    One might also consider highlighting the importance (particularly for the ICF/HEDP applications) of the “secondary” KHI, which develops at highly nonlinear stages of RTI/RMI and leads to roll-up.

    3. I do not see how the derivation of the Orr-Sommerfeld equation is related to the rest of the text. This equation has been derived for a 2D linear stability analysis of a viscous flow for which the background velocity is horizontal, constant as a function of the horizontal coordinate but sheared in the vertical direction. The perturbed stream function depends on both coordinates and could be chosen periodic in the horizontal direction, just as in Fig. 1. It would be natural to use (16) for generalization of the dispersion relation (7) to take the viscosity into account. Could it be done? If it could, then showing such generalization would be worthwhile. If it could not, then why is this equation discussed here at all? This is not clear because, apparently, none of the work described in the next section “Roll-up in viscous flow” is based on the Orr-Sommerfeld equation.

    Reply to Reviewer C (Dr. Velikovich)

    Thank you for detailed comments. I modified the manuscript accordingly.

    1. Introduction

    (i) I added some references for KHI.

    (ii) I rewrote the second paragraph and added some references including the internal references in Scholarpedia [W. Cook Andrew and Youngs David (2009, RTI), and Oleg Schilling and Jeffrey W. Jacobs (2008, RMI)].

    2. \(\rho_2 = 0\) corresponds to \(A=-1\) in this article [because the definition of the Atwood number is \(A=(\rho_2 - \rho_1)/(\rho_1 + \rho_2)\)]. For this case, the system is always stable when \(U_1 = U_2\) (capillary-gravity wave). However, if \(U_1 \ne U_2\) and \(|U_1 - U_2|\) is sufficiently large, the system can become unstable. [Figure 4 corresponds to this case. Also, for other parameters including \(A=-1\), I confirmed this by numerical calculations, although the roll-up did not appear for large \(|A|\) (instead, a cusp appeared).] It is unclear whether this instability is called as KHI or not (it seems to depend on researchers); therefore, I only mentioned this case as linearly "unstable" in the section "Linear stability theory".

    As pointed out by Dr. Velikovich, I have in mind RTI and RMI. However, for these topics , there have already existed excellent articles in Scholarpedia by W. Cook Andrew and Youngs David (RTI), and Oleg Schilling and Jeffrey W. Jacobs (RMI) (these articles are cited in this article); therefore, I would like to avoid to describe these instabilities in detail here. Instead, I rewrote the item of the linear stability theory (in the section "Linear stability theory") more precisely.

    3. The subsection "Orr-Sommerfeld equation" has not exsisted in the original article and I have added that following the comment of Reviewer B. However, as pointed out by Dr. Velikovich, I feel that this subsection drifts away from other descriptions in this article; therefore, I deleted this subsection in the revised article. Instead, I rewrote the section "Effects of viscosity and roll-up" including its title and added the section "Interfacial instabilities in magnetohydrodynamic and electro-magnetohydrodynamics flows" in order to describe KHI and roll-up in MHD flows or secondary KHI in plasmas.

    4. I cited all references introduced by Dr. Velikovich in the revied article except the one by Shumlak et al. [U. Shumlak and N. F. Roderik, Phys. Plasmas 5, 2384 (1998)]. This study is interesting as the one describing the stabilizing effect of sheared axial flows on RTI in Z-pinch implosions; however, I consider that it should be referred in "Rayleigh-Taylor instability and mixing" (online) rather than this article. Therefore, I did not cite this reference here.

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