Euler and Navier-Stokes equations in a new time-dependent helically symmetric system
(1) Misleading is the following statement: "For three-dimensional time-dependent fluid flows, CLs [conservation laws] were studied in very much detail in Cheviakov & Oberlack (2014). Therein, they considered higher-order CL multipliers and obtained an infinite family of vorticity CLs" (p.345). What is misleading here is that it is not said that all these new CLs obtained in this way are ultimately all trivial in having no physical significance. The reason is that this infinite set of new conservation laws obtained by Cheviakov & Oberlack (2014) involve arbitrary functions of all independent variables (careful: not dependent, but independent ones), making them thus physically trivial. This was clearly shown by Rosenhaus & Shankar (Second Noether theorem for quasi-Noether systems, J. Phys. A: Math. Theor., 49, 175205, 2016). Hence, also the CLs EV5 (3.17) and NSV5 (4.7) derived in this new study by Dierkes & Oberlack are of a trivial nature without having any physical significance.
(2) The statement "By assuming $\alpha(t) = \text{const.} = 1/a$ one easily retains the classical helical velocity components, pressure, similarity variable and form function B(r) given in KCO [Kelbin et al. (2013)]" (p.348), is wrong. If we denote the two parameters $(a,b)$ in KCO as $a=:a_K$, $b=:b_K$, and the two parameters $(\alpha,b)$ in Dierkes & Oberlack as $\alpha(t)=\text{const.}=:a_D$ and $b=:b_D$, then we do not get the relation $a_D=1/a_K$ as claimed, but rather the quite different connection $a_D=b_K$ and $b_D=1/a_K$. That means one has two different helical variables, $\xi_D$ in the "Dierkes" system and $\xi_K$ in the "Kelbin" system, which then are related by $\xi_D=(\xi_K)/(a_K b_K)$ or $\xi_K=(\xi_D)\cdot (a_D/b_D)$. Hence the asymptotic limits discussed in Dierkes & Oberlack do not correspond to the same limits in Kelbin et al., as incorrectly claimed e.g. on p.348. To note is also the difference that $\xi_D$ is dimensionless, while $\xi_K$ carries the physical dimension of length.
(3) In Appendix B it's said: "It is well known that for the three-dimensional time-dependent Euler system, the kinetic energy is conserved in every coordinate system" (p.362). This statement is incorrect. For closed systems, the kinetic energy is directly conserved only for coordinate transformations which define and connect inertial frames of reference, but not for non-inertial (accelerated) frames. This is a standard result in mechanics, both for the dynamics in the discrete (point) as well as in the continuum case. Illustrative examples for a non-conserved kinetic energy is the change into a uniform rotating frame or into a linearly accelerating frame (non-uniform boost). All these frames generate inertial, so-called fictitious forces which collectively cannot be written in a spatial divergence form when considering the temporal change of the kinetic energy only, correctly defined as being only the sum of the squared velocity components in each frame respectively.
However, when weakening this constraint of a direct conservation of kinetic energy, namely by allowing the quantity which is called the density of the local conservation law (quantity $\Theta$ in eqn.(1.1), p.345) to depend not only on the kinetic energy but also on all other dependent and independent variables of the considered system, then with the tools of differential geometry it is straightforward to show that the equation for kinetic energy can always be written in conservative form, independent of the coordinate transformation considered, i.e., independent of whether an inertial or non-inertial frame of reference is induced. But then it is not the sole kinetic energy which is conserved. Instead of the kinetic energy $K$, only some functional relation $F$ involving $K$ ,i.e., $F = F(K,\mathbf{u},\mathbf{x})$, where $\mathbf{u}$ and $\mathbf{x}$ collectively denote all systems' dependent and independent variables, will be conserved.
To note here is that for time-dependent coordinate transformations the kinetic energy in a 3D framework is not transforming as a tensorial scalar. Only in a true 4D or higher dimensional framework it is transforming as such. Hence, to achieve the above mentioned goal in writing the kinetic energy equation into a conservative form for all coordinate transformations, it is necessary to change at least into a 4D framework. Of course this change has to occur such that the underlying physics is not changed. Since we are dealing here with classical Newtonian mechanics (and not for example with relativistic Einsteinian mechanics), the restrictions for the change into 4D are to only allow for coordinate transformations where the time coordinate transforms invariantly (up to some constant scaling or translation). Choosing an even higher dimensional framework than 4D one can drop this restriction. See for example the work of M. Vinokur (Conservation Equations of Gasdynamics in Curvilinear Coordinate Systems, J. Comp. Phys., 14, 105-125, 1974), who re-wrote the classical gasdynamic equations into conservative form for all coordinate transformations, including all time-dependent transformations by using a 5D framework.
Coming back to the work of Dierkes & Oberlack, it is important to observe that the so-called "generalized Galilean transformation" $X_G$ (eqn.(A1b)) is not a true coordinate transformation, since the pressure is not transforming as a tensorial scalar in any dimension. In fact, $X_G$ is a composed transformation where one element represents a time-dependent coordinate transformation and the other one a re-gauging of the pressure field. In the sense that one usually associates a "Galilean transformation" with a true coordinate transformation, it can be misleading to call $X_G$ a "generalized Galilean transformation", since this transformation not only changes the frame of reference but simultaneously also re-gauges the pressure field in each space-time point. In other words, $X_G$ is more than just the accelerated version of the usual inertial Galilei coordinate transformation. Hence, this transformation $X_G$ does not belong to the class of transformations discussed before. Because the discussion given above only refers to transformations which constitute true coordinate transformations, namely where the transformation of all dependent variables cannot be chosen freely as they are merely induced by the given transformation of their independent variables, their coordinates. $X_G$ is thus a special transformation, particularly arranged such that it is admitted as a symmetry of the incompressible Euler and Navier-Stokes momentum equations. As can be easily shown, $X_G$ is also admitted as a symmetry transformation of the Euler kinetic energy equation, that is, only in the special case of the time-dependent accelerated transformation $X_G$, the Euler kinetic energy is automatically conserved. But as already said, this transformation is special and does not belong to the discussed problem class of coordinate transformations, and hence it is incorrect to say that the "[Euler] kinetic energy is conserved in every coordinate system".
(4) The second bracket in the fourth term of (B7) is incorrect. This can already be seen by checking the physical dimensions, which are incompatible. Instead of the independent variable $\eta$, a term proportional to the dependent variable $u^\varphi$ must appear. This has consequences for the central result (B8), which itself cannot be correct (see the next comment).
(5) Independent of the form of the $\eta$-independent term $R$, i.e., independent of whether $R$ just stands for an arbitrary collection of all $\eta$-independent terms or for an $\eta$-independent structure being organized in conservative form (which, by the way, is not clear from the text on p.363 which case applies), relation (B8) obviously shows that the Euler kinetic energy equation cannot be reduced by the underlying transformation $X = (1/b)X_R - X_G$ (eqn.A3), when equivalently re-written as a pure translation $X = \partial_\eta$ with the help of the defining canonical variables as given by eqn.(2.3). But this result (B8) cannot be correct, since $X$ is a symmetry of the Euler kinetic energy equation and thus by definition must be reducible and therefore independent of $\eta$ when applying $X$. This fact can be straightforwardly shown with the help of the reduced continuity and momentum equations. However, careful, although the kinetic energy equation (B1) is reducible by default when applying symmetry $X$, this does not automatically mean that this reduced equation can then also be written in conservative form. The actual fact that the reduced Euler kinetic equation cannot be written in conservative form has already been shown correctly by Dierkes & Oberlack in Section 3, in that it does not appear as a result of the performed analysis. But this fact is definitely not shown through the analysis as done in Appendix B, by using the argument of relation (B8) with the intention to give an explanation of why the reduced Euler kinetic energy equation cannot be written in conservative form.
(6) As a final remark for the discussion given between eqns.(1.1) and (1.3) on pp.345-346, it is helpful to recall the study of W.-H. Steeb et al. (A Comment on Conservation Laws and Constants of Motion, Foundations of Physics, Vol.12, No.7, 1982) to avoid wrong conclusions when trying to determine the conserved quantity (constant of motion) from a conservation law.
Rosenhaus & Shankar (2016): http://iopscience.iop.org/article/10.1088/1751-8113/49/17/175205/meta
Vinokur (1974): http://www.sciencedirect.com/science/article/pii/0021999174900084
W.-H. Steeb et al. (1982): https://link.springer.com/article/10.1007/BF00729809