On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation
(1) The interpretations and conclusions in this study are not justified by the results given. Particularly, Fig.7 cannot be reproduced from the given DNS data shown in Fig.4(a-c) as claimed. The analytic asymptotic behaviours of the underlying equations (4.10-12) do not match the corresponding numerical asymptotic behaviours as shown in Fig.7: Firstly, when proposing the scaling (4.10-12), a collapse of the profiles for large $\tilde{x}_2$ cannot be confirmed and, secondly, in this regime the profiles also do not tend to zero as incorrectly shown in Fig.7.
That the profiles for the invariantized diagonal Reynolds stresses do not collapse and do not tend to zero in the asymptotic regime when scaled as (4.10-12) can be easily seen by considering e.g. the case $\tilde{R}\vphantom{R}^0_{22}$ (4.11) — the reasoning for the other two cases (4.10) and (4.12) is analogous: Since in free planar jet flow all fields, including $R_{22}$, decay for large lateral distances $x_2$ and therefore also for large invariant distances $\tilde{x}_2\propto x_2/\sqrt{t-t_0}$ for any fixed finite time $t$, say $t=20$, the first term $-D R^0_{22}(t-t_0)$ on the right-hand side of (4.11) tends to zero. Note that $D$ and $t_0$ are empirically fixed constants, taking the global values $D=-7.57$ (p.249) and $t_0=8.64$ (p.247). Now, since this first term is negligibly small compared to the second one on the right-hand side $a_{H_{22}}t$, which, according to Tab.1(p.252) takes for $t=20$ the finite and non-zero value $a_{H_{22}}t = 0.0706\cdot 20 = 1.4$, the asymptotic value for the invariantized Reynolds stress $\tilde{R}\vphantom{R}^0_{22}$ does not tend to zero, as incorrectly shown in Fig.7(b), but rather tends to the value $1.4$. This is true also for any other profile, say $t=30$, which would tend to the even higher value $a_{H_{22}}t = 0.0706\cdot 30 = 2.1$, being even a different value than for the profile $t=20$. Hence, for large $\tilde{x}_2$ the profiles for different $t$ do not only go to a non-zero value but also do not collapse. Please see Fig.4 in https://www.researchgate.net/publication/328080340, which shows the correct profiles for this scaling (4.10-12). The discrepancy to Fig.7 is clearly visible.
(2) The reason for this failure described above is that in this study again, as already in all previous studies from the group of Oberlack et al., nonphysical statistical symmetries are being employed that violate the classical principle of cause and effect. If the statistical symmetries of the system are chosen as given in Appendix A, then, for example, the translation group $a_{H_{22}}$ of the double-velocity moment in (A5) is inconsistent to the symmetry for the triple-velocity moment $H^0_{122}$ as given in (A8) or (A9), which, in order to be consistent with the lower moment, needs to contain a term that is at least proportional to the mean velocity $\overline{U}_{\!1}$. See https://www.researchgate.net/publication/328080340 for a possible choice of symmetries that do not violate the causality principle and which lead to a more convincing and robust collapsing of all profiles up to second order.
(3) The result of their performed Lie-group symmetry analysis, as presented in (A1-A11), not only contains inaccuracies (for example, if the symmetries for the double-velocity moments are chosen as given by (A4-A7), then the symmetry of each triple-velocity moment $H^0_{ij2}$ misses the vital term proportional to $H^0_{ij}$), but their analysis is also spectacularly incomplete. Irrespective of whether performing a 1-point symmetry analysis, or a more general 2-point symmetry analysis within the 1-point limit, in both cases, when done properly and correctly, will give an infinitely larger symmetry group than the one presented in Appendix A, not only for all dependent variables but also for the independent variable $x_2$: Any correctly performed symmetry analysis for the inviscid ($\nu=0$) free planar jet flow case will not lead to the linear $x_2$-specification from the outset as given in (A2), but rather to an arbitrary function in $x_2$ and $t$. Even when including the lateral mass flux constraint (2.4), the symmetry generator $\xi_{x_2}$ stays arbitrary and will not be restricted in a certain way — the mass flux constraint (2.4), also in its original form $\int(d\overline{U}_{\!1}/dt)dx_2=0$, only restricts the symmetry of the mean velocity $\overline{U}_{\!1}$ and not that of $x_2$, since all the restricting terms for $x_2$ will cancel exactly (after partial integration).
Of course, it is clear that one aims to make a connection to the classical $x_2$-invariant scaling $\tilde{x}_2\propto x_2/\sqrt{t-t_0}$ and that for this very reason one has to choose a linear ansatz for the symmetry generator of $x_2$, but this particular choice is not given as an explicit result from Lie-group analysis itself. In other words, this particular choice is not given by theory from within, as misleadingly claimed in their study; instead it is put as an external condition to match the collapsing DNS profiles of the mean velocity field. The same is true for all other variables, except for $\xi_t$ (A1) which from the outset is already in its general form.
Hence, presenting the symmetry result in the reduced form as given in (A1-A11) is misleading, as the reader might think that a Lie-group symmetry analysis for the inviscid free planar jet will only lead to this particular result (A1-A11) without any further intervention by the user. But this is not the case: A consistent and complete symmetry analysis gives a highly arbitrary result, which, if one aims to match with experimental or numerical data, needs to be arranged and specified externally. Due to this arbitrariness in the symmetries, all derived scalings in this study (irrespective of their higher-order inconsistencies as described in (1) and (2) above) are thus only a-posteriori scalings and not a-priori scalings as wanted. Under such conditions as given in the theory of turbulence, a systematic Lie-group analysis cannot make and give any analytical prediction as to how turbulence scales statistically, due to that not only unknown parameters but also an abundance of arbitrary functions get induced. It is easy to show (https://www.researchgate.net/publication/328080340) that a collapse of all profiles can also be obtained when choosing a different set of symmetries than the one incorrectly proposed in their study, i.e., when choosing any set of symmetries which does not violate the classical principle of cause and effect. There is no prediction or no unique choice in the scaling as how to make the profiles collapse; any other choice of symmetries which have a cause will also do the job. Hence, it is incorrect to state that "within the present work, we develop a theoretical basis using a Lie symmetry group that predicts such behaviour for the flow evolution, as an exact solution of the two- and multi-point correlation equations, which can be an important key in 'filling the gaps' of our understanding of self-similarity" (p.251). Instead of theoretically predicting flow evolution, this study only presents and proposes a sophisticated post-processing scheme. Nothing more!
The real problem simply is that the underlying statistical transport equations are unclosed, and so are their symmetries. The closure problem of turbulence cannot be circumvented by just employing the method of Lie-group symmetry analysis alone. Hence, without modelling these unclosed equations, an a-priori prediction as how turbulence scales is and will not be possible. Only a-posteriori, by anticipating what to expect from numerical or experimental data, the adequate invariant scalings can be generated through an iterative trial-and-error process.
(4) For the reason mentioned above in the beginning of (3), namely in presenting an incomplete symmetry analysis, it seems that the 1-point limit $\mathbf{r}\rightarrow 0$ in their symmetry analysis was not done correctly, thus leading to that overly restricted symmetry group. An indication for this is given on p.242, when saying "we currently devote ourselves only to generating invariant scaling laws for the mean velocity and Reynolds stresses, and therefore, the $r$-dependency of the terms is skipped". Doing the 1-point limit, the $r$-dependency may not be simply dropped, as this may lead to missing terms. The problem here is that when performing the 1-point limit one has to respect the non-commutivity of this limit: For example $\lim_{\mathbf{x}_{(2)}\to \mathbf{x}_{(1)}}d\overline{\mathbf{U}(\mathbf{x}_{(1)})\mathbf{U}(\mathbf{x}_{(2)})}/d\mathbf{x}_{(1)}$ is simply not just $d\overline{\mathbf{U}(\mathbf{x}_{(1)})\mathbf{U}(\mathbf{x}_{(1)})}/d\mathbf{x}_{(1)}$; it is more than that if one chooses this particular form — for a detailed explanation, see e.g. (C27) on p.40 in https://arxiv.org/abs/1412.3061.
(5) The derivation of the invariant scaling (B3) from (B2) and (B1) is a deception. The first step to (B1) is that $U_1 x_2$ is being identified as the invariant integration constant $\tilde{U}_{\!1}\tilde{x}_2$ in order to reduce the integration effort of equation (B1). This step is correct and not to complain about. However, this constant $\tilde{U}_{\!1}\tilde{x}_2$ is also multiplied by the factor $dF_2(t)/dt$, which from the calibration done in Sec.4 has to be zero, since $F_2(t)$ is fixed as the global constant $F_2(t) = -D\cdot n$ (4.9), i.e., $dF_2(t)/dt=0$. This results into an equation (B1) which does not contain the term involving $\tilde{U}_{\!1}\tilde{x}_2$, and thus when integrated to (B2) should not contain this term either, because it is zero. Nevertheless, this term appears as an overall non-zero constant in (B2), because $F_2(t) = -D\cdot n$ is a non-zero constant. To solve this contradiction such that the integrated solution (B2) is consistent with its underlying equation (B1), which again does not contain the term involving $\tilde{U}_{\!1}\tilde{x}_2$, this term appearing on the right-hand side of (B2) has to be transported and to be absorbed into the invariant tilde-expression on the left-hand side of (B2), which is the collection pool of all integration constants. But definitely not as done in (B3), by re-identifying the invariant constant $\tilde{U}_{\!1}\tilde{x}_2$ back to the non-invariant expression $U_1 x_2$ and then by rewriting $F_2(t)$ as an expression of $F_1(t)$ using relation (3.11). Hence, the derivation of (B3) is not mathematically sound since the round-bracketed term on the right-hand side actually belongs on the left-hand side of (B3). In fact, when comparing the classical scaling in Fig.4(d) with the corresponding new scaling in Fig.6 described by (B3), no improvement can be seen. In fact, the new scaling for the Reynolds stress $R^0_{12}$ is even weaker than the classical scaling, since next to the already existing problematic region around $\tilde{x}_2=1$, a new problematic region around $\tilde{x}_2=1.5$ is induced which does not arise in classical scaling.
Besides this issue, it is further to note that also the plot for this newly scaled Reynolds stress $\tilde{R}\vphantom{R}^0_{12}$ as shown in Fig.6 cannot be fully reproduced too. It is off by a global scaling factor of nearly 3. The correct plot for $\tilde{R}\vphantom{R}^0_{12}$ according to the final scaling (B5) is shown in Fig.4 in https://www.researchgate.net/publication/328080340, revealing that the maximum value for $\tilde{R}\vphantom{R}^0_{12}$ is nearly three times higher than shown in Fig.6.
In addition, their study also faces the following minor problem:
(6) The title chosen for this study is misleading. Nowhere throughout this study any scaling laws are derived. Instead, scaling relations are derived with the aim to let numerical profiles of a certain field variable collapse. The functional structure of these collapsing profiles, however, remain unknown. Thus no prediction on the scaling behaviour of the statistical solutions is made, as incorrectly claimed on p.251 and already discussed above in (3). For example, when integrating the invariant surface condition for the mean velocity profile, i.e., when integrating the first three terms of (3.1), the solution will be an arbitrary function in terms of an arbitrary invariant variable involving next to $x_2$ the arbitrary temporal functions $F_1$, $F_2$ and $F_3$. And knowing from the above discussion (3) that the invariant surface condition (3.1) is only a specification and not in its most general form, the arbitrariness in the set of invariant solutions is even higher from the outset than given by (3.1). To obtain invariant scaling laws, a systematic Lie-group symmetry analysis is not of much help here, since it just shifts the arbitrariness from one function to another. For a further discussion of this point, see e.g. https://arxiv.org/abs/1412.3069 and https://arxiv.org/abs/1609.08155.