On symmetries and averaging of the G-equation for premixed combustion

Published in Combustion Theory and Modelling, 2001 by Oberlack, Wenzel & Peters.

This publication is misleading and flawed.

Although first correctly revealed as flawed by V.A. Sabel'nikov & A.N. Lipatnikov in https://doi.org/10.1260/1756-8277.2.4.301, I want to lay open a further and even more fundamental problem of this study by Oberlack et al. which has been overlooked and not addressed so far:

The generalized scaling symmetry (15a), or in its finite form (21a), is physically void, i.e., it cannot be exploited or used as a guideline for dynamical modelling, as it is misleadingly claimed in Section 3. The reason is that this symmetry does not participate in the dynamics of the flame surface. It is just an irrelevant relabelling symmetry without having any influence or impact on the dynamics of the $G$-equation. Sure, this symmetry defines the $G$-equation, as was shown in Section 2.3, but it has no consequences on the dynamics (i.e. on the solutions) of this equation, simply because this symmetry does not map existing solutions to new solutions, but only every existing solution into itself, however, such that only the dependent variable $G$ gets transformed while all dynamics-defining variables as time, space, velocity and pressure fields stay unchanged. Hence, this generalized scaling symmetry (15a) effectively acts like an identity transformation, and therefore can be safely neglected when modelling turbulent combustion. In the following, I will briefly show why:

For convenience, let's take the specified scaling symmetry $G^*=\exp(G)$ as given in Section 3 (p.374). Since this transformation $G\rightarrow G^*=\exp(G)$  is a symmetry of the $G$-equation, it will leave it invariant. But not only the equation, also any initial condition of the $G$-equation will be left invariant.

Proof: Any initial condition of the $G$-equation consists of two parts: (A) Specification of the surface: $G\vert_{t=0} = g_0(x,y,z)$, and (B) specification of the level set: $G\vert_{t=0} = G_0=\text{const.}$  Without restricting the general case, let's choose e.g. a sphere around the origin as the initial surface: $g_0(x,y,z)=x^2+y^2+z^2$, and e.g. the value $G_0=1$ as the level set. The initial condition of the $G$-equation is then given as the iso-surface $x^2+y^2+z^2=1$. Now, if this iso-surface is transformed solely by the scaling symmetry $G^*=\exp(G)$, it will stay invariant, because all other variables like e.g. space and time are not transformed. The initial iso-surface will thus just be invariantly transformed to $x^{*2}+y^{*2}+z^{*2}=1$. 

This result can also be derived by using the scaling symmetry explicitly: First by transforming the surface condition (A) $$G\vert_{t=0} = x^2+y^2+z^2\;\: \Leftrightarrow\;\: \ln(G^*)\vert_{t^*=0} = x^{*2}+y^{*2}+z^{*2},$$ which then leads to the transformed initial surface $G^*\vert_{t^*=0} = \exp(x^{*2}+y^{*2}+z^{*2})$, and secondly by transforming the level set condition (B) $$G\vert_{t=0} = 1\;\: \Leftrightarrow\;\: \ln(G^*)\vert_{t^*=0} = 1,$$ which then leads to the transformed level set $G^*\vert_{t^*=0} = \exp(1)$. Combining these two results, we finally get the initial condition of the transformed $G^*$-equation, namely $\exp(x^{*2}+y^{*2}+z^{*2}) = \exp(1)$, which is equivalent to $x^{*2}+y^{*2}+z^{*2}=1$ and thus identical to the result obtained before.

Hence, the generalized scaling symmetry (15a) transforms any initial value problem, namely the $G$-equation plus any initial condition exactly into itself again (i.e. with same functional structure $g_0$ and value $G_0$): $$\frac{\partial G}{\partial t}+(\mathbf{u}\cdot\nabla)G=s_L|\nabla G| \;\: \Leftrightarrow\;\; \frac{\partial G^*}{\partial t^*}+(\mathbf{u}^*\cdot\nabla^*)G^*=s_L|\nabla^* G^*|,$$ $$\text{and}\qquad G\vert_{t=0}=g_0(x)=G_0 \;\: \Leftrightarrow\;\; G^*\vert_{t^*=0}=g_0(x^*)=G_0.$$ In other words, the generalized scaling symmetry (15a) acts like an identity transformation on the solutions of the $G$-equation, because it not only leaves one particular but eventually all initial conditions for the $G$-equation invariant, with the effect that before and after the transformation we have or get the very same instantaneous dynamics and finally therefore, as a consequence, also the same averaged dynamics: $\langle G^*\rangle =\langle G\rangle$. Hence, no information at all on the solution space of the $G$-equation is revealed when applying this generalized scaling symmetry, and therefore can be safely ignored in any dynamical or turbulent modelling process.

To present this reasoning as clearly as possible, let it be repeated once again in different words: If the generalized scaling symmetry (15a) of the $G$-equation would only leave one particular initial condition (IC) invariant, then, of course, one may not conclude that it behaves as an identity transformation, simply because this particular IC chosen is just special in the sense that it is compatible with this symmetry. But if all possible ICs of the $G$-equation stay invariant under this symmetry, as it is the case here, then one can obviously only conclude that this symmetry is just a disguised identity transformation in solution space and thus physically trivial and empty.

Hence, the key conclusions by Oberlack et al. on p.380 that "a new generalized scaling symmetry is obtained which is of considerable importance for a variety of different purposes" and "that the generalized scaling symmetry cannot under any circumstances be neglected to derive statistical quantities for turbulent combustion" cannot be confirmed due to that they are based on misleading and flawed derivations.