Revisiting the Lie-group symmetry method for turbulent channel flow with wall transpiration (original) (raw)
Related papers
A fact-check is presented to a Webinar recently held at the Australasian Fluid Mechanics Society (AFMS) on symmetries and its applications in turbulence [M. Oberlack, Oct. 14th, 2020: https://www.afms.org.au/events.html\]. More than a dozen of statements made therein prove to be misleading and partly even mathematically incorrect. No knowledge in Lie-group symmetry theory is required to recognize, for example, that the presented scaling theory for turbulent channel and pipe flow is misleading: Instead of matching the determined scaling laws to the relevant fluctuation correlations, they are matched to the irrelevant correlations of the full velocity field, which in configurations with a strong mean flow, as in channel or pipe flow, effectively just equate to the exponentiated mean velocity field. In other words, the presented approach only proves to be a curve-fitting process for the exponentiated mean velocity, and not for the physically interesting and relevant fluctuation correlations. But not only misleading, also mathematically incorrect statements are made: For example, two nonphysical invariants are used, which generate the scalings not only in the wrong way, but also give a false picture of the intermittency and the non-Gaussian behavior in turbulence. Ultimately, the main message with the fact-check here is that the Lie-group symmetry method alone, like any other analytical method, cannot bypass the closure problem of turbulence, as is falsely claimed.
2014
A detailed theoretical investigation is given which demonstrates that a recently proposed statistical scaling symmetry is physically void. Although this scaling is mathematically admitted as a unique symmetry transformation by the underlying statistical equations for incompressible Navier-Stokes turbulence on the level of the functional Hopf equation, by closer inspection, however, it leads to physical inconsistencies and erroneous conclusions in the theory of turbulence. The new statistical symmetry is thus misleading in so far as it forms within an unmodelled theory an analytical result which at the same time lacks physical consistency. Our investigation will expose this inconsistency on different levels of statistical description, where on each level we will gain new insights for its non-physical transformation behavior. With a view to generate invariant turbulent scaling laws, the consequences will be finally discussed when trying to analytically exploit such a symmetry. In fact, a mismatch between theory and numerical experiment is conclusively quantified. We ultimately propose a general strategy on how to not only track unphysical statistical symmetries, but also on how to avoid generating such misleading invariance results from the outset. All the more so as this specific study on a physically inconsistent scaling symmetry only serves as a representative example within the broader context of statistical invariance analysis. In this sense our investigation is applicable to all areas of statistical physics in which symmetries get determined in order to either characterize complex dynamical systems, or in order to extract physically useful and meaningful information from the underlying dynamical process itself.
Symmetries and scaling-laws in turbulence
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1999
A new turbulence approach based on Lie-group analysis is presented. It unifies a large set of self-similar solutions for the mean velocity of stationary parallel turbulent shear flows. The theory is derived from the Reynolds averaged Navier-Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. For the plane case the results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile that corresponds to the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law conforms to both the centre and the near wall regions of turbulent channel flows. FOT a non-rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both laws having equal scaling exponents. I n case of a rapidly rotating pipe a new logarithmic scaling law for the axial velocity is developed.
2022
A recent Letter by Oberlack et al. [Phys. Rev. Lett. 128, 024502 (2022)] claims to have derived new symmetry-induced solutions of the non modelled statistical Navier-Stokes equations of turbulent channel flow. A high accuracy match to DNS data for all streamwise moments up to order 6 is presented, both in the region of the channel-center and in the inertial sublayer close to the wall. Here we will show that the findings and conclusions in that study are highly misleading, as they give the impression that a significant breakthrough in turbulence research has been achieved. But, unfortunately, this is not the case. Besides trivial and misleading aspects, we will demonstrate that even basic turbulence relevant correlations as the Reynolds-stress cannot be fitted to data using the proposed symmetry-induced scaling laws. The Lie-group symmetry method as used by Oberlack et al. cannot bypass the closure problem of turbulence. It is just another assumption-based method that requires modelling and is not, as claimed, a first-principle method that leads directly to solutions. Next to PRL, two more papers by Oberlack et al. are called out for correction or a retraction.
Symmetries and turbulence modeling. A critical examination
The recent study by Klingenberg, Oberlack & Pluemacher (2020) proposes a new strategy for modeling turbulence in general. A proof-of-concept is presented therein for the particular flow configuration of a spatially evolving turbulent planar jet flow, coming to the conclusion that their model can generate scaling laws which go beyond the classical ones. Our comment, however, shows that their proof-of-concept is flawed and that their newly proposed scaling laws do not go beyond any classical solutions. Hence, their argument of having established a new and more advanced turbulence model cannot be confirmed. The problem is already rooted in the modeling strategy itself, in that a nonphysical statistical scaling symmetry gets implemented. Breaking this symmetry will restore the internal consistency and will turn all self-similar solutions back to the classical ones. To note is that their model also includes a second nonphysical symmetry. One of the authors already acknowledged this fact for turbulent jet flow in a formerly published Corrigendum (Sadeghi, Oberlack & Gauding, 2020). However, the Corrigendum is not cited and so the reader is not made aware that their method has fundamental problems that lead to inconsistencies and conflicting results. Instead, the very same nonphysical symmetry gets published again. Yet, this unscientific behaviour is not corrected, but repeated and continued in the subsequent and further misleading publication Klingenberg & Oberlack (2022), which is examined in this update in the appendix.
The study by Oberlack et al. (2006) consists of two main parts: a direct numerical simulation (DNS) of a turbulent plane channel flow with streamwise rotation and a preceding Lie-group symmetry analysis on the two-point correlation equation (TPC) to analytically predict the scaling of the mean velocity profiles for different rotation rates. We will only comment on the latter part, since the DNS result obtained in the former part has already been commented on by Recktenwald et al. (2009), stating that the observed mismatch between DNS and their performed experiment is possibly due to the prescription of periodic boundary conditions on a too small computational domain in the spanwise direction. By revisiting the group analysis part in Oberlack et al. (2006), we will generate more natural scaling laws describing better the mean velocity profiles than the ones proposed. However, due to the statistical closure problem of turbulence, this improvement is illusive. As we will demonstrate, any arbitrary invariant scaling law for the mean velocity profiles can be generated consistent to any higher order in the velocity correlations. This problem of arbitrariness in invariant scaling persists even if we would formally consider the infinite statistical hierarchy of all multi-point correlation equations. The closure problem of turbulence simply cannot be circumvented by just employing the method of Lie-group symmetry analysis alone: as the statistical equations are unclosed, so are their symmetries! Hence, an a priori prediction as how turbulence scales is thus not possible. Only a posteriori by anticipating what to expect from numerical or experimental data the adequate invariant scaling law can be generated through an iterative trial-and-error process. Finally, apart from this issue, also several inconsistencies and incorrect statements to be found in Oberlack et al. (2006) will be pointed out.
Turbulent Scaling Laws and What We Can Learn From the Multi-Point Correlation Equations
We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Furthermore, we consider a rotating channel flow, whose scaling behavior can only be described using the new statistical symmetries. It can be seen that the direction of rotation axes plays an important role, because different axes result in very different scaling laws.
Theories of Turbulence, 2002
First a short introduction to the notion of symmetries of differential equations is given including infinitesimal transformations, invariant functions and invariant solutions. Then it is shown that the symmetry properties i.e. invariant transformations of the Navier-Stokes equations are pivotal to understand the physics of fluid flow. We demonstrate that all common symmetries "transfer" to the statistical equations such as the Reynolds stress transport equations or the multi-point correlation equations. From the knowledge of the symmetries we derive from the latter equations a broad variety of invariant solutions (scaling laws) using only first principles. These solutions comprise classical results such as the logarithmic-law-of-the-wall and other wall bounded shear flows. Also homogeneous and inhomogeneous time-dependent flows are analyzed and solutions are discussed. Since the symmetries of fluid motion are admitted by all statistical quantities of turbulent flows we give necessary conditions on turbulence models such that they "capture" the proper physics i.e. the symmetries and their corresponding invariant solutions. Particularly we will investigate two-equation models such as the kmodel as well as Reynolds stress transport models with respect to their symmetry properties. Finally we give conditions for the sub-grid scale model in large-eddy simulation of turbulence to obey the proper symmetries. For all of the latter turbulence models it is demonstrated that symmetry violation gives rather disadvantageous prediction capabilities of the model under investigation.
This is an informal collection of comments, remarks and discussions which I received and which I had with various experts who interacted with my Fact-Check of the AFMS 2020-Webinar: “Symmetry induced turbulent scaling laws for arbitrary moments and their validation with DNS and experimental data”. This supplement will provide additional information as well as some explanations to certain statements I made but which I did not explicate in the Fact-Check because I wanted to keep it concise and not lose focus on the central issues of the Webinar. In my opinion, all comments, explanations and remarks presented are valuable additions and provide useful insights to be shared publicly. Also included will be the correct answers to questions the participants had in the Q&A round of the Webinar.
Symmetries and their importance for statistical turbulence theory
Mechanical Engineering Reviews, 2015
The present article is intended to give a broad overview and present details on the Lie symmetry induced statistical turbulence theory put forward by the authors and various other collaborators over the last twenty years. For this is crucial to understand that our present textbook knowledge proclaims that Lie symmetries such as Galilean transformation lie at the heart of fluid dynamics. These important properties also carry over to the statistical description of turbulence, i.e. to the Reynolds stress transport equations and its generalization, the multi-point correlation equations (MPCE). Interesting enough, the MPCE admit a much larger set of symmetries, in fact infinite dimensional, subsequently named statistical symmetries. Apart from the MPCE also the two other known complete theories of turbulence, the Lundgren-Monin-Novikov (LMN) hierarchy of probability density functions and the Hopf functional theory, share this property of admitting both classical mechanical and statistical Lie symmetries. As the Galilean transformation illuminates fundamental properties of classical mechanics, the new statistical symmetries mirror key properties of turbulence such as intermittency and non-gaussianity. After an introduction to Lie symmetries have been given, these facts will be detailed for all three turbulence approaches i.e. MPCE, LMN and Hopf approach. From a practical point of view, these new symmetries have important consequences for our understanding of turbulent scaling laws. The symmetries form the essential foundation to construct exact solutions. Presently we detail this only for the infinite set of MPCE, which in turn are identified as classical and new turbulent scaling laws. Examples on various classical and new shear flow scaling laws including higher order moments will be presented. Even new scaling have been forecasted from these symmetries and in turn validated by DNS.