# FY2020 Annual Report

**Continuum Physics Unit
Professor Gustavo Gioia**

## Abstract

The Continuum Physics Unit carried out experimental, computational and theoretical research on rough-walled turbulent friction in two and 3 dimensions, on the nature of "slugs" in transitional pipe flow, and on stratified-flow analogues of the law of the wall, the log law and the defect law.

## 1. Staff

- Gustavo Gioia, Professor
- Tomoe Owan, Research Unit Administrator

## 2. Collaborations

### 2.1 Theory of spectral link in turbulent flows

- Type of collaboration: Scientific collaboration
- Researchers:
- Prof. Pinaki Chakraborty, OIST

### 2.2 **Frictional drag in two-dimensional flows **

- Type of collaboration: Scientific collaboration
- Researchers:
- Prof. Pinaki Chakraborty, OIST
- Dr. Alexandre Vilquin, University of Bordeaux, France
- Prof. Hamid Kellay, University of Bordeaux, France
- Prof. Charles-Henri Bruneau, University of Bordeaux, France
- Prof. Patrick Fischer, University of Bordeaux, France

## 3. Activities and Findings

### 3.1 Rough-walled turbulent friction in 2 and 3 dimensions

During 2020 we completed this multi-year research project in collaboration with the fluid mechanics unit and researchers at the University of Bordeaux, France, and the results were published in Sci Adv on 29 January 2021. The project involved carrying out extensive, unprecedented experiments and direct numerical computations on rough-walled 2D flows (a kind of flow that may be realized experimentally in soap films), with the aim of verifying the predictions of the spectral theory of turbulent fluid friction in rough-walled flows. These predictions concern the asymptotic (Reynolds number-independent) relation between the fluid friction f and the relative roughness r/R (where r is the roughness lengthscale of the wall and R is the characteristic lengthscale of the flow), which according to the spectral theory depends on the exponent of the spectrum of turbulent energy, the "spectral exponent," which in turn depends on the dimesionality of the flow, being 5/3 in 3D (corresponding to the "energy cascade") and 3 in 2D (corresponding to the "enstrophy cascade"). Thus the spectral theory predicts f \propto (r/R)^1/3 (the classical Strickler scaling) and f \propto (r/R) in 3D and 2D, respectively, contrary to the classical theory of turbulent friction, which is constitutionally incapable of predicting what kind of difference, if any, should be expected between the f--r/R relation in 2D and the f--r/R relation in 3D. Both the experimental measurements and the numerical computations yielded results in full agreement with these predictions of the spectral theory. In view of these results, the disparity between the turbulent friction in 3D and 2D must be interpreted as a spectral phenomenon. A change in the spectrum brings about a predictable change in the attendant friction, which is the ultimate reason why the f--r/R relation for soap film flows turns out to differ from the f--r/R relation for pipe flows. The same can be said about the relevant scalings in smooth-walled flows (as we showed in previous work on the spectral theory). Our findings reveal that the long-standing, standard theory of friction is incomplete because, being predicated on Buckingham’s Pi theorem and plausible assumptions of similarity, it entails a noncommital, or indifferent, attitude toward the physical sources of friction. In contrast to the standard theory, the spectral theory singles out the turbulent eddies, and with them the spectrum, as the effective agents of shear-stress production at the interface between the flow and the wall, with the implication that spectrum and friction become linked to one another. In rough- and smooth-walled flows alike, friction proclaims the spectrum.

### 3.2 The etiology of certain hitherto ignored features of the turbulent mean velocity profile of turbulent pipe flow

In this project, we are using the spectral theory of the mean velocity profile (MVP) to elucidate a number of features of the MVPs of turbulent pipe flow. These features, which may be clearly revealed by suitable analysis of direct numerical simulations (and verified, if with some difficulty due to the presence of noise, in experimental measurements of the MVPs) have never been discussed, let alone explained, in the pertinent literature. Preliminary results indicate that they are closely connected to---and may in fact be said to be remarkable manifestations of---certain well-known but poorly understoond features of the fabric of turbulence as embodied in the spectrum of turbulent energy.

## 4. Publications

### 4.1 Journals

- A. Vilquin, J. Jagielka, S. Djambov, H. Herouard, P. Fisher, C-H Bruneau, P. Chakraborty, G. Gioia, and H. Kellay. 2021. Asymptotic turbulent friction in 2D rough-walled flows.
*Science Advances*, Vol. 7, no. 5, eabc6234. PDF

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

Nothing to report

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

Nothing to report

## 7. Other

Nothing to report.