# FY2017 Annual Report

**Fluid Mechanics Unit
Associate Professor Pinaki Chakraborty**

## Abstract

The Fluid Mechanics unit pursued research on turbulent flows, atmospheric flows, planetary flows, and granular flows, and continued to develop a joint fluid mechanics--continuum physics laboratory.

## 1. Staff

**Pinaki Chakraborty,**Associate Professor**Julio Manuel Barros Junior**, Staff Scientist**Rory Cerbus**, Postdoctoral Scholar**Tinihau Meuel**, Postdoctoral Scholar**Christian Butcher**, Research Unit Technician**Yuna Hattori**, PhD Student**Lin Li**, PhD Student**William Powell**, PhD Student**Kaori Egashira**, Research Unit Administrator

## 2. Collaborations

### 2.1 Theory of spectral link in turbulent flows

- Type of collaboration: Scientific collaboration
- Researchers:
- Professor Gustavo Gioia, OIST

### 2.2 Granular cratering

- Type of collaboration: Scientific collaboration
- Researchers:
- Professor Gustavo Gioia, OIST
- Dr. Tapan Sabuwala, OIST

### 2.3 Experiments on turbulent pipe flows

- Type of collaboration: Scientific collaboration
- Researchers:
- Professor Gustavo Gioia, OIST
- Jun Sakakibara, Meiji University, Japan

### 2.4 Experiments on Taylor-Couette flows

- Type of collaboration: Scientific collaboration
- Researchers:
- Dr. Yasuo Higashi, OIST
- Professor Gustavo Gioia, OIST

## 3. Activities and Findings

### 3.1 Spectral derivation of the classic laws of wall-bounded turbulent flows

We formulated a new derivation for the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law.’ Although these classic laws constitute standard discussion in textbooks of turbulent flows, their derivation either makes crucial assumptions about the MVP itself (the very topic of investigation) or invokes a specific model of turbulent eddies thereby inextricably tying the laws to the specific model. In our derivation, we mathematically linked the classic laws to the turbulent eddies without having recourse to any specific model of those eddies. In particular, building on our previous research on the spectral analogues and the spectral link, we showed that the classic laws can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also showed that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.

### 3.2 The third-order structure function in two-dimensional turbulent flows

In a celebrated paper from 1941, Andre Kolmogorov derived an exact relation for the third-order structure function, *S*_{3}(*r*), in three-dimensional (3D) turbulent flows. This relation is the arch-famous "4/5th law," which has been extensively verified by empirical data from experiments and computational simulations. By contrast, the situation for two-dimensional (2D) turbulent flows is starkly different. In 2D turbulent flows, two disparate cascades, changed dissipation anomalies, a large-scale drag, and other factors conspire to create several versions of the *S*_{3}(*r*) “law.” We study *S*_{3}(*r*) in 2D turbulence in the spectral space and show that *S*_{3}(*r*) generically embodies a mixture of energy and enstrophy fluxes. Building on this result, we derive *S*_{3}(*r*) laws for freely decaying and forced two-dimensional turbulent flows, where we also account for the effects of a large-scale drag, an inextricable feature of quasi two-dimensional turbulence in experimental and atmospheric flows.

### 3.3 Laws of resistance in transitional pipe flows

We continued to conduct experiments and direct numerical simulations of transitional pipe flows, training our focus on the laws of resistance: the law governing how the flow friction, *f*, varies with the Reynolds number, Re. This problem dates back to 1883, when Osborne Reynolds discovered the laws of resistance for laminar and turbulent flows, but for transitional flows he concluded that the laws were "either indefinite or very complex," and to date these laws remain unknown. Our work showed that the search for these laws have been hitherto unsuccessful because the studies lumped together the *f* from two distinct elements of transitional flows: the laminar plugs and the flashes. By properly distinguishing between them, we showed experimentally and numerically that the law of resistance for laminar plugs corresponds to the laminar law and the law of resistance for flashes is identical to that of turbulence.

### 3.4 Granular crater ray system: experiments and simulations

We continued to conduct experiments and direct numerical simulations of transitional pipe flows, training our focus on the laws of resistance: the law governing how the flow friction, *f*, varies with the Reynolds number, Re. This problem dates back to 1883, when Osborne Reynolds discovered the laws of resistance for laminar and turbulent flows, but for transitional flows he concluded that the laws were "either indefinite or very complex," and to date these laws remain unknown. Our work showed that the search for these laws have been hitherto unsuccessful because the studies lumped together the *f* from two distinct elements of transitional flows: the laminar plugs and the flashes. By properly distinguishing between them, we showed experimentally and numerically that the law of resistance for laminar plugs corresponds to the laminar law and the law of resistance for flashes is identical to that of turbulence.

## 4. Publications

### 4.1 Journals

- Gioia, G, Chakraborty, P. (2017) Spectral derivation of the classic laws of wall-bounded turbulent flows.
*Proceedings of the Royal Society, Mathematical, Physical, and Engineering Sciences*, vol. 473, PP.326-343, DOI: 10.1098/rspa.2017.0354 - Cerbus, R., Chakraborty, P. (2017) The third-order structure function in two dimensions: The Rashomon effect.
*Physics of Fluids*, vol 29, PP. 111110-1 to 111110-9, DOI: http://dx.doi.org/10.1063/1.5003399 - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P. (2018) The laws of resistance in transitional pipe flows.
*Physical Review Letters, American Physical Society*, vol. 120, DOI: https://doi.org/10.1103/PhysRevLett.120.054502

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

- Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Flashes in Reynolds's Turbulence*, Bethel University Physics Department, Minneapolis, MN USA, May 2 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Flashes in Reynolds's Turbulence*, Hong Kong University of Science and Technology Physics Department, Hong Kong, May 19 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Flashes in Reynolds's Turbulence*, Academia Sinica Department of Physics, Taipei, Taiwan, June 12 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Flashes in Reynolds's Turbulence*, National Central University Department of Physics, Taoyuan, Taiwan, June 14 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Turbulence in Transitional Pipe Flows*, OIST Mini-vortex Meeting, OIST, Japan, July 5 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*Turbulence in Transitional Pipe Flows*, Kyoto University RIMS Conference on the Mathematics of Inhomogeneous Turbulence, Kyoto, Japan, July 26 (2017) - Zhang, D., Gioia, G., Chakraborty, P.,
*Spectral theory of the mean-velocity profile in thermally stratified plane-Couette flows*, Kyoto University RIMS Conference on the Mathematics of Inhomogeneous Turbulence, Kyoto, Japan, July 26 (2017) - Gioia, G, Chakraborty, P.
*Inferring the turbulent-energy spectrum from the mean-velocity profile*, Kyoto University RIMS Conference on the Mathematics of Inhomogeneous Turbulence, Kyoto, Japan, July 27 (2017) - Zhang, D., Gioia, G., Chakraborty, P.,
*Spectral theory of the mean-velocity profile in thermally stratified plane-Couette flows*, 2nd International Conference in Aerospace for Young Scientists, Beijing, Beihang university, Beijing, China, September 7 (2017) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Hokkaido University, Sapporo, Japan, October 16 (2017) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Tohoku University, Sendai, Japan, October 19 (2017) - Chakraborty, P., Gioia, G.,
*The spectral link in turbulent frictional drag*, Tokyo Institute of Technology, Tokyo, Japan, October 20 (2017) - Chakraborty, P., Gioia, G.,
*The spectral link in turbulent frictional drag*, Keio University, Tokyo, Japan, November 9 (2017) - Chakraborty, P., Gioia, G.,
*The spectral link in turbulent frictional drag*, University of Tokyo, Tokyo, Japan, November 10 (2017) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Harvard University, Cambridge, USA, November 13 (2017) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Massachusetts Institute of Technology, Cambridge, USA, November 14 (2017) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Johns Hopkins University, Baltimore, USA, November 17 (2017) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*The laws of resistance in transitional pipe flows*, 70th Annual Meeting of American Physical Society Division of Fluid Dynamics (APS-DFD), Denver, Colorado, USA, November 19 (2017) - Chakraborty, P.,
*The spectral link in turbulent frictional drag*, Tata Institute of Fundamental Research, Colaba, Mumbai, Maharashtra, India, January 29 (2018) - Chakraborty, P., Gioia, G., Sabuwala, T.,
*Effect of rainpower on hurricane intensity,*Indian Institute of Tropical Meteorology, Pune, Maharashtra, India, February 2 (2018) - Chakraborty, P., Cerbus, R., Liu, C., Gioia, G.,
*The states of flow in transitional pipes*, Indian Institute of Technology, Mumbai, Maharashtra, India, February 7 (2018) - Cerbus, R., Liu, C., Gioia, G., Chakraborty, P.,
*The laws of resistance in transitional pipe flows*, 2018 International Conference on Mechanical and Aerospace Systems, Chengdu, China, March 17 (2018) - Zhang, D., Gioia, G., Chakraborty, P.,
*Spectral theory of the mean-velocity profile in thermally stratified plane-Couette flows*, 2018 International Conference on Mechanical and Aerospace Systems, Chengdu, China, March 17 (2018)

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

### 6.1 Seminars

1. **Skin-friction and vorticity fields in wall-bounded flows and the attached eddy hypothesis**

- Date: September 7
^{th}, 2017 - Venue: OIST Campus Lab1
- Speaker: Prof. M.S. Chong (University of Melbourne)

2. **Hierarchy of vortices in turbulence at high Reynolds numbers**

- Date: October 31
^{st}, 2017 - Venue: OIST Campus Lab1
- Speaker: Prof. Susumu Goto (Osaka University)

3. **Sustaining mechanism of turbulence in a precessing sphere**

- Date: October 31
^{st}, 2017 - Venue: OIST Campus Lab1
- Speaker: Yasuhumi Horimoto, Prof. Susumu Goto (Osaka University)

## 7. Other

Nothing to report.