# FY2021 Annual Report

**Continuum Physics Unit
Professor Gustavo Gioia**

## Abstract

The Continuum Physics Unit carried out research related to the mean-velocity profile and the turbulent-energy spectrum of both thermally-stratified plane-Couette flows and pipe flows.

## 1. Staff

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

## 2. Collaborations

### 2.1 The mean-velocity profile of thermally-stratified flows

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

## 3. Activities and Findings

### 3.1 The mean-velocity profile of thermally-stratified plane-Couette flows

During 2021 we completed this multi-year research project in collaboration with the fluid mechanics unit; the latest findings were published in Fluid Dynamics Research on December 30, 2021. The mean-velocity profile (MVP) of a wall-bounded flow, u(y), is the time-averaged streamwise velocity u expressed as a function of the distance to the wall, y. In canonical uniform-density, wall-bounded turbulent flows (including channel flow, plane-Couette flow, and boundary layer flow), the classic scaling laws of the MVP are the law of the wall, the log law, and the defect law. Furthermore, the MVP may be computed using the spectral-link theory of the MVP (Gioia et al. 2010 Phys. Rev. Lett. 105 184501), in which the agents of mass and momentum transfer are the turbulent eddies of Kolmogorov's turbulent energy spectrum E(k). The purpose of the present research is to develop a cognate theory of the MVP of thermally-stratified, wall-bounded turbulent flows, which theory may be used to predict complete MVP for such flows. Examples of thermally-stratified, wall-bounded turbulent flows include the flow of air in the atmospheric boundary layer, the approximately 1 km thick region enveloping the surface of Earth. Interaction with this surface, which entails the no-slip condition and is subject to daily cycles of heating and cooling, make the flow turbulent via two concurrent effects: (1) frictional drag associated with shear-stress production at the wall and (2) heat transfer, which makes the flow thermally stratified and induces buoyancy. It has long been known from measurements that the MVP of thermally-stratified, wall-bounded turbulent flows deviated significantly from its uniform-density counterpart. Theoretical analyses have been circumscribed to the 'intermediate layer' of the MVP, equivalent to the celebrated 'log layer' of the classic log law. In contrast to such analyses, we seek to compute the shape and scaling of the MVP of thermally-stratified plane-Couette flows over the whole extent of the flows. To develop a spectral-link theory, we consider the whole structure of E(k)—inertial range plus the energetic-range and the dissipative-range corrections—as well as the whole extent of the MVP—from the wall to the centerline. Noting that very close to the wall shear dominates over buoyancy, we are able to derive a simplified form of the heat equation. This simplified form of the heat equation may be solved along with the momentum equation to compute a complete MVP that captures all of the salient qualitative features of the empirical MVP over its whole extent---that is to say, from wall to centerline. It is thus demonstrated that the distinctive shape of the MVP can be understood in terms of the energetics of the turbulent eddies (described by E(k)). The predictions of the theory are put to use in successfully testing generalized scalings obtained via dimensional analysis and the assumption of complete similarity. Our findings are in good accord with currently available empirical data and provide guidance as to the type of empirical data that would be most useful to gather in future.

### 3.2 Features of the turbulent mean velocity profile of pipe flow

This is a project started during f.y. 2020 aimed at elucidating the etiology of certain hitherto ignored features of the turbulent mean velocity profile (MVP) of turbulent pipe flow. Preliminary results indicated that these features of the MVP (which are distinct in computer-simulation data and also, as we have recently been able to show, in experimental data, albeit substantially obscured, in this latter case, by noise) may be closely connected to certain well-known but poorly understood features of the fabric of turbulence as embodied in the spectrum of turbulent energy. The latest development in connection to this project is the formulation of a technique which allows us to revert the spectral link of the MVP, which starts with a model of the spectrum and yields the MVP. We are now able to extract data on the viscous-range correction, for example, directly from computational data on the mean-velocity profile or on the frictional drag. By virtue of this new technique, much currently available data that are in principle unrelated to the spectrum becomes relevant to it, with implications beyond the scope of this particular project.

## 4. Publications

### 4.1 Journals

Nothing to report

### 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.