Power plant engineers cannot predict when supercritical fluids—used at high pressure and temperature in clean energy systems—will abruptly switch from smooth to turbulent flow, because the underlying physics remains a mathematical blind spot. This project tackles a gap in fundamental fluid dynamics. Standard models assume density variations are small, but in strongly stratified flows—where viscosity and density change sharply—those models miss newly discovered instabilities that can trigger turbulence. Without a framework to capture these effects, engineers cannot know whether a given flow will remain laminar or become turbulent, nor how to encourage or suppress that transition. The researchers will develop a mathematical approach called resolvent analysis, adapted for flows with large density gradients. They will derive the governing equations, build a numerical tool to solve them, and validate it against existing data. This is fundamental science: the immediate goal is understanding, not a product. If successful, the framework could eventually help design more efficient power plants that use supercritical fluids. Because these fluids transfer heat far more effectively when turbulent, engineers could operate at lower flow rates while maintaining high heat transfer—potentially improving the efficiency of clean energy systems. The work may also open new avenues for modelling turbulent heat transfer in industrial energy applications.
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Flows with large gradients of viscosity and density have recently been found to exhibit new unstable behaviours. New linear energy growths have been detected, suggesting the existence of novel routes leading to turbulence. Fundamental understanding and mathematical models of these new phenomena are currently missing: the physical origin of some instabilities is unclear as well as the conditions under which they may prevail. Hence, the range of physical parameters for which these flows are laminar or turbulent is unknown. Even less understood is how we may act on these flows in order to enhance or impede transition to turbulence. Such questions have great practical implications in engineering. Strongly stratified flows are found, for example, in power plants that use supercritical fluids – fluids at high pressure and temperature. Bridging those gaps will eventually enable the design of more efficient clean energy systems by leveraging the potential of supercritical fluids to exist in a turbulent state (thereby ensuring high heat transfer) even at relatively low flow rates. This project aims to develop a mathematical framework, based on resolvent analysis, that models the linear dynamics of strongly stratified flows. Resolvent analysis is a modern approach in fluid dynamics. It is particularly well suited to study transition to turbulence, as it allows the largest linear energy growths to be identified whilst informing on the optimal conditions that trigger these growths. No such framework currently exists for stratified flows beyond the Boussinesq approximation. Under this assumption, inertial baroclinic effects are neglected, making such models unable to represent instabilities in strongly stratified flows, as recently shown in the literature. The present project addresses this issue by deriving the resolvent analysis using the low-Mach approximation. Based on an asymptotic expansion in terms of the vanishingly small Mach number, a framework modelling arbitrary large density variations can be obtained. Establishing this set of equations for an arbitrary equation of state constitutes the first objective of this project. Developing a numerical tool that solves it is the second objective. This will be achieved by employing a classical spectral method and a singular value decomposition of the resolvent matrix. A thorough validation of this new code will be conducted against data from the literature. The third objective is to obtain results that will demonstrate the applicability of the approach to identify the non-modal mechanisms at play in a laminar strongly stratified flow. Preliminary results will also be obtained on turbulent mean flows. The development of this novel theoretical and numerical framework will benefit the fluid dynamics community interested in stratified flows. Beyond transition to turbulence, the versatility of resolvent analysis can unlock new avenues of research regarding turbulence modelling, as it provides an efficient approach to model coherent structures. These structures, which underpin the statistics of turbulence, have never been studied in strongly stratified flows. This could yield new insights into the modelling of turbulent heat transfer in supercritical fluids which, along with the development of a novel operational computational tool, have the potential to foster technological and knowledge transfers to industrial partners in the energy sector.
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