Small-scale two-dimensional turbulence shaped by bulk viscosity


The Stokes’ hypothesis that bulk viscosity can be set to zero in turbulent flows continues to be applied in various applications as it is the basis of current continuum-medium models i.e. the Navier-Stokes equations. In past reviews, evidences have emerged that carbon dioxide can exhibit bulk- to shear-viscosity ratios of more than three orders of magnitude. Consequently, research findings support the assumption that bulk viscosity has no effects on the solenoid kinetic turbulence energy. With the rapid increase in the applications of high speed and turbulent flows of carbon dioxide, the need for accurate prediction tools for aerothermal loads and turbulent mixing is very crucial. As a result, there is a need to address the growing concerns regarding the efficiency and reliability of Stokes’ hypothesis and a justification of the surprising earlier findings.

In a recent paper published in the Journal of Fluid Mechanics, Dr. Emile Touber who is currently an honorary senior lecturer at the Imperial College London, Department of Mechanical Engineering argued that such possible findings and conclusions may have emanated from configurations limitations.  Dr. Touber looked carefully at the effects of bulk viscosity on the turbulence kinetic energy redistribution based on the exact solution of linearized Navier-Stokes equations. The illustrations were provided in two-dimensional turbulence.

Two distinct asymptotic regimes in the eigenmodes of Navier-Stokes equations: Stokes-Newton and Euler-Landau were utilized. Euler-Landau reported minor effects on the kinetic turbulence energy upon damping of the acoustic and entropy waves. However, small-scale enstrophy was highly promoted by the prevailing diffused and isothermal compressions in the Stokes-Newton regime. Whereas the transition of the Stokes-Newton regime occurred in the absence of bulk viscosity, it was worth noting that it was not observed in practice as opposed to the effects of the high bulk viscosities. Thus, turbulence flows with large bulk- to shear-viscosity and substantive inertial ranges should experience improved transfers to small-scale solenoid kinetic energy. This could further enhance the rate of dissipation and modifications of the heat transfer properties.

Based on the experimental results, Dr. Touber presented scaling arguments bringing out the importance of bulk viscosity in turbulence kinetic energy flow and thus stating the reason as to why it should not be ignored in predicting of aerothermal loads in external and internal turbulent flows. Despite being formulated on one-dimensional and linearized flow systems, the Stokes-Newton and Landau-Euler regimes remained relevant for three-dimensional flows. The author reported that the two-dimensional turbulence presented exhibited features that showed excellent agreement with the predictions obtained from the linearized one-dimensional flow analysis.

This work has been identified by the Advances in Engineering selection committee as a key contributor to the study of bulk-viscosity effects on turbulence. In a statement to AIE, Dr. Touber highlighted that the proposed theory together with the insightful interpretation of the past results offers useful guidelines to the design of experimental studies that can as well be extended to advanced three-dimensional turbulence studies.

Small-scale two-dimensional turbulence shaped by bulk viscosity - Advances in Engineering

About the author

Professor Emile Touber leads the Shocks, Solitons and Turbulence research unit at the Okinawa Institute of Science and Technology (Japan). Prior to this, he was a Lecturer and Senior Lecturer at the Department of Mechanical Engineering at Imperial College London (2012-2019).

Emile is a fluid dynamicist concerned with compressible and turbulent flows. His research unit studies how the kinetic energy of turbulence is redistributed in space and time in compressible and shocked substances with applications in both industrial (aero-thermal loads, mixing properties, flow control) and natural (geophysical flows, cosmology, vegetation-pattern dynamics) environments.


Touber, E. (2019). Small-scale two-dimensional turbulence shaped by bulk viscosity. Journal of Fluid Mechanics, 875, 974-1003.

Go To Journal of Fluid Mechanics

Check Also

Transition behavior from Mode I cracking to crystallographic cracking in a Ni-base single crystal superalloy - Advances in Engineering

Transition behavior from Mode I cracking to crystallographic cracking in a Ni-base single crystal superalloy