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On the Boussinesq approximation in arbitrarily accelerating frames of reference

Published online by Cambridge University Press:  11 August 2021

Hugh M. Blackburn Open the ORCID record for Hugh M. Blackburn [Opens in a new window] ,
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window] ,
Jagmohan Singh Open the ORCID record for Jagmohan Singh [Opens in a new window]  and
Alexander J. Smits Open the ORCID record for Alexander J. Smits [Opens in a new window]
Hugh M. Blackburn*
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia
Juan M. Lopez
Affiliation:
School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Jagmohan Singh
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia
Alexander J. Smits
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

This work examines consequences of modelling approximation errors made within the context of the Navier–Stokes–Boussinesq system. Starting from a canonical Boussinesq model, where density fluctuations are allowed to interact with all accelerative terms of the incompressible Navier–Stokes equations in arbitrarily accelerating reference frames, a unified treatment is developed that provides a straightforward way to identify buoyancy forcing associated with gravitational effects, centrifugal forcing associated with frame rotation, as well as centrifugal-type forcing due to variations in flow kinetic energy. The results of the cases studied in inertial, rotating and mixed reference frames demonstrate that in general it may be important to apply buoyancy effects to all non-local accelerative terms, including non-gradient terms such as Coriolis acceleration. Additionally, it is shown that the common practice of ignoring terms representing interaction between density fluctuation and local fluid acceleration can lead to non-negligible error in Boussinesq modelling of highly unsteady flows. These findings have special significance for accurate simulation of flows with density variations in which there may be both background rotation and localised regions of strong swirl, but are also relevant for studies conducted in the inertial frame of reference.

JFM classification

Convection: Buoyancy-driven instability Geophysical and Geological Flows: Rotating flows Mathematical Foundations: General fluid mechanics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 924 , 10 October 2021 , R1
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Barcilon, V. & Pedlosky, J. 1967 On the steady motions produced by a stable stratification in a rapidly rotating fluid. J. Fluid Mech. 29, 673–690.CrossRefGoogle Scholar
Blackburn, H.M., Lee, D., Albrecht, T. & Singh, J. 2019 Semtex: a spectral element–Fourier solver for the incompressible Navier–Stokes equations in cylindrical or Cartesian coordinates. Comput. Phys. Commun. 245, 106804.CrossRefGoogle Scholar
Brummell, N., Hart, J.E. & Lopez, J.M. 2000 On the flow induced by centrifugal buoyancy in a differentially-heated rotating cylinder. Theor. Comput. Fluid Dyn. 14, 39–54.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press. Dover edn, 1981.Google Scholar
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, 2nd print.Google Scholar
Hart, J.E. 2000 On the influence of centrifugal buoyancy on rotating convection. J. Fluid Mech. 403, 133–151.CrossRefGoogle Scholar
Lopez, J.M., Marques, F. & Avila, M. 2013 The Boussinesq approximation in rapidly-rotating flows. J. Fluid Mech. 737, 56–77.CrossRefGoogle Scholar
Marques, F., Mercader, I., Batiste, O. & Lopez, J.M. 2007 Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities. J. Fluid Mech. 580, 303–318.CrossRefGoogle Scholar
Pitz, D.B., Marxen, O. & Chew, J.W. 2017 Onset of convection induced by centrifugal buoyancy in a rotating cavity. J. Fluid Mech. 826, 484–502.CrossRefGoogle Scholar
Tritton, D.J. 1988 Physical Fluid Dynamics, 2nd edn. Oxford University Press.Google Scholar