Fluid Mechanics at Interfaces 2

1 Cover

2 Title Page

3 Copyright First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2022 The rights of Roger Prud’homme and Stéphane Vincent to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021949304 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-817-7

4 Preface

5 1 Turbulent Channel Flow to Reτ = 590 in Discrete Mechanics 1.1. Introduction 1.2. Discrete mechanics formulation 1.3. Turbulent flow in channel 1.4. Conclusion 1.5. References

6 2 Atomization in an Acceleration Field1 2.1. Introduction 2.2. Generation of droplets through vibrations normal to the liquid layer 2.3. Rayleigh–Taylor instability at the crest of an axial wave 2.4. Recent work 2.5. Conclusion 2.6. References

7 3 Numerical Simulation of Pipes with an Abrupt Contraction Using OpenFOAM 3.1. Introduction 3.2. Modeling an abrupt contraction in a pipe 3.3. Numerical results 3.4. Conclusion and future prospects 3.5. References

8 4 Vaporization of an Equivalent Pastille 4.1. Introduction 4.2. Equations for the problem 4.3. Linear analysis of the liquid phase 4.4. Some results 4.5. Conclusion 4.6. References

9 5 Thermal Field of a Continuously-Fed Drop Subjected to HF Perturbations 5.1. Drops in a liquid-propellant rocket engine 5.2. A continuously fed droplet 5.3. Equations of the problem 5.4. Linearized equations 5.5. Linearized equations for small harmonic perturbations 5.6. Thermal field in the drop when neglecting internal convection 5.7. Conclusion 5.8. Appendix 1: Coefficients that come into play in linearized equations 5.9. Appendix 2: Solving the thermal equation 5.10. Appendix 3: The case of the equivalent pastille 5.11. Appendix 4: 2D representation for the spherical drop 5.12. References

10 6 Study of the Three-Dimensional and Non-Stationary Flow in a Rotor of the Savonius Wind Turbine 6.1. Introduction 6.2. Mathematical modeling of the problem 6.3. Numerical resolution 6.4. Validation of the results 6.5. Results and discussion 6.6. Conclusion 6.7. Acknowledgments 6.8. References

11 List of Authors

12 Index

13 Summary of Volume 1

14 End User License Agreement

1 Chapter 1 Table 1.1. Curvature in the zone y+ > δn calculated by model-free simulati...

2 Chapter 4Table 4.1. Six types of configuration when there is an exchange coefficient at t...

1 Chapter 1Figure 1.1. Discrete geometric structure: a set of primitive planar facets S are... Figure 1.2. Turbulent channel with Reτ = 590; average velocity profiles in reduc... Figure 1.3. (a) Turbulent channel with Reτ = 590 (Denaro et al. 2011), r = f(y+)... Figure 1.4. Turbulent channel with Reτ = 590, results of unit-2 (Denaro et al. 2... Figure 1.5. Turbulent channel with Reτ = 590, comparison between the results of ...Figure 1.6. Turbulent channel with Reτ = 590. Results of the TDM model (dots) co...Figure 1.7. Turbulent channel with Reτ = 590. The reduced turbulent viscosity μd...Figure 1.8. Turbulent channel with Reτ = 590, results from MKM99 (Moser et al. 1...

2 Chapter 2Figure 2.1. (a) Rayleigh–Taylor instability produced by carefully injecting hot,...Figure 2.2. The atomization of a solution of distilled water and 80% glycerin, a...Figure 2.3. Liquid mass excited by ultrasonic oscillation. The Oz axis is orient...Figure 2.4. Stability diagram of a liquid layer subject to forced oscillation: s...Figure 2.5. Diagram of the device used by Ben Rayana, Cartellier and Hopfinger (...Figure 2.6. Study of the crest of the axial wave resulting from a convective Kel...Figure 2.7. Transverse waves with the wavelength λTR ∝ UG −1/4 (UG − UC)−1 (seen...Figure 2.8. Rupture regimes in space for the parameters ReL/We. The lines with c...

3 Chapter 3Figure 3.1. Comparison of the three solvers. For a color version of this figure,...Figure 3.2. (a) Cartesian domain; (b) cylindrical domainFigure 3.3. (a) Relaxed Cartesian mesh; (b) refined Cartesian mesh; (c) relaxed ...Figure 3.4. Boundary and initial conditions IFigure 3.5. Boundary conditions and initial conditions IIFigure 3.6. Upstream and downstream positionsFigure 3.7. Different sections showing the upstream and downstream positions 1, ...Figure 3.8. (a) Mean flow rate over space during the outward movement; (b) mean ...Figure 3.9. (a) Mean flow rate for position 1 during the outward movement; (b) m...Figure 3.10. (a) Mean flow rate for position 1 during the outward movement; (b) ...Figure 3.11. Left: volume contained between the upstream position 1 and downstre...Figure 3.12. Variation of the mass m(Vi, t) over time. For a color version of th...Figure 3.13. (a) Velocity vector field for a relaxed mesh; (b) velocity vector f...Figure 3.14. (a) Contour lines for a relaxed mesh; (b) contour lines for a refin...Figure 3.15. (a) Isosurfaces for a relaxed mesh; (b) isosurfaces for a refined m...Figure 3.16. Calculation cost in minutes. For a color version of this figure, se...Figure 3.17. Calculation cost ratio. For a color version of this figure, see www...Figure 3.18. (a) Mean flow rate for position 1 during an outward movement associ...Figure 3.19. (a) Vector field for a non-refined mesh; (b) vector field for a ref...

4 Chapter 4Figure 4.1. Geometric configuration of a pastille equivalent to a sphere with di...Figure 4.2. Droplet with radius r̄s fed with liquid with a flow rate and the e...Figure 4.3. Influence of the exchange coefficient with the values being identica...Figure 4.4. Influence of the Péclet number (inverse of θ) with the values being ...Figure 4.5. Curves showing the reduced response factor N /α as a function of the...Figure 4.6. Differences between the temperature field in the liquid for two cond...

5 Chapter 5Figure 5.1. Physical phenomena brought into play in a cryogenic, coaxial injecto...Figure 5.2. (a) Packet of drops in combustion in a schematized combustion chambe...Figure 5.3. Spherical drop with adiabatic feeding regime: the influence of θ on ...Figure 5.4. Influence of the Péclet number PeL (inverse of θ) with all other par...Figure 5.5. Pastille with adiabatic feeding regime (h = 0). For a color version ...Figure 5.6. Pastille with isothermal feeding regime (h is infinite). For a color...Figure 5.7. Spherical drop with adiabatic feeding regime: the influence of θ on ...

6 Chapter 6Figure 6.1. Description of a Savonius rotor and the calculation field. For a col...Figure 6.2. Presentation of the three frames used. For a color version of this f... Figure 6.3. Meshes in the blade-to-blade plane and meridian plane. For a color v...