The role of boundary conditions in scaling laws for turbulent heat transport

Camilla Nobili; Camilla Nobili

doi:10.3934/mine.2023013

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-41. doi: 10.3934/mine.2023013

Previous Article Next Article

Review Special Issues

The role of boundary conditions in scaling laws for turbulent heat transport

Camilla Nobili ^,

Department of Mathematics, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Received: 05 November 2021 Revised: 31 December 2021 Accepted: 31 December 2021 Published: 15 February 2022

In most results concerning bounds on the heat transport in the Rayleigh-Bénard convection problem no-slip boundary conditions for the velocity field are assumed. Nevertheless it is debatable, whether these boundary conditions reflect the behavior of the fluid at the boundary. This problem is important in theoretical fluid mechanics as well as in industrial applications, as the choice of boundary conditions has effects in the description of the boundary layers and its properties. In this review we want to explore the relation between boundary conditions and heat transport properties in turbulent convection. For this purpose, we present a selection of contributions in the theory of rigorous bounds on the Nusselt number, distinguishing and comparing results for no-slip, free-slip and Navier-slip boundary conditions.
Citation: Camilla Nobili. The role of boundary conditions in scaling laws for turbulent heat transport[J]. Mathematics in Engineering, 2023, 5(1): 1-41. doi: 10.3934/mine.2023013

Related Papers:

Abstract

In most results concerning bounds on the heat transport in the Rayleigh-Bénard convection problem no-slip boundary conditions for the velocity field are assumed. Nevertheless it is debatable, whether these boundary conditions reflect the behavior of the fluid at the boundary. This problem is important in theoretical fluid mechanics as well as in industrial applications, as the choice of boundary conditions has effects in the description of the boundary layers and its properties. In this review we want to explore the relation between boundary conditions and heat transport properties in turbulent convection. For this purpose, we present a selection of contributions in the theory of rigorous bounds on the Nusselt number, distinguishing and comparing results for no-slip, free-slip and Navier-slip boundary conditions.

References

[1]	C. R. Doering, Turning up the heat in turbulent thermal convection, PNAS, 117 (2020), 9671–9673. http://dx.doi.org/10.1073/pnas.2004239117 doi: 10.1073/pnas.2004239117
[2]	F. Otto, S. Pottel, C. Nobili, Rigorous bounds on scaling laws in fluid dynamics, In: Mathematical thermodynamics of complex fluids, Cham: Springer, 2017,101–145. http://dx.doi.org/10.1007/978-3-319-67600-5_3
[3]	C. R. Doering, J. D. Gibbon, Applied analysis of the Navier-Stokes equations, Cambridge university press, 1995. http://dx.doi.org/10.1017/CBO9780511608803
[4]	W. V. Malkus, The heat transport and spectrum of thermal turbulence, Proc. R. Soc. Lond. A, 225 (1954), 196–212. http://dx.doi.org/10.1098/rspa.1954.0197 doi: 10.1098/rspa.1954.0197
[5]	R. H. Kraichnan, Turbulent thermal convection at arbitrary Prandtl number, Phys. Fluids, 5 (1962), 1374–1389. http://dx.doi.org/10.1063/1.1706533 doi: 10.1063/1.1706533
[6]	E. A. Spiegel, Convection in stars I. Basic Boussinesq convection, Annu. Rev. Astron. Astr., 9 (1971), 323–352. http://dx.doi.org/10.1146/annurev.aa.09.090171.001543 doi: 10.1146/annurev.aa.09.090171.001543
[7]	E. D. Siggia, High Rayleigh number convection, Annu. Rev. Fluid Mech., 26 (1994), 137–168. http://dx.doi.org/10.1146/annurev.fl.26.010194.001033 doi: 10.1146/annurev.fl.26.010194.001033
[8]	G. Ahlers, S. Grossmann, D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection, Rev. Mod. Phys., 81 (2009), 503. http://dx.doi.org/10.1103/RevModPhys.81.503 doi: 10.1103/RevModPhys.81.503
[9]	P. P. Vieweg, J. D. Scheel, J. Schumacher, Supergranule aggregation for constant heat flux-driven turbulent convection, Phys. Rev. Research, 3 (2021), 013231. http://dx.doi.org/10.1103/PhysRevResearch.3.013231 doi: 10.1103/PhysRevResearch.3.013231
[10]	J. Serrin, Mathematical principles of classical fluid mechanics, In: Fluid dynamics I/Strömungsmechanik I, Berlin, Heidelberg: Springer, 1959,125–263. http://dx.doi.org/10.1007/978-3-642-45914-6_2
[11]	C. Amrouche, P. Penel, N. Seloula, Some remarks on the boundary conditions in the theory of Navier-Stokes equations, Annales Mathématiques Blaise Pascal, 20 (2013), 37–73. http://dx.doi.org/10.5802/ambp.321 doi: 10.5802/ambp.321
[12]	J. P. Whitehead, C. R. Doering, Rigid bounds on heat transport by a fluid between slippery boundaries, J. Fluid Mech., 707 (2012), 241–259. http://dx.doi.org/10.1017/jfm.2012.274 doi: 10.1017/jfm.2012.274
[13]	C. L. M. H. Navier, Sur les lois de l'équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 6 (1827), 1827.
[14]	X. Wang, J. P. Whitehead, A bound on the vertical transport of heat in the 'ultimate' state of slippery convection at large Prandtl numbers, J. Fluid Mech., 729 (2013), 103–122. http://dx.doi.org/10.1017/jfm.2013.289 doi: 10.1017/jfm.2013.289
[15]	T. Clopeau, A. Mikelic, R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions, Nonlinearity, 11 (1998), 1625. http://dx.doi.org/10.1088/0951-7715/11/6/011 doi: 10.1088/0951-7715/11/6/011
[16]	M. L. Filho, H. N. Lopes, G. Planas, On the inviscid limit for two-dimensional incompressible flow with Navier friction condition, SIAM J. Math. Anal., 36 (2005), 1130–1141. http://dx.doi.org/10.1137/S0036141003432341 doi: 10.1137/S0036141003432341
[17]	L. N. Howard, Heat transport by turbulent convection, J. Fluid Mech., 17 (1963), 405–432. http://dx.doi.org/10.1017/S0022112063001427 doi: 10.1017/S0022112063001427
[18]	F. H. Busse, On Howard's upper bound for heat transport by turbulent convection, J. Fluid Mech., 37 (1969), 457–477. http://dx.doi.org/10.1017/S0022112069000668 doi: 10.1017/S0022112069000668
[19]	C. R. Doering, P. Constantin, Variational bounds on energy dissipation in incompressible flows. III. Convection, Phys. Rev. E, 53 (1996), 5957. http://dx.doi.org/10.1103/PhysRevE.53.5957 doi: 10.1103/PhysRevE.53.5957
[20]	G. Fantuzzi, A. Arslan, A. Wynn, The background method: Theory and computations, 2021, arXiv: 2107.11206.
[21]	S. I. Chernyshenko, P. Goulart, D. Huang, A. Papachristodoulou, Polynomial sum of squares in fluid dynamics: a review with a look ahead, Phil. Trans. R. Soc. A, 372 (2014), 20130350. http://dx.doi.org/10.1098/rsta.2013.0350 doi: 10.1098/rsta.2013.0350
[22]	S. Chernyshenko, Relationship between the methods of bounding time averages, 2017, arXiv: 1704.02475.
[23]	D. Goluskin, Bounding averages rigorously using semidefinite programming: mean moments of the Lorenz system, J. Nonlinear Sci., 28 (2018), 621–651. http://dx.doi.org/10.1007/s00332-017-9421-2 doi: 10.1007/s00332-017-9421-2
[24]	G. Fantuzzi, D. Goluskin, D. Huang, S. I. Chernyshenko, Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization, SIAM J. Appl. Dyn. Syst., 15 (2016), 1962–1988. http://dx.doi.org/10.1137/15M1053347 doi: 10.1137/15M1053347
[25]	I. Tobasco, D. Goluskin, C. Doering, Optimal bounds and extremal trajectories for time averages in dynamical systems, In: APS Division of Fluid Dynamics Meeting Abstracts, 2017, M1-002.
[26]	F. Otto, C. Seis, Rayleigh–Bénard convection: improved bounds on the Nusselt number, J. Math. Phys., 52 (2011), 083702. http://dx.doi.org/10.1063/1.3623417 doi: 10.1063/1.3623417
[27]	C. Seis, Scaling bounds on dissipation in turbulent flows, J. Fluid Mech., 777 (2015), 591–603. http://dx.doi.org/10.1017/jfm.2015.384 doi: 10.1017/jfm.2015.384
[28]	X. Wang, Infinite Prandtl number limit of Rayleigh‐Bénard convection, Commun. Pure Appl. Math., 57 (2004), 1265–1282. http://dx.doi.org/10.1002/cpa.3047 doi: 10.1002/cpa.3047
[29]	X. Wang, Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number, Commun. Pure Appl. Math., 60 (2007), 1293–1318. http://dx.doi.org/10.1002/cpa.20170 doi: 10.1002/cpa.20170
[30]	C. R. Doering, P. Constantin, Energy dissipation in shear driven turbulence, Phys. Rev. Lett., 69 (1992), 1648. http://dx.doi.org/10.1103/PhysRevLett.69.1648 doi: 10.1103/PhysRevLett.69.1648
[31]	C. R. Doering, P. Constantin, On upper bounds for infinite Prandtl number convection with or without rotation, J. Math. Phys., 42 (2001), 784–795. http://dx.doi.org/10.1063/1.1336157 doi: 10.1063/1.1336157
[32]	C. R. Doering, F. Otto, M. G. Reznikoff, Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection, J. Fluid Mech., 560 (2006), 229–241. http://dx.doi.org/10.1017/S0022112006000097 doi: 10.1017/S0022112006000097
[33]	C. Nobili, F. Otto, Limitations of the background field method applied to Rayleigh-Bénard convection, J. Math. Phys., 58 (2017), 093102. http://dx.doi.org/10.1063/1.5002559 doi: 10.1063/1.5002559
[34]	C. Nobili, Rayleigh-Bénard convection: bounds on the Nusselt number, PhD thesis of Leipzig University, 2015.
[35]	G. R. Ierley, R. R. Kerswell, S. C. Plasting, Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory, J. Fluid Mech., 560 (2006), 159–227. http://dx.doi.org/10.1017/S0022112006000450 doi: 10.1017/S0022112006000450
[36]	P. Constantin, C. R. Doering, Infinite Prandtl number convection, Journal of Statistical Physics, 94 (1999), 159–172. http://dx.doi.org/10.1023/A:1004511312885 doi: 10.1023/A:1004511312885
[37]	X. Wang, Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number, Commun. Pure Appl. Math., 61 (2008), 789–815. http://dx.doi.org/10.1002/cpa.20214 doi: 10.1002/cpa.20214
[38]	X. Wang, Bound on vertical heat transport at large Prandtl number, Physica D, 237 (2008), 854–858. http://dx.doi.org/10.1016/j.physd.2007.11.001 doi: 10.1016/j.physd.2007.11.001
[39]	A. Choffrut, C. Nobili, F. Otto, Upper bounds on Nusselt number at finite Prandtl number, J. Differ. Equations, 260 (2016), 3860–3880. http://dx.doi.org/10.1016/j.jde.2015.10.051 doi: 10.1016/j.jde.2015.10.051
[40]	Y. Cao, M. S. Jolly, E. S. Titi, J. P. Whitehead, Algebraic bounds on the Rayleigh–Bénard attractor, Nonlinearity, 34 (2021), 509. http://dx.doi.org/10.1088/1361-6544/abb1c6 doi: 10.1088/1361-6544/abb1c6
[41]	J. Otero, Bounds for the heat transport in turbulent convection, PhD thesis of University of Michigan, 2002.
[42]	J. P. Whitehead, C. R. Doering, Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries, Phys. Rev. Lett., 106 (2011), 244501. http://dx.doi.org/10.1103/PhysRevLett.106.244501 doi: 10.1103/PhysRevLett.106.244501
[43]	B. Wen, D. Goluskin, M. LeDuc, G. P. Chini, C. R. Doering, Steady Rayleigh–Bénard convection between stress-free boundaries, J. Fluid Mech., 905 (2020), R4. http://dx.doi.org/10.1017/jfm.2020.812 doi: 10.1017/jfm.2020.812
[44]	S. C. Plasting, G. R. Ierley, Infinite-Prandtl-number convection. part 1. conservative bounds, J. Fluid Mech., 542 (2005), 343–363. http://dx.doi.org/10.1017/S0022112005006555 doi: 10.1017/S0022112005006555
[45]	T. Drivas, H. Nguyen, C. Nobili, Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries, 2021, arXiv: 2109.13205.
[46]	C. R. Doering, S. Toppaladoddi, J. S. Wettlaufer, Absence of evidence for the ultimate regime in two-dimensional Rayleigh-Bénard convection, Phys. Rev. Lett., 123 (2019), 259401. http://dx.doi.org/10.1103/PhysRevLett.123.259401 doi: 10.1103/PhysRevLett.123.259401
[47]	C. R. Doering, Absence of evidence for the ultimate State of turbulent Rayleigh-Bénard convection, Phys. Rev. Lett., 124 (2020), 229401. http://dx.doi.org/10.1103/PhysRevLett.124.229401 doi: 10.1103/PhysRevLett.124.229401
[48]	K. P. Iyer, J. D. Scheel, J. Schumacher, K. R. Sreenivasan, Classical $1/3$ scaling of convection holds up to $ {\rm{Ra}} = 10^15$, PNAS, 117 (2020), 7594–7598. http://dx.doi.org/10.1073/pnas.1922794117 doi: 10.1073/pnas.1922794117
[49]	I. Tobasco, C. R. Doering, Optimal wall-to-wall transport by incompressible flows, Phys. Rev. Lett., 118 (2017), 264502. http://dx.doi.org/10.1103/PhysRevLett.118.264502 doi: 10.1103/PhysRevLett.118.264502
[50]	C. R. Doering, I. Tobasco, On the optimal design of wall‐to‐wall heat transport, Commun. Pure Appl. Math., 72 (2019), 2385–2448. http://dx.doi.org/10.1002/cpa.21832 doi: 10.1002/cpa.21832
[51]	P. Hassanzadeh, G. P. Chini, C. R. Doering, Wall to wall optimal transport, J. Fluid Mech., 751 (2014), 627–662. http://dx.doi.org/10.1017/jfm.2014.306 doi: 10.1017/jfm.2014.306
[52]	S. Wagner, O. Shishkina, Heat flux enhancement by regular surface roughness in turbulent thermal convection, J. Fluid Mech., 763 (2015), 109–135. http://dx.doi.org/10.1017/jfm.2014.665 doi: 10.1017/jfm.2014.665
[53]	D. Goluskin, C. R. Doering, Bounds for convection between rough boundaries, J. Fluid Mech., 804 (2016), 370–386. http://dx.doi.org/10.1017/jfm.2016.528 doi: 10.1017/jfm.2016.528
[54]	P. E. Roche, B. Castaing, B. Chabaud, B. Hébral, Observation of the $1/2$ power law in Rayleigh-Bénard convection, Phys. Rev. E, 63 (2001), 045303. http://dx.doi.org/10.1103/PhysRevE.63.045303 doi: 10.1103/PhysRevE.63.045303
[55]	S. Toppaladoddi, S. Succi, J. S. Wettlaufer, Roughness as a route to the ultimate regime of thermal convection, Phys. Rev. Lett., 118 (2017), 074503. http://dx.doi.org/10.1103/physrevlett.118.074503 doi: 10.1103/physrevlett.118.074503
[56]	Y. Zhu, S. Granick, Limits of the hydrodynamic no-slip boundary condition, Phys. Rev. Lett., 88 (2002), 106102. http://dx.doi.org/10.1103/PhysRevLett.88.106102 doi: 10.1103/PhysRevLett.88.106102
[57]	R. P. Feynman, R. B. Leighton, M. Sands, The Feynman lectures on physics; Vol. I, Am. J. Phys., 33 (1965), 750–752. http://dx.doi.org/10.1119/1.1972241 doi: 10.1119/1.1972241
[58]	C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal., 43 (1992), 245–296. http://dx.doi.org/10.1080/00036819208840063 doi: 10.1080/00036819208840063

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)