Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number

Abstract

The vertical heat transfer in Bénard-Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number Nu as a function of the Marangoni number Ma. Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that Nu 0.838Ma 2/7 . In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on Nu, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering's formulation at a given Ma. Using a piecewise-linear, monotonically decreasing profile we then show that Nu 0.803Ma 2/7 , lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering's original formulation. We subsequently utilise convex optimisation to optimise the bound on Nu over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that Nu O(Ma 2/7 (ln Ma) −1/2 ) when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

Pressure-driven flows in helical pipes: bounds on flow rate and friction factor

“…Since the work of Doering and Constantin, this method has been applied to a wide variety of problems in fluid dynamics. Examples include upper bounds on the rate of energy dissipation in surface-velocity-driven flows (Doering & Constantin 1992, 1994Marchioro 1994;Wang 1997;Plasting & Kerswell 2003), pressure-driven flows (Constantin & Doering 1995) and surface-stress-driven flows (Tang, Caulfield & Young 2004;Hagstrom & Doering 2014); upper bounds on the heat transfer in different configurations of Rayleigh-Bénard convection (Doering & Constantin 1996Otero et al 2002;Plasting & Ierley 2005;Wittenberg 2010;Whitehead & Doering 2011;Whitehead & Wittenberg 2014;Goluskin 2015;Goluskin & Doering 2016;Fantuzzi 2018) and Bénard-Marangoni convection (Hagstrom & Doering 2010;Fantuzzi, Pershin & Wynn 2018;Fantuzzi, Nobili & Wynn 2020); and upper bounds on buoyancy flux in stably stratified shear flows (Caulfield & Kerswell 2001;Caulfield 2005).…”

Bound on the drag coefficient for a flat plate in a uniform flow

“…Until now, all the applications of the background method have focused on flows confined between solid boundaries. Examples include bounds on the rate of energy dissipation in surface-velocity-driven flows (Doering & Constantin 1992, 1994; Marchioro 1994; Nicodemus, Grossmann & Holthaus 1997; Wang 1997; Hoffmann & Vitanov 1999; Plasting & Kerswell 2003), pressure-driven flows (Constantin & Doering 1995) and surface-stress-driven flows (Tang, Caulfield & Young 2004; Hagstrom & Doering 2014); bounds on the heat transfer in Rayleigh–Bénard convection in various settings (Doering & Constantin 1996, 2001; Otero et al 2002; Plasting & Ierley 2005; Wittenberg 2010; Whitehead & Doering 2011 b ; Whitehead & Wittenberg 2014; Goluskin 2015; Goluskin & Doering 2016; Fantuzzi 2018) and Bénard–Marangoni convection (Hagstrom & Doering 2010; Fantuzzi, Pershin & Wynn 2018; Fantuzzi, Nobili & Wynn 2020); and bounds on buoyancy flux in stably stratified shear flows (Caulfield & Kerswell 2001; Caulfield 2005).…”

Time evolution equation for advective heat transport as a constraint for optimal bounds in Rayleigh-Bénard convection

“…While the same systematic procedure is in principle possible for the Navier-Stokes equation 10 , it becomes unpractical for Rayleigh-Bénard convection even at modest Rayleigh numbers because all energy unstable modes need to be retained which leads to a large SDP. Fantuzzi et al 11 study bounds for Bénard-Marangoni convection and in this context discuss in depth limitations of the SDP approach and possible future lines of investigation. At present, the most straightforward approach remains to formulate an optimization problem in the form of an SDP with many decoupled linear matrix inequalities rather than a problem with a few, but large linear matrix inequalities.…”