On the General Properties of Steady Gravitational Thermal Flows of Liquid Metals in Variable Cross-section Containers

Abstract

The rmogravitational flows of liquid me tals in ope n or close d ducts and containe rs play a re le vant role in a varie ty of applications in mechanical, mate rials and nuclear e ngineering. Such flows are known to be very sensitive to the e ffective shape of the containe r use d to host the fluid and its thermal boundary conditions. For the case of tempe rature gradie nts having the main compone nt dire cte d along the horizontal dire ction, re late d convective phe nome na fall unde r the ge neral heading of "Hadle y flow". He re we introduce a ge neral frame work for the de te rmination of the properties of these flows in the case of domains having converging or diverging top and bottom walls. The frame work is built via a hybrid approach in which typical techniques of CFD are use d in synergy with analytical solutions of the e nergy e quation. The prope r use of initial and boundar y conditions results in algorithm convergence acce le ration. The role playe d by the top and bottom wall inclination with re spect to the horizontal is assessed through parametric inve stigation.

On the oscillatory hydrodynamic instability of gravitational thermal flows of liquid metals in variable cross-section containers

“…As preliminarily shown by Lappa and Ferialdi, 24 both the streamfunction and the nondimensional shear stress being associated with such unicellular flows attain a minimum when the top and bottom walls are perfectly horizontal (the canonical rectangular enclosures that have attracted so much attention in past studies on the subject), i.e., for η = 1. Any departure from this ideal condition, however, can cause an increase in the shear stress and ψ (the related dependence is relatively complex and non-monotone 24 ).…”

“…Moreover, most conveniently (to speed up algorithm convergence), for each simulation the initial temperature field has been initialized using the analytic solution determined by Lappa and Ferialdi 24 in the limit as Ra → 0 (diffusive conditions),…”

“…The specific case of the thermogravitational flow of liquid metals in differentially heated "converging or diverging" containers, encompassing both the direct (from the hot side to the cold side) and "return" currents (in the opposite direction), has been addressed most recently by Lappa and Ferialdi 24 on the basis of a hybrid approach relying on the numerical integration of the IBVP problem in conjunction with analytic solutions of the temperature field (used as initial conditions).…”

Gravitational thermal flows of liquid metals in 3D variable cross-section containers: Transition from low-dimensional to high-dimensional chaos

“…For all cases, we assume the top and bottom walls to be adiabatic (no heat exchange). As preliminarily shown by Lappa and Ferialdi 28 , the properties of this unicellular basic flow are very sensitive to the value of the parameter . More specifically, both the streamfunction and the nondimensional shear stress attain a minimum when =1, i.e.…”

An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows

“…In turn, this requires implicitly the adoption of a given solution strategy, be it analytical, approximate or "numerical". By analytical approach we mean one of the standard methods for classical partial differential equations (the obvious outcome of such a process being an algebraic expression relating the dependent variables to the independent variables, see, e.g., Ostroumov [6], Birikh [7,8], Gershuni and Zhukhovitskii [9], Belghazi et al [10], Lappa [11], Lappa and Ferialdi [12]). An approximate method results when the governing boundary value problem is solved using a series-expansionbased technique (see, e.g., Jane [13], Jebari et al [14], Al-Saif et al [15]) or a transformation is used, based on the introduction of a similarity variable, by which the original partial differential equations are replaced by a set of coupled nonlinear ordinary differential equations (e.g., Makinde and Olanrewaju [16]).…”