Use of a Fictitious Marangoni Number for Natural

a Use of a fictitious Marangoni number for natural convection simulation Francisco J. Arias∗ and Geoffrey T. Parks Department of Engineering, University of Cambridge Trumpington Street, Cambridge, CB2 1PZ, United Kingdom In this paper, a method based on the use of a fictitious Marangoni number is proposed for the simulation of natural thermocapillary convection as an alternative to the traditional effective diffusivity approach. The fundamental difference between these two methods is that the new method adopts convective mass flows in simulating natural convection. Heat transfer in the natural convection simulation is calculated through the mass transport. Therefore, empirical Nusselt numbers correlations required in the effective diffusivity method are eliminated. This represents a clear advantage in the context of design studies where flexibility in varying the geometry unconstrained by the availability of appropriate correlations is highly desirable. The new method is demonstrated using a simple geometrical model. An analytical expression of the fictitious Marangoni number associated with convection between vertical plates is derived and a computational fluid dynamics (CFD) simulation is performed to study the efficacy of the proposed method. The results show that the new method can approximate real natural convection quite accurately and can be used to simulate the convective flow in complex, obstructed or finned structures where the specific Nusselt correlation is not known. aKeywords. Natural convection, Effective thermal conductivity, Marangoni convection, Compu- tational Fluid Dynamics (CFD) I. INTRODUCTION ship for the specific geometry, such as finned structures; however, these correlations are often not available. This Natural convection in enclosed cavities is of great im- then motivates us to find an alternative approach that portance in many engineering and scientific applications does not require knowledge of Nusselt number correla- such as energy transfer, boilers, nuclear reactor systems, tions. energy storage devices, etc. In the design of such systems In this paper, an alternative approach is proposed for numerical simulation using computational fluid dynamics natural convection simulations in which the momentum (CFD) and experimental testing of prototypes are exten- equation is modified and then the mass flow represented sively used. However, these methods are not well suited by using a fictitious Marangoni term in the stress ten- to activities such as parametric analysis due to their time- sor inducing thermocapillary currents. The heat trans- consuming nature and high cost [1]. fer is then the result of this mass flow. In the next Natural convection analysis often involves complex section the theoretical background behind the proposed simulations. Such simulations entail a set of relaxation approach will be presented. Although prior knowledge factors to converge and no easy way to find the relax- of Marangoni convection is not essential to understand ation factors except through continuous trials demanding the material presented in the next section, the interested significant computational resources. This frequently dis- reader is referred to the text by Kuhlmann and Rath suades thermal analysts and designers from attempting [4] for further information about fundamental Marangoni 3D simulations. In order to overcome this problem, the theory and to recent research outputs [5{14] and the book traditional approach is to use an effective diffusivity term by Lappa [15] to obtain an overview of thermal convec- (effective thermal conductivity) to convert effects of con- tion and the state of the art. vection into pure conduction [2, 3]. The fluid within an enclosure behaves like a fluid the thermal conductivity κ of which is modified by an effective thermal conductivity II. THEORETICAL BACKGROUND κeff as κeff = κ · Nu, with the Nusselt number Nu being determined by an appropriate correlation. This provides A. The fictitious Marangoni approach (FMA) a challenge to engineers when they are designing a complex or novel system. The engineers must have knowledge of the appropriate Nusselt number correlation relation- Let us start by considering the Navier-Stokes equation, which has, in presence of a gravitational field, the following tensorial form ∗Corresponding author: Tel.: +32 14 33 21 94; Electronic address: @vi @vi 1 @p @ 0 + vk = − + fσikg + gi (1) [email protected] @t @xk ρ @xi @xk 2 0 where v is the velocity, p is the static pressure, σik is the or, considering the velocity profile between the parallel viscous stress tension (described below), and gi is the plates [17], gravitational body force per unit volume. The viscous stress tensor is given by dp ηw = −12 3 (8) dxi loss ρs @v @v 2 @v σ0 = η i + k − δ l (2) ik @x @x 3 ik @x where w is the mass flow rate per unit of width. k i l For natural convection flow, this flow resistance is bal- where η is the dynamic viscosity and δik is the Kronecker anced by the buoyant potential [17] given by delta. dp Now let us derive the equation describing the natural = −ρβdT gi (9) convection. For the sake of simplicity, we will assume the dxi buoy fluid is incompressible. This assumption implies that the Equating the dp=dx terms in Eq. (8) and Eq. (9) we variation of density due to variation in pressure may be i obtain the well-known solution given by Bar-Cohen and neglected. We can express the variations in temperature, Rohsenow [18] for the mass flow rate per unit width in density and pressure as functions of small variations dT , the channel: dρ and dp, respectively. This is the well-known Boussi- nesq approximation (for buoyancy). Introducing this into ρ2g βs3dT w = i (10) the Navier-Stokes equation (Eq. (1)), results in the fol- 12η lowing expression [16]: As mentioned, we want to find an appropriate ficti- @vi @vi 1 @dp @ 0 tious surface-tension gradient @γ=@xi. In this case, using + vk = − + fσikg − βdT gi (3) @t @xk ρ @xi @xk Eqs. (5) and (7), we can write the stress tensor as @v @γ∗ where, with our assumption that the fluid is incompress- σ0 = η i = (11) ik @x @x ible, i.e. div · u = 0, the stress tensor is simplified as k xk=0 i 0 @vi @vk where @γ∗=@xi is the fictitious surface-tension gradient σik = η + (4) @xk @xi associated with the wall. Taking into account Eqs. (6), (9) and (11) we find that Let us now consider the situation of a boundary con- the fictitious surface-tension gradient must be given by dition that must be satisfied at the boundary between @γ∗ ρβg sdT the fluid and the walls, when surface-tension forces are = i (12) taken into account. If we assume that the surface-tension @xi 2 coefficient γ is not constant over the surface (in our case Given that because of temperature variation), then a force tangen- tial to the surface is developed @γ , and the stress tensor @γ∗ @γ∗ @xi = @sT (13) then becomes @xi @Ts @v @v @γ where @sT = @Ts=@xi and the subscript s means that σ0 = η i + k + (5) ik @x @x @x the temperature is evaluated at the wall, Eqs. (12) and k i i (13) can be combined to give Now, our objective in this paper is to define a ficti- @γ∗ ρβgis dT tious surface-tension gradient, @γ=@xi, which emulates = (14) @T 2 @ T the buoyancy potential, βdT gi. Although there are sev- s s eral ways in which to do this, perhaps the following anal- Using the definitions of the Rayleigh number Ras and ogy is easiest in applying this approach to cases involving the Marangoni number Ma: plates and finned structures. In laminar, fully developed, two-dimensional (2D) flow ρβg s3 Ra = i dT (15) between parallel plates (see Fig. 1), the pressure drop is s ηα given by [16] dp σ0 @γ∗ hdT = −2 ik (6) Ma = − (16) @Ts ηα dxi loss s where α is the thermal diffusivity and h is a characteristic where s is the distance between the plates and the stress length, here the length of the channel, Eq. (14) can then tensor is given by be rewritten as @vi C @ T σ0 = η (7) Ma = 1 f Ra (17) ik @x s k xk=0 2 @sT 3 Combining Eq. (19) with Eq. (14) then yields 2 @γ∗ ρβgis ≈ C1 (20) @Ts 2 All parameters in this equation are known once the geometry is specified. This then enables natural convection to be simulated without knowledge of the appropriate Nus- selt number correlation. B. The effective diffusivity approach (EDA) In the previous section we propose a new approach for natural convection simulation based on the introduction of a “fictitious” Marangoni term in the stress tensor in the momentum equations. In this proposed approach heat transfer will be obtained as a result of the induced mass flow. In contrast, the problem of computational natural con- FIG. 1: Simulated cavity model with a bottom wall temper- vection simulation is often these days tackled from a to- ature of Th, a top wall temperature of Tc and adiabatic side tally different point of view using more traditional ap- walls. proaches. Instead of modifying the momentum equations (and then considering mass transport) it is the energy equation that is modified. There are several approaches where @f T = dT=h is the linear average fluid temperature to representing the effects of convection by modifying gradient and C1 is a geometry-dependent constant. the energy equation: for example, by introducing em- Eq. (17) presents a relationship between the fictitious pirical velocities into the convection terms in the energy Marangoni number and the Rayleigh number. The dif- equation [19]. However, undoubtedly the most commonly ference between these two dimensionless numbers relates used approach of this kind is the so-called effective diffu- to mechanism by which natural convection is initiated.

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