A level set method and a heat transfer model implemented into FLUENT for modeling of microscale two phase flows

Thermal design and simulation of two-phase cooling of avionics onboard military aircraft requires extensive analysis to better understand the effects of channel orientation, gravitational effects and transients on thermal performance. Such analysis requires a robust numerical code that can model the fundamental phenomena, while also benchmarked versus known solutions and published experimental data. This paper thus describes the successful implementation of a level set method and a heat transfer model into FLUENT for modeling of adiabatic and evaporating two phase flows. The level set method belongs to the so called 'one' fluid methods where a single set of conservation equations are solved for the whole domain and the interface between the two phases is tracked or captured. FLUENT has already a 'one' fluid method for modeling two phase flows, namely volume-of-fluid, which is then coupled here with the level set method. The resulting coupled level set volume-of-fluid method is superior to both standalone methods by retaining the advantages of each method (better accuracy in computing curvature and normal to the interface for the level set method and precise mass conservation for the volume-of-fluid method). The level set function is used to compute the surface tension contribution to the momentum equations, via curvature and its normal to the interface, using the Brackbill method [1], while the volume of fluid function is used to capture the interface itself. A re-initialization equation is implemented and solved at every time step using a fifth order weighted essentially non-oscillatory (WENO) scheme for the spatial derivative, and a first order Euler method for time integration. The coupling effect is introduced by solving at the end of each time step an equation which connects the volume fractions with the level set function. The detail description is presented in [2]. A heat transfer model was presently implemented into this FLUENT code for solving problems involving boiling or condensation. Besides the Navier-Stokes equations, a energy equation was solved in the whole domain, including the gas phase, and the temperature for the cells containing the interface was set to the saturation one. In boiling or condensation a mass transfer across the interface occurs and this has been handled in the model. This was specified using User Defined Functions (UDFs) as a mass source in the continuity equation. As a first step, a vapor bubble growing inside a microchannel during flow boiling of water is simulated in the present study. A parametric study is performed by varying the static contact angle. 1.0 INTRODUCTION In the last decade or so, micro two-phase flows have become an alternative to the existing cooling solutions in industry, due to its capacity to remove large thermal densities using a relatively small quantity of refrigerant. Consequently, nowadays it is studied intensively, both experimentally and numerically. Two approaches exist for modeling of two-phase flows: ‘one’ fluid methods and ‘two’ fluid methods. In ‘one’ fluid methods, a single set of conservation equations is solved and the interface is tracked or captured. In ‘two’ fluid methods, a set of conservation equations is solved for each phase and the A LS method and a HT model implemented into FLUENT for modeling of two phase flows 26 2 RTO-MP-AVT-178 NATO/PFP UNCLASSIFIED NATO/PFP UNCLASSIFIED interaction between phases is given by some correlations. Interface tracking methods explicitly track the interface with marker particles like in MAC (marker and cell) methods [3], arbitrary Lagrangian-Eulerian (ALE) methods [4-6], or front tracking methods [7-8]. These methods are very accurate and efficient for flexible moving boundaries with small deformations. However, their main drawback is that they are difficult to use in cases where the interface breaks up or coalesces with another interface. Also additional re-meshing is needed when a large deformation of the interface occurs. In interface capturing methods, an auxiliary function is needed. These methods are very robust and have a wide range of applicability. However, they require higher mesh resolution. Examples include volume-offluid (VOF) [9-12] and level set (LS) methods [13-16]. In VOF methods, the interface is given implicitly by a color function, which is defined as the volume fraction of one of the fluids within each cell. From this function, a reconstruction of the interface is made and the interface is then propagated implicitly by updating the color function. VOF methods are conservative and can deal with topological changes of the interface. However, VOF methods cannot accurately compute several important properties, such as curvature and normal to the interface. In LS methods, the interface is represented by the zero contour of a continuous signed distance function; this is known as the level set function. The movement of the interface is governed by a transport equation for the level set function. To keep the level set function as a signed distance function, a re-initialization process is needed. LS methods automatically deal with topological changes. It is generally easy to obtain a high order of accuracy just by using an ENO (Essentially non-Oscillatory) or WENO (Weighted Essentially non-Oscillatory) scheme. However, LS methods are not conservative, so that a significant, physically incorrect loss or gain of mass occurs for incompressible two phase flow. Several authors have overcome this problem by coupling the LS and VOF methods [17–24]. In such coupled level set volume of fluid (CLSVOF) techniques, the level set function is used to accurately compute the curvature and the normal to the interface while the volume of fluid function is used to capture the interface. Normally, a CLSVOF method is more accurate than both the standalone LS and VOF methods. In this paper, we implemented a LS method inside FLUENT and coupled it with the existing VOF method to create a CLSVOF method which improved FLUENT capabilities to simulate two-phase flows. A heat transfer with phase change was also implemented inside FLUENT. The rest of the paper is organized as follows: Section 2 describes briefly the mathematical models, Section 3 presents several numerical results and conclusions are given in Section 4. 2. MATHEMATICAL MODELS 2.1 CLSVOF method Since VOF method is already implemented and well documented inside FLUENT, in this section the focus will be on the LS method. In this work we perform a direct numerical simulation, so we need to solve the continuity equation and Navier-Stokes equations: 0 t        u (2.1) ( ) ( ) (2 ) st p t              u u u D g F (2.2) A LS method and a HT model implemented into FLUENT for modeling of two phase flows RTO-MP-AVT-178 26 3 NATO/PFP UNCLASSIFIED NATO/PFP UNCLASSIFIED where in Eq. 2.1 and 2.2  is the density, t is the time,  is the dynamic viscosity, u is the velocity vector, g is the gravity vector, D is the rate of deformation tensor and st F is the surface tension force. It is worth mentioning that  and  are discontinuous. To express them we used a smoothed Heaviside function: 0 if ( ) ( + )/(2 )+sin( / )/(2 ) if 1 if > H                      (2.3)