A Third Order Front-Tracking model for sharp interfaces and its application to phase change

Transcription

1 20th AIAA Computational Fluid Dynamics Conference June 2011, Honolulu, Hawaii AIAA A Third Order Front-Tracking model for sharp interfaces and its application to phase change S.N. Guignard 1 and L. Tachon. 2 Université de Provence - IUSTI, Marseille, FRANCE. Osama A.Shawky 3 AlAzhar University Cairo, EGYPT A new 3D third Order Front-Tracking method and its application to 2D flow with phase change is presented. The originality of this method lies in the meshing of the computational domain it generates: At each time step, the highly deformed interface is a grid surface. This allows an accurate computation of the physical field in the vicinity of the interface as an exact quantification of their eventual jumps at the interface. To step toward the understanding of the physical processes involved in liquid film evaporation, the 2D formulation of this new front tracking method is used by merging it in a numerical model solving the conservation equations in a quadratic finite element formulation. The heat transport equation is solved in the three different phases (solid, liquid and vapor) as the Navier-Stokes equations are solved in the two fluids. The temperature gradient discontinuity at the liquid vapor interface provides local value of the evaporative flux density that is directly linked to the interface velocity jump through mass conservation principle and used as boundary conditions for the two fluid flow computations. Testing on academic cases and application to axysimmetric film evaporation including comparison with experiments are shown. Nomenclature = Temperature of liquid phase [K] = Temperature of vapour phase [K] = Latent heat of evaporation [J/kg] = Thermal conductivity of liquid phase [W/m.K] = Thermal conductivity of vapour phase [W/m.K] = Mass flux density [kg/m²] = Velocity of interface [m/s] = Velocity in liquid phase [m/s] = Velocity in vapour phase [m/s] = Density of liquid phase [kg/m³] = Density of vapour phae [kg/m³] 1 Associate Professor, IUSTI, Technopôle de Château-Gombert, 5 rue Enrico Fermi, Marseille cedex 13, FRANCE. guignard@polytech.univ-mrs.fr, AIAA Member as of PhD, IUSTI, Technopôle de Château-Gombert, 5 rue Enrico Fermi, Marseille cedex 13, FRANCE. loic.tachon@polytech.univ-mrs.fr 3 Master Student, AlAzhar University Cairo, EGYPT. 1 Copyright 2011 by the, Inc. All rights reserved.

2 I. Introduction Thermo-fluid study of film evaporation is an important subject for various scientific, engineering and medical fields. Virtually, many processing technology must deal with film evaporation, from saltwater distillation, and critical biological processes like laser surgery cavitation, to Micro electro mechanical systems (MEMS). A lab-on-a-chip (LOC), which is subset of MEMS, is another example of technologies dealing with film evaporation. It is a device that integrates one or several laboratory functions on a single chip of only millimetres to a few square centimetres in size, dealing with the handling of extremely small fluid volumes. Clearly, the ability to predict the fluid behaviour of these processes is central to efficiency and effectiveness of those processes. While evaporating, the film produces a triple line; it is a place where large heat and mass fluxes occurs. It has become apparent that an improvement in our knowledge of the film evaporation process is so important. Because of the complexity of film evaporation phenomena, application codes rely heavily on empirical correlations. This approach has a number of serious shortcomings. Advances in parallel computing and continuing improvements in computer speed and memory have stimulated the development of numerical simulation tools that rely less on empirical correlations and more on fundamental physics. The most recent numerical modelling of thin film evaporation 1,2,3 use accurate modelling of the physical phenomenon (curvature, contact angle, disjunction pressure ), however, the details of the fluid flow in the liquid or vapour are not described as the models are generally integrated along the film width. In this paper, we present a two-dimensional axisymmetric film evaporation model to predict interface deformation and topology change, interface mass, momentum and energy transfer. This includes calculating heat fluxes at solid-liquid, and liquid-vapour interfaces, and comparing our results to experimental results. In addition, the experimental results are used to feed the numerical model with initial conditions for the interface shape and with boundary conditions at the triple line (contact angle and velocity). The model uses a coupled, quadratic finite element formulation of the conservation equation. The heat transport equation is solved in the three different phases (solid, liquid and vapour). The gradient discontinuity at the liquid vapour interface provides local value of the evaporative flux density that is directly linked to the interface velocity jump the velocity jump through the mass conservation principle. The Navier- Stokes equations are solved in the fluid phase. A special feature of this model is the interface modelling: The interface is a moving grid line that allows an accurate application of the interfacial boundary conditions, like velocity and pressure jump. The numerical results concerning interface shape evolution and heat flux distribution at the interfaces will be shown and then compared to experiments. Before presenting the 2D simulations applied to phase change, the three-dimensional version of the interface tracking method is presented. A. Method principle II. 3 rd order three dimensional interface tracking In this method, the interface is discretized using 3D surface quadratic elements (9 and 6 nodes). These interface nodes are moved in a Lagrangian way according to the velocity field and appropriated time scheme. The interface intersection with the edges of a the rectangular reference mesh is then computed. Then, based on the interface-mesh intersection analysis, mesh node are projected on the interface that produces the interface regridding and imposes the interace as a mesh internal boundary. The non interface grid nodes are then displaced by a simple smoothing algorithm to obtain a computation ready grid. Figure 1 illustrates the time marching of the interface tracking method. Figure 1. Sharp interface tracking method principle 2

3 B. Convergence order and accuracy This section presents the application of our method to kinematic test case. No conservation equation is solved in this section, the interface nodes are moved according to an imposed solenoidal velocity fields. 1) Sphere movements and order evaluation This subsection presents the cases of a sphere moving in two different divergence free velocity fields, one deforming the sphere (a flow perpendicular to a wall), the other one conserving it s shape (solid rotation). The aim of those test are to evaluate the convergence order of the method, in other words, the error decrease with the grid size. The two velocity fields are periodic in time so that the sphere has to return at its initial position and shape after one period. Figure 2. The interface shape of a sphere deformed by à velocity field of a solenoidal axisimetric flow perpendicular to an horizontal wall The error is computed after one period by comparing the sphere shape (the max (Lmax) and average (L1) node distance to the theoretical center). Figure 3 and 4 show the log-log representation of the errors after one period on the position, the normal to the interface and the curvature. These graphs definitely exhibits a convergence order close to 3 for the position, 2 for the normal, and 1 for the curvature. Figure 3. Log of the errors on position, interface normal vector and curvature in function of log of the cell size for the deforming velocity field (left) and non deforming velocity field (right). 3

4 2) The Zalesak test : the rotation of a solid of spatially highly varying curvature A solid rotation velocity field moves the shape presented on fig. 4. The quantification of the error is not available yet as its evolution with the cell size. However as can be seen on fig. 4, the shape after two rotations has been only slightly changed. Here there is 50 cells in the diameter of the sphere. Figure 4. The Zalesak shape at instant 0 and 2T 3) The Enright test : Periodic extrem deformation of a sphere In order to show the capability of the interface tracking method in dealing with highly evolving curvature surface, let s move our attention on a well-known 3D test case. Originally introduced in 4 the problem has been named the Enright test 5,6 and consists of a sphere immersed in a velocity field given by: u(x,y,z,t) =2sin 2 (πx)sin(2πy)sin(2πz)cos(πt/t) v(x,y,z,t) = sin 2 (πy)sin(2πx)sin(2πz)cos(πt/t) w(x,y,z,t) = sin 2 (πz)sin(2πx)sin(2πy)cos(πt/t) (1) C. Algorythm complexity Figure 5: The interface shape of a sphere deformed by a velocity field of a the Enright test. The sphere initial contains 38 cells in its diameter. The problem here is to evaluate the evolution of the execution time with the grid size in order to know if this method will represent an blocking strait in a CFD computation or not. Figures 6 and 7 show the execution time evolution with the total number of cells and with the number of cells containing the interface respectively. It shows clearly that the complexity of the algorithm is of the order of the interface cells number and not of the total grid cell number Numerical Results y = 0,73 x 7, Numerical Results y = 1,08 x Log(Time step) Log(Time step) Log(Volume elements number) Log(Surface elements number) Figures 6&7. Log-log representation of the computation time for different grid size versus the volume elements number (left) and the surface elements number (right). 4

5 III. Numerical model representation This numerical model is especially dedicated for solving full conservation equations in the presence of moving and deforming interfaces, taking into account discontinuous boundary conditions at the interfaces. At each time step, the interfaces are gridlines around which the computational grid is adapted. The energy, mass and momentum equations are successively solved. The phase change phenomenon is modelled as normal heat flux density jump (HDJ) at the interface: (2) Where is positive in the direction normal to interface towards the vapour phase. The local value of HDJ is result of the energy conservation equation solving for which the interface temperature is given as boundary condition. Eq.(2) gives the value of mass flux density at the interface from which the normal velocity jump at the interface can be deduced as the interface velocity, by applying simple mass conservation of incompressible flow: (3) That turns into: (4) and For the tangential component of the velocities at the interface, no special boundary conditions are applies as they are supposed to be all the same (tangential velocity continuity). The interface velocity jump of equation is imposed as a boundary condition of the interface in the momentum conservation equation. The momentum equation solver uses an iterative projection algorithm inspired from the Ref.7 and Ref.8 that ensures the incompressibility. Knowing the velocity field, each interface node is displaced according to equation Eq.(5). The grid is adapted to the new interface shape in an ALE manner. This means that no grid node can travel through the interface. This produces a displacement for each node of the grid that is taken into account in the ALE solver of all conservation equations. At this sub-step of the main time step, velocity, pressure and temperature fields are known on a grid called G1. In order to avoid this grid to be too strongly deformed, another grid G2 is created by adapting the original grid to the interface shape, but allowing grid nodes to travel through the interface. The physical fields are interpolated from G1 to G2 in order to be used in the upcoming timestep. Depending on the physical case simulated, suitable boundary conditions are applied on the boundaries of the computation domain as on the eventual solid-fluid interfaces. IV. Numerical model testing This part of the paper is dedicated to test cases that will validate the model. The first one will show the capability of the model to take into account surface tension, and the second one will take into account phase change. In each case, tests on volume conservation are shown that quantify the model quality. The computational domain has open boundaries that are naturally obtained by the finite element formulation when applying a constant pressure on the boundaries. (5) 5

6 A. Oscillating drops In this case, we consider a 2D liquid drop in gas of initial shape non-circular and thus oscillating under surface tension effect only (gravity forces and phase change are not taken into account). The computational domain is 5x5mm² with a grid of 101x101 nodes. Simulations have been performed for different capillary numbers but with the same initial conditions. As shown in Fig. 8, very good volume conservation can be observed: the volume variation is lower than 0.1%. Figure 8: Plot of bubble volume vs. time Figure 9 shows the evolution of the drop diagonal for different capillary numbers. We can observe here the damping due to the increase in capillary number. This test shows the capability of the model to simulate two-phase flows with deforming interface under the effect of surface tension. B. Free-growing bubbles In this case, we consider a 2D vapour bubble in its liquid of circular initial shape and growing under the effect of interfacial vapour production due to thermal boundary conditions on the interface (gravity forces are not taken into account). The computational domain is 10x10mm² with a grid of 101x101 nodes. As thermal boundary conditions, a constant temperature is imposed on the interface lower that the temperature imposed on the four sides of the computational domain. This leads to a temperature for the liquid adjacent to the interface that is higher that the interface temperature, and thus to heat flux towards the interface causing phase change quantified by equation Eq.(2). Figure 10 : Plot of bubble mass vs. time Figure 9: Plot of bubble diagonal vs. time 6 In order to test the ability of our model to simulate phase change, we compare the bubble mass variation due to the interface normal heat flux discontinuity (theoretical mass evolution) and the one computed by the numerical model (actual mass evolution). The comparison is plotted in Fig. 10. As we can see in fig.10, the error is lower than 2% while the bubble volume has been multiplied by 20. In fig.11, we can see the interface shape at t=0, 10, 20, 30 and 40s, as in Fig. 12, the velocity and the temperature fields are shown for t=10s.

7 Figure 11: Plot of interface at different time instants Figure 12 : Velocity and temperature fields at t=10s V. Thin film evaporation: Results and comparison with experiments We present here result of the simulation of thin film evaporation. The computational domain is 3.6x4.2mm² with a grid of 61x133 nodes. The domain is a replication of an experiment described into details in Ref.9 and summarized below: A millimetric evaporating well is constituted by a (Teflon made) circular ring of rectangular cross section laid on a glass blade. The sizes of these materials are described on Fig. 13. This well is poured with a highly evaporating liquid (HFE 7100). The evaporation process leads to the generation of a dry spot in the middle of the well that is delimited by the triple-line. After this step, the liquid evaporates as a toroidal meniscus. A laser probing system allows measuring instantly the interface shape of the meniscus in a radial cross section. Figure 13: The computational domain of the evaporating film case This 2D axisymmetric simulation starts right after the apparition of the triple line. The heat transport equation is solved in the three phases (solid, liquid and vapour). The temperature is imposed on the interface as every other boundaries of the domain are adiabatic. To initialize the temperature field, the temperature is uniform in the entire domain except the lower one imposed on the interface (T domain -T interface = -20K). The initial interface shape is taken more or less similar to the one measured in the experiment in the Ref.9. The triple line (one grid point) velocity is imposed equal to 5e-5 m.s -1 that is an average value deduced from the experiment. The liquid velocity at the solidliquid boundaries is imposed as zero. The vapour velocity at the solid-vapour boundaries is imposed as zero except at the triple line where it is free. All the physical parameters used in the computation are the real one for each material. In this simulation, surface tension and gravity are taken into account. 7

8 In Fig. 14, are plotted simulated (asterisk lines) and experimental (solid lines) interface shape of the film during its evaporation with triple line every 20 seconds. First of all, the initial conditions are not exactly the same, and the triple line velocity in the numerical model is not exactly the experimental one. In addition to this, no special triple line model is used here: there is no correlation between the local heat flux density and the evaporation rate. This explains the discrepancy to the observed between the simulated and experimental profiles. However, it is to be observed, that the general shapes of the interfaces tend to be similar as time increases, this is a first encouraging result. Figure 14. Simulated and Experimental interfaces shapes. As we can see on fig. 15 and fig. 16, we can deduce from the temperature field, the heat flux densities distribution at various interfaces (Solid-Liquid and Liquid-Vapour respectively). Here appears nearly the heat flux density increases in the vicinity of the triple line. Figure 15: Heat flux distribution at Liquid- Solid interface Figure 16: Heat flux distribution at Liquid-Vapour interface VI. Conclusion We presented a new numerical model for the simulation of two-phase flow with phase change and surface tension able to deal with highly deformed moving interfaces. This model has been applied successfully on simple test cases. That shows its capability to take into account phase change and surface tension for incompressible flows. This model is finally applied to the case of an axisymmetric evaporating meniscus. The results on the interface shape in time have been compared with experimental results obtained under similar conditions. Despite some discrepancies, the comparison is encouraging. This work is still in progress, and has to be improved in different ways: better initial conditions, better thermal boundary conditions and the addition of a triple line subscale modelling.. Acknowledgments The authors gratefully appreciate the support of the National Agency for Research (ANR) with project ANR-06- BLAN INTENSIFILM. 8

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