atw 2017-07

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atw Vol. 62 (2017) | Issue 7 ı July 478 the wetting history of all cell faces on the plate, denoted by f. AMNT 2017 ı COMPETENCE PRIZE | | Fig. 1. Pictures of lab tests with 2° inclination and mass flow of 11 _ g s (see [4]) after 2 min and 15 min. Pictures taken during the experiments show the flow patterns and run down behavior of the water on the plates and can be used to compare the simulations to the experiments. 3 Empirical contact angle model In this section an empirical model, which accounts for the wetted history and therefore allows for a stable rivulet flow, is presented. In the lab experiments rivulets, once they are formed, rarely change their path ( Figure 1). Prior simulations with constant contact angles did not show this behavior even over a large contact angle range [3]. The run down behavior was always volatile and no pseudo-stationary state was reached. A new, empirical contact angle model was developed to enforce a spatially and temporally stable rivulet behavior. 3.1 Theory of empirical contact angle model The contact angle forms between the liquid, the solid and the gas phase and is a measure for the interplay of the materials in use but can also be influenced by the dynamics of the system [5]. The contact angle has a huge impact on the formation of rivulets and their stability as shown in previous studies [3]. The idea of the empirical model is to considerably lower the contact angle on wet surfaces to create a preferred flow path down the plate for the water. Hence the wetted parts of the surface become more hydrophilic than the dry ones and the water is more likely to flow down the same path as before, because the contact angle on this path is significantly smaller than the contact angle on the rest of the dry plate. The wetting function w(f,t) ∈ (0,1) keeps track of with a water (f,t n ) ∈ (0,1) being the volume fraction of water in face f at time t n and θ (f,t n ) ∈ (θ wet , θ dry ) is the contact angle for each face f and time t n . At first the whole plate is dry, thus the wetting function on every face equals zero and each face on the surface, where the empirical contact angle condition holds, is initialized with θ dry . With increasing time, the wetting function takes the maximum value of volume fraction of water from the past to the current time step on this face. Thus the wetting function can only take values between zero and one and is monotonically increasing. The resulting contact angle θ is then a linear interpolation of the contact angle for wet surface θ wet and the contact angle for dry surface θ dry . The lower limit θ wet is reached for faces that at some point in the past were completely covered by water. If the contact line lies within a certain face the contact angle θ of this face is assigned to the contact line and is used to adjust the curvature of the phase interface. A contact angle field representing the wetted surface on the floor develops during the course of the simulations (Figure 2). The empirical model can also be combined with dynamical contact angle models and hysteresis. 3.2 Simulation setup The computational domain is a trapezoid-shaped geometry (length = | | Fig. 2. Contact angle distribution after 5 s using the empirical contact angle model and the wetting function to keep track of the wetted surface. | | Fig. 3. Schematic of computational area of inclined trapezoid, all dimensions in mm. 1.215 m, small base = 0.09 m, large base = 0.475 m, Figure 3), as used in the laboratory experiments con ducted by Becker Technologies [2]. The simulations are carried out with the Software OpenFOAM using the standard two-phase solver InterFoam. The Navier-Stokes equations for isothermal and incompressible multiphase flow are solved and the phase interface is captured by the Volumeof-Fluid method. The time step is adjusted such that the maximum interface Courant number is below 0.4 to guarantee a good resolution in time for the transient process; this will also ensure a sufficient level of accuracy. The time is discretized via Euler implicit. At the upper most part of the initially dry plate water enters the computational domain through a rectangular inlet extending over the entire upper boundary with a velocity parallel to the surface such that a mass flow rate of 11 g _ s , according to the experiments, is achieved. The flat plate is bounded by vertical sidewalls and has an inclination angle a. Two cases are considered: one with inclination a = 20˚ and one with inclination a = 2˚. Material properties of water and air are used. A no-slip boundary condition holds on walls. On the floor, the implemented empirical contact angle model takes effect with θ dry = 70˚ and θ dry = 15˚ for the water phase and on the sidewalls a zero gradient con ditions holds which matches a constant contact angle of 90˚. The top and the lower end of the plate are openings and the pressure is specified. Water can drain away freely through these outlets. The mesh is refined in z direction towards the floor, where the wetting and many of the relevant processes take place. A thin rivulet is resolved with approximately five to eight cells AMNT 2017 Modeling and Simulation of Water Flow on Containment Walls with Inhomo geneous Contact Angle Distribution ı Katharina Amend and Markus Klein
atw Vol. 62 (2017) | Issue 7 ı July | | Fig. 4. Mesh resolution and thin rivulet. standard deviation and amplitude and then interpolated onto the mesh. This initial field (Figure 6) is then superposed with the contact angle model. 4.1 Theory of filtered randomized initial contact angle field In the previous simulations, the decisive influence of the contact angle has become clear. Experimental surfaces are imperfect and therefore this step is necessary for a more realistic description of the experiments. The procedure is as follows: before the simulation starts, a virtual field is generated in the x-y-plane which is later rotated and mapped onto the mesh of the computational boundary. Every cell in that virtual field contains a random number generated by a preassigned distribution (uniform distribution or normal distribution on a given interval symmetrical around zero). This random field is filtered with a box or Gaussian filter with a predefined filter width or standard deviation (see Figure. 6). 479 AMNT 2017 ı COMPETENCE PRIZE | | Fig. 5. Comparison of simulations with empirical contact angle model and the laboratory experiments by Becker Technologies [2] (in false color representation) with mass flow rate ṁ = 11 _ g s , two different inclinations (left 2°, right 20°) and without aerosol loading. Gaussian filter kernel: –3σ ≤ i,j ≤ 3σ in height and four to five cells in width (Figure 4). Overall the unstructured mesh has 3.8 million cells (99 % hexahedrons, 1 % prisms). 3.3 Simulation results The simulations with the empirical contact angle model (with a constant contact angle for dry and wetted surfaces) show a very good qualitative agreement with the laboratory experiments (see Figure 5). The rivulets do not change their path and a pseudo-stationary state is reached. The deviations in the area of the inlet at the top of the plate can be explained by the fact that the scattering nozzles used in the experiment are not exactly modeled in the simulations, but a slotted nozzle was used instead. The influence of the inclination on the water is clearly visible. Very similar to the laboratory experiments, for a bigger inclination (20˚) several rivulets (up to eleven) with a small width (one to two centimeters) and relatively high run down velocities form in the simulations. On the almost horizontal plate (2˚ inclination), the water runs down in only few (about six), wider (about three to four centimeters) rivulets. The velocity of the water is also significantly reduced in this case. These results agree qualitatively very well with the observations of the experiments. 4 Filtered randomized initial contact angle field An initialization of filtered random contact angles on walls, based on [6], is introduced to mimic realistic surfaces with impurities and contact angle inhomogeneities. Randomized contact angles on a virtual field are filtered by a Gaussian filter with given After initializing the filtered random field the simulation starts and during the simulation the initial contact angle field is superposed with the contact angle model in use (e.g. with the empirical contact angle model). The goal is to investigate to which extent the water behavior is disturbed by such inhomogeneities and to conduct several variations of one experiment and extract statistical quantities for several characteristics of the flow field. σ = 0, σ = 2, σ = 8, σ = 16, | | Fig. 6. Filtered randomized initial contact angle field, filtered with increasing standard deviation from left (σ = 0) to right (σ = 16). Normal distribution and Gaussian filter were applied. AMNT 2017 Modeling and Simulation of Water Flow on Containment Walls with Inhomo geneous Contact Angle Distribution ı Katharina Amend and Markus Klein
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