4.4 Summary
5.1.1 Laminar shear flow
Drops in laminar shear flow can be found for low Reynolds numbers or for small drops within the viscous turbulent sublayer. An accurate description of drops in laminar shear flow is a mile-stone to tackle turbulent shear flow. Table 5.1 gives an overview of selected literature describing incipient motion in laminar shear flow experimentally, theoretically and in simulations. Most relevant here is the work by Milne and Amirfazli (2009) and Milne (2013), who present in-cipient motion for a wide range of drop volumesVD=0.5−100µl. The drops are placed on an elliptic profile within a channel which operates with mean velocities of up toUChannel≤30 m/s. The corresponding drop Reynolds numbers of ReD ≤3000and the Reynolds numbers calculated by the curved boundary layer length of up toRe≤2×105are well below the turbulent transition of a flat plate or a cylinder in free stream conditions. The dimensions of the channel are given in Figure 5.1a. The water drops are placed on three different surfaces, PMMA (θA=76.3◦±3.3◦,θR=53.1◦±3.8◦), Teflon (θA =124.3◦±0.7◦,θR=108.2◦±3.4◦) and SHS (θA=161.4◦±0.8◦,θR=129.8◦±8.9◦). Refer to Milne and Amirfazli (2009) for details on the surface preparation. The drops are exposed to both the boundary layer and the free
Literature Methodology Note Schleizer and Bonnecaze (1999) Experiments
Simulation
2D Boundary Element Method simulations of drops in Stokes shear flow (Re=0.01−0.03)
Dimitrakopoulos and Higdon (2001) Simulation 3D simulations with Spectral Boundary Element method minimizing the free surface energy of drops for low Reynolds numbers.
Spelt (2005) Simulation 2D Level-Set simulation of shear flow (Re≤25) of the setup by Schleizer and Bonnecaze (1999)
Ding and Spelt (2008) Simulation 3D Diffusive Interface simulation for moderate shear flow (Re≤222.5).
Focuses on the influence of the Weber number and also on the difference between 2D and 3D solutions
Milne and Amirfazli (2009) Experiments Theory
Investigation of drops (0.5−100µl) incipient motion on an elliptical profile within a channel (0−30 m/s,Re≤2×105)
Dupont and Legendre (2010) Simulation 2D VOF simulation of a Couette-Stokes-Flow (Re =0.01−0.03) of the setup by Schleizer and Bonnecaze (1999) and a Poiseuille-Stokes-Flow
Seevaratnam et al. (2010) Experiments Simulation
3D Diffusive Interface method simulating drops in cross-flowRe ≤700 for very low Bond numbersBo≤10−4. Three different drop motions are found: sliding, crawling, and detachment
Fan et al. (2011) Experiments
Theory Drops (≤50µl) in channel flow (0−10 m/s)
Milne (2013) Experiments
Theory In addition to the first publication of this table, the drag force on the drop is measured
Linder (2015) Simulation
Experiments
3D VOF simulation of drop (50µm) incipient motion on a spherical profile within a channel (0−10 m/s)
Roisman et al. (2015) Theory
A force balance for drops of the height of the viscous boundary layer yields a relation between wetting properties, the drop volume, and the velocity gradient at the wall.
This study Simulation 3D VOF simulation with enhanced interface handling of the setup described by Milne and Amirfazli (2009)
Table 5.1:Literature selection on experiments, theory and simulations of incipient drop motion in laminar shear flow and laminar boundary layers.
68.58 119.38
344.17
34.28
U0 19◦ 469.90
(a)Dimensions (b)
Figure 5.1:a) A drop sits on a surface probe enclosed by a profile with an elliptical nose that is placed in the center of a channel.
The mean channel velocityU0is increased over time to investigate the incipient motion of the drop. Experiments with water drops on three surfaces with this setup are performed by Milne and Amirfazli (2009) used for the validation of the simulation framework.
b) Simulation domain initialized with a velocity profile (U0=2 m/s) and a drop volume ofVD=58µl. The drop is refined in three levels down to∆x=250µm. The numerical domain-depth measures30 mm.
2 2.5 3 3.5 4 4.5 5 U0in [m/s]
-0.5 0 0.5 1 1.5 2
x/D0
0 0.1 0.2 0.3
Fin[N]
×10−3
advancing CL receding CL
Fadh
−Fs
Ucr- Exp.
Figure 5.2:The incipient motion of a water drop (VD=30µl) on SHS in laminar shear flow. The contact line position is shown over the ramped inlet velocity as well as the magnitude of the adhesion and shear force.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Lb(cosθmin−cosθmax) ×10-3 0
5 10 15 20 25 30
Ucrin[m/s]
Teflon SHS
PMMA
100µl-Exp.
100µl-Sim.
58µl-Exp.
58µl-Sim.
30µl-Exp.
30µl-Sim 13µl-Exp.
3.8µl-Exp.
2.1µl-Exp.
0.5µl-Exp.
Figure 5.3:The critical velocity of water drops of different sizes and the three surfaces (PMMA, Teflon, SHS) is compared be-tween experiments by Milne and Amirfazli (2009) and simula-tions.
stream. Their height ranges from approximately 0.9 to 2.5 times the boundary layer thicknessδ.
Besides the wall boundary and the in and outflow conditions, a free flow condition is set at the top patch of the domain and symmetry at the bottom patch. The numerical discretization is realized with a block-mesh of high quality which is refined dynamically at the drop in three levels. At the drop, the resolution reaches∆x=125µmwhich leads to a total of about one million cells. In contrast to Linder et al. (2015), the drop is fully resolved in 3D to account for asymmetric effects at the receding contact line of large drops. A velocity profile is initialized with an inflow velocity of2.0 m/s.
The drop is placed with the shape of a spherical cap (see equations (4.5) - (4.7)) according to the drop volume and the initial contact angles ofθinit =76◦, 124◦, 161◦for the surfaces PMMA, Teflon and SHS, respectively. In contrast to an acceleration of0.5 m/s2in the experiment, the velocity is ramped up in simulations by2 m/s2to avoid high computa-tional costs. Most challenging is the time-step restriction by the Courant number ofC o=0.4, which limits the time step to∆t=2×10−7sfor higher velocities. The Courant number couples the mesh size with the time step. Therefore, in this study drop volumes ofVD = 100µl, 58µland30µlare simulated, avoiding the smaller drops for which a higher resolution would be necessary due to the higher curvature.
In Figure 5.2 the displacement of the drop is shown exemplarily in terms of the contact line position at both extrema of the drop in up- and downstream direction while ramping up the velocity. At the beginning, the drop adjusts from the ini-tialized state to an equilibrium in the flow field. With increasing velocity the adhesion and the shear force increase. Both, the advancing and receding contact lines move, when the integral shear force on the drop surpasses the counteracting adhesion force.
In Figure 5.3 the critical channel velocitiesUcrfor the three surfaces and different drop volumes are compared between experimental results by Milne and Amirfazli (2009) and simulations. As given in equation (4.1), the contact line diameter of the initialized spherical capLb, combined with the difference of the cosine of advancing and receding contact angle, are a measure of the adhesion force acting on the drop which increases with drop volume and hysteresis. For PMMA there is a larger discrepancy between simulations and experiments. Since the simulation results match very well for the two other surfaces, the contact angle measurement in the experiments could be a reason for this discrepancy. For instance, an earlier incipient motion of the drop in simulation suggests that the advancing contact angle is expected to be much greater than that measured by Milne and Amirfazli. In Figure 2 of Milne and Amirfazli (2009) a downstream contact
Literature Methodology Note
White and Schmucker (2008) Exp. Incipient motion of drops in stagnation-point boundary layer flow inspired by aircraft foils.
Moghtadernejad et al. (2013, 2015a,b) Exp./Sim.
Experiments on drops moving in shear flow of up toU0≤90 m/s. Simula-tions with Smooth Particle Hydrodynamics method. No study on a critical shear velocity.
Maurer and Janoske (2015) Exp. Drops (VD=5−12µl) in turbulent shear flowRe=6000−16000in a channel with variable height. Includes critical shear velocities.
Maurer (2017) Exp./Sim.
Drops (VD=5−20µl) in turbulent shear flowRe=6000−16000in a chan-nel. VOF simulation without specification of the used turbulence model.
Focus on the influence of surface vibration on the incipient motion.
Barwari et al. (2018) Exp. Water and glycerin drops (VD=8−40µl) in turbulent shear flowRe= 2000−14000in a channel with variable height. Includes critical shear velocities.
Seiler et al. (2018a,b) Exp. Experiments with water and glycerin drops (VD=5−40µl) in turbulent shear flowRe=3000−32000on flat and complex surfaces.
This study Sim. 3D VOF simulation using theζ−f-VLES hybrid turbulence model. Water drops (VD=15−30µl) in turbulent shear flowRe=3000−15000.
Table 5.2:Literature on experiments and simulations of incipient drop motion in turbulent shear flow.
angle of aboutθmax =76◦ is visible in the third row depicting a drop at the critical velocityUcr for incipient motion.
In the fourth row, however, a contact angle aboveθmax =90◦ is unmistakable for a drop that just started moving and has consequentially a relatively low contact line velocity. Hence, the current dynamic contact angle should not be much apart of the advancing contact angle, which is stated to beθmax=76◦and simulated here as such. In conclusion, the real advancing contact angle for PMMA is expected to be much higher. Despite this discrepancy, the simulation results using PMMA, the simulations on Teflon and SHS match the experiments very well, following the characteristic dependencies on wetting properties and the drop volume.
Oscillations of drops in laminar shear flow are observed by Milne and Amirfazli as well as predicted by Thoroddsen et al. (2008) who relate the natural oscillation frequency to density, surface tension and drop size. An analysis of oscillations for drops in turbulent shear flow is given in Section 5.2. Oscillations also occur hand in hand with the in-dependent contact line movement of the advancing and receding front, manifested also in the caterpillar-like motion of the drop for shear velocities higher than the critical value. After having validated the methodology by a good match of experiments and simulations, the next level of complexity, embodied by turbulent shear flow, is analyzed in the following.