4.5 Results and Discussion
4.5.4 Distributions of Temperature and Heat Flux at the Wall Surface108
0.0 0.2 0.4 0.6 0.8 1.0 r
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 0.45 ms
0.0 0.2 0.4 0.6 0.8 1.0
r 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 2.95 ms
0.0 0.2 0.4 0.6 0.8 1.0
r 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 0.5 ms
0.0 0.2 0.4 0.6 0.8 1.0
r 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 3.5 ms
0.0 0.2 0.4 0.6 0.8 1.0
r 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 2.85 ms
0.0 0.2 0.4 0.6 0.8 1.0
r 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
h
t = 5 ms
Figure 4.10: Free surface morphology variation illustrating the entrapped bubble oscillation and its consequent breakup in the case F.
model for combined simulations of free-surface flow and conjugate heat transfer, the hydrodynamics of the oscillating bubble is out of scope and is not further analyzed.
4.5.4 Distributions of Temperature and Heat Flux at the Wall
0 1 2 3 r, mm
80 90 100 110 120 130
T w, °C
t = 0.5 ms t = 1 ms t = 3 ms
0 1 2 3
r, mm 0.0
2.0 4.0 6.0 8.0 10.0
q· w , MW/m2
t = 0.5 ms t = 1 ms t = 3 ms
Figure 4.11: Comparison of the present predictions for the radial distribution of temperature (left) and heat flux (right) at the solid surface (lines) with numerical results from Pasandideh-Fard et al. [74] (symbols).
the presence of the afore-mentioned air ring (Mehdi-Nejad et al. [58]), where higher temperature and correspondingly lower heat flux are computed, before the ring joins and collapses at the axis of symmetry. The temperature and heat flux distributions are rather smooth underneath the lamella in the reminder of the observed area up to the rim region, within which, domains of higher and lower temperatures are created due to the complex local flow field.
The theoretical solution for the hydrodynamics and heat transfer in drop im-pact onto a solid substrate was obtained by Roisman [97]. In this study a similar-ity solution for the Navier-Stokes and energy equations is found for the case of a nonisothermal flow configuration, including the phenomena of near-wall phase tran-sition. For the particular case of drop impact without phase change and constant thermophysical properties of the fluid and the wall, the theoretically predicted wall heat flux is expressed in the form
˙
qw,theor = eles(Ts,0−Td,0) [el+esI(Pr,0,∞)]√
π√
t at t >1. (4.24) where el and es are the thermal effusivities of liquid and solid substrate materials and I(Pr,0,∞) is a dimensionless function of the Prandtl number, the numerical values of which are provided by Roisman [97]. The theoretically predicted contact temperature is constant and uniform
Tc = elTd,0+esI(Pr,0,∞)Ts,0
el+esI(Pr,0,∞) . (4.25) Comparison of the present numerical predictions for the averaged heat flux at the solid surface with the analytical result is given in Fig. 4.12. In the simulations, the heat fluxes, averaged over the wall surface in one time step, are determined from the following expression
˙ qw =
∫
Sw( ˙qs·dSs)
∫
Sw|dSs| ≈
∑N
i=1[−kw,i(∇Tw,i)|⊥|Sf,i|]
∑N
i=1|Sf,i| , (4.26)
where the summation is performed over all cell-faces N at the solid-fluid interface.
Although heat transfer in the air is accounted for in simulations, it is much smaller compared to heat removed from the substrate by the liquid. Therefore, only those cell-faces belonging to cells filled with liquid are taken into account.
0 1 2 3 4 5
t 0.0
0.5 1.0 1.5 2.0
q· sim
/
q· theorcase A case B case C case D case E case F
Figure 4.12: Comparison of the present numerical predictions for the averaged heat flux at the solid surface with the analytical result from Roisman [97].
According to the results shown in Fig. 4.12, the theory and the numerics quickly converge at the dimensionless time t= 1. At the earlier times, not considered theo-retically, the average heat flux is influenced by the edge effects in the neighborhood of the contact line. The predicted average heat fluxes are slightly smaller for higher impact parameters in the short time during the initial spreading phase up to t ≈1, as can be seen in Fig. 4.12. This is found to be in accordance with the results from Pasandideh-Fard et al. [74], where similar behavior was obtained indicating that the drop’s cooling effectiveness is independent of the impact velocity only for large Weber numbers, We≫Re0.5.
In the present study the Weber numbers are much lower compared to this criterion resulting in the more pronounced dependence of the heat transfer rates on the impact velocity. This can be seen in Fig. 4.13, where the computed overall heat transfer versus time is shown. The averaged overall heat transfer from the wall surface to the drop is evaluated during the computations at every time instant from the expression
Q=
∫ t 0
[∫
Sw
( ˙qs,j·dSs) ]
dt≈
∑n j=1
[ ˙qw,jSw] ∆tj, (4.27) where the term in the brackets represents the heat transfer rate corresponding to the time step ∆tj and the summation is performed over all time steps n. The wall heat flux ˙qw,j on the r.h.s. of Eq. (4.27) corresponding to the time step ∆tj is calculated from Eq. (4.26) by assuming that the obtained value prevails during the time step. Thus, the amount of heat transferred from the wall surface is obtained by accumulating the heat fluxes from all time steps up to the given time tj. Since the
0 1 2 3 4 5 t, ms
0 0.1 0.2 0.3 0.4 0.5
Q , J
case A case B case C case D case E case F
Figure 4.13: Predicted averaged heat transfer at the solid surface.
mesh is a two-dimensional slice, representing a part of the cylinder with an angle of 5◦ in the azimuthal direction, the total heat transfer from the substrate to the drop is evaluated by multiplying the values obtained from Eq. (4.27) by 360/5 = 72 and these values are plotted in Fig. 4.13. It can be seen that the amount of heat transferred from the substrate to the spreading drop increases with increasing the impact velocity in all cases in the spreading phase, tending to decay in the receding phase due to lower velocities.
4.5.5 Mean Temperature in the Spreading Drop
In addition to the distributions of the temperature and the heat flux at the wall surface, it is interesting to examine the rise of the mean temperature in the spreading drop. The flow in the spreading lamella is fast, inertia-dominated, and it takes only a few milliseconds for the lamella to start receding. However, it is found that even in the relatively small temperature ranges considered here, the temperature in the spreading drop may increase significantly. The mean droplet temperature is determined in the simulation from the expression similar to Eq. (3.26)
Tmean =
∑N
i=1Ti∆Vi
∑N
i=1∆Vi , (4.28)
and the rise in the mean drop temperature over time is given in Fig. 4.14. It can be observed that the mean temperature within the spreading drop may rise up to≈50% of the initial temperature difference even during this very short initial spreading phase. This supports the observation that heat transfer may be significant already at the initial stage of spreading.
0 1 2 3 4 5 t, ms
10 20 30 40 50 60
Tmean , °C
case A case B case C case D case E case F
Figure 4.14: Predicted time variation of the mean drop temperature.