Performance of passive scalar method in heat transfer simulation of a two-dimensional droplet focusing on the parasitic temperatures

Abstract

The pseudopotential models are very convenient, flexible and robust for practical purposes of two-phase dynamics, but the presence of spurious velocities is inevitable due to the limited discretization of gradient terms, and large amount of intermolecular forces at the interface of liquid and gas phases. The existence of such velocities, in addition to creating counterfeit flows, causes errors in obtaining the temperature field and solving the energy equation. Here, two models (Li (Phys Rev 89(5):053022, 2014) and Kupershtokh (Phys Rev E 98(2):023308, 2018)) are examined to show their capability in weakening and reducing the temperature parasites arising from the spurious velocities. Results are presented for a stationary droplet, which show the good performance of the exact difference method in reducing the parasitic temperatures. Also, the Marangoni phenomenon is simulated to validate the solution of the temperature field in the two-phase flow, which expresses the movement of droplet caused by the surface tension gradient. For more complicated problem, the Rayleigh–Bénard convection is simulated and effects of density ratio, Rayleigh number, and contact angle on the average wall Nusselt number and the average temperature of the droplet and vapor phases are investigated. The results of this work show the capability of the improved passive scalar method in simulating the two-phase heat transfer problems.

Appendices

Appendix 1: Laplace test

Laplace test, confirms the linear changes of the pressure difference between the inside and outside of the droplet with its radius, and determines the surface tension:

$$\Delta p = p_{{{\text{in}}}} - p_{{{\text{out}}}} = \frac{\sigma }{R}$$

(40)

where σ is the surface tension. Therefore, a droplet with four different radii is generated in a 200 × 200 computational domain. Then the pressure difference between the inside and outside of the droplet is calculated. After fitting a line from the created points, surface tension is obtained from the slope of the line according to Fig. 

Fig. 14

Linear changes of the pressure difference between the inside and outside of the droplet with its radius and computation of the surface tension

14. The surface tension for the three density ratios of γ = 5.64, γ = 31.35, and γ = 111.42 are obtained equal to σ = 0.0404, σ = 0.0917, and σ = 0.1485, respectively.

Appendix 2: Thermal performance of the single-phase

To ensure the accuracy of the passive scalar method, the Rayleigh–Bénard convection is investigated for the single-phase flow. To do this, a viscous liquid is heated in a rectangular reservoir from the bottom, and at the same time, the up boundary is kept at the low temperature. The Prandtl number is 0.71 and in all simulations, the condition $\beta gH\Delta T < 0.1$ is applied to ensure the incompressibility of the flow. The calculations are done in a grid of 100 × 50. The periodic boundary condition is applied for the lateral walls and the constant temperature is assumed for the upper and lower walls. The initial conditions for the temperature and pressure field are applied according to the following equations:

$$T(x,y) = T_{0} - \Delta T\frac{y}{H}$$

(41)

$$p(x,y) = \left[ {1 + \frac{{\rho {\beta g}_{0} \Delta Ty}}{2}\left(1 - \frac{y}{H}\right)} \right]\left[ {1 + .001\cos \left(\frac{2\pi x}{L}\right)} \right]$$

(42)

The single-phase temperature contours for the three Rayleigh numbers 5000, 10,000, and 50,000 are presented in Fig. 

Fig. 15

Temperature contours for the single-phase flow for a Ra = 5000, b Ra = 10,000 and c Ra = 50,000

15.

The average Nusselt numbers in this work are computed and compared with those of He et al. [19], as well as the correlation of ${\text{Ra}} = 1.56\left( {{\text{Ra}}/{\text{Ra}}_{c} } \right)^{0.296}$. Figure 

Fig. 16

Variation of averaged Nusselt numbers in terms of the Rayleigh number in the present work and available numerical and correlated results for the single flow

16 shows that the average Nusselt number gives a maximum error of 5.06% for the large Rayleigh number Ra = 10⁵. The discrepancy between the average Nusselt numbers in the present work and the correlated relation only appears for the Rayleigh numbers less than 3000, which is in agreement with the results of He et al. [19].