**3.1. Donor‐acceptor method**

( ) ( ) (( ) ) ( )

a r

T

222 2 2

**u u g F**

+Ñ× - +Ñ = - + +

a r

¶ ¶ (8)

=× +- × 1 2 ( ) 1 (9)

=× +- × 1 2 ( ) 1 (10)

(6)

(7)

1 1

where indices 1 and 2 show first and second fluids properties, and *α* is a scalar phase indica‐

1 Control volume is filled only with phase 1 0 Control volume is filled only with phase 2

This phase indicator function is the fluid property or volume fraction, which moves with it

0 *<sup>i</sup> i U t x* ¶ ¶ a

This function can be used to calculate the fluid properties in each phase as a weight function. In order to use a set of governing equations using the weight function, each fluid property should be calculated based on the volume occupied by this fluid in the surface cell as ex‐

> a r

 a m

Free surfaces considered here are those on which discontinuities exist in one or more varia‐ bles. This has been the challenge for researchers to omit or reduce this problem as much as possible. The transient state as well as phenomena such as surface tension, changing of fluid phase and Kelvin‐Helmholtz instability makes numerical simulation of such problems cumbersome. It is expected that methods used to simulate interface of fluids have a number of characteristics. These include mass conservation, simulating the interface as thin as possible,

 a

+ =

*<sup>P</sup> <sup>s</sup> <sup>t</sup>*

( ) ( )

Ñ× - Ñ +Ñ é ù ë û

22 2

**u u**

0 1 Interface present

r ar

m am

2 2

a

**u**

1

¶ -

a r

368 Numerical Simulation - From Brain Imaging to Turbulent Flows

¶

a m

1

tor function which is defined as follows:

ì ï = í ï î < <

a

and can be derived as follows:

pressed in Eqs. (9) and (10) [2]:

The main idea of donor‐acceptor approach is that the value of volume fraction in downwind cell, the acceptor cell, is used for anticipation of transferring fluid in each time step. The problem in this approach is that using downwind cell in calculations may lead to unreal situations which are values out of zero and unity domain in surface cells. **Figure 3a** shows this method with the first fluid with gray color and volume of fluid equals to unity. It could be seen that using donor‐acceptor approach with downwind differencing scheme results in values greater than unity in donor cell. It is because the second fluid in the acceptor cell is greater than the value needed in the donor cell. Similarly in **Figure 3b**, using downwind differencing scheme leads to negative values for volume of fluid, which is because the needed fluid in acceptor cell is more than what is in the donor cell [3].

**Figure 3.** Schematic view of donor‐acceptor approach [4].

In order to be assured that volume of fluid is between zero and one, the amount of fluid or volume of fluid in donor cell should be used to regulate the estimated fluid transferring between two adjacent cells [5].

One drawback of donor‐acceptor method is that this method changes any finite gradient into step, and consequently increases the slope of the surface model in the direction of flow. This problem was alleviated by proposing a method to consider the slope of interface for flux transferring in adjacent cells by Hirt and Nichols [6]. For this purpose, a donor‐acceptor equation was proposed so that it could detect the direction of the flow in interface and then define the upwind and downwind cells accordingly. Thereafter, this model was expanded

for 3D domains by Torrey et al. [7]. The Surfer method is one version of volume of fluid which deals with merging and fragmenting of interfaces in multiphase flows [8].

The volume of fluid method is one of the most popular methods for anticipation of interfa‐ ces, and many researches have been conducted based on this method including dam break, Rayleigh‐Taylor instability, wave generation and bubble movement [6, 9–12]. This method was modified in 2008 to get more accurate results by considering diagonal changes in fluxes of adjacent cells for structured grid domains [13, 14].
