Related Studies

The computation of flows with free surfaces is difficult in the prediction of the shape and the position of the interface between two immiscible fluids. Hirt and Nichols [4] illustrated that three types of problems arise in the numerical treatment of free surfaces: (1) their discrete representation, (2) their evolution in time, and (3) the manner in which boundary conditions are imposed. Therefore, it is important to develop robust methods to capture interfaces. Such methods in finite volume approach can be classified into two typical categories: surface methods and volume methods [5]. In the following, these methods are briefly reviewd.

(a) (b)

Figure 1.1 (a) surface methods (b) volume methods

Surface methods, which are called interface-tracking method, mark and track the interface with calculation only on one phase based on the satisfaction of two conditions. First, the free surface is a sharp interface between the two fluids and there is no flow across the interface. Second, forces acting on the interface are in equilibrium. These methods only compute the liquid flow and the computational grids vary with the shape and position of the free surface. The differences between surface methods and volume methods can be shown as Figure 1.1. By surface methods the grid fits the interface and moves with it. The free surface is treated as a boundary of the computation domain. The advantage of this approach is that the interface remains sharp as it is across the mesh and the surface tension force is included in the subsequent implement without be coped specially. On the other hand, interface-tracking or moving meshes methods can not be used if the interface changes significantly. There are several interface-tracking (surface) methods which mark and track the interface explicitly, (i) with a set of marker particles or line segments on interface [6], or (ii) height functions [7], (iii) level set method [8], and (iv) surface fitted methods. [5]

In volume methods, the different fluids are marked either by massless particles or by an indicator function which may turn to be volume fraction. The advantage of these methods is their ability to cope with the arbitrary shaped interface and large deformation. One of the main difficulties associated with these methods is the advection of the interface without

The earliest numerical technique designed for simulating free surface flows was the well-known marker and cell (MAC) method [9] .The MAC method use massless marker particles which spread over the volume occupied by a fluid with a free surface (see Figure 1.2). A cell with no marker particles is considered to be empty. This method can treat complex phenomena like wave breaking. Although the MAC method can tackle arbitrary unstructured grids, there two problems arise in further simulation. First, the disadvantage of this method is that the computer storage will increase significantly and time consuming is largely increased in three-dimension of calculations. The other is that it is also difficult to obtain quantitative information on interface orientation. However, some improved versions have been presented, for example by Chen et al. [10]

Figure 1.2 Schematic representation of marker and cell mesh layout.

Several volume tracking methods for finite volume and arbitrary meshes have been developed with the aim of maintaining sharp interfaces. The better known methods are Simple Line Interface Calculation (SLIC) method [11], the volume-of-fluid (VOF) method [5] and the method of Youngs[12].

(a) Actual fluid configuration (b) SLIC (x-direction)

(c) SLIC (y-direction) (d) Youngs’ VOF

Figure 1.3 Interface reconstructions of (a) actual fluid configuration, (b) SLIC in x-direction, (c) SLIC in y-direction (d) Youngs’ VOF.

The SLIC method [11] approximates interfaces as piecewise constant, where interfaces within each cell are reconstructed using straight lines parallel to one of the coordinate directions. It is a direction-split algorithm and during each direction sweep, only cell neighbors in the sweep direction are used to determine the interface reconstruction. (Figure 1.3(b) and 1.3(c)). An extensive review of this type of methods can be found in [13].

Youngs’ VOF (Y-VOF) [12] method uses a more accurate interface reconstruction than the SLIC method. Youngs give a useful refinement to the SLIC method with the use of oblique lines to approximate the interface in the cell (Figure 1.3(d)). Unlike the SLIC method, the neighbor cells are used to approximate the slope of the interface in Youngs’ VOF. The SLIC method and Youngs’ VOF can be classified as line technique and these methods use structured grids and are therefore restricted to such mesh topologies.

The VOF method uses donor-acceptor approximation and is one of the most effective

volume tracking methods in the simulation and prediction of two-fluid system with interfaces where density and viscosity change abruptly. The VOF method became popular in the 80s though the development of this method occurred earlier [9,14]. Its appealing feature is its volume fluxes can be formulated algebraically without reconstructing the interface. The interfaces are represented by the value of the VOF function which is a volume fraction of one of the fluids. Another benefit of using volume fractions is that only a scalar convection equation, which is called indicator equation, need to be solved to propagate the volume fractions through the computational domain. In the calculation of convective fluxes, two of the main errors in numerical modeling of convection-diffusion transport problems are numerical diffusion and numerical dispersion. Numerical diffusion causes smearing of predicted profile, and numerical dispersion results in non-physical phenomenon such as oscillation or over/under shoots produced in the solution.

Because the first-order upwind scheme is too diffusive, differencing schemes of order higher than unity are therefore required. Second-order accuracy can be achieved by linear interpolation from two upwind values, yield so-called linear upwind schemes (LUDS) of Shyy et al. [15]. Third-order accuracy is achieved by a quadratic line through those two upwind values and one downwind, the QUICK scheme of Leonard [16]. These schemes get more accurate results than first-order upwind, but they are not bounded which may give rise to oscillations in the region where there are strong gradients of the variable being solved.

There are two major categories to handle these numerical dispersion problems, known as flux-blending and composite flux-limiter methods. The flux-blending method can be divided into two classes. The first class is added an anti-diffusion flux to a first order upwind scheme, for example, Zalesak presented the flux corrected transport (FCT) method [17]. In the second class, a smoothing diffusive flux is introduced into an unbounded higher order schemes. In generally speaking, due to the nature of flux-blending method, the numerical dispersion can be reduced but it will be very expansive computationally and the optimal blending factor can not be obtained easily.

Rudman used the flux corrected transport (FCT) method and developed the FCT-VOF

[18] to enhance the accuracy of simulation. The first order upwind is diffusive and stable. The first order downwind is unstable, but has the advantage of maintain sharp interface. If a suitable combination of upwind and downwind fluxes can be formulated, a volume tracking algorithm can be designed. Hence, this method combines the low order scheme and higher order scheme. Although the FCT-VOF method is non-diffusive in nature, but create unphysical flotsam and jetsam.

As everyone knows, only the first order upwind difference scheme satisfies the sufficient condition of the Convective Boundedness Criteria (CBC) [23]. In order to fit in with the CBC, a higher order difference scheme has to be a non-linear combined framework. In the composite flux-limiter method, the numerical flux at a consider face is adjusted by the flux-limiter function. Normalized Variable (NV) and Normalized Variable Diagram (NVD) which presented by Leonard [19] are important implements for the composite flux-limiter method and simplify the connection between high accuracy and bounedness in high resolution schemes (HRs) [20].

At first, the flux limiter function is presented by Van Leer [21]. Sweby [22] developed the Total Variation Diminishing (TVD) approach for high resolution schemes. Flux limiter functions were introduced to guarantee that values of a conserved property remain within the bounds. The accurate simulation of convection continues to attract many workers due to the many challenges it still offers. The difficulty in devising an effective scheme lies in the conflicting requirements of accuracy, stability and boundedness. There are many high resolution schemes developed in these years, such as SMART of Gaskell and Lau [23], GAMMA of Jasak [24], SUPERBEE of Roe [25], STOIC of Darwish [26], MUSCL and Van Leer of Van Leer [27]

Numerical diffusion, a significant source of error in numerical solution of conservation equations, can be separated into two components, namely cross-stream and stream-wise numerical diffusions. The former, the cross-stream numerical diffusion, occurs in a multi-dimensional flow when the flow direction is perpendicular to the grid lines. The latter,

lines.

In addressing aforementioned issue, many researchers try to improve the accuracy by reducing streamwise and cross-stream numerical diffusion. Hence, the blending strategy that depends on the angle between the flow direction and the grid lines was developed. The best approach is to have a continuous switching function whereby the values of compressive and high resolution schemes are blended together with a blending factor. This general approach has been followed in the derivation of composite schemes and is utilized in the HRIC of Muzaferija [28], STACS of Darwish [29] and CICSAM of Ubbink [30].