Multiphase Modelling

Impulse turbines are naturally multiphase and within the Pelton turbine two primary phases are to be modelled; air and water. The air and water interact with each other with high complexity through the interface, often referred to as the ‘free-surface’. In cases where the flow is highly turbulent, or where surface tension is no longer dominant for example, this interface can break down leading to air entrainment and droplet formation respectively. The entrained air alters the properties of the flow, namely the density and compressibility and as a consequence the turbulent nature of flow, and the droplet formation requires the modelling of the discrete particles as an additional third phase.

In most CFD solvers the free surface is modelled by approximation, this is due to the computational effort required to completely resolve the 3D Navier-Stokes equations. The nature of time marching simulations is that numerical viscosity (often called numerical or false

61 diffusion) smears sharp interfaces between phases. Consequently, in cases that involve complex fluid dynamics, the solution of the complete Navier-Stokes is requisite and approximation techniques to track or capture the interface are needed. Several methods are available to predict the location of the free-surface using both the Eulerian frame (in the case of static meshes and dynamic meshes) and also mesh free methods in the Lagrangian frame, summarised in Fig. 2-44.

Fig. 2-44 demonstrates how the interface can be constructed using two sub-techniques; 1) interface tracking, where the interface is modelled as a sharp boundary between the two fluids and the motion is explicitly tracked through the domain (b, c) and 2) interface capturing, where the interface is implicitly represented on a regular fixed mesh by marking the fluid on both sides of the interface (d, e, f).

The moving mesh method offers the advantage of treating the interface as a sharp discontinuity, which can lead to a more accurate representation, however when the interface undergoes large deformations significant errors can be introduced due to the correspondingly large mesh distortions. Furthermore the requirement for remeshing introduces significant simulation time. The front tracking method, where the flow is computed on a stationary mesh while the dynamic evolution of the interface (front) is explicitly tracked using Lagrangian-based approach also is hindered by computational time due to the many necessary operations to delete, insert reordering of the interface particles to preserve the resolution of the interface.

Interface-capturing methods offer significant advantages since they implicitly represent the interface on a regular fixed mesh by marking the fluids on either side of the interface using the following methods; d) virtual particles (marker and cell method), e) and indicator function usually volume fraction (Volume of Fluid (VOF) method), and f) a signed distance function (level set method). As the name implies the marker and cell method uses particles to mark the fluids, these are coupled and advected by the flow, the downside of this method is the high cost of the additional transport equations for the particles, and nevertheless this method does enforce mass conservation. The VOF method uses an indicator function, which physically represents the volume fraction of the one of the two phases, where the free surface is represented with the cell volume occupied by exactly half of each phase. The VOF method ensure mass conservation and since only one phase needs to be advected through the domain is more computationally efficient, however in coarser meshes the interface sharpness can become diffused over time [92]. Lastly, the level set method uses a signed distance function to represent the interface as a sharp boundary and therefore captures the interface more accurately than the VOF method. However, since it is not inherently mass conservative additional calculations are necessary to correct this leading to additional effort [93]. Based on the above descriptions a decision was

62 made to use the VOF method for multiphase modelling since it combines the best compromise between accuracy and simulation time.

Fig. 2-44 A schematic representation of a portion of a fluid-fluid interface and related methods used to model its shape and motion. (a) Interface between two fluids. Interface tracking methods such as (b) the interface-fitted moving mesh method, and (c) the front tracking method. Interface capturing methods such as (d) the marker-and-cell method, (e) the volume-of-fluid method, and (f) the level-set method [94]

2.7.4.1 Volume of Fluid

Volume methods the entire domain is marked by an indicator function. An indicator function is defined on a set X (in this case the mesh cells) that indicates membership of an element in a subset of A of X, having the value 1 for all elements of A (those where one phase is totally present) and value 0 for all elements of X not in A (those where the same phase is totally not present). In the Volume method the exact position of the interface is a posteriori and therefore special techniques that form part of the solution algorithm, need to be applied to capture the interface. In the case for FLUENT the indicator function is known as the volume fraction and it is implemented by Hirt and Nichols’ Volume of Fluid (VOF) algorithm [95]. Fig. 2-44 gives an example of how the VOF records the interface in the cell as indicated by the volume fraction distinguishing the presence of the phase fluid. The volume fraction of one of the two fluids can

63 be defined as, 𝛼𝑛, which varies smoothly between 0 and 1. The sum of the volume fraction of the n phases must therefore be:

∑ 𝛼𝑛

2

𝑛=1

= 1 (2.49)

The free surface interface can be defined when the value of 𝛼𝑛= 0.5. The indicator function is evolved using the following advection equation:

𝜕𝛼

𝜕𝑡 + ∇ ∙ (𝛼𝑛𝐕) = 0 (2.50)

The VOF has a clear advantage over a number of other techniques as the volume of fluid occupying the cell enforces continuity as that volume cannot be occupied by the other phase. The flow properties (i.e. density, viscosity) are a weighted function of the properties of both phases and since these properties from the continuity equation are then used to solve a single momentum equation through the domain, the attained velocity field is shared among the fluids. The precision of the VOF model is therefore dependant on the interface reconstruction technique and the advection scheme selected, which will be explored in more detail in 3.3.2.2. In summary the VOF offers the best compromise between the accuracy of the reconstruction of the interface and simulation time.