Two-phase models

Two-phase models can be grouped into models that either use: (1) the Eulerian-Eulerian approach whereby the fluid phase and sediment phase are treated as inter-penetrating continua; or, (2) the Eulerian-Lagrangian approach, by which the motion of the fluid phase is

treated as a continuous medium, whilst the sediment phase is treated as a discrete phase by tracking the motion of individual sediment grains. There are also three-phase models, in which air, water, and sediment are considered for shallow water conditions (e.g. [53]). However, a three-phase model is not considered in this work to model scour in very shallow waters (e.g. h/D < 2), because scour beneath subsea pipelines is the primary focus in this work. Therefore, a two-phase Eulerian-Eulerian model is employed, in which the fluid phase is modelled as a Newtonian fluid with a constant viscosity, whilst the sediment phase is also modelled as a “fluid”, but with a non-uniform viscosity; further details are presented in Chapter 3.

2.2.2.1 Eulerian-Eulerian models

An example of an output of a two-phase Eulerian-Eulerian model is shown in Figure 2.4, whereby tunnel erosion beneath a pipeline in a unidirectional flow is simulated. As seen in Figure 2.4, a non-dimensional volume fraction is used in Eulerian-Eulerian models to represent the dimensionless sediment concentration in every cell. The continuity and momentum equations for both the fluid phase and sediment phase are also typically solved using either the RANS or LES approach; though the models employed for turbulence modelling and the inter-phase momentum transfer would differ among different models. It seemed that the stresses of the sediment phase can either be modelled as: (1) a function of the sediment concentration, fluid properties and flow velocity (e.g. [54]); or, (2) via kinetic theory for granular flow [55]. The inter-phase momentum transfer is generally computed using an empirical drag model (e.g. [56]), since the drag force is the dominant contributor to the inter-phase momentum transfer term in dense fluid-solid systems [57]. A comprehensive review of CFD models for fluid-solid interaction is available in Van Wachem et al. [58].

Zhao and Fernando [59] appeared to be the first to use a two-phase model to simulate scour beneath a pipeline, whereby the stresses of the sediment phase are modelled using kinetic theory for granular flow. The Eulerian-Eulerian approach is employed, and the predicted equilibrium scour depth seemed to be in good agreement with the experimental measurement from Mao [3]. However, the scour depth has been over-predicted during the tunnel erosion stage, which is thought to be caused by having an initial gap between the pipe and the seabed at the beginning of the simulation to initialise the scour process. The gap is generated using a

sinusoidal function with an amplitude of 0.1D, whereas the pipe in the experiment [3] is bottom-seated (i.e. eo/D = 0).

Figure 2.4: The seabed profile represented by the dimensionless sediment volume fraction during tunnel erosion, which is modelled using a two-phase Eulerian-Eulerian model with a steady current flowing from left to right; figure reproduced from Yeganeh-Bakhtiary et al. [60].

In addition, an unrealistic accumulation of sediment near the pipe has been reported in Zhao and Fernando [59] when the two-phase model is employed. Hence, a different numerical solution procedure has been implemented, in which the velocity of the fluid phase is first solved without the influence of the sediment, and the solution of the fluid phase is then used to calculate the resulting sediment motion. Nonetheless, the unrealistic accumulation of sediment near the pipe may be caused by the use of the frictional viscosity model from Schaeffer [61], which is developed for “quasi-static flow”. Savage [62] stated that there are strain rate fluctuations which exist even in purely quasi-static flow, which will reduce the shear stress, but has not been considered in the model from Schaeffer [61]. It is hypothesised that this issue can be addressed by using the JohnsonJacksonSchaeffer frictional stress model [57], in which the frictional viscosity model proposed by Schaeffer [61] is combined with the normal frictional stress model from Johnson and Jackson [63]. This new model has been implemented in recent two-phase models for modelling sediment transport (e.g. [64]).

More recently, Yeganeh-Bakhtiary et al. [60] successfully simulated tunnel erosion underneath a pipeline using a two-phase Eulerian-Eulerian model (Figure 2.4). Several notable differences as compared to Zhao and Fernando [59] include: (1) only tunnel erosion

is modelled, and not scour equilibrium; (2) a “pure” two-phase calculation is performed, as opposed to constraining the sediment phase to first obtain a solution for the fluid velocities; (3) instead of adopting kinetic theory for granular flow, the inter-granular stresses are modelled as a function of the sediment concentration, fluid properties and flow velocity, where it seemed that the effects of the particle velocity fluctuations are ignored; (4) a fully- orthogonal mesh is used, which is trimmed at the surface of the pipe, and thus the boundary layer and flow separation may not be adequately modelled due to the non-uniform mesh around the pipe; and, (5) the initial seabed gap is smaller (i.e. one cell height), and the initial seabed profile is flat, instead of having a small sinusoidal profile beneath the pipe. A more accurate seabed profile during the tunnel erosion stage is predicted, and it is reported that having an initial sinusoidal profile has led to an over-predicted scour depth.

Unfortunately, it is unclear whether modelling scour equilibrium is not pursued due to a longer computation time, or the results would not be accurate. Furthermore, both Zhao and Fernando [59] and Yeganeh-Bakhtiary et al. [60] used the k-ε turbulence model [36] for the fluid phase, which is notorious for its poor performance in the near-wall region, and this would affect flow separation at the surface of the pipe. Therefore, in this work, a two-phase Eulerian-Eulerian model is employed, in which the k-ω SST turbulence model [65] is used for the fluid phase, while the JohnsonJacksonSchaeffer frictional stress model [57] is employed for the sediment phase; further details are presented in Chapter 3.

2.2.2.2 Eulerian-Lagrangian models

In regard to Eulerian-Lagrangian models, individual sediment grains can be included in the computational domain, and the trajectory of each grain would be calculated. For Eulerian- Eulerian models, the calculation of particle-particle interactions is influenced by the mesh resolution. Having the fluid phase represented by a continuous medium, and the sediment phase represented by Lagrangian particles, can potentially provide a more physically accurate solution. However, this also translates to longer computational run times and the need for an immensely larger amount of memory [66]. A well-written review on discrete particle simulations has been published in Zhu et al. [67].

Figure 2.5: Piping at different upstream current velocities which is modelled using CFD-DEM; figure extracted from Zhang et al. [68].

Figure 2.5 depicts an example of an Eulerian-Lagrangian simulation of the onset of scour beneath a pipeline, where a coupled computational fluid dynamics-discrete element method (CFD-DEM) model [68] is employed. For this CFD-DEM model, turbulence associated with the fluid phase is modelled using the standard k-ε turbulence model [36], while the drag force acting on the particles is modelled using the drag model from Di Felice [69], and a linear spring model [70] is employed to account for inter-particle collisions. It is stated that periodic boundaries, which are parallel to the flow direction in Figure 2.5, had to be implemented in between 10 layers of particles, to limit the total number of particles in the simulation. When a particle exits one of the periodic boundaries, it would enter through the other periodic boundary which is on the opposite side of the domain. In addition, the computational time step had to be very small (i.e. 10-6 s), to ensure that every inter-particle collision is modelled whilst maintaining numerical stability.

Yeganeh-Bakhtiary et al. [66] successfully implemented a two-dimensional CFD-DEM model to predict scour equilibrium underneath a pipeline, wherein the particles are represented by rigid disks with uniform diameters. The particle diameter had been set to 2 mm, instead of 0.36 mm which is used in the experiment conducted in Mao [3]. This greatly reduced the number of particles needed (i.e. 6,240), as compared to having 110,000 particles in Zhang et al. [68]. However, the equilibrium scour depth had been under-predicted, which might be a result of having inaccurate sediment properties, or further calibration of the coefficients associated with DEM may be required. In a more recent study which involved models similar to that in Zhang et al. [68], Yang et al. [71] adopted a small computational domain and large sediment particle size (i.e. 1 mm in diameter) to reduce the computational

load. Scour would occur at the upstream edge of the sediment container, and this may affect the scour profile around the pipe when the computational domain is not sufficiently large (discussed in Chapter 7). Therefore, an Eulerian-Lagrangian model is not employed in this work because it might not be more accurate than Eulerian-Eulerian models, yet it requires more computational resources as compared to Eulerian-Eulerian models [72].