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Solving the Navier-Stokes equations when the density field contains a large sharp discontinuity---such as a water/air free surface---is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that...
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... first two-phase flow example is a classic a dam break in a box simulation as illustrated in figure 4 based on the experiments by Martin and Moyce (1952). This is one of the most commonly used test cases in the high Reynolds number, low surface tension regime. This is the regime most interesting for studying marine and offshore structures, which is the ultimate goal of the method. The total kinematic energy, figure 5 as a function of time for the slope limited and the non-limited methods. From the start, with both fluids at rest, the kinetic energy increases as the water mass starts to flow down and towards the right wall. There is a slight reduction as the water hits this wall at approximately t = 0.19 s. The Gibbs oscillations start to dominate the non-limited solution after a short time and from t = 0.04 s the solution is non-physical and completely dominated by numerical errors. In the rest of this paper we will remove the non-limited method from the results as it leads to non-physical solutions in all cases. The results of the two slope limited methods are compared to the experimental results in figure 6. The experimental data points are the ensemble averages of the results reported for the same geometry by Martin and Moyce (1952). We have employed free slip boundary conditions, and hence the surge front moves slightly faster than in the experimental results, but the qualitative behaviour is correct. The maximum height of the water column matches very well until the water hits the domain boundary and creates a jet shooting up along the right wall at t ≈ 0.19 s which corresponds to T ≈ 3.5 and τ ≈ 2.8. The right wall was placed much further away in the experimental ...
Context 2
... is the regime most interesting for studying marine and offshore structures, which is the ultimate goal of the method. The total kinematic energy, figure 5 as a function of time for the slope limited and the non-limited methods. From the start, with both fluids at rest, the kinetic energy increases as the water mass starts to flow down and towards the right wall. ...
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Solving the Navier-Stokes equations when the density field contains a large sharp discontinuity---such as a water/air free surface---is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that...
High-order solvers are accurate but computationally expensive as they require small time steps to advance the solution in time. In this work, we include a corrective forcing to a low-order solution to achieve high accuracy while advancing in time with larger time steps and achieving fast computations. This work is a continuation of our previous res...
Citations
... Their idea is to transform an incompressible flow system into a series of Poisson and convection-diffusion subproblems that are simpler to solve. Especially for multi-phase flows, which feature conservation laws of different nature, split-step methods are more popular [5,[9][10][11][12][13][14][15][16][17][18][19][20] than monolithic schemes [21][22][23]. Among the most common time-splitting schemes are so-called projection or pressure correction methods, in which incompressibility is enforced by projecting the velocity onto a weakly divergence-free space. ...
... since ∇ · (∇ × v) = 0 for any vector v. We are then left with only first-and second-order derivatives in (22). Now, the purpose of the χν∇(∇ · u) term artificially (but consistently) added in Eq. (12) becomes clear, and it is in fact twofold: it adds diffusion to the divergence equation (17) and allows us to eliminate the third-order derivatives in the PPE by simply setting χ = 2. ...
... The second-order velocity derivatives in (22) cannot be handled by standard Lagrangian finite elements, so we need a weak formulation containing only first-order derivatives. This is not straightforward, and in fact Li et al. [13] used finite differences to evaluate the higher-order derivatives at the nodal points, then interpolated those values in a continuous source term-which is hardly practical in unstructured meshes. ...
For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first-and second-order stepping, as well as flows with one, two or three phases.
... In recent decades, the discontinuous Galerkin (DG) method [28][29][30][31][32][33][34][35][36][37] has become a popular method for solving a system of conservation laws. In this study, we first develop a new modal discontinuous Galerkin method, for solving the two-fluid model of dusty gas flows. ...
The present work investigates the bubble formation and vortex shedding phenomena in the viscous flow of a compressible gas seeded with dust particles. A new modal discontinuous Galerkin method was developed for solving the two-fluid model of dusty gas flows. Most previous studies have been limited to flows with low Mach numbers without the presence of shock waves. This study considered a wider Mach number range, from subsonic to supersonic, in the presence of shock waves. We also investigated in detail the effects of the presence of solid particles on flow properties such as bubble size and frequency and the amplitude of the Bérnard-von Kármán vortex street. A novel approach was employed to circumvent the non-strictly hyperbolic nature of the equations of the dusty-gas flow model caused by the non-existence of the pressure term. This allowed the same inviscid numerical flux functions to be applicable for both the gaseous Euler and solid pressureless-Euler systems. The simulation results revealed that the transition from stationary flow to unsteady flow is dependent on both the Reynolds and Mach numbers of the flow. Moreover, it was shown that in stark contrast with the pure gas case above the critical Reynolds number in the supersonic regime, where no flow instability was observed, in the multiphase flows, adding particles produced flow instability. This unusual behavior is because the two-way coupling effects between the gas phase and solid phase override the compressibility effect and cause severe flow instability and spontaneous symmetry breaking in the coherent dynamics of the vortices.
A time-explicit Runge–Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection–reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ⩾ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results indicate that the proposed scheme is more robust and accurate in comparison to existing approaches based on the continuous Galerkin finite element method or the fast Fourier transform method.
