Two-phase Incompressible Navier Stokes Equations Research Articles

Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counterpart of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior calculus operators. We prove the boundedness of the method for the forward Euler and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme to handle large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

Discrete Exterior Calculus Discretization of Two-Phase Incompressible Navier-Stokes Equations with a Conservative Phase Field Method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counter part of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior operators. We prove the boundedness of the method for the first order Euler forward and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme towards handling large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

A thermodynamically consistent model and its conservative numerical approximation for moving contact lines with soluble surfactants

We derive a continuum sharp-interface model for moving contact lines with soluble surfactants in a thermodynamically consistent framework. The model consists of the isothermal two-phase incompressible Navier–Stokes equations for the fluid dynamics and the bulk/surface convection–diffusion equations for the surfactant transportation. The interface condition, the slip boundary condition, the dynamic contact angle condition, and the adsorption/desorption condition are derived based on the principle of the total free energy dissipation. In particular, we recover classical adsorption isotherms from different forms of the surface free energy. The model is then numerically solved in two spatial dimensions. We present an Eulerian weak formulation for the Navier–Stokes equations together with an arbitrary Lagrangian–Eulerian weak formulation for the surfactant transport equations. Finite element approximations are proposed to discretize the two weak formulations on the moving mesh. The resulting numerical method is shown to conserve the total mass of the surfactants exactly. By using the proposed model and its numerical method, we investigate the droplet spreading and migration in the presence of surfactants and study their dependencies on various dimensionless adsorption parameters.

Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier–Stokes equations

We present a numerical method for two-phase incompressible Navier–Stokes equation with jump discontinuities in the normal projection of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuities sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure using recently introduced techniques by Egan and Gibou [9]. Numerical results support the inference that the proposed method converges in L∞ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.

A finite element method for electrowetting on dielectric

We consider the problem of electrowetting on dielectric. The system involves the dynamics of a conducting droplet, which is immersed in another dielectric fluid, on a dielectric substrate under an applied voltage. The fluid dynamics is modeled by the two-phase incompressible Navier-Stokes equations with the standard interface conditions, the Navier slip condition on the substrate and a contact angle condition which relates the dynamic contact angle and the contact line velocity, as well as the kinematic condition for the evolution of the interface. The electric force acting on the fluid interface is modeled by the Maxwell's equations in the domain occupied by the dielectric fluid and the dielectric substrate. We develop a numerical method for the model based on its weak form. This method combines the finite element method for the Navier-Stokes equations on a fixed bulk mesh with a parametric finite element method for the dynamics of the fluid interface, and the boundary integral method for the electric force along the fluid interface. Numerical examples are presented to demonstrate the accuracy and convergence of the numerical method, the effect of various physical parameters on the interface profile and other interesting phenomena such as the transportation of droplet driven by applied non-uniform voltage.

Water wave transmission and energy dissipation by a floating plate in the presence of overwash

A numerical model, based on the two-phase incompressible Navier–Stokes equations, is used to study transmission of regular water waves by a thin floating plate in two dimensions. The model is shown to capture the phenomenon of waves overwashing the plate, and the generation of turbulent bores on the upper plate surface. It is validated against laboratory experimental measurements, in terms of the transmitted wave field and overwash depths, for a set of incident wave periods and steepness values. Corresponding simulations are performed for a thick plate that does not experience overwash, which are validated using experiments where an edge barrier prevents thin-plate overwash. The model accurately reproduces (i) the linear relationship between the transmitted and incident amplitudes for the thick plate, and (ii) the decrease in proportion of incident-wave transmission for the thin plate, as incident steepness increases. Model outputs are used to link the decreasing transmission to wave-energy dissipation in the overwash, particularly where bores collide, and in the surrounding water, particularly at the plate ends. It is shown that most energy dissipation occurs in the overwash for the shortest incident waves tested, and in the surrounding water for the longer incident waves. Further, evidence is given that overwash suppresses plate motions, and causes asymmetry in plate rotations.

Pressure reconstruction for weak solutions of the two-phase incompressible Navier–Stokes equations with surface tension

For the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.

Numerical analysis on the secondary load cycle on a vertical cylinder in steep regular waves

The secondary load cycle induced by regular waves of large amplitude has an important influence on the ringing response of offshore structures. To understand the mechanism of the effect of local violent water deformation of the free surface around a vertical cylinder on the wave loads, the numerical simulations are carried out for the cases of the steep regular waves. The numerical wave tank is established by solving the two-phase incompressible Navier–Stokes equations. The VOF method is applied to capture the violent flow with the free surface near the cylinder. The numerical model is validated by experimental measurement and theoretical result. New characteristic parameters of the secondary load cycle are defined in terms of the amplitude, duration and phase. The effects of wave steepness, relative diameter and relative depth on these characteristic parameters are presented and discussed. The results show that the characteristic parameters depend mainly on relative wave height and relative diameter of cylinder while the dependence on relative depth is weak. Using the fitting method, the empirical formulas are proposed to calculate the characteristic parameters of the secondary load cycle in terms of different wave steepness and relative diameter.

Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

Analyses of Wave Forces on Surface Piercing Vertical Cylinders of Intermediate Scale

Maximum horizontal forces by regular waves on a surface piercing vertical cylinder of intermediate scales are computed by solving the two-phase incompressible Navier-Stokes equations with a VOF method. The incident waves are generated by the fifth order Stokes wave and cnoidal wave. The numerical flume is verified and validated by comparison to the second order Stokes theory and by grid-independence. A variety of scales and wave steepness are computed in deep water and finite depth. In fully nonlinear wave condition, the viscosity and diffraction effects are discussed. A more accurate demarcation of intermediate scales is given by scales (the ratio of the diameter to the wavelength) from 0.1 to 0.125.

Forcing of a bottom-mounted circular cylinder by steep regular water waves at finite depth

AbstractForcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

Numerical solution of the two-phase incompressible Navier–Stokes equations using a GPU-accelerated meshless method

Abstract This paper presents the development and implementation of a Meshless two-phase incompressible fluid flow solver and its acceleration using the graphics processing unit (GPU). The solver is formulated as a Localized Radial-Basis Function Collocation Meshless Method and the interface of the two-phase flow is captured using an implementation of the Level-Set method. The Compute Unified Device Architecture (CUDA) language for general-purpose computing on the GPU is used to accelerate the solver. Through the combined use of the LRC Meshless method and GPU acceleration this paper seeks to address the issue of robustness and speed in computational fluid dynamics. Traditional mesh-based methods require extensive and time-consuming user input for the generation and verification of a computational mesh. The LRC meshless method seeks to mitigate this issue through the use of a set of scattered points that need not meet stringent geometric requirements like those required by finite-volume and finite-element methods, such as connectivity and poligonalization. The method is shown to render very accurate and stable solutions and the implementation of the solver on the GPU is shown to accelerate the solution by several orders.

Solving incompressible two-phase flows on multi-GPU clusters

We present a fully multi-GPU-based double-precision solver for the three-dimensional two-phase incompressible Navier–Stokes equations. It is able to simulate the interaction of two fluids like air and water based on a level-set approach. High-order finite difference schemes and Chorin’s projection approach for space and time discretization are applied. An in-depth performance analysis shows a realistic speed-up of the order of three by comparing equally priced GPUs and CPUs and more than a doubling in energy efficiency for GPUs. We observe profound strong and weak scaling on two different multi-GPU clusters.

Photorealistic visualization and fluid animation: coupling of Maya with a two-phase Navier-Stokes fluid solver

We have coupled the three-dimensional solver for the two-phase incompressible Navier-Stokes equations NaSt3DGPF with Autodesk Maya. Maya is the industry standard software framework for the creation of three-dimensional animations. The parallel level-set-based fluid solver NaSt3DGPF simulates the interaction of two fluids like air and water. It uses high-order finite difference discretization methods that are designed for physics applications. By coupling both applications, we are now able to set up scientific fluid simulations in an easy-to-use user interface. Moreover, the rendering techniques provided by Maya allow us to create photorealistic visualizations for computational fluid dynamics problems and support the creation of highly visually realistic fluid simulations for animation movies. Altogether, we obtain an easy usable and fully coupled fluid animation toolkit for two-phase fluid simulations. These are the first published results of the full integration of a physics-oriented, high-order grid-based parallel two-phase fluid solver in Maya, at least to our knowledge.

A conservative phase field method for solving incompressible two-phase flows

In this paper a conservative phase-field method based on the work of Sun and Beckermann [Y. Sun, C. Beckermann, Sharp interface tracking using the phase-field equation, J. Comput. Phys. 220 (2007) 626–653] for solving the two- and three-dimensional two-phase incompressible Navier–Stokes equations is proposed. The present method can preserve the total mass as the Cahn–Hilliard equation, but the calculation and implementation are much simpler than that. The dispersion-relation-preserving schemes are utilized for the advection terms while the Helmholtz smoother is applied to compute the surface-tension force term. To verify the proposed method, several benchmarks are examined and shown to have good agreements with previous results. It also shows that the satisfactions of mass conservations are guaranteed.

A multi-GPU accelerated solver for the three-dimensional two-phase incompressible Navier-Stokes equations

The use of graphics hardware for general purpose computations allows scientists to enormously speed up their numerical codes. We presently investigate the impact of this technology on our computational fluid dynamics solver for the three-dimensional two-phase incompressible Navier-Stokes equations, which is based on the level set technique and applies Chorin’s projection approach. To our knowledge, this is the first time, that a two-phase solver for the Navier-Stokes equations profits from the computation power of modern graphics hardware. As part of our project, a Jacobi preconditioned conjugate gradient solver for the pressure Poisson equation and the reinitialization of the level set function of our CPU based code were ported to the graphics processing unit (GPU). They are implemented in double precision and parallelized by the Message Passing Interface (MPI). We obtain speedups of 16.2 and 8.6 for the Poisson solver and the reinitialization on one GPU in contrast to a single CPU. Our implementation scales close to perfect on multiple GPUs of a distributed memory cluster. This results in excellent speedups of 115.8 and 53.7 on eight GPUs of our cluster. Furthermore our whole multi-GPU accelerated solver achieves an impressive speedup of 69.6 on eight GPUs/CPUs.