Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.

NUMERICAL INVESTIGATION OF THREE-DIMENSIONAL VISCOUS INCOMPRESSIBLE FLOWS IN DIVERGENT CURVED CHANNELS AND TURBULENT MODEL STUDY

In order to make the numerical calculation of viscous flows more convenient for the flows in channel with complicated profile governing equations expressed in the arbitrary curvilinear coordinates were derived by means of Favre density-weighted averaged method, and a turbulent model with effect of curvature modification was also derived. The numerical calculation of laminar and turbulent flown in divergent curved channels was carried out by means of parabolizeil computation method. The calculating results were used to analyze and investigate the aerodynamic performance of talor cascades in compressors preliminarily.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by join- ing the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an im- plicit second-order scheme is utilized to enhance the com- putational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method （FVM） and the pressure Poisson equation is solved by the Galerkin finite el- ement method （FEM）. The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both veloc- ity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

An Eulerian SPH method with WENO reconstruction for compressible and incompressible flows

While Eulerian smoothed particle hydrodynamics(SPH)method has received increasing attention in scientific and industrial communities owing to its high spatial accuracy,it exhibits excessive numerical dissipation due to the fact that the flux is derived in particle pair pattern.In this paper,we adopt a one-dimensional weighted essentially non-oscillatory(WENO)reconstruction to reduce the numerical dissipation and improve the overall accuracy particularly in capturing the contact discontinuity.The underlying principle is to construct a 4-point stencil along the interacting line of each particle pair and then the WENO scheme is applied to reconstruct the initial states of the Riemann problem which determines the flow flux.A set of benchmark tests for both compressible and incompressible flows are studied to investigate the accuracy,robustness and versatility of the proposed Eulerian SPH method with the WENO reconstruction(ESPH-WENO).

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations,we show that the simple bound-preserving limiter in Li et al.(SIAM J Numer Anal 56:3308–3345,2018)can enforce the strict bounds of the vorticity,if the velocity field satisfies a discrete divergence free constraint.For reducing oscillations,a modified TVB limiter adapted from Cockburn and Shu(SIAM J Numer Anal 31:607–627,1994)is constructed without affecting the bound-preserving property.This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.

A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows

In this work,we develop a novel high-order discontinuous Galerkin(DG)method for solving the incompressible Navier-Stokes equations with variable density.The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term.Then,since the hyperbolic nature of the governing equations is recovered,the simple and robust Harten-Lax-van Leer(HLL)flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations.The viscous term is discretized by the direct DG(DDG)method,the construction of which was initially inspired by the weak solution of a scalar diffusion equation.In addition,in order to eliminate the spurious oscillations around sharp density gradients,a local slope limiting operator is also applied during the highly stratified flow simulations.The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases.Due to its advantages of simplicity and robustness in implementation,the present method offers an effective approach for simulating the variable density incompressible flows.

A Numerical Comparison of Outflow Boundary Conditions for Spectral Element Simulations of Incompressible Flows

Outflow boundary conditions(OBCs)are investigated for calculation of incompressible flows by spectral element methods.Several OBCs,including essentialtype,natural-type,periodic-type and advection-type,are compared by carrying out a series of numerical experiments.Especially,a simplified form of the so-called Orlanski’s OBCs is proposed in the context of spectral element methods,for which a new treatment technique is used.The purpose of this paper is to find stable low-reflective OBCs,suitable and flexible for use of spectral element methods in simulation of incompressible flows in complex geometries.The computation is firstly carried out for a 2D simulation of Poiseuille-B´enard channel flow with Re=10,Ri=150 and Pr=2/3.This flow serves as a useful example to demonstrate the applicability of the proposed OBCs because it exhibits a feature of vortex shedding propagating through the outflow boundary.Then a 3D flow around an obstacle is computed to show the efficiency in the case of more general geometries.Among the tested OBCs,the advection-type OBCs are proven to have better behavior as compared with the others.

On the modeling of viscous incompressible flows with smoothed particle hydrodynamics

Smoothed particle hydrodynamics （SPH） is a Lagrangian, meshfree particle method and has been widely applied to diffe- rent areas in engineering and science. Since its original extension to modeling free surface flows by Monaghan in 1994, SPH has been gradually developed into an attractive approach for modeling viscous incompressible fluid flows. This paper presents an overview on the recent progresses of SPH in modeling viscous incompressible flows in four major aspects which are closely related to the computational accuracy of SPH simulations. The advantages and disadvantages of different SPH particle approximation sche- mes, pressure field solution approaches, solid boundary treatment algorithms and particle adapting algorithms are described and analyzed. Some new perspectives and fuRtre trends in SPH modeling of viscous incompressible flows are discussed.

Physics-informed deep learning for incompressible laminar flows

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems,whose basic concept is to embed physical laws to constrain/inform neural networks,with the need of less data for training a reliable model.This can be achieved by incorporating the residual of physics equations into the loss function.Through minimizing the loss function,the network could approximate the solution.In this paper,we propose a mixed-variable scheme of physics-informed neural network(PINN)for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers.A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy.The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions.Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.

THE FINITE VOLUME PROJECTION METHOD WITH HYBRID UNSTRUCTURED TRIANGULAR COLLOCATED GRIDS FOR INCOMPRESSIBLE FLOWS

In this article a finite volume method is proposed to solve viscous incompressible Navier-Stokes equations in two-dimensional regions with corners and curved boundaries. A hybrid collocated-grid variable arrangement is adopted, in which the velocity and pressure are stored at the centroid and the circumcenters of the triangular control cell, respectively. The cell flux is defined at the mid-point of the cell face. Second-order implicit time integration schemes are used for convection and diffusion terms. The second-order upwind scheme is used for convection fluxes. The present method is validated by results of several viscous flows.

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver(LBFS)is presented in this work for simulation of incompressible viscous and inviscid flows.The new solver is based on Chapman-Enskog expansion analysis,which is the bridge to link Navier-Stokes(N-S)equations and lattice Boltzmann equation(LBE).The macroscopic differential equations are discretized by the finite volume method,where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers.The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.LBFS is validated by its application to simulate the viscous decaying vortex flow,the driven cavity flow,the viscous flow past a circular cylinder,and the inviscid flow past a circular cylinder.The obtained numerical results compare very well with available data in the literature,which show that LBFS has the second order of accuracy in space,and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.

LPS'S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper we derive LPS＇s criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in R^3. We show that if 0 〈 T 〈 ＋∞ is the maximal time interval for the unique smooth solution u ∈ C^∞（[0, T）,R^3）,then ｜u｜＋｜△d｜∈L^q（[0,T],L^p（R^3））,where p and q satisfy the Ladyzhenskaya-Prodi-Serrin＇s condition：3/p＋2/q=1 and p∈（3,＋∞].

Multi-scale Equations for Incompressible Turbulent Flows

The short-range property of interactions between scales in incompressible turbulent flow was examined. Some formulae for the short-range eddy stress were given. A concept of resonant-range interactions between extremely contiguous scales was introduced and some formulae for the resonant-range eddy stress were also derived. Multi-scale equations for the incompressible turbulent flows were proposed. Key words turbulence - incompressible flow - interactions between scales - multi-scale equations MSC 2000 76F70

STABLE FOURTH-ORDER STREAM-FUNCTION METHODS FOR INCOMPRESSIBLE FLOWS WITH BOUNDARIES

Fourth-order stream-function methods are proposed for the time dependent, incom- pressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.

An Immersed Method Based on Cut-Cells for the Simulation of 2D Incompressible Fluid Flows Past Solid Structures

We present a cut-cell method for the simulation of 2D incompressible flows past obstacles.It consists in using the MAC scheme on cartesian grids and imposing Dirchlet boundary conditions for the velocity field on the boundary of solid structures following the Shortley-Weller formulation.In order to ensure local conservation properties,viscous and convecting terms are discretized in a finite volume way.The scheme is second order implicit in time for the linear part,the linear systems are solved by the use of the capacitance matrix method for non-moving obstacles.Numerical results of flows around an impulsively started circular cylinder are presented which confirm the efficiency of the method,for Reynolds numbers 1000 and 3000.An example of flows around a moving rigid body at Reynolds number 800 is also shown,a solver using the PETSc-Library has been prefered in this context to solve the linear systems.

CORRECTION METHODS FOR STEADY INCOMPRESSIBLE FLOWS

Correction methods for the steady semi-periodic motion of incompressible fluid are investigated. The idea is similar to the influence matrix to solve the lack of vorticity boundary conditions. For any given boundary condition of the vorticity, the coupled vorticity-stream function formulation is solved. Then solve the governing equations with the correction boundary conditions to improve the solution. These equations are numerically solved by Fourier series truncation and finite difference method. The two numerical techniques are employed to treat the non-linear terms. The first method for small Reynolds number R equals 0-50 has the same results as that in M. Anwar and S.C.R. Dennis' report. The second one for R greater than 50 obtains the reliable results. (Author abstract) 4 Refs.

A hybrid scheme for computing incompressible two-phase flows

We propose a hybrid scheme for computations of incompressible two-phase flows. The incompressible constraint has been replaced by a pressure Poisson-like equation and then the pressure is updated by the modified marker and cell method. Meanwhile, the moment equations in the incompressible Navier-Stokes equations are solved by our semidiscrete Hermite central-upwind scheme, and the interface between the two fluids is considered to be continuous and is described implicitly as the 0.5 level set of a smooth function being a smeared out Heaviside function. It is here named the hybrid scheme. Some numerical experiments are successfully carried out, which verify the desired efficiency and accuracy of our hybrid scheme.

An analysis of the incompressible viscous flows problem by meshless method

Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.

GENERAL EXACT SOLUTION OF INCOMPRESSIBLE POTENTIAL FLOWS AROUND TWO CIRCLES

Three exact solutions are obtained for 2-D incompressible potential flows around two moving circles in three cases: (i) expansion (or contraction) of themselves, (ii) approaching (or departing from) each other, (iii) moving perpendicularly to the line connecting the centres in opposite directions. Meanwhile, an- other set of two exact solutions is obtained for 2-D incompressible potential flows between two moving eccen- tric circles in two cases: moving parallelly or perpendicularly to the line connecting the centres.