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.

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.

Combined adaptive meshing technique and characteristic-based split algorithm for viscous incompressible flow analysis

A combined characteristic-based split algorithm and all adaptive meshing technique for analyzing two-dimensional viscous incompressible flow are presented. Tile method uses the three-node triangular element with equal-order interpolation functions for all variables of tile velocity components and pressure. The main advantage of the combined nlethod is that it inlproves the sohltion accuracy by coupling an error estinla- tion procedure to an adaptive meshing technique that generates small elements in regions with a large change ill sohmtion gradients, mid at the same time, larger elements in the other regions. The performance of the combined procedure is evaluated by analyzing one test case of the flow past a cylinder, for their transient and steady-state flow behaviors.

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.

A Staggered Grid Method for Solving Incompressible Flow on Unstructured Meshes

A finite volume method based unstructured grid is presented to solve the two dimensional viscous and incompressible flow.The method is based on the pressure-correction concept and solved by using a semi-staggered grid technique.The computational procedure can handle cells of arbitrary shapes,although solutions presented in this paper were only involved with triangular and quadrilateral cells.The pressure or pressure-correction value was stored on the vertex of cells.The mass conservation equation was discretized on the dual cells surrounding the vertex of primary cells,while the velocity components and other scale variables were saved on the central of primary cells.Since the semi-staggered arrangement can’t guarantee a strong coupling relationship between pressure and velocity,thus a weak coupling relationship leads to the oscillations for pressure and velocity.In order to eliminate such an oscillation,a special interpolation scheme was used to construct the pressure-correction equation.Computational results of several viscous flow problems show good agreement with the analytical or numerical results in previous literature.This semi-staggered grid method can be applied to arbitrary shape elements,while it has the most efficiency for triangular cells.

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.

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).

Impact of Artificial Compressibility on the Numerical Solution of Incompressible Nanofluid Flow

The numerical solution of compressible flows has become more prevalent than that of incompressible flows.With the help of the artificial compressibility approach,incompressible flows can be solved numerically using the same methods as compressible ones.The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations.Any numerical method highly depends on its accuracy and speed of convergence.Although the artificial compressibility approach is utilized in several numerical simulations,the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies.Therefore,this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow.To improve the stability and convergence speed of time discretizations,the fifth-order Runge-Kutta method is applied.A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids.The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation.The optimum artificial compressibility is found to be between 1 and 5.These findings can be utilized to enhance the performance of commercial numerical simulation tools,including ANSYS and COMSOL.

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.

A MATRIX-FREE IMPLICIT METHOD FOR INCOMPRESSIBLE FLOW AROUND 3D COMPLEX UNDERWATER BODIES

A fast, matrix-free implicit method based on the Lower-Upper Symmetric Gauss-Seidel （LU-SGS） method was developed to solve the three-dimensional incompressible Reynolds-averaged Navier-Stokes equations on multi-block structured grids. The method was applied to the simulations of a variety of flowfields around 3D complex underwater bodies with different appendages. The numerical procedure and results show that the method is efficient, reliable and robust for steady viscous flow simulations.

Numerical Simulation of an Aortic Flow Based on a HLLC Type Incompressible Flow Solver

In this study,a three-dimensional artificial compressibility solver based on the average-state Harten-Lax-van Leer-Contact(HLLC)[13]type Riemann solution is first proposed and developed to solve the time-dependent incompressible flow equations.To implement unsteady flow calculations,a dual time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow term.Also a third-order accurate HLLC numerical flux is derived for approximating the inviscid terms.To verify numerical accuracy,flows over a lid-driven cavity and an oscillating flat plate are chosen as the benchmark tests.In addition,the current solver is extended to solve blood flows in a realistic human aorta measured from MRI(Magnetic Resonance Imaging).The simulation geometry was derived from a three-dimensional reconstruction of a series of two-dimensional slices obtained in vivo.Numerical results demonstrate wall stresses were highly dynamic,but were generally high along the outer wall in the vicinity of the branches and low along the inner wall,particularly in the descending aorta.The maximum wall stress distribution is presented on the aortic arch in the systole.In addition,extensive counter-clockwise secondary flows and three-dimensional helical vortex influenced considerably by the presence of vessel contraction,torsion and the branches were shown in the descending aorta in the late systole and early diastolic cycles.

NONLINEAR GALERKIN METHOD AND CRANK-NICOLSONMETHOD FOR VISCOUS INCOMPRESSIBLE FLOW

In this article we discuss a new full discrete scheme for the numerical solution of the Navier-Stokes equations modeling viscous incompressible flow. This scheme consists of nonlinear Galerkin method using mixed finite elements and Crank-Nicolson method. Next, we provide the second-order convergence accuracy of numerical solution corresponding to this scheme. Compared with the usual Galerkin scheme, this scheme can save a large amount of computational time under the same convergence accuracy. (Author abstract) 8 Refs.

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.

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.

USE THE EQUATIONS OF TOTAL PRESSURE AND VELOCITY TO SOLVE THE VISCOUS INCOMPRESSIBLE FLOW IN THE CHANNEL

This paper presents a new method to numerically solve the viscous incompress- ible steady flow in a channel with non-uniform section by use of the equations of total pressure and velocity.This method is to solve the equations of total pressure and velocity in the non-or- thogonai curvilinear mesh.The advantages of this method is discussed.

FOURIER PSEUDOSPECTRAL-FINITE DIFFERENCE METHOD FOR INCOMPRESSIBLE FLOW

A Fourier pseudospectral-finite difference scheme is proposed for unsteady Navier-Stokes equation. It is showed that the numerical solution keeps semi-discrete conservation. The strict error estimation is established. The numerical results are presented.

Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow

Fixed-interval smoothing,as one of the most important types of state estimation,has been concerned in many practical problems especially in the analysis of flight test data.However,the existing sequential filters and smoothers usually cannot deal with nonlinear or high-dimensional systems well.A state-of-the-art technique is employed in this study to explore the fixed-interval smoothing problem of a conceptual two-dimensional airfoil model in incompressible flow from noisy measurement data.Therein,the governing equations of the airfoil model are assumed to be known or only partially known.A single objective optimization problem is constructed with the classical Runge–Kutta scheme,and then estimations of the system states,the measurement noise and even the unknown parameters are obtained simultaneously through minimizing the objective function.Effectiveness and feasibility of the method are examined under several simulated measurement data corrupted by different measurement noises.All the obtained results indicate that the introduced algorithm is applicable for the airfoil model with cubic or free-play structural nonlinearity and leads to accurate state and parameter estimations.Besides,it is highly robust to Gaussian white and even more complex heavy-tailed measurement noises.It should be emphasized that the employed algorithm is still effective to high-dimensional nonlinear aeroelastic 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.