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 The incompressible constraint has been replaced by a pressure Poisson-like equation and then the pressure is updated by the modified marker and cell method.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 all-speed relaxation scheme for interface flows with surface tension

We consider interface flows where compressibility and capillary forces (surface tension) are significant. These flows are described by a non-conservative, unconditionally hyperbolic multiphase model. The numerical approximation is based on finite-volume method for unstructured grids. At the discrete level, the surface tension is approximated by a volume force (CSF formulation). The interface physical properties are recovered by designing an appropriate linearized Riemann solver (Relaxation scheme) that prevents spurious oscillations near material interfaces. For low-speed flows, a preconditioning linearization is proposed and the low Mach asymptotic is formally recovered. Numerical computations, for a bubble equilibrium, converge to the required Laplace law and the dynamic of a drop, falling under gravity, is in agreement with experimental observations.     

An unconditionally gradient stable numerical method for solving the Allen-Cahn equation

We consider an unconditionally gradient stable scheme for solving the Allen-Cahn equation representing a model for anti-phase domain coarsening in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. We also show the pointwise boundedness of the numerical solution for the Allen-Cahn equation. We describe various numerical experiments we performed to study properties of the Allen-Cahn equation.     

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics.Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives.This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.     

A massively parallel fractional step solver for incompressible flows

This paper presents a parallel implementation of fractional solvers for the incompressible Navier–Stokes equations using an algebraic approach. Under this framework, predictor–corrector and incremental projection schemes are seen as sub-classes of the same class, making apparent its differences and similarities. An additional advantage of this approach is to set a common basis for a parallelization strategy, which can be extended to other split techniques or to compressible flows.     

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-by-step 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.     

Pressure boundary conditions for computing incompressible flows with SPH

In Smoothed Particle Hydrodynamics (SPH) methods for fluid flow, incompressibility may be imposed by a projection method with an artificial homogeneous Neumann boundary condition for the pressure Poisson equation. This is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries. We resolve this problem by utilizing a “rotational pressure-correction scheme” with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity. We show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects, and extends the scope of SPH simulation beyond the usual periodic boundary conditions.     

Provably unconditionally stable,second-order time-accurate,mixed variational methods for phase-field models

We introduce provably unconditionally stable mixed variational methods for phase-field models. Our formulation is based on a mixed finite element method for space discretization and a new second-order accurate time integration algorithm. The fully-discrete formulation inherits the main characteristics of conserved phase dynamics, namely, mass conservation and nonlinear stability with respect to the free energy. We illustrate the theory with the Cahn–Hilliard equation, but our method may be applied to other phase-field models. We also propose an adaptive time-stepping version of the new time integration method. We present some numerical examples that show the accuracy, stability and robustness of the new method.     

An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows

The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [14] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [14], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [14], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [14] for 3D flows with curved boundaries, the flows over a sphere and torus are simulated. The obtained numerical results compare very well with available data in the literature.     

A remark on the Hamiltonian formalism for incompressible flows

We revisit the Hamiltonian formalism for incompressible flows as introduced by Oseledets. Our aim is to clarify some ambiguities in this formalism due to the nonintegrable singularity of the kernel which defines the Hamiltonian.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

耦合不可压流场输运方程的格子Boltzmann方法研究

基于格子Boltzmann方法,提出了求解耦合不可压缩流场输运方程的一种改进数值方法. 该方法使用格子Boltzmann方法求解流场方程,并根据流场格子模型的密度分布函数构建了输运方程的二阶离散格式. 通过二维平板通道流场输运系统验证了该方法的有效性.数值结果表明,该方法可以有效地减少计算过程中出现的非物理耗散, 并克服了传统模型所需巨大存储量的缺点.     

隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI-LU,LGS,LU-SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

Stochastic least-action principle for the incompressible Navier-Stokes equation

We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton’s principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in the framework of the stochastic variational principle. In particular, we discuss the connection of the stochastic Kelvin Theorem with our previous “martingale hypothesis” for fluid circulations in turbulent solutions of the incompressible Euler equations.     

一种隐式预处理方法及其在定常和非定常流动数值模拟中的应用

将Choi-Merkle矩阵预处理方法与LU-SGS隐式方法、双时间法以及多重网格方法结合,发展适用于绕飞行器定常和非定常粘性流动的高效隐式预处理计算方法和程序.介绍一种针对定常和非定常流动的LU-SGS隐式预处理方法的统一表述方法.在不改变流动解的前提下,对Navier-Stokes方程的伪时间导数项实施Choi-Merkle矩阵预处理,从而改善可压缩控制方程在低速情况下的系统刚性,使基于LU-SGS时间推进格式的数值模拟方法同时适用于从极低马赫数到可压缩范围内的数值模拟.对Jameson中心格式的人工粘性进行相应的修改,以提高低速流动的计算精度.翼型、机翼以及翼身组合体绕流的数值模拟研究表明,隐式预处理方法获得了很高的计算效率,可使马赫数0.1左右的低速流动计算时间减少50%以上;通过对现有可压缩计算程序进行小量改动,便可使其均匀覆盖整个低速流动范围,提高CFD程序在飞行器绕流数值模拟中的实用性.     

A class of Cartesian grid embedded boundary algorithms for incompressible flow with time-varying complex geometries

We present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures. The proposed Cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct-forcing immersed boundary methods: the proposed algorithms calculate and apply the force density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid-body surface. The principle of the embedded boundary algorithm allows us to solve the fluid equations on a Cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure. The proposed algorithms use the MAC (marker and cell) algorithm to solve the incompressible Navier-Stokes equations. Unlike projection methods, the MAC scheme incorporates the gradient of the force density in solving the pressure Poisson equation, so that the dipole force, due to the jump of pressure across the solid-fluid interface, is directly balanced by the gradient of the force density. We validate the proposed algorithms via the classical benchmark problem of flow past a cylinder. Our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid-fluid interface. Finally, we consider the problem of a cylinder moving between two parallel plane walls. Numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions. We show that the two solutions are in good agreement.     

A improved incompressible lattice Boltzmann model for time-independent flows

It is well known that the lattice Boltzmann equation method (LBE) can model the incompressible Navier-Stokes (NS) equations in the limit where density goes to a constant. In a LBE simulation, however, the density cannot be constant because pressure is equal to density times the square of sound speed, hence a compressibility error seems inevitable for the LBE to model incompressible flows. This work uses a modified equilibrium distribution and a modified velocity to construct an LBE which models time-independent (steady) incompressible flows with significantly reduced compressibility error. Computational results in 2D cavity flow and in a 2D flow with an exact solution are reported.     

An adaptive central-upwind weighted essentially non-oscillatory scheme

In this work, an adaptive central-upwind 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. The scheme adapts between central and upwind schemes smoothly by a new weighting relation based on blending the smoothness indicators of the optimal higher order stencil and the lower order upwind stencils. The scheme achieves 6th-order accuracy in smooth regions of the solution by introducing a new reference smoothness indicator. A number of numerical examples suggest that the present scheme, while preserving the good shock-capturing properties of the classical WENO schemes, achieves very small numerical dissipation.     

Force-coupling method for flows with ellipsoidal particles

The force-coupling method, previously developed for spherical particles suspended in a liquid flow, is extended to ellipsoidal particles. In the limit of Stokes flow, there is an exact correspondence with known analytical results for isolated particles. More generally, the method is shown to provide good approximate results for the particle motion and the flow field both in viscous Stokes flow and at finite Reynolds number. This is demonstrated through comparison between fully resolved direct numerical simulations and results from the numerical implementation of the force-coupling method with a spectral/hp element scheme. The motion of settling ellipsoidal particles and neutrally buoyant particles in a Poiseuille flow are discussed.