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.     

基于直角网格法的单个和阵列布置下柔性水翼绕流数值模拟

研究柔性水翼在不可压缩流体中的水动力特性,对于船舵和减摇鳍等海洋结构物的设计和性能优化具有重要意义.本文将自主开发的径向基函数虚拟网格法求解器扩展到模拟绕单个或多个柔性水翼的不可压缩流动问题.数值模型基于虚拟网格有限差分法考虑浸入边界对流场的影响,引入紧支径向基函数(compact supported radial basis function,CSRBF)以物面Lagrangian质点追踪复杂的柔性动边界.基于该方法,首先模拟了均匀流中主动拍动的柔性水翼,升阻力系数良好的网格收敛性结果验证了本文方法的精度和可靠性.并研究了柔性水翼在不同振荡频率下的水动力特性,阐述了柔性水翼的推力生成机制.然后模拟了绕阵列布置柔性水翼的流动现象,研究了不同间距和不同振荡频率下水翼表面的升阻力系数变化规律和尾涡特性,观察到紧密布置的柔性水翼在高频振荡下推力系数存在显著的放大效应,同时推力为零时的临界频率提前.     

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 kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.     

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.     

A note on pressure accuracy in immersed boundary method for Stokes flow

In this short note, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure or indicator function in immersed boundary calculations is first-order accurate in L₁ norm, half-order accurate in L₂ norm, but has O(1) error in L_∞ norm. Despite the pressure has O(1) error near the interface, the velocity field still has the first-order convergence in immersed boundary calculations.     

The theoretical research of basic function method in incompressible viscous flow and its simulations in three-dimensional aneurysms

Basic function method is developed to treat the incompressible viscous flow. Artificial compressibility coefficient, the technique of flux splitting method and the combination of central and upwind schemes are applied to construct the basic function scheme of trigonometric function type for solving three-dimensional incompressible Navier-Stokes equations numerically. To prove the method, flows in finite-length-pipe are calculated, the velocity and pressure distribution of which solved by our method quite coincide with the exact solutions of Poiseuille flow except in the areas of entrance and exit. After the method is proved elementary, the hemodynamics in two- and three-dimensional aneurysms is researched numerically by using the basic function method of trigonometric function type and unstructured grids generation technique. The distributions of velocity, pressure and shear force in steady flow of aneurysms are calculated, and the influence of the shape of the aneurysms on the hemodynamics is studied. Supported by the National Natural Foundation of China (Grant Nos. 40874077, 40504020, and 40536029) and the National Basic Research Program of China (Grant No. 2006CB806304)     

Heat transfer of an oscillating cylinder in a viscous incompressible fluid flow

Unsteady flow and heat transfer of the heated cylinder with transverse and longitudinal oscillations in unbounded flow of viscous incompressible heat conducting fluid are considered. Influence of amplitude, frequency and direction of harmonic oscillations of the cylinder on the near-wake structure and heat transfer are investigated.     

Integro-differential equation method in the hydrodynamics of an incompressible viscous fluid

An integro-differential equation method is proposed to describe the motion of an incompressible viscous fluid. The method uses an analogy between the hydrodynamic equations for an incompressible viscous fluid and the magnetostatic equations. An analysis is made of the flow of an incompressible viscous fluid round an object as a specific application. The solution automatically satisfies the boundary conditions at the surface of the object and at infinity. Zh. éksp. Teor. Fiz. 112, 1332–1339 (October 1997)     

The Ghost Cell Method and Its Applications for Inviscid Compressible Flow on Adaptive Tree Cartesian Grids

In this paper, an immersed boundary algorithm is developed by combining the ghost cell method with adaptive tree Cartesian grid method. Furthermore, the proposed method is successfully used to evaluate various inviscid compressible flow with immersed boundary. The extension to three dimensional cases is also achieved. Numerical examples demonstrate the proposed method is effective.     

Shape oscillations of a droplet in an Oldroyd-B fluid

We present a Navier-Stokes/Oldroyd-B immersed boundary algorithm that captures the interaction of a flexible structure with a viscoelastic fluid. In particular, we study the effects of bulk viscoelasticity on freely decaying shape oscillations of an Oldroyd-B fluid droplet suspended in an Oldroyd-B matrix. Our numerical data indicate that if the fluid viscosity is low, viscoelasticity plays a modulating role in the drop shape relaxation; specifically, it increases the oscillation frequency and decreases the decay rate when the fluid relaxation time is above a critical value. In the high viscosity limit, i.e., when a Newtonian droplet is expected to return to a spherical shape with an aperiodic decay, an increase in the relaxation time eventually results in the reappearance of the oscillations. Both these results are in line with the prediction of small deformation theory for viscoelastic droplet oscillations. The algorithm was also validated by direct comparison with linear asymptotics.     

粘性不可压流体流动问题用直角坐标网格的贴体解法

研究一种新的全贴体的求解粘性不可压流体流动问题的非结构化直角坐标网格方法.该方法在于利用直角坐标网格但通过在边界附近保留不规则控制体,使得算法是完全贴体的.这有别于目前流行的各种非结构化直角坐标网格方法.通过对两个典型流动问题的计算对该数值方法进行验证.对比结果表明,本方法计算的结果与精确解和STAR-CD的结果在一定Re数和网格数时是很接近的,可以满足一定的精度要求,说明该数值计算方法是可行的.还对二维钝头体周围的流场进行了计算,计算的流场与STAR-CD的结果相当吻和,说明该算法还可计算较复杂的流动现象.     

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.     

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier–Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.     

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.     

Boundary data immersion method for Cartesian-grid simulations of fluid-body interaction problems

A new robust and accurate Cartesian-grid treatment for the immersion of solid bodies within a fluid with general boundary conditions is described. The new approach, the Boundary Data Immersion Method (BDIM), is derived based on a general integration kernel formulation which allows the field equations of each domain and the interfacial conditions to be combined analytically. The resulting governing equation for the complete domain preserves the behavior of the original system in an efficient Cartesian-grid method, including stable and accurate pressure values on the solid boundary. The kernel formulation allows a detailed analysis of the method, and it is demonstrated that BDIM is consistent, obtains second-order convergence relative to the kernel width, and is robust with respect to the grid and boundary alignment. Formulation for no-slip and free slip boundary conditions are derived and numerical results are obtained for the flow past a cylinder and the impact of blunt bodies through a free surface. The BDIM predictions are compared to analytic, experimental and previous numerical results confirming the properties, efficiency and efficacy of this new boundary treatment for Cartesian grid methods.