Characteristic-based Split Method Research Articles

Numerical and Experimental Study on the Nonlinear Liquid Sloshing in Cassini Tank

Liquid sloshing within propellant tanks of space vehicles has been a major concern in aerospace engineering. The aim of the work in this paper is to develop a flexible computational framework with high precision to simulate three-dimensional large-amplitude liquid sloshing in Cassini tanks. The finite element method is adopted to solve the fluid equations of motion in an arbitrary Lagrangian–Eulerian (ALE) framework, where the characteristic-based split method is combined with a fractional step method for solving the control equations in which the ALE kinematic description is incorporated to track the free liquid surface flexibly and effectively for large-amplitude liquid sloshing in Cassini tanks. In addition, an experimental platform is set up to verify the reliability and effectiveness of the presented method. The numerical results obtained from computer programming by using the ALE finite element method proposed in this paper are compared with the experimental results, proving the model to be successful. The nonlinear phenomena of large-amplitude liquid sloshing, especially rotary sloshing (steady-state swirling and non-steady-state swirling), in Cassini tanks under horizontal harmonic excitation are investigated by both the ground physical experiments and numerical simulations.

Physics-informed neural network combined with characteristic-based split for solving forward and inverse problems involving Navier–Stokes equations

This paper presents a novel approach for solving the shallow-water transport equation using a physics-informed neural network (PINN) combined with characteristic-based split (CBS). Simulation of the tide in East Coast of China is performed to verify the applicability of the present method in a practical problem. Furthermore, we propose a boundary condition that accounts for the second-order partial derivative term, which is more appropriate for solving the diffusion equation with open domains than the commonly used assumption of zero boundary values. Our numerical results demonstrate that this boundary condition leads to improved convergence of the network. In addition, we introduce a parameter estimation method that requires information from the field at only two different times, yet yields accurate parameter estimates. We observe that excessive participation of variables in gradient backpropagation can lead to neural networks getting trapped in local optima. We use PINN combined with CBS method to solve 3-D incompressible flow. As the number of variables involved in gradient backpropagation increases, the accuracy of the solution decreases, which can partially support our viewpoint. The source codes for the numerical examples in this work are available at https://github.com/double110/PINN-cbs-.git.

Effect of spacing ratios on coupling mechanism of three tandem cylinders in planar shear flow

Using four-step semi-implicit characteristic-based split method, flow-induced vibrations of three tandem cylinders in planar shear flow are numerically simulated at Re = 100. The vibration responses are more complicated than that of a single cylinder due to the gap development and mutual interference with each other. The spacing ratio (L/D) between three tandem cylinders of equal size ranges from 2.5 to 5.5, with a mass ratio (mr) of 2.0 attributed to each one. The reduced velocity (Ur) spans from 3 to 30, and the shear ratio (k) varies as 0, 0.05 and 0.1 respectively. The numerical results show that spacing ratio and shear ratio have an influence on the dynamic responses, performances of frequency, phase and energy for three tandem cylinders. The vibration responses of each cylinder can be clearly divided in three major branches, which first increase then decrease with increasing Ur. The gap flow inside cylinders at small L/D heightens the probability of vortex merging, intensifying mutual interference and consequently amplifying the amplitude. It is worth noting that as k increases to 0.1, downstream cylinders exhibit different vibration responses at L/D ≥ 4.0. Specifically, a secondary increase occurs after Ur > 18 at L/D = 4.0, and the amplitude is completely suppressed after Ur > 25 at L/D = 5.5. With the help of other performances, the differences in vibration responses of three tandem cylinders at different k and L/D are revealed.

Physics-informed neural network combined with characteristic-based split for solving Navier–Stokes equations

This paper presents a novel approach for solving the shallow-water transport equation using a physics-informed neural network (PINN) combined with characteristic-based split (CBS). Our method separates the variables, so it is no need to consider the weight coefficients between different loss functions. As not all partial derivatives are involved in the gradient backpropagation, our method can save half of the memory occupation without losing accuracy, and resulting in a significant reduction in computation time compared to traditional PINN. We solving the progress of the dispersion of hot water under known flow fields. Furthermore, we propose a boundary condition that accounts for the second-order partial derivative term, which is more appropriate for solving the diffusion equation with open domains than the commonly used assumption of zero boundary values. Our numerical results demonstrate that this boundary condition leads to improved convergence of the network. In addition, we introduce a parameter estimation method to estimate the diffusion coefficient of hot water flow, which requires information from the field at only two different times. We observe that excessive participation of variables in gradient backpropagation can lead to neural networks getting trapped in local optima. We use PINN combined with CBS method to solve 3-D incompressible flow. As the number of variables involved in gradient backpropagation increases, the accuracy of the solution decreases, which can partially support our viewpoint. The source codes for the numerical examples in this work are available at https://github.com/double110/PINN-cbs-.git.

A nodal position finite element model for fluid-structure interaction analysis of floating bodies with mooring

A numerical model based on the Nodal Position Finite Element Method (NPFEM) is proposed in this work for fluid-structure interaction (FSI) analysis of moored bodies subject to multiphase free surface flows. The semi-implicit CBS (Characteristic-Based Split) method is utilized here for discretization of the flow fundamental equations in the context of the Finite Element Method, where linear tetrahedral elements are adopted. Turbulence is analyzed using Large Eddy Simulation (LES) with the dynamic sub-grid scale model and the Level Set method is utilized for simulation of multiphase free surface flows. The structural system is modeled using a three-dimensional rigid body approach for the floating object, while the mooring is discretized using a geometrically nonlinear cable formulation based on the NPFEM. A partitioned coupling scheme is adopted for fluid-structure interaction analysis taking into account the fluid-structure and cable-structure couplings, where the flow equations are kinematically described employing an Arbitrary Lagrangean-Eulerian (ALE) formulation with a numerical scheme for mesh motion. The algorithms constituting the numerical model are verified first considering classical applications on fluid dynamics and FSI, in addition to applications involving the dynamic analysis of cables. Finally, problems involving floating bodies with and without mooring are simulated to demonstrate the applicability and accuracy of the proposed numerical model.

Isogeometric analysis with level set method for large-amplitude liquid sloshing

Large-amplitude sloshing in partially filled tanks is an important topic in the field of ship dynamics research, which has a great impact on the stability of carrier ships. As the most intuitive method, the computational fluid dynamics method has been proposed to simulate large-amplitude liquid sloshing for decades. In this paper, isogeometric analysis is used to model and simulate the large-amplitude liquid sloshing in closed tanks, considering the motions of both liquid and gas. The characteristic-based split method is applied to decompose the Navier-Stokes equations into an explicit scheme to improve the calculation efficiency. The level set method is used to track the interface between gas and liquid, also known as the liquid free surface. A mass correction method is proposed to correct the volume errors caused by the level set method in tracking the liquid free surface. By comparing with the experimental, analytical and FLOW-3D software results, the effectiveness and accuracy of the proposed method is verified, and the mass correction method can significantly improve the stability of liquid sloshing and the accuracy of free liquid surface tracking.

A three-dimensional hybrid immersed smoothed point interpolation method for fluid-structure interactions

A three-dimensional hybrid immersed smoothed point interpolation method (3D HIS-PIM) has been developed to handle fluid-structure interactions (FSI) with simple geometries. Incompressible viscous fluid is solved by semi-implicit characteristic-based split method with standard Galerkin discretization, smoothed point interpolation method is employed to solve the motion and deformation of nonlinear solids based on gradient smoothing technique, and the virtual fluid is introduced into the entire solid domain to calculate the FSI force. The proposed method can avoid the fluid mesh adjustments, and allows the use of four-node tetrahedral elements for both fluid and solid domains to simplify the pre-processing for mesh generations. The FSI force condition is imposed with a hybrid force, which uses a form of body force for pressure term and boundary force for shear force. The boundary force accords with the physical law and the body force can ensure the numerical stability. The proposed method works well for FSI problems with moving or largely deformable solids, and can achieve more accurate results using a hybrid force than the fully body force in the original method. A large range of mesh size ratios between fluid and solid meshes is allowed due to the treatment of FSI boundary conditions.

A modified Taylor Galerkin scheme for turbulent compressible flow simulation

As already demonstrated by different authors, the Taylor-Galerkin (TG) scheme, in the context of the Finite Element Method (FEM), is particularly suitable for the solution of supersonic flows. The TG scheme, using hexahedral finite elements with analytical evaluation of element matrices, is applied in this work. Tools to avoid locking and a shock capturing technique for the solution of supersonic viscous and non-viscous compressible flows are also employed. However, TG scheme usually presents instabilities in subsonic flows. Even in cases in which the free stream Mach number corresponds to supersonic flows, there are always flow regions, specifically near the walls of the immersed obstacles, where the speed is lower and the local Mach number corresponds to a subsonic flow. The Characteristic Based Split (CBS) scheme was developed in order to obtain a single method to improve the behavior with respect to TG method in subsonic and supersonic regimes. In the last two decades some works have shown advantages in convergence rates of the CBS method when compared to the TG algorithm. However, simulation time increases in the CBS method since split operations, typical of this algorithm, imply in additional element loops. In this paper a hybrid algorithm called Modified-Taylor-Galerkin scheme (MTG) is proposed. This algorithm presents advantages with respect to TG and CBS schemes in terms of convergence properties and computational processing time. In order to get an efficient algorithm, the element matrices are analytically integrated. This is performed with two different approaches. In the first approach the inverse matrix and the determinant of the Jacobian matrix at element level are evaluated with a reduced integration form, using the point located in the center of the element for mass, convective, diffusive and stabilization element matrices; all these matrices are integrated analytically. In the second approach, mass and convective matrices are calculated by a complete integration scheme (including the inverse matrix and the determinant of the Jacobian matrix at element level in the analytical expression to be integrated) and the diffusive and stabilization matrices are calculated with a reduced integration form, using the point located at the center of the element to calculate the inverse matrix and the determinant of the Jacobian matrix at element level. Finally, this work incorporates the Spalart-Allmaras (S-A) turbulence model using a conservative version of the transport equation, as proposed by the authors of the original S-A model in a later paper. Algorithms are tested to determine convergence rate improvements in both laminar and turbulent cases and for different Mach numbers (supersonic, transonic and subsonic flows).&nbsp

A sharp-interface immersed smoothed finite element method for interactions between incompressible flows and large deformation solids

A sharp-interface algorithm developed for immersed smoothed finite element method (Sharp-ISFEM) is proposed for fluid–structure interaction (FSI) problems. In Sharp-ISFEM, a line normal approach is used for unstructured meshes to reconstruct the solutions near interface. Advanced smoothed finite element models describe the finite deformation of immersed solid structures. Well-developed characteristic-based split method is employed to simulate the incompressible laminar flow. The coupling of fluid and solid is treated by the modified weak partitioned fictitious domain method with different time steps for two phases. To handle large-scale FSI problems, octree and neighbor-neighbor searching algorithms have been implemented. Meanwhile, parallelization has been implemented for the time-consuming parts of in-house codes. The effectiveness of proposed Sharp-ISFEM is validated by several benchmarking problems, including: fixed rigid body, flow over square cylinder; moving rigid body, flow induced by 2D hovering wing; large deformed solid, a beam with large deformation induced by flow; 3D deformable solid–fluid interaction, and lid-driven cavity flow with a rubber wall. Good agreements are shown in all test cases, comparing with existed experimental data and reference solutions. With the improvement of local velocity reconstruction, advanced searching algorithms and code parallelization, the proposed method has shown great promises for complex engineering flows with higher Reynolds number.A large-scale FSI problem of a hummingbird with flexible wing in hovering is finally simulated using more than one million nodes.

A generalized Characteristic-Based Split projection method for Navier-Stokes with real fluids

A new computational approach for the general numerical simulation of fluid flows is presented and demonstrated for compressible natural convection flows with real fluid properties. The Characteristic-Based Split finite element method is generalized, and abbreviated as GCBS, in the present work to introduce both pressure-expansion effects in addition to thermal-expansion effects on density within the projection step, in contrast to recent works that preserve only one of those two expansion effects. Moreover, we keep the fluid in thermodynamic equilibrium to support a real equation of state and fluid model. The method retains the full Navier-Stokes equations, transitioning smoothly between compressible and incompressible flows, without special artificial-compressibility methods, special low-Mach-number methods, or other modeling treatments. Neither isothermal or isobaric assumptions are made. The non-linear solution obtained is validated by the linearized solution during the early time period of the simulation. Natural convection in a cavity is explored for its convenience in testing across known benchmark flow regimes, from classical Boussinesq flow to the full Navier-Stokes equations with real fluid models. Results for natural convection in a cavity across Rayleigh numbers, non-dimensional adiabatic temperature gradients, and temperature differences, are obtained across flow regimes with the GCBS method. Classical Boussinesq, Thermodynamic Boussinesq, low-Mach number, and real-gas flows are presented, demonstrating the generalization of the method across flow regimes. Thus, our contribution is to generalize the Characteristic-Based Split method to capture the full Navier-Stokes equations while demonstrating the capability with compressible natural convection.

A comparative study of fractional step method in its quasi-implicit, semi-implicit and fully-explicit forms for incompressible flows

Purpose – The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared. Design/methodology/approach – This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split. Findings – In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations. Originality/value – A comprehensive comparison between three versions of the CBS method is provided for the first time.

Numerical analysis of single pad of thrust bearing with the energy equation solved by the characteristic-based split method

The solution of the energy equation of thermo-elasto-hydrodynamic analysis for bearings by the finite element method usually leads to convergence difficulties due to the presence of convection terms inherited from the Navier–Stokes equations. In this work, the numerical analysis is performed with finite element method universally by adopting the characteristic-based split method to solve the energy equation. Five case studies of fixed pad thrust bearings have been set up with different geometries, loads, and lubricants. The two-dimensional film pressure is obtained by solving the Reynolds equation with pre-defined axial load on the pad. The energy equation of the lubricant film and the heat transfer equation of the bearing pad are handled by characteristic-based split method and conventional finite element method in three-dimensional space, respectively. Hot oil carry-over effect and variable lubricant viscosity are considered in the simulations. The results of the temperature distributions in the lubricant film and the bearing pad are presented. The possible usability of characteristic-based split method for future thermo-elasto-hydrodynamic analysis is discussed.

Characteristic-Based Split (CBS) Finite Element Method for Incompressible Viscous Flow with Moving Boundaries

This paper gives an insight into the existing ALE (arbitrary Lagrangian Eulerian)-CBS (characteristic-based split) method and proposes an alternative CBS scheme for incompressible viscous flow with moving boundaries. First, the existing CBS scheme in ALE framework is reformulated from the Navier-Stokes (NS) equations in Eulerian description, and its advantages and drawbacks are discussed. Then, the classical semi-implicit CBS scheme is combined with the segment spring analogy method, and a new CBS solution procedure in Eulerian framework is developed for moving mesh. By the proposed CBS scheme, the influence of the mesh velocity in the existing ALE-CBS method is circumvented, and the extension of the existing CBS schemes and programs for fixed boundary problems to moving boundary problems seems to be more convenient. Finally, two moving boundary problems, namely the broken dam problem and the flow around an oscillating circular cylinder are presented. Numerical results from the proposed CBS scheme agree very well with available results in literature and show a slightly higher accuracy than those from the existing ALE-CBS method. Combined with other classical CBS schemes involving compressibility, turbulence, viscoelasticity, etc. and suitable moving mesh techniques, the proposed CBS solution procedure could be applied widely to the moving boundary problems in engineering.

An artificial compressibility CBS method for modelling heat transfer and fluid flow in heterogeneous porous materials

AbstractThis work is concerned with the development of an artificial compressibility version of the characteristic‐based split (CBS) method proposed by Zienkiewicz and Codina (Int. J. Numer. Meth. Fluids 1995; 20:869–885). The technique is applied to modelling both forced convection as well as heat transfer and fluid flow through heterogeneous saturated porous materials via an edge‐based finite volume discretization scheme. A volume‐averaged set of local thermal disequilibrium governing equations is employed to describe the general case which allows for the modelling of effects such as wall channelling and wall‐bed radiative heat transfer. The resulting set of coupled non‐linear partial differential equations is solved in a matrix‐free manner with spatial discretization being effected with a compact vertex‐centred finite volume edge‐based discretization scheme. The latter was done in the interest of efficiency and accuracy. The developed scheme is validated via application to problems ranging from forced convection to natural convection in heterogeneous materials, and shown to be stable, robust and accurate. Copyright © 2011 John Wiley & Sons, Ltd.

The characteristic-based split (CBS) meshfree method for free surface flow problems in ALE formulation

A semi-implicit characteristic-based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.

Convective heat transfer from a rotating cylinder with inline oscillation

In this study convective heat transfer from a rotating cylinder with inline oscillation is studied using a finite element method based on the Characteristic Based Split method (CBS) to solve governing equations consisting of continuity, full Navier–Stokes, and energy equations. Employing the Arbitrary Lagrangian-Eulerian (ALE) formulation, the dynamic unstructured triangular grid used here is accompanied by lineal and torsional spring analogy to consider large boundary movements. Simulations are conducted to study convective heat transfer past a rotating cylinder with inline oscillation at Reynolds numbers of 100, 200 and 300. Different rotational speeds of the cylinder in the range of 0–2.5 are considered at various oscillating amplitudes and frequencies with three different Prandtl numbers of 0.7, 6 and 20. Effects of oscillation and rotation of cylinder on the temperature and flow field, vortex lock-on, mean Nusselt number, and the pattern of vortex shedding are investigated in detail at constant temperature boundary condition on the cylinder surface. It is found that similar to the fixed cylinder, beyond a critical rotating speed, the vortex shedding is strongly suppressed. Furthermore, as the rotational speed of the cylinder increases, both the Nusselt number and the drag coefficient decrease rapidly. In the vortex lock-on region, the Nusselt number increases rapidly.

A numerical study of convective heat transfer from a rotating cylinder with cross-flow oscillation

This article presents a numerical investigation of convective heat transfer from a rotating cylinder with cross-flow oscillation. A finite element analysis using Characteristic Based Split method (CBS) is developed to solve governing equations involving continuity, Navier–Stokes, and energy equations. Dynamic unstructured triangular grid is used employing improved lineal and torsional spring analogy which is coupled with the solver by the Arbitrary Lagrangian–Eulerian (ALE) formulation. After verifying the numerical code accuracy, simulations are conducted to study convective heat transfer past a rotating cylinder with cross-flow oscillation at Reynolds numbers of 50, 100, and 200. Different rotational speeds of the cylinder normalized by free stream velocity, in the range of 0–2.5 are considered at various oscillating amplitudes and frequencies and three different Prandtl numbers of 0.7, 6, and 20. Effects of oscillation and rotation of cylinder on the temperature and flow field, vortex lock-on, mean Nusselt number, and the pattern of vortex shedding are investigated in detail considering iso-temperature and iso-flux boundary conditions on the cylinder surface. It is found that similar to the fixed cylinder, beyond a critical rotating speed, vortex shedding is mainly suppressed. Also by increasing the non-dimensional rotational speed of the cylinder, both the Nusselt number and the drag coefficient decrease rapidly. However, in vortex lock-on region, the Nusselt number increases in a large amount.

Numerical simulation of tsunami propagation using the characteristic-based split method

Numerical simulation of tsunami propagation using the characteristic-based split method

On the use of characteristic‐based split meshfree method for solving flow problems

AbstractThis study presents characteristic‐based split (CBS) algorithm in the meshfree context. This algorithm is the extension of general CBS method which was initially introduced in finite element framework. In this work, the general equations of flow have been represented in the meshfree context. A new finite element and MFree code is developed for solving flow problems. This computational code is capable of solving both time‐dependent and steady‐state flow problems. Numerical simulation of some known benchmark flow problems has been studied. Computational results of MFree method have been compared to those of finite element method. The results obtained have been verified by known numerical, analytical and experimental data in the literature. A number of shape functions are used for field variable interpolation. The performance of each interpolation method is discussed. It is concluded that the MFree method is more accurate than FEM if the same numbers of nodes are used for each solver. Meshfree CBS algorithm is completely stable even at high Reynolds numbers. Copyright © 2007 John Wiley & Sons, Ltd.

Editorial

This special issue on the ‘Characteristic-Based Split (CBS)’ scheme was conceived from the experience of different people working in different areas of computational mechanics. The objective in the main is the presentation of the CBS scheme in its very general form to accommodate general engineering problems of interest to the engineering community. Thus, we have included papers on general fluid dynamics, shallow-water flows, solid dynamics, porous media and heat transfer. A total of eight papers have been included into this special issue. The first paper by Nithiarasu et al. describes the CBS scheme in detail with an explanation on the origin of the characteristic-based procedures and their extension to solve compressible and incompressible flows. This paper also describes the merger of the characteristic-based procedure with a fractional step algorithm to develop the CBS procedure. Several examples are also presented to demonstrate the robustness and accuracy of this unified approach to compressible and incompressible flows. Though only fluid dynamics examples are presented in the first paper, the suitability of the method to solve solid dynamic problems is demonstrated in the second paper by Rojek et al. This paper again demonstrates the novelty of the proposed approach. The presented idea immediately forms the basis for coupled fluid-solid problems using a single approach with different constitutive equations. The third paper is presented by Ortiz et al. which describe the CBS approach to shallow-water problems. Once again the versatility of the method is demonstrated for an entirely different kind of incompressible flow equations. The shallow-water equations presented in this paper resembles the inviscid compressible flows. Thus, it is obvious that the same methodology used to solve the compressible flows can be extended to the shallow-water equations. The next paper is a recent work on porous media problems using the CBS approach. The authors, Salomoni and Schrefler, have demonstrated the use of the CBS scheme to a complicated porous medium problem. Once again this proves that the application of the CBS scheme is not limited to fluid dynamics problems. The next three papers by Massarotti et al., Morandi Cecchi and Venturin and Kulkarni et al. are the applications of the CBS scheme to different engineering problems. In Massarotti et al., the method has been extensively used to steady and unsteady incompressible flows with heat transfer to make a general assessment of the method in its fully explicit and semi-implicit forms. This paper gives an in-depth comparison between the two variations. In Morandi Cecchi and Venturin, the method is applied to solve a practical shallow-water problem of the Venice Lagoon. The third paper in the application series by Kulkarni et al. deals with a practical application of under-fill simulation process in electronic packaging. All these three papers clearly demonstrate the flexibility of the CBS method for different applications of practical importance. This special issue is concluded by a paper on a comparative study between the CBS scheme and other recent methods. This last paper is presented by Codina et al. and compares the CBS method against the family of recent subgrid scale (SGS) methods. Finally, we would like to thank Prof. O. C. Zienkiewicz for supporting this special issue and writing a foreword. We are also grateful to Professor R. W. Lewis, editor of the International Journal for Numerical Methods in Engineering for permitting us to produce this special issue. We hope this special issue will be of great use to the readers to learn about the CBS scheme and its development over the last 10 years.