Six convective difference schemes on different grid systems for fluid flow and heat transfer with SIMPLE algorithm

This paper compares the finite difference and finite volume methods for solving time-fractional diffusion equations. These methods are widely known for diffusion equations with integer order, but their effectiveness for time-fractional diffusion equations has not been sufficiently studied. The definition of the Grunwald-Letnikov fractional derivative is used to approximate the equation. An explicit difference scheme for the finite difference method is obtained and a stability condition for the fractional time order difference scheme is derived, which is also a generalisation for parabolic and hyperbolic type equations, which was previously unknown for schemes with a fractional time order. An explicit discrete form for solving subdiffusion equations in two-dimensional space with fractional time order by the finite volume method is presented. Numerical results show that the finite difference method demonstrates high accuracy, while the finite volume method is better suited for complex geometries. These findings provide insights for future developments in anomalous diffusion modeling.

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

A Boussinesq-type wave model is developed in this paper to simulate the interaction of coastal waves with bottom-mounted porous structures. The governing equations are rewritten in the conservative form to facilitate the use of hybrid finite volume (FV) and finite difference (FD) method. Higher-order slope terms are also inserted into the equations to account for rapidly varying bathymetry. The convective flux is approximated using the FV method, while the remaining terms are discretized using the FD method in a uniform rectangle grid system. The time integration is implemented using the third order Runge–Kutta method with an adaptive time step. A single GPU parallel computation is also implemented to save computation costs. The numerical model is validated against a series of experimental datasets, including data acquired in a new laboratory experiment. The predictions are in overall agreement with the measurements, proving that the model is capable of handling wave interaction with porous structures in the coastal region for a wide range of scenarios.

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Nonstandard finite difference scheme for a Tacoma Narrows Bridge model

Quantifying Uncertainty in CFD

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

Los métodos en volúmenes finitos son técnicas bien conocidas usualmente aplicadas a la simulación de problemas basados en leyes de conservación. Presentan, además, particular interés para el tratamiento de geometrías no triviales y condiciones de contorno complejas. Además se pueden aplicar tanto a mallados no estructurales como a mallados regulares, como los que se utilizan habitualmente en el ámbito de las diferencias finitas. Estas técnicas numéricas son además particularmente eficientes para la resolución de las ecuaciones difusivo-viscosas (DVWpor sus siglas en inglés), de interés en esta tesis, dado que permiten resolver de manera muy precisa los fenómenos de propagación de ondas en un medio concreto con condiciones de contorno complejas. En esta tesis se explora la aplicación de la aproximación por volúmenes finitos a la simulación de fenómenos basados en ondas difusivo-viscosas. Se analiza, además la estabilidad numérica en dominios tridimensionales mediante la técnica de von Neumann. La simulación de las ondas sísmicas se realiza, en esta tesis, mediante modelos de ondas difusivo-viscosas en dos y tres dimensiones, que se resuelven mediante métodos en volúmenes finitos, en un medio fluido saturado. Los resultados de esta simulación se comparan con los obtenidos con modelos basados en ondas acústicas, así como con modelos visco-acústicos. La simulación numérica realizada pone de manifiesto que se produce una mayor atenuación en la propagación de ondas que cuando se emplean modelos basados en ondas acústicas y en ondas visco-acústicas. También se emplea el denominado factor de calidad, Q, con la finalidad de obtener los Parámetros de Atenuación Activa (AAP por sus siglas en inglés), que básicamente describen la atenuación de las DVWs y se comparan con los que corresponden a las ondas visco-acústicas. ----------ABSTRACT---------- Finite volume methods have currently been demonstrated to be a practicable solution to the problem of complex boundary conditions over non-trivial geometries, and they also permit the application of energy methods for numerical stability analyses. Finite volume methods accommodate themselves naturally to fully unstructured grids and they can simplify to the types of grids usually employed in finite difference methods. This allows for diffusive-viscous wave (DVW) simulation models that balance the simplicity of finite difference methods for wave propagation in a medium with the detail of finite volume methods for complex wave phenomena as well as complex boundaries. This thesis is an exploration of the finite volume approach to the simulation of the diffusiveviscous wave equation. The underlying theme in this research is to balance between numerical simulation, accuracy, and numerical stability. Second-order space and time optimised schemes in two and three spatial dimensions are developed towards the objective of discovering accurate and efficient finite volume schemes. Numerical stability is analysed in three dimensions using von Neumann’s stability analyses, allowing for stable numerical scheme appropriate for three-dimensional diffusive-viscous wave modelling. Also, the accuracy of the numerical scheme is investigated for the two-dimensional and three-dimensional diffusive-viscous wave models. Seismic waves are then simulated in a two-dimensional and three-dimensional fluid-saturated medium and the results compared with those of the acoustic and visco-acoustic cases, respectively. The numerical results indicate stronger attenuation when compared to the acoustic and visco-acoustic case accordingly. The Quality Factor, Q is used to obtain the Active Attenuation Parameters (AAP), which basically describes the attenuation of DVWs, and they are compared to those of the viscoacoustic wave.

Diffusive-Time-of-Flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast marching method (FMM). However, application of the finite difference based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM.The finite volume form of the Eikonal equation is derived by differential manipulation, volume integration and the divergence theorem. Using product rule, the differential term is first converted to the divergence form. Then volume integrals that contain divergence terms are converted to surface integrals using the divergence theorem. Consequently, the spatial coordinates are replaced by cell volumes and transmissibilities which are universal for both structured and unstructured grids in finite volume simulators. When applied with the upstream scheme, the finite volume form evolves into a set of quadratic equations, and fast marching method is implemented to solve these equations.The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFM), dual-porosity dual-permeability models, and unstructured PEBI grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics such as evolution of the drainage volume of the wells and well interactions.The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.

A comparative analysis of thermal models in the lattice Boltzmann method(LBM) for the simulation of laminar natural convection in a square cavity is presented. A HYBRID method, in which the thermal equation is solved by the Navier-Stokes equation method while the mass and momentum conservation are resolved by the lattice Boltzmann method, is introduced and its merits are explained. All the governing equations are discretized on a cell-centered, non-uniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with a deferred correction method to ensure stability of the solutions. The HYBRID method and the double-population method are applied to the simulation of natural convection in a square cavity and the predicted results are compared with the benchmark solutions given in the literatures. The predicted results are also compared with those by the conventional Navier-Stokes equation method. In general, the present HYBRID method is as accurate as the Navier-Stokes equation method and the double-population method. The HYBRID method shows better convergence and stability than the double-population method. These observations indicate that this HYBRID method is an efficient and economic method for the simulation of incompressible fluid flow and heat transfer problem with the LBM.

High-order cell-vertex finite difference schemes applied to multi-block structured grids are used widely in computational aeroacoustics for their low-dispersion and low-dissipation properties. Structured grids for complex geometries may contain discontinuous grid metrics at multi-block interfaces. In this work it is demonstrated that the grid-induced errors from such interfaces can be reduced by applying finite difference schemes in the cell-centered space. Further reduction of these grid-induced errors can be achieved by applying an additional finite volume method, which serves as an interface condition. In this paper, the development of a hybrid cell-centered finite difference and finite volume method is demonstrated. An interpolation scheme is derived from a high-order finite difference scheme to apply the finite volume method at interfaces. The order of accuracy of this hybrid method is demonstrated and the method is used to simulate the flow around a single cylinder, tandem cylinders, and a complex isolated wheel. Comparisons with experimental measurements and numerical predictions show that the hybrid method can provide accurate results at block interfaces and can be applied to high-order simulations of complex geometries.

This textbook covers fundamental and advanced concepts of computational fluid dynamics, a powerful and essential tool for fluid flow analysis. It discusses various governing equations used in the field, their derivations, and the physical and mathematical significance of partial differential equations and the boundary conditions. It covers fundamental concepts of finite difference and finite volume methods for diffusion, convection-diffusion problems both for cartesian and non-orthogonal grids. The solution of algebraic equations arising due to finite difference and finite volume discretization are highlighted using direct and iterative methods. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding. The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering, for a course on computational fluid dynamics and heat transfer. The textbook will be accompanied by teaching resources including a solution manual for the instructors. Written clearly and with sufficient foundational background to strengthen fundamental knowledge of the topic. Offers a detailed discussion of both finite difference and finite volume methods. Discusses various higher-order bounded convective schemes, TVD discretisation schemes based on the flux limiter essential for a general purpose CFD computation. Discusses algorithms connected with pressure-linked equations for incompressible flow. Covers turbulence modelling like k-ε, k-ω, SST k-ω, Reynolds Stress Transport models. A separate chapter on best practice guidelines is included to help CFD practitioners.

Mesoscopic methods serve as a pivotal link between the macroscopic and microscopic scales, offering a potent solution to the challenge of balancing physical accuracy with computational efficiency. Over the past decade, significant progress has been made in the application of the discrete Boltzmann method (DBM), which is a mesoscopic method based on a fundamental equation of nonequilibrium statistical physics (i.e., the Boltzmann equation), in the field of nonequilibrium fluid systems. The DBM has gradually become an important tool for describing and predicting the behavior of complex fluid systems. The governing equations comprise a set of straightforward and unified discrete Boltzmann equations, and the choice of their discrete format significantly influences the computational accuracy and stability of numerical simulations. In a bid to bolster the reliability of these simulations, this paper utilizes the finite volume method as a solution for handling the discrete Boltzmann equations. The finite volume method stands out as a widely employed numerical computation technique, known for its robust conservation properties and high level of accuracy. It excels notably in tackling numerical computations associated with high-speed compressible fluids. For the finite volume method, the value of each control volume corresponds to a specific physical quantity, which makes the physical connotation clear and the derivation process intuitive. Moreover, through the adoption of suitable numerical formats, the finite volume method can effectively minimize numerical oscillations and exhibit strong numerical stability, thus ensuring the reliability of computational results. Particularly, the MUSCL format where a flux limiter is introduced to improve the numerical robustness is adopted for the reconstruction in this paper. Ultimately, the DBM utilizing the finite volume method is rigorously validated to assess its proficiency in addressing flow issues characterized by pronounced discontinuities. The numerical experiments encompass scenarios involving shock waves, Lax shock tubes, and acoustic waves. The results demonstrate the method's precise depiction of shock wave evolution, rarefaction waves, acoustic phenomena, and material interfaces. Furthermore, it ensures the conservation of mass, momentum, and energy within the system, as well as accurately measures the hydrodynamic and thermodynamic nonequilibrium effects of the fluid system. Compared with the finite difference method, the finite volume method is also more convenient and flexible in dealing with boundary conditions of different geometries, and can be adapted to a variety of systems with complex boundary conditions. Consequently, the finite volume method further broadens the scope of DBM in practical applications.

Forward seismic modeling is a useful tool for simulating the response of the low velocity zones near the ground surface and delicate structures at a deeper depth in northwestern China. The main objective of forward seismic modeling is to solve the wave equation accurately. The finite difference method (FDM) is a commonly used numerical method for solving the equation. It can obtain very high accuracy with high‐order difference scheme if the grid configuration is appropriate. However it shows numerical dispersion and its stability criteria is also very strict. In order to deal with these problems, a new precise integration method is advanced in this paper, using an integration scheme in the temporal domain rather than a traditional low‐order difference scheme. The precise integration method employs a difference approximation in the spatial domain, converting the acoustic or elastic wave equation into a set of differential equations with respect to time and then solves this equation system using an integration method. The final solution is attributed to an integration of a matrix of exponential functions. Theoretically this solution is accurate in the temporal domain, hence increasing the total accuracy. The whole spatial domain can be divided into several sub‐domains, considering computer capacity and speed and then the solutions put together. This revised sub‐domain precise integration method (SPIM) is more stable than the FDM and more accurate as is tested in the text. Besides it can greatly reduce the numerical dispersion with the same grid configuration as the FDM, though it may take a little more CPU time. A numerical test is presented to justify this method and then an application is shown to indicate that the SPIM is of potential value in the geophysical exploration field.

Colloidal suspensions flowing through microchannels were studied for the effects of both the shear flow and the proximity of walls on the particles' self-diffusion. Use of hydrostatic pressure to pump micron-sized silica spheres dispersed in water-glycerol mixture through poly(dimethylsiloxane) channels with a cross section of 30×24μm(2), allowed variation in the local Peclet number (Pe) from 0.01 to 50. To obtain the diffusion coefficients, image-time series from a confocal scanning laser microscope were analyzed with a method that, after finding particle trajectories, subtracts the instantaneous advective displacements and subsequently measures the slopes of the mean squared displacement in the flow (x) and shear (y) directions. For dilute suspensions, the thus obtained diffusion coefficients (D(x) and D(y)) are close to the free diffusion coefficient at all shear rates. In concentrated suspensions, a clear increase with the Peclet number (for Pe > 10) is found, that is stronger for D(x) than for D(y). This effect of shear-induced collisions is counteracted by the contribution of walls, which cause a strong local reduction in D(x) and D(y).