Function-based Gas Kinetic Scheme Research Articles

Hydrodynamic performance of manta rays swimming in staggered arranged group

To investigate the hydrodynamic performance of manta rays swimming in staggered arranged group, a morphological and kinematic model of manta rays is developed based on biological observations, and then a numerical calculation method is established for group swimming of manta rays based on the Immersed Boundary Method and the Sphere function-based Gas Kinetic Scheme (IB-SGKS). The group swimming of two manta rays with a fixed vertical spacing of 0.1 times the body thickness, and a flow direction spacing of 0—1.5 times the body length is systematically investigated. The average thrust/efficiency of the group system and each individual in the group are analyzed by combining the global three-dimensional (3D) vortex structure and the characteristic cross-section two-dimensional (2D) vortex structure. The numerical results are shown below. When the streamwise spacing between individuals is small, the propulsive performance decreases sharply compared with swimming alone; as the streamwise spacing increases, the propulsive performance of the leader manta ray is consistently better than that of swimming alone, with the maximum thrust enhanced up to 11.24% when <i>D</i><sub><i>x</i></sub> = 0.4BL, and the maximum efficiency is enhanced up to 3.58% when <i>D</i><sub><i>x</i></sub> = 0.3BL; with the increase of the streamwise spacing, in the thrust/efficiency curves of the follower manta ray appears volatility, with the maximum thrust enhanced to 48.14% when <i>D</i><sub><i>x</i></sub> = 0.4BL and the maximum efficiency reached to 12.39% when <i>D</i><sub><i>x</i></sub> = 0.5BL; the system average thrust and efficiency enhancement both reach their corresponding maximum values, specifically, 29.69% and 6.77%, when <i>D</i><sub><i>x</i></sub> = 0.4BL, which is because the tail vortex of the leading manta ray just passes through the front edge of the follower manta ray and directly acts on the tip vortex that initially falls off from the follower manta rays, thus substantially increasing their vortex energy.

Development of explicit formulations of G45-based gas kinetic scheme for simulation of continuum and rarefied flows.

In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.

A generalized sphere function-based gas kinetic scheme for simulation of compressible viscous flows

In this paper, a generalized sphere function-based gas kinetic scheme (GKS) is developed for simulation of two-dimensional compressible viscous flows. This work aims to improve the existing simplified GKS, in which a simple two-dimensional/three-dimensional distribution function is used as the equilibrium state. The present scheme applies the finite volume method to discretize Navier–Stokes equations and the numerical flux at cell interface is evaluated by the local reconstruction of solution for the continuous Boltzmann equation with [Formula: see text]-dimensional sphere function. This local solution contains both the equilibrium part and nonequilibrium part, and the contribution of the nonequilibrium part is controlled by a switch function. It is found that the present scheme has the same expression as the circular function-based GKS for the two-dimensional case and the sphere function-based GKS for the three-dimensional case. Therefore, these two GKSs can be unified, and the developed method provides another way for the simple distribution function.

Computational Model Construction and Analysis of the Hydrodynamics of a Rhinoptera Javanica

In this study, numerical simulations are employed to investigate the hydrodynamic performance and wake topology of a swimming Rhinoptera javanica. The study is motivated by the quest to understand the hydrodynamics of median and paired fin(MPF) mode with both spanwise and chordwise flexibility. The simulations employ an immersed boundary(IB)-simplified sphere function-based gas kinetic scheme(SGKS) method that allows us to simulate flows with complex moving boundaries on fixed Cartesian grids. A computational model is constructed based on biological data. The evolution of the hydrodynamic force and 3D vortex structures are presented. Besides, the effect of frequency and amplitude are also discussed to explain some behaviors of the actual Rhinoptera javanica. This work can provide a baseline for the design of a bio-inspired underwater vehicle.

Improvement of computational efficiency of circular function-based gas kinetic scheme by using Jacobian-free Newton–Krylov method

The gas-kinetic BGK scheme is a promising method for simulation of inviscid and viscous flows. Different from conventional Navier–Stokes (N-S) solvers, it evaluates the inviscid and viscous fluxes simultaneously by reconstructing the local solution of BGK Boltzmann equation at the cell interface. Due to its inherently superior dissipation property, it usually gives accurate and robust numerical results. However, a notable drawback of the BGK-type scheme is the low computational efficiency. Recently, aimed at reducing the computational effort, a circular function-based BGK (CBGK) scheme was developed. Nevertheless, it is still time consuming because the original scheme used an explicit way in the time integration as in most existing BGK schemes and the convergence speed can be slow. To improve the computational efficiency, an implicit CBGK scheme is developed in this paper by incorporating the Jacobian-free Newton–Krylov (JFNK) method into the scheme. Particularly, the generalized minimal residual (GMRES) approach is employed to iteratively solve the large linear equation system. With the help of the Jacobian-free approach, a faster convergence speed can be achieved without explicitly computing and storing the flux Jacobian, which is usually a large and sparse matrix. In order to reduce the number of GMRES iterations, the preconditioning is also adopted and the Lower-Upper symmetric Gauss-Seidel (LUSGS) scheme is employed as a preconditioner. For validation of the present CBGK-JFNK method, several two-dimensional inviscid and viscous test cases are investigated. The numerical results show that the needed computational time is significantly reduced as compared with the original explicit CBGK scheme and the CBGK-LUSGS method.

An implicit simplified sphere function-based gas kinetic scheme for simulation of 3D incompressible isothermal flows

In this work, an implicit simplified sphere function-based gas kinetic scheme (SGKS) is presented for simulation of 3D incompressible isothermal flows. At first, the numerical fluxes of governing equations are reconstructed by the local solution of Boltzmann equation with sphere function distribution. Due to incompressible limit, the sphere at cell interface can be approximately considered to be symmetric as shown in the work. Besides that, the energy equation is usually not needed for simulation of incompressible isothermal flows. With all these simplifications, the formulations of the simplified SGKS can be expressed concisely and explicitly. Secondly, the commonly-used implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method is adopted to further improve the computational efficiency and numerical stability of present scheme. In LU-SGS method, only a forward and a backward sweep are needed for marching the conservative variables in time. As a result, the simplified SGKS with the LU-SGS method can be implemented easily. Numerical experiments, including the 3D lid-driven cavity flow and flow over a backward-facing step, showed that the incompressible isothermal flows can be well simulated by the developed scheme and its computational efficiency is significantly higher than that of the original SGKS and the lattice Boltzmann flux solver (LBFS). In addition, it was found that the present scheme with the LU-SGS method is more efficient than that with the explicit Euler method, and the speedup ratio is about 2 to 5.

An immersed boundary-simplified sphere function-based gas kinetic scheme for simulation of 3D incompressible flows

In this work, an immersed boundary-simplified sphere function-based gas kinetic scheme (SGKS) is presented for the simulation of 3D incompressible flows with curved and moving boundaries. At first, the SGKS [Yang et al., “A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows,” J. Comput. Phys. 295, 322 (2015) and Yang et al., “Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows,” J. Comput. Phys. 319, 129 (2016)], which is often applied for the simulation of compressible flows, is simplified to improve the computational efficiency for the simulation of incompressible flows. In the original SGKS, the integral domain along the spherical surface for computing conservative variables and numerical fluxes is usually not symmetric at the cell interface. This leads the expression of numerical fluxes at the cell interface to be relatively complicated. For incompressible flows, the sphere at the cell interface can be approximately considered to be symmetric as shown in this work. Besides that, the energy equation is usually not needed for the simulation of incompressible isothermal flows. With all these simplifications, the simple and explicit formulations for the conservative variables and numerical fluxes at the cell interface can be obtained. Second, to effectively implement the no-slip boundary condition for fluid flow problems with complex geometry as well as moving boundary, the implicit boundary condition-enforced immersed boundary method [Wu and Shu, “Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications,” J. Comput. Phys. 228, 1963 (2009)] is introduced into the simplified SGKS. That is, the flow field is solved by the simplified SGKS without considering the presence of an immersed body and the no-slip boundary condition is implemented by the immersed boundary method. The accuracy and efficiency of the present scheme are validated by simulating the decaying vortex flow, flow past a stationary and rotating sphere, flow past a stationary torus, and flows over dragonfly flight.

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21–23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.

A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

AbstractIn this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows

The sphere function-based gas kinetic scheme (GKS), which was presented by Shu and his coworkers [23] for simulation of inviscid compressible flows, is extended to simulate 3D viscous incompressible and compressible flows in this work. Firstly, we use certain discrete points to represent the spherical surface in the phase velocity space. Then, integrals along the spherical surface for conservation forms of moments, which are needed to recover 3D Navier–Stokes equations, are approximated by integral quadrature. The basic requirement is that these conservation forms of moments can be exactly satisfied by weighted summation of distribution functions at discrete points. It was found that the integral quadrature by eight discrete points on the spherical surface, which forms the D3Q8 discrete velocity model, can exactly match the integral. In this way, the conservative variables and numerical fluxes can be computed by weighted summation of distribution functions at eight discrete points. That is, the application of complicated formulations resultant from integrals can be replaced by a simple solution process. Several numerical examples including laminar flat plate boundary layer, 3D lid-driven cavity flow, steady flow through a 90° bending square duct, transonic flow around DPW-W1 wing and supersonic flow around NACA0012 airfoil are chosen to validate the proposed scheme. Numerical results demonstrate that the present scheme can provide reasonable numerical results for 3D viscous flows.

A simple distribution function-based gas-kinetic scheme for simulation of viscous incompressible and compressible flows

In this work, a simple distribution function-based gas-kinetic scheme for simulation of viscous flows is presented. The work applies the finite volume method to discretize the governing differential equations, and inviscid and viscous fluxes at the cell interface are evaluated simultaneously by local reconstruction of solution for the continuous Boltzmann equation. Differently from the conventional gas-kinetic scheme [13–15], in the present work, the Maxwellian distribution function is simplified by a simple distribution function, and integrals in the infinity domain of phase space are reduced to integrals around a circle. As a consequence, the computational efficiency is greatly improved. Since the simple distribution function is defined on the circle, for simplicity, it is termed as circular function hereafter. The present work is the extension of our previous work [20], where the circular function-based gas-kinetic scheme is presented to simulate inviscid flows. Only the equilibrium distribution function is considered in [20]. To solve viscous flows, the non-equilibrium part of density distribution function has to be considered. One of major contributions in this work is to present a simple way to compute the non-equilibrium part of the distribution function. It can be calculated by the difference of equilibrium distribution functions at the cell interface and its surrounding point. As a result, the formulations for computing the conservative flow variables and fluxes at the cell interface can be given explicitly. The present solver can simulate both incompressible and compressible viscous flows. To validate the proposed new gas-kinetic scheme, several incompressible and compressible viscous flows are simulated. Numerical results showed that the circular function-based gas-kinetic scheme can provide accurate numerical results with the same computational cost as that needed by conventional Navier–Stokes solver.

Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows

This paper presents a new gas-kinetic scheme for simulation of compressible inviscid flows. It starts to simplify the integral domain of Maxwellian distribution function over the phase velocity ξ and phase energy ζ to the integral domain of modified Maxwellian function over the phase velocity ξ only. The influence of integral over phase energy ζ is embodied as the particle internal energy ep. The modified Maxwellian function is further simplified to a circular function with the assumption that all the particles are concentrated on a circle. Then two circular function-based gas-kinetic schemes are presented for simulation of compressible inviscid flows. In the new schemes, no error and exponential functions, which are often appeared in the Maxwellian function-based gas-kinetic schemes, are involved. As a result, the new schemes can be implemented in a more efficient way. To validate the proposed new gas-kinetic schemes, test examples in the transonic flow, supersonic flow and hypersonic flow regimes are solved. Numerical results showed that the solution accuracy of the circular function-based gas-kinetic schemes is comparable to that of corresponding Maxwellian function-based gas-kinetic schemes. However, the circular function-based gas-kinetic schemes need less computational effort.