K. S. Yeo

Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations

Local radial basis function-based differential quadrature method is presented in detail in this paper. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with “truly” mesh-free property. In this paper, we mainly concentrate on the multiquadric RBFs since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to the simulation of natural convection in a square cavity. Excellent numerical results are obtained on an irregular knot distribution.

Ghost fluid method for strong shock impacting on material interface

It is found that the original ghost fluid method (GFM) as put forth by Fedkiw et al. [J. Comp. Phys. 152 (1999) 457] does not work consistently and efficiently using isentropic fix when applied to a strong shock impacting on a material interface. In this work, the causes for such inapplicability of the original GFM are analysed and a modified GFM is proposed and developed for greater robustness and consistency. Numerical tests also show that the modified GFM has the property of reduced conservation error and is less problem-related.

Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity

Abstract In this paper, two-dimensional mesh-free finite-difference schemes for solving incompressible viscous flows are presented. The method is based on the use of a weighted least-square approximation procedure together with a Taylor series expansion of the unknown function. Discretization error for derivatives is investigated analytically on the uniform mesh and the convergence property of the method is numerically tested. The role of the weighting function playing in the method is studied. Neumann-type boundary condition is treated by applying locally orthogonal boundary grids. Application to a problem of natural convection in a cavity is demonstrated on three different types of point distribution.

The simulation of compressible multi-medium flow: II. Applications to 2D underwater shock refraction

Abstract In this work, the methodology developed in Part I [Comp. Fluids (2000), submitted for publication] is applied to study underwater shock refracting at a gas–water interface. The reflected wave is always a shock (rarefaction) wave if a shock (rarefaction) wave enters from a gas medium into water, while the reflected wave is always a rarefaction (compression) wave if the incident shock (rarefaction) wave enters from water into a gas medium. In the first study of a vertical planar underwater shock interacting with a cylindrical gas bubble, regardless of the strength of incident shock, shock refraction at the gas bubble surface is regular initially and transforms into the irregular type before the incident shock reaches the top/bottom section of the bubble. In the second study of an underwater explosion near the free surface, the dominant physical phenomena in the earlier stages of explosion consist of the outward propagation of an underwater shock, the symmetrical expansion of a gas bubble and the possible generation of a second shock inside the expanding gas bubble. At a later stage, the underwater shock refraction at the free surface begins resulting in the generation of a pair centered Prandtl–Meyer waves, and the latter interacts with the expanding gas bubble. The numerical results exhibit all the physical phenomena described by Ballhaus and Holt [Phys. Fluids 17 (1974) 1069].

Wake-Structure Formation of a Heaving Two-Dimensional Elliptic Airfoil

This paper is prompted by a recent numerical study that shows that for a two-dimensional (2-D) elliptic airfoil undergoing prescribed heaving motion in a viscous fluid, both leading-edge vortices and trailing-edge vortices contributed to the formation of the wake structures. However, an earlier dye-visualization study on a heaving NACA 0012 airfoil appears to show that the wake structures were derived from trailing-edge vortices only. The dissimilarity in the two studies remains unclear because there is no corresponding experimental data on a 2-D heaving elliptic airfoil. In this study, digital particle image velocimetry technique was used to investigate the wake-structure formation of a 2-D elliptic airfoil undergoing simple harmonic heaving motion. For the range of flow conditions investigated here, our results show that the type of wake structures produced is controlled by when and how the leading-edge vortices interact with the trailing-edge vortices

The merging of two gaseous bubbles with an application to underwater explosions

A numerical study of two gaseous bubbles merging into a single coalesced bubble as in underwater explosions is investigated in this paper. This explosive phenomenon is modeled using a boundary integral method. Two configurations, which are in-phase and out-of-phase explosions, are simulated and compared with available experimental results. The thickness of the liquid film between the two bubbles determines our coalescence criterion. Bubble shapes and periods of oscillation are predicted well, compared to those of the experiments.

SIMULATION OF NATURAL CONVECTION IN ECCENTRIC ANNULI BETWEEN A SQUARE OUTER CYLINDER AND A CIRCULAR INNER CYLINDER USING LOCAL MQ-DQ METHOD

ABSTRACT A numerical study of the free convective flow in a horizontal eccentric annulus between a square outer cylinder and a heated circular inner cylinder is undertaken by using a local multiquadrics-based differential quadrature (MQ-DQ) method. The method combines the advantages of the conventional differential quadrature (DQ) method for derivative approximation and the mesh-free nature of the multiquadrics (MQ) method. It is capable of simulating practical problems with much larger discretization systems compared to the traditional global MQ method. In this article, it is shown the local MQ-DQ method can accurately simulate the natural-convection problem at large Rayleigh number (106). Numerical simulations are also carried out to study the effect of geometric parameters, such as eccentricities and angular positions, on the mean and local heat transfer rates.

The simulation of compressible multi-medium flow. I. A new methodology with test applications to 1D gas-gas and gas-water cases

Abstract A technique to simulate the flow field near a moving material interface is developed for multi-material compressible flow, in particular, for compressible gas–water flow. This technique can be conveniently applied with a well-established conservative scheme to solve for the regions away from the interface. Material interfaces are captured using the level set technique with minimum or no smearing. To treat wave interaction with the interface, an implicit characteristic method is developed. In this paper, the method is described in detail and tested extensively for several one-dimensional gas–gas and gas–water cases. Application to multi-dimensional shock–free surface interaction and shock–gas bubble interaction are presented in Part II [Liu TG, Khoo BC, Yeo KS. The simulation of compressible multi-medium flow. Part II: Applications to 2D underwater shock refraction, submitted for publication].

Block-marching in time with DQ discretization: an efficient method for time-dependent problems

Abstract In this paper, an efficient method is presented to solve time-dependent problems. The proposed method marches in the time direction block by block. In each block, there are several time levels, and the numerical results at these time levels are obtained simultaneously. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. The proposed method is validated by a sample problem, which has an exact solution for easy comparison. It was found that as compared to the conventional 4-stage Runge–Kutta method, the present method could give much higher order of accuracy for numerical results and require much less computational effort.

AN EFFICIENT APPROACH TO SIMULATE NATURAL CONVECTION IN ARBITRARILY ECCENTRIC ANNULI BY VORTICITY-STREAM FUNCTION FORMULATION

The global method of polynomial-based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) is applied in this work to simulate the natural convection in an annulus between two arbitrarily eccentric cylinders. The vorticity-stream function formulation in the curvilinear coordinate system is taken as the governing equation, and the pressure single value condition is converted to an explicit formulation to update the stream function value on the inner cylinder wall. The present approach is very efficient, which combines the high efficiency and accuracy of the differential quadrature (DQ) method with simple implementation of pressure single value condition. When the present approach is applied to the concentric case, it was found that the computed stream function on the inner cylinder is almost zero and the flow field is symmetric. The computed average equivalent conductivity for the concentric case also agrees very well with available data in the literature. For the eccentr...The global method of polynomial-based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) is applied in this work to simulate the natural convection in an annulus between two arbitrarily eccentric cylinders. The vorticity-stream function formulation in the curvilinear coordinate system is taken as the governing equation, and the pressure single value condition is converted to an explicit formulation to update the stream function value on the inner cylinder wall. The present approach is very efficient, which combines the high efficiency and accuracy of the differential quadrature (DQ) method with simple implementation of pressure single value condition. When the present approach is applied to the concentric case, it was found that the computed stream function on the inner cylinder is almost zero and the flow field is symmetric. The computed average equivalent conductivity for the concentric case also agrees very well with available data in the literature. For the eccentric case, it was found that the computed stream function on the inner cylinder is not zero and there is a global circulation. The present result confirms the findings by Guj and Stella (Numer. Heat Transfer, vol. 27, pp. 89-105, 1995).