C. Shu

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.

A lattice Boltzmann model for multiphase flows with large density ratio

a lattice Boltzmann model for simulating multiphase flows with large density ratios is described in this paper. The method is easily implemented. It does not require solving the Poisson equation and does not involve the complex treatments of derivative terms. The interface capturing equation is recovered without any additional terms as compared to other methods [M.R. Swift, W.R. Osborn, J.M. Yeomans, Lattice Boltzmann simulation of liquid-gas and binary fluid systems, Phys. Rev. E 54 (1996) 5041-5052; T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628-644; T. Lee, C.-L. Lin, A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys. 206 (2005) 16-47]. Besides, it requires less discrete velocities. As a result, its efficiency could be greatly improved, especially in 3D applications. It is validated by several cases: a bubble in a stationary flow and the capillary wave. The numerical surface tension obtained from the Laplace law and the interface profile agrees very well with the respective analytical solution. The method is further verified by its application to capillary wave and the bubble rising under buoyancy with comparison to other methods. All the numerical experiments show that the present approach can be used to model multiphase flows with large density ratios.

Diffuse interface model for incompressible two-phase flows with large density ratios

We investigate the applicability of an incompressible diffuse interface model for two-phase incompressible fluid flows with large viscosity and density contrasts. Diffuse-interface models have been used previously primarily for density-matched fluids, and it remains unclear to what extent such models can be used for fluids of different density, thereby potentially limiting the application of these models. In this paper, the convective Cahn-Hilliard equation and the condition that the velocity field is divergence-free are derived from the conservation law of mass of binary mixtures in a straightforward way, for fluids with large density and viscosity ratios. Differences in the equations of motion with a previously derived quasi-incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species. The convergence properties of the model are investigated for cases with large density ratio. Quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation. Tests show that the variation in volume during the computation is of the order of machine accuracy, which is consistent with our use of a conservative discretization scheme (finite volume methods) for the Cahn-Hilliard equation. Results of the method are compared with previous work for the change in topology of rising bubbles and Rayleigh-Taylor instability. Additional results are presented for head-on droplet collision and the onset of droplet entrainment in stratified flows.

Application of lattice Boltzmann method to simulate microchannel flows

Microflow has become a popular field of interest due to the advent of microelectromechanical systems. In this work, the lattice Boltzmann method, a particle-based approach, is applied to simulate the two-dimensional isothermal pressure driven microchannel flow. Two boundary treatment schemes are incorporated to investigate their impacts to the entire flow field. We pay particular attention to the pressure and the slip velocity distributions along the channel in our simulation. We also look at the mass flow rate which is constant throughout the channel and the overall average velocity for the pressure-driven flow. In addition, we include a simulation of shear-driven flow in our results for verification. Our numerical results compare well with those obtained analytically and experimentally. From this study, we may conclude that the lattice Boltzmann method is an efficient approach for simulation of microflows.

Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications

A version of immersed boundary-lattice Boltzmann method (IB-LBM) is proposed in this work. It is based on the lattice Boltzmann equation with external forcing term proposed by Guo et al. [Z. Guo, C. Zheng, B. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E 65 (2002) 046308], which can well consider the effect of external force to the momentum and momentum flux as well as the discrete lattice effect. In this model, the velocity is contributed by two parts. One is from the density distribution function and can be termed as intermediate velocity, and the other is from the external force and can be considered as velocity correction. In the conventional IB-LBM, the force density (external force) is explicitly computed in advance. As a result, we cannot manipulate the velocity correction to enforce the non-slip boundary condition at the boundary point. In the present work, the velocity corrections (force density) at all boundary points are considered as unknowns which are computed in such a way that the non-slip boundary condition at the boundary points is enforced. The solution procedure of present IB-LBM is exactly the same as the conventional IB-LBM except that the non-slip boundary condition can be satisfied in the present model while it is only approximately satisfied in the conventional model. Numerical experiments for the flows around a circular cylinder and an airfoil show that there is no any penetration of streamlines to the solid body in the present results. This is not the case for the results obtained by the conventional IB-LBM. Another advantage of the present method is its simple calculation of force on the boundary. The force can be directly calculated from the relationship between the velocity correction and the force density.

Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates

Abstract In this paper, a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature (GDQ) method. The proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as SBCGE approach. The SBCGE approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions. A comparison of the SBCGE approach with the method of modifying weighting coefficient matrices (MWCM) is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions. Some details of the GDQ method are also described in the paper.

A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate flow past a circular cylinder

A novel immersed boundary velocity correction-lattice Boltzmann method is presented and validated in this work by its application to simulate the two-dimensional flow over a circular cylinder. The present approach is inspired from the conventional immersed boundary method (IBM). In the conventional IBM, the effect of rigid body on the surrounding flow is modeled through a forcing term, which is in turn used to correct the surrounding velocity field. It was found that this process is actually an iterative procedure, trying to satisfy the non-slip boundary condition at the solid wall. In this work, a new concept of immersed boundary velocity correction approach is proposed, which directly corrects the velocity to enforce the physical boundary condition. The main advantage of the new method is that it is simple in concept and easy for implementation, and the convergence of numerical computation is faster and more stable than the conventional IBM. One challenging issue of conventional IBM is that some streamlines may pass through the solid body since there is no mechanism to enforce the non-slip condition at the boundary. As shown in the present numerical results, this unphysical phenomenon is avoided in our new method since the non-slip condition is enforced. The present results for the steady and unsteady flows compare very well with available data in the literature.

A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates

Abstract A new approach for implementing general boundary conditions in the GDQ free vibration analysis of rectangular plates is presented in this paper. The proposed approach directly couples the boundary conditions with the governing equations and is referred to as the CBCGE approach. It can be applied to any combination of simply supported, clamped, and free boundary conditions. The generality and accuracy of the CBCGE approach are demonstrated through its application to the vibration analysis of a rectangular plate with various combinations of free edges and corners. In particular, the effect of grid point distribution on the numerical results for plates with free boundary conditions is discussed, and a new grid point distribution is suggested.

Numerical study of natural convection in an eccentric annulus between a square outer cylinder and a circular inner cylinder using DQ method

Abstract In this paper, natural convective heat transfer in a horizontal eccentric annulus between a square outer cylinder and a heated circular inner cylinder is numerically studied using the differential quadrature (DQ) method. The vorticity–stream function formulation is taken in the governing equation. To take the global circulation flow into consideration, the pressure single-value condition is applied, an explicit formulation is derived and the stream function value on the inner cylinder wall is updated from the values at all the interior points. To apply the DQ method, the coordinate transformation is performed. A super elliptic function is introduced in this paper for approximating the square outer boundary located eccentrically to the inner boundary. As a result, the coordinate transformation from the physical domain to the computational domain is set up by an analytical expression. It is demonstrated in this paper that the DQ method is an efficient approach in computing the weak global circulation in the domain. The present method is validated by comparing its numerical results with available data in the literature and very good agreement has been achieved. A systematic study is conducted for the analysis of flow and thermal fields at different eccentricities and angular positions.

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.