A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems

Radial basis function networks (RBFNs) have been widely used in solving partial differential equations as they are able to provide fast convergence. Integrated RBFNs have the ability to avoid the problem of reduced convergence-rate caused by differentiation. This paper is concerned with the use of integrated RBFNs in the context of control-volume discretisations for the simulation of fluid-flow problems. Special attention is given to (i) the development of a stable high-order upwind scheme for the convection term and (ii) the development of a local high-order approximation scheme for the diffusion term. Benchmark problems including the lid-driven triangular-cavity flow are employed to validate the present technique. Accurate results at high values of the Reynolds number are obtained using relatively-coarse grids

Kinetics of in situ epoxidation of hemp oil under heterogeneous reaction conditions: an overview with preliminary results

Epoxidised hemp oil (EHO) was synthesised in the laboratory by reacting hemp oil (HO) with peroxyacetic acid (PA) in a batch reactor. The peroxyacetic acid was formed in situ from acetic acid (AA) and hydrogen peroxide (H2O2) in the presence on an acidic ion exchange resin (Amberlite IR-120) as catalyst. The overall reaction can be thought of as having two components. The first being epoxidation, a homogenous reaction which occurs at the interface of the aqueous phase and the HO phase while the second is the formation of PA, a heterogeneous reaction at the interface of the aqueous phase and the solid catalyst phase. The overall reaction kinetics were modelled by applying the Langmuir-Hinshelwood-Hougen-Watson (LHHW) model to heterogeneous reactions. Of the steps in the reaction it is postulated that the formation of PA is rate limiting, while the epoxidation occurs comparatively fast negating the requirement for an additional homogenous model. The diffusion steps in the reaction are also ignored in the kinetic model as it is believed that their effects are negligible due to intensive mixing in the batch reactor. Experiments were used to determine the optimal molar ratios of reactants and it was found that at these conditions 88% conversion of double bonds to epoxy groups occurred. The kinetic model was found to be in good agreement with the experimental results

A continuum-microscopic method based on IRBFs and control volume scheme for viscoelastic fluid flows

A numerical computation of continuum-microscopic model for visco-elastic flows based on the Integrated Radial Basis Function (IRBF) Control Volume and the Stochastic Simulation Techniques (SST) is reported in this paper. The macroscopic flow equations are closed by a stochastic equation for the extra stress at the microscopic level. The former are discretised by a 1D-IRBF-CV method while the latter is integrated with Euler explicit or Predictor-Corrector schemes. Modelling is very efficient as it is based on Cartesian grid, while the integrated RBF approach enhances both the stability of the procedure and the accuracy of the solution. The proposed method is demonstrated with the solution of the start-up Couette flow of the Hookean and FENE dumbbell model fluids

Bounding film drainage in common thin films

A review of thin film drainage models is presented in which the predictions of thinning velocities and drainage times are compared to reported values on foam and emulsion films found in the literature. Free standing films with tangentially immobile interfaces and suppressed electrostatic repulsion are considered, such as those studied in capillary cells. The experimental thinning velocities and drainage times of foams and emulsions are shown to be bounded by predictions from the Reynolds and the theoretical MTsR equations. The semi-empirical MTsR and the surface wave equations were the most consistently accurate with all of the films considered. These results are used in an accompanying paper to develop scaling laws that bound the critical film thickness of foam and emulsion films

Imposition of physical parameters in dissipative particle dynamics

In the mesoscale simulations by the dissipative particle dynamics (DPD), the motion of a fluid is modelled by a set of particles interacting in a pairwise manner, and it has been shown to be governed by the Navier–Stokes equation, with its physical properties, such as viscosity, Schmidt number, isothermal compressibility, relaxation and inertia time scales, in fact its whole rheology resulted from the choice of the DPD model parameters. In this work, we will explore the response of a DPD fluid with respect to its parameter space, where the model input parameters can be chosen in advance so that (i) the ratio between the relaxation and inertia time scales is fixed; (ii) the isothermal compressibility of water at room temperature is enforced; and (iii) the viscosity and Schmidt number can be specified as inputs. These impositions are possible with some extra degrees of freedom in the weighting functions for the conservative and dissipative forces. Numerical experiments show an improvement in the solution quality over conventional DPD parameters/weighting functions, particularly for the number density distribution and computed stresses

BEM-RBF approach for viscoelastic flow analysis

A new BE-only method is achieved for the numerical solution of viscoelastic flows. A decoupled algorithm is chosen where the fluid is considered as being composed of an artificial Newtonian component and the remaining component is accordingly defined from the original constitutive equation. As a result the problem is viewed as that of solving for the flow of a Newtonian liquid with the non-linear viscoelastic effects acting as a pseudo body force. Thus the general solution is obtained by adding a particular solution to the homogeneous one. The former is obtained by a BEM for the base Newtonian fluid and the latter is obtained analytically by approximating the pseudo body force in terms of suitable radial basis functions (RBFs). Embedded in the approximation of the pseudo body force is the calculation of the polymer stress. This is achieved by solving the constitutive equation using RBF networks (RBFNs). Both the calculations of the particular solution and the polymer stress are therefore meshless and the resultant BEM-RBF method is a BE-only method. The complete elimination of any structured domain discretisation is demonstrated with a number of flow problems involving the Upper Convected Maxwell (UCM) and the Oldroyd-B fluids

High-order fluid solver based on a combined compact integrated RBF approximation and its fluid structure interaction applications

In this study, we present a high-order numerical method based on a combined compact integrated RBF (IRBF) approximation for viscous flow and fluid structure interaction (FSI) problems. In the method, the fluid variables are locally approximated by using the combined compact IRBF, and the incompressible Navier-Stokes equations are solved by using the velocity-pressure formulation in a direct fully coupled approach. The fluid solver is verified through various problems including heat, Burgers, convection-diffusion equations, Taylor-Green vortex and lid driven cavity flows. It is then applied to simulate some FSI prob- lems in which an elastic structure is immersed in a viscous incompressible fluid. For FSI simulations, we employ the immersed boundary framework using a regular Eulerian computational grid for the fluid mechanics together with a Lagrangian representation of the immersed boundary. For the immersed fibre/membrane FSI problems, although the order of accuracy of the present scheme is generally similar to FDM approaches reported in the literature, the present approach is nonetheless more accurate than FDM approaches at comparable grid spacings. The numerical results obtained by the present scheme are highly accurate or in good agreement with those reported in earlier studies of the same problems