Numerical simulation of the continuous media problems on hybrid computer systems

The paper presents some results of modeling continuous media problems on computer system with hybrid architecture on the base of Quasi Gas Dynamic (QGD) equations system. The successful experience in solving a wide variety of gas dynamic problems by means of QGD based schemes showed that they describe viscous heat conducting flows as good as schemes for Navier–Stokes equations, where the latter are applicable. The explicit scheme described here has a Courant stability condition even for very low Mach numbers. So, it is very convenient for computer systems with the hybrid architecture, in particular for GPU-based computers. Parallel realization is based on shmem programming technology. The calculations results show good parallelization efficiency.     

Calculation of gas flow parameters around a reentry vehicle

An original numerical algorithm for calculating the gas parameters around a reentry vehicle was described and tested. It is based on a system of Quasi Gas Dynamics (QGD) equations and its approximations on hybrid meshes, which suppose adaptive, locally thickening meshes including both rectangular and triangular cells. The results of the calculations on a sequence of condensed meshes for Mach number values of 2, 5, and 12 are presented. These results correspond to well-known data and demonstrate the high efficiency of the algorithm.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

An efficient solver integrating the restarted simpler generalized minimal residual method (SGMRES(m)) with finite volume method (FVM) on triangular grid is developed to simulate the viscoelastic fluid flows. In particular, the SGMRES(m) solver is used to solve the large-scale sparse linear systems, which arise from the course of FVM on triangular grid for modeling the Newtonian and the viscoelastic fluid flows. To examine the performance of the solver for the nonlinear flow equations of viscoelastic fluids, we consider two types of numerical tests: the Newtonian flow past a circular cylinder, and the Oldroyd-B fluid flow in a planar channel and past a circular cylinder. It is shown that the numerical results obtained by the SGMRES(m) are consistent with the analytical solutions or empirical values. By comparing CPU time of different solvers, we find our solver is a highly efficient one for solving the flow equations of viscoelastic fluids.     

MHD flow of radiative micropolar nanofluid in a porous channel: optimal and numerical solutions

The flow of a radiative and electrically conducting micropolar nanofluid inside a porous channel is investigated. After implementing the similarity transformations, the partial differential equations representing the radiative flow are reduced to a system of ordinary differential equations. The subsequent equations are solved by making use of a well-known analytical method called homotopy analysis method (HAM). The expressions concerning the velocity, microrotation, temperature, and nanoparticle concentration profiles are obtained. The radiation tends to drop the temperature profile for the fluid. The formulation for local Nusselt and Sherwood numbers is also presented. Tabular and graphical results highlighting the effects of different physical parameters are presented. Rate of heat transfer at the lower wall is seen to be increasing with higher values of the radiation parameter while a drop in heat transfer rate at the upper wall is observed. Same problem has been solved by implementing the numerical procedure called the Runge–Kutta method. A comparison between the HAM, numerical and already existing results has also been made.

     

Conservative Models and Numerical Methods for Compressible Two-Phase Flow

The paper presents the computational framework for solving hyperbolic models for compressible two-phase flow by finite volume methods. A hierarchy of two-phase flow systems of conservation-form equations is formulated, including a general model with different phase velocities, pressures and temperatures; a simplified single temperature model with equal phase temperatures; and an isentropic model. The solution of the governing equations is obtained by the MUSCL-Hancock method in conjunction with the GFORCE and GMUSTA fluxes. Numerical results are presented for the water faucet test case, the Riemann problem with a sonic point and the water-air shock tube test case. The effect of the pressure relaxation rate on the numerical results is also investigated.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Exact and approximate Riemann solvers for compressible two-phase flows

Numerical methods for solving equations of two-phase hydrodynamics, which describe the flow of a dispersed solid and gas mixture are considered. The Godunov method is applied as the main approach to approximate numerical fluxes in solutions of the relevant Riemann problems. The formulations of these problems for the solid and gas phases are given, their exact analytical solution is described, and possible simplified approximate solutions are discussed. The obtained theoretical results are applied to the construction of a discrete model, which results in the generalization of the well-known Godunov-type and Rusanov-type methods to the case of nonequilibrium two-phase media. The numerical results involve the verification of the constructed methods on the analytical solutions of two-phase equations.     

Forced convection of air through networks of square rods or cylinders embedded in microchannels

Forced convection of air at near-standard conditions through periodical networks of micrometer square rods or cylinders is investigated. The Navier–Stokes equations subjected to first-order velocity-slip condition, and energy equations for the fluid and solid phases are numerically solved for two-dimensional structures. The flow is created by imposing pressure gradients, and heat is volumetrically generated inside the solid rods. Assuming periodicity in the direction transverse to the pressure gradient, the computational domain consists in long open channels, partially filled with solid rods placed regularly. The various structures considered are then modeled as porous media. For the permeability calculations, both volume averaging technique and multiple scale expansion technique are employed, and the results are favorably compared. The slip effect on permeability is highlighted for Knudsen number of about 0.05. On the other hand, the use of a periodic approach in the flow direction for heat transfer calculations is demonstrated not to be based on realistic assumptions. In addition, the importance of axial heat diffusion in channels of width close to one micrometer is emphasized. The reason is found in the low Péclet numbers typically encountered when the incompressible approximation is invoked. Based on numerical solutions at the microscopic scale, a new macroscopic modeling is suggested. Comparisons between numerical solutions and analytical predictions for various networks of rods are discussed. A very good agreement is shown.     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

     

The application of compact residual distribution schemes to two-phase flow problems

In this work, the application of residual distribution schemes (RDS) to two-phase flow problems is described. These schemes have been previously applied to system of conservation laws. However, its implementation in two-phase problems is not straightforward. Different numerical difficulties have been encountered: model hyperbolicity, non-conservative form of the equations, presence of source terms and degenerate cases have been analyzed. Within this context, two different models have been studied. Firstly, we present the Cortes model, where the inclusion of the interface pressure term makes the original system of equations hyperbolic, but there is no analytical expression of the eigenvalues, so linear perturbation methods are applied to obtain the approximate eigenstructure of the system. Secondly, we present the Staedtke model. This model has been designed to be hyperbolic, and analytical expressions of the eigenvectors can be computed. Different one and two dimensional tests have been computed to check the validity of the approach.     

A coarse-grid projection method for accelerating incompressible MHD flow simulations

Coarse grid projection (CGP) is a multiresolution technique for accelerating numerical calculations associated with a set of nonlinear evolutionary equations along with stiff Poisson’s equations. In this article, we use CGP for the first time to speed up incompressible magnetohydrodynamics (MHD) flow simulations. Accordingly, we solve the nonlinear advection–diffusion equation on a fine mesh, while we execute the electric potential Poisson equation on the corresponding coarsened mesh. Mapping operators connect two grids together. A pressure correction scheme is used to enforce the incompressibility constrain. The study of incompressible flow past a circular cylinder in the presence of Lorentz force is selected as a benchmark problem with a fixed Reynolds number but various Stuart numbers. We consider two different situations. First, we only apply CGP to the electric potential Poisson equation. Second, we apply CGP to the pressure Poisson equation as well. The maximum speed-up factors achieved here are approximately 3 and 23, respectively, for the first and second situations. For the both situations, we examine the accuracy of velocity and vorticity fields as well as the lift and drag coefficients. In general, the results obtained by CGP are in an excellent to reasonable range of accuracy. The CGP results are significantly more accurate compared to the numerical simulations of the advection–diffusion and electric potential Poisson equations on pure coarse scale grids.

     

Theoretical evaluation of the influence of geometric parameters and materials on the behavior of the airflow in a solar chimney

An analytical and numerical study of the unsteady airflow inside a solar chimney was performed. The conservation and transport equations that describe the flow were modeled and solved numerically using the finite volumes technique in generalized coordinates. The numerical results were physically validated through comparison with the experimental data. The developed model was used for airflow simulation in solar chimneys with operational and geometric configurations different from those found in the experimental prototype. Analysis showed that the height and diameter of the tower are the most important physical variables for solar chimney design.     

The extension of incompressible flow solvers to the weakly compressible regime

Numerical simulation schemes for incompressible flows such as the Simple scheme are extended to weakly compressible fluid flow. A single time scale, multiple space scale asymptotic analysis is used to gain insight into the limit behavior of the compressible flow equations as the Mach number vanishes. Motivated by these results, multiple pressure variables (MPV) are introduced into the numerical framework. These account separately for thermodynamic effects, acoustic wave propagation and the balance of forces. Discretized analogues of the averaging and large scale differencing procedures known from multiple scales asymptotics allow accurate capturing of various physical phenomena that are operative on very different length scales. The MPV approach combines the explicit numerical computation of global compression from the boundary and the long wavelength acoustics on coarse grids with an implicit pressure or pressure correction equation that formally converges to the corresponding incompressible one when the Mach number tends to zero.     

Numerical and theoretical considerations for sensitivity calculation of discontinuous flow

Linearization of 1-D Euler equations about a discontinuous solution is discussed from both the theoretical and numerical point of view. Estimates for the norm of the solution of the linearized system are shown to be valid for the case presented. Numerically, the linearization is performed following the guidelines of tangent linear model and sensitivities with respect to a flow parameter are computed, being in better agreement with the analytical value when compared with previously reported numerical results.     

基于速率变化的最优参数化

该文改进衡量参数优劣的标准,利用分段线性变换,得到了一种新的最优参数化方 法。以分段节点作为自由变量,以曲线参数速率的变化率为目标函数进行优化。通过求解一个方 程组,得到了所求节点的显式解。与以往利用Möbius变换的最优参数化不同,该文得到的曲线仍 为Bézier曲线。最优参数化后的参数接近弧长参数,文末的数值实例验证了本算法的有效性。     

A Transformation-free HOC Scheme for Incompressible Viscous Flow Past a Rotating and Translating Circular Cylinder

In the present work, a numerical study is made using a recently developed Higher Order Compact (HOC) finite difference scheme to test its capacity in capturing the very complex flow phenomenon of unsteady flow past a rotating and translating circular cylinder. The streamfunction-vorticity formulation of the Navier-stokes equations in cylindrical polar coordinate are considered as the governing equations. In the present investigation, flow is computed for a fixed Reynolds number (Re) 200 and rotational parameter values 0.5, 1.0, 2.07 and 3.25 are considered. Firstly, the flow patterns for different α values and for long time range are computed and qualitative comparisons are made with existing experimental and numerical results. Then, as a further check on the consistency of the experimental and present numerical results, quantitative comparisons are made for the velocity profiles at several locations. All these qualitative and quantitative comparisons show excellent agreements with existing experimental and numerical results.