Interfacial dynamics‐based modelling of turbulent cavitating flows,Part‐2: Time‐dependent computations

The interfacial dynamics‐based cavitation model, developed in Part‐1, is further employed for unsteady flow computations. The pressure‐based operator‐splitting algorithm (PISO) is extended to handle the time‐dependent cavitating flows with particular focus on the coupling of the cavitation and turbulence models, and the large density ratio associated with cavitation. Furthermore, the compressibility effect is important for unsteady cavitating flows because in a water–vapour mixture, depending on the composition, the speed of sound inside the cavity can vary by an order of magnitude. The implications of the issue of the speed of the sound are assessed with alternative modelling approaches. Depending on the geometric confinement of the nozzle, compressibility model and cavitation numbers, either auto‐oscillation or quasi‐steady behaviour is observed. The adverse pressure gradient in the closure region is stronger at the maximum cavity size. One can also observe that the mass transfer process contributes to the cavitation dynamics. Compared to the steady flow computations, the velocity and vapour volume fraction distributions within the cavity are noticeably improved with time‐dependent computations. Copyright © 2004 John Wiley & Sons, Ltd.     

PISO算法在求解方柱绕流非定常尾迹中的应用

应用PISO算法及非交错斜交网格技术,编制了求解二维定常、非定常流动的通用程序。计算分析了定常不可压缩Couette流动以及水平通道内方柱绕流尾迹的非定常流动,并详细分析了不同时刻方柱后尾迹的旋涡结构及发展过程,得到了合理的结果。验证了非迭代PISO算法及非交错斜交网格在分析非定常流动中的有效性,为进一步开展非定常流动的数值分析奠定了基础．     

Performance analysis of IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems

Recently, an efficient segregated algorithm for incompressible fluid flow and heat transfer problems, called inner doubly iterative efficient algorithm for linked equations (IDEAL), has been proposed by the present authors. In the algorithm there exist inner doubly iterative processes for pressure equation at each iteration level, which almost completely overcome two approximations in SIMPLE algorithm. Thus, the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of solution process. However, validations have only been conducted for two‐dimensional cases. In the present paper the performance of the IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems is analyzed and a systemic comparison is made between the algorithm and three other most widely used algorithms (SIMPLER, SIMPLEC and PISO). By the comparison of five application examples, it is found that the IDEAL algorithm is the most robust and the most efficient one among the four algorithms compared. For the five three‐dimensional cases studied, when each algorithm works at its own optimal under‐relaxation factor, the IDEAL algorithm can reduce the computation time by 12.9–52.7% over SIMPLER algorithm, by 45.3–73.4% over SIMPLEC algorithm and by 10.7–53.1% over PISO algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Parallel numerical simulation for the coupled problem of continuous flow electrophoresis

The performance of parallel subdomain method with overlapping is analysed in the case of the 3D coupled boundary‐value problem of continuous flow electrophoresis which is governed by Navier–Stokes equations coupled with convection–diffusion and potential equations. Convergence of parallel synchronous and asynchronous iterative algorithms is studied. Comparison between implemented explicit and implicit schemes for the transport equation is made using these algorithms and shows that both methods provide similar results and comparable performances. Copyright © 2007 John Wiley & Sons, Ltd.     

Variable density flow in porous media. A study by means of pore level numerical simulations

This paper is devoted to the modelling of isothermal low Reynolds and Mach numbers transient compressible flow through porous media. Traditionally, this type of flow at the macroscopic level is described by the classical Darcy's law combined with a mass balance that includes the transient term. This model is called the ‘classic model’. The aim of this paper is to explore the validity of this classic model. To this end, the flow of an ideal gas is considered within two‐dimensional model porous media. The flow is due to the imposed pressure variations at the outlet of the fluid domain. At the microscopic level, the flow is computed by solving the full compressible Navier–Stokes equations in two dimensions. Special attention is given to the outlet boundary conditions. The analysis is based on the comparison between the macroscopic data, obtained on the one hand by spatially averaging the microscopic results, and on the other hand by solving the problem directly at the macroscopic level. Situations for which a good agreement is found between the two series of data and situations for which discrepancies are observed are exhibited. These various behaviours are discussed in terms of the various time scales controlling the flow and are explained by analysing the flow structure at pore level. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparison of SIMPLE- and PISO-type algorithms for transient flows

Various pressure-based schemes are proposed for transient flows based on well-established SIMPLE and PISO algorithms. The schemes are applied to the solution of unsteady laminar flow around a square cylinder and steady laminar flow over a backward-facing step. The implicit treatment and the performance of the various schemes are evaluated by using benchmark solutions with a small time step. Three different second-order-accurate time derivatives based on different time levels are presented. The different time derivatives are applied to the various schemes under consideration. Overall the PISO scheme was found to predict accurate results and was robust. However, for small time step values, alternative schemes can predict accurate results for approximately half the computational cost. The choice of time derivative proved to be very significant in terms of the accuracy and robustness of a scheme. Significantly, the one-sided forward differencing scheme was the most successful used in conjunction with a strongly implicit-based algorithm. However, a greater degree of accuracy was achieved using the standard PISO algorithm with the Crank–Nicolson time derivative. Recommendations for future work are discussed. © 1998 John Wiley & Sons, Ltd.     

A pressure‐based method with AUSM‐type fluxes for MHD flows at arbitrary Mach numbers

In this paper, we propose an extension of a PISO method, previously developed to solve the Euler equations, and which is here extended to the ideal magnetohydrodynamic (MHD) equations. By following a pressure‐based approach, we make use of the flexibility given by pressure equation for calculating flows at arbitrary Mach numbers. To handle MHD discontinuities, we have adapted the MHD‐Advection Upstream Splitting Method for our pressure‐based formulation. With the purpose of validation, four sets of test cases are presented and discussed. We start with the circularly polarized Alfvén waves that serves as a smooth flow validation. The second case is the 1‐D Riemann problem that is calculated using both 1‐D and 2‐D formulation of the MHD equations. The third and fourth problems are the Orszag–Tang vortex and the supersonic low‐ β cylinder allowing validation of the method in complex 2‐D MHD shock interaction. Copyright © 2013 John Wiley & Sons, Ltd.     

A parallel pressure implicit splitting of operators algorithm applied to flows at all speeds

A parallel implementation of the pressure‐based implicit splitting of operators (PISO) method is described and applied to both compressible and incompressible flows. The treatment of variables at the interfaces between adjacent blocks is highlighted, and, for compressible flow, a straightforward method for the implicit handling of density is described. Steady state and oscillatory flow through a sudden expansion are considered at low speeds for both two‐ and three‐dimensional geometries. Extension of the incompressible method to compressible flow is assessed for subsonic, transonic and supersonic flow through a two‐dimensional bump. Although good accuracy is achieved in these high‐speed flows, including the automatic capturing of shock waves, the method is deemed unsuitable for simulating steady state high‐speed flows on fine grids due to the requirement of very small time steps. Copyright © 2001 John Wiley & Sons, Ltd.     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.