A preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

A heterogeneous modeling method for porous media flows

This paper presents a new heterogeneous multiscale modeling method for porous media flows. Physics at the global level is governed by one set of PDEs, while features in the solution that are beyond the resolution capacity of the global model are accounted for by the next refined set of governing equations. In this method, the global or coarse model is given by the Darcy equation, while the local or refined model is given by the Darcy–Stokes equation. Concurrent domain decomposition where global and local models are applied to adjacent subdomains, as well as overlapping domain decomposition where global and local models coexist on overlapping domains, is considered. An interface operator is developed for the case where global and local models commute along the common interface. For the overlapping decomposition, a residual‐based coupling technique is developed that consistently facilitates bottom‐up embedding of scale effects from the local Darcy–Stokes model into the global Darcy model. Numerical results are presented for nonoverlapping and overlapping domain decompositions for various benchmark problems. Computed results show that the hierarchically coupled models accurately account for the heterogeneity of the medium and efficiently incorporate local features into the global response. Copyright © 2014 John Wiley & Sons, Ltd.     

On the discretization of gradient and curl in atmospheric models using height‐based terrain‐following coordinates

Different discretizations of the gradient and curl operators are considered for a staggered grid in a height‐based terrain‐following coordinate system. A combination of discrete operators is identified that guarantees the mimetic property that the curl of the gradient of any scalar vanishes identically. The result is illustrated with some numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A segregated method for compressible flow computation. Part II: general divariant compressible flows

Typically, segregated methods have been used for the computation of incompressible flows whereas coupled solvers, for compressible flows. Compared to coupled solvers, segregated methods present the advantage of computational savings in RAM memory and CPU time, although at the cost of an inferior robustness. However, previously published segregated algorithms for general compressible flows are known to present pitfalls, like convergence to wrong solutions, lack of robustness in the presence of strong discontinuities, such as normal and oblique shocks, and complicated boundary condition imposition. Therefore, in this paper a segregated method for non‐isothermal compressible flows is proposed that preserves the thermodynamic coupling and overcomes the criticisms of existing methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A segregated method for compressible flow computation Part I: isothermal compressible flows

Traditionally, coupled methods have been employed for the computation of compressible flows, whereas segregated methods have been preferred for the computation of incompressible flows. Compared to coupled methods, segregated solvers present the advantage of reduced computer memory and CPU time requirements, although at the cost of an inferior robustness. Therefore, in a series of papers we present unified computational techniques to compute compressible and incompressible flows with segregated stabilized methods. The proposed algorithms have an increased robustness compared to existing techniques, while possessing additional benefits such as employing standard pressure boundary conditions. In this first part, the thermodynamics of isothermal, thermally perfect compressible flows is set up in the framework of symmetric systems and the corresponding segregated algorithms are introduced. Copyright © 2005 John Wiley & Sons, Ltd.     

Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparing vortex methods and finite difference methods in a homogeneous turbulent shear flow

The vortex method is applied to the calculation of a homogeneous shear turbulence, and compared with a finite difference code using identical calculation conditions. The core spreading method with spatial adaptation is selected as the viscous diffusion scheme of the vortex method. The shear rate is chosen so that it matches the maximum value observed in a fully developed channel flow. The isosurface, anisotropy tensors, and joint probability density functions reflect the ability of the present vortex method to quantitatively reproduce the anisotropic nature of strongly sheared turbulence, both instantaneously and statistically. Copyright © 2009 John Wiley & Sons, Ltd.     

Investigation of a Gas Flow with Solid Particles in a Supersonic Nozzle

A two-phase flow with high Reynolds numbers in the subsonic, transonic, and supersonic parts of the nozzle is considered within the framework of the Prandtl model, i.e., the flow is divided into an inviscid core and a thin boundary layer. Mutual influence of the gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by the Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by the Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The effect of various boundary conditions for the flow of particles in the inviscid region is considered. For an axisymmetric nozzle, the influence of the contour of the subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 65–77, November–December, 2005.     

Wave propagation in a fluid flowing through a curved thin-walled elastic tube

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.     

简单剪切流动中转捩的低维模式与数值模拟

回顾了过去10年在壁湍流和自 由剪切流转捩问题的数值研究中取得的重要进展, 介绍了数值方法和模式研究方面 的进展, 以及由此带来的关于转捩理论认识上的进展. 对于壁面流动, 文中主要介 绍了渐进稳定流动中``跨越(bypass)转捩'研究中的各种观点. 本文也简要介绍了 对感受性和转捩控制方面的研究.     

Finite volume method for three-dimensional transonic potential flow through turbomachinery blade rows

A finite-volume method has been developed for the calculation of transonic, potential flows through 3-D turbomachinery blades with complex geometries. The exact transonic potential flow equation is solved on a mesh constructed from small volume elements. A transformation is introduced through which cuboids of the physical plane are mapped into computational cubes. Two sets of overlapping volumes are used. While the thermodynamic properties are calculated at the primary volume centres, the flux balance is established on the secondary volumes. For transonic flows an artificial compressibility term (upwind density gradient) is added to density to produce the necessary directional bias in the hyperbolic region. The successive point over-relaxation Gauss-Seidel method has been used to solve the non-linear partial differential equations. Comparisons with experiments and/or other numerical solutions for various turbomachinery configurations show that the 3-D finite-volume approach is a relatively accurate, reliable and fast method for inviscid, transonic flow predictions through turbomachinery blade rows     

Design of S-shaped diffusers in incompressible flow

The hodograph method, in conjunction with a numerical form of the Schwarz-Christoffel transformation, is applied to the determination of the shape of an S-shaped diffuser subject to certain prescribed characteristics in incompressible flow. It is shown how the resulting diffuser of infinite length can be modified to one of finite length by limiting the upstream and downstream velocities to within a small percentage of their normal asymptotic values     

A finite volume method for two-dimensional transonic potential flow through turbomachinery blade rows

To predict inviscid transonic flow through turbomachinery blade rows, the exact transonic potential flow equation is solved on a mesh constructed from small area elements. A transformation is introduced through which distorted squares of the physical plane are mapped into computational squares. Two sets of overlapping elements are used; while the thermodynamic properties are calculated at the primary element centres, the flux balance is established on the secondary elements. For transonic flows an artificial compressibility term (upwind density gradient) is added to density in order to produce the desired directional bias in the hyperbolic region. while the entropy does not increase across mass conservative shock jump regions. Comparisons withexperiments and with other numerical and analytical solutions for various turbomachinery configurations show that this approach is comparatively accurate, reliable, and fast.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

AN UNFACTORED IMPLICIT MOVING MESH METHOD FOR THE TWO-DIMENSIONAL UNSTEADY N–S EQUATIONS

An unfactored implicit time-marching method for the solution of the unsteady two-dimensional Reynolds-averaged thin layer Navier–Stokes equations is presented. The linear system arising from each implicit step is solved by the conjugate gradient squared (CGS) method with preconditioning based on an ADI factorization. The time-marching procedure has been used with a fast transfinite interpolation method to regenerate the mesh at each time step in response to the motion of the aerofoil. The main test cases examined are from the AGARD aeroelastic configurations and involve aerofoils oscillating rigidly in pitch. These test cases have been used to investigate the effect of various parameters, such as CGS tolerance and laminar/turbulent transition location, on the accuracy and efficiency of the method. Comparisons with available experimental data have been made for these cases. In order to illustrate the application of the mesh generator and flow solver to more general flows where the aerofoil deforms, results for an NACA 0012 aerofoil with an oscillating trailing edge flap are also shown.     

Derivation of a spectral pressureless formulation for direct numerical simulation of incompressible channel flows based on a functional formalism

In this article, a new computational spectral algorithm is developed for simulation of general three-dimensional, time-dependent, incompressible channel flow. The development is based on a general functional formalism of non-equilibrium thermodynamics, and, although it is illustrated here for a Newtonian fluid, it is easily adapted to non-Newtonian fluids. The advantage of this algorithm is that the scalar pressure is eliminated from the discrete spectral analog to the equations of motion, which are expressed solely in terms of the spectral coefficients of the velocity vector field. This alleviates the need for the application of boundary conditions on the pressure, the specification of which can be a major source of difficulty in direct numerical simulations. At the same time, the velocity spectrum is quite general, and not subject to any a priori constraints. Thus, it is anticipated that the ideas exposed in the present algorithm can lead to the development of better numerical simulation techniques for complicated three-dimensional and turbulent flows.     

Extension of some numerical schemes to the analysis of gas and particle mixtures

In this paper, several numerical schemes are extended to obtain approximate solutions to the system of equations encountered in the analysis of multiphase mixtures of gas and particles. Both dense and dilute mixtures are studied, the gas is modelled as a perfect gas and the solid is considered incompressible. Although the tests employed throughout this work for studying the behaviour of the schemes are essentially one dimensional, the finite volume method developed permits its application to multidimensional problems in unstructured grids. Copyright © 2007 John Wiley & Sons, Ltd.     

A filtered mass density function approach for modeling separated two-phase flows for LES I: Mathematical formulation

The overall objective of this study is to develop a full velocity-scalar filtered mass density function (FMDF) formulation for large eddy simulation (LES) of a separated two-phase flow. Required in the development of the two-phase FMDF transport equation are the local instantaneous equations of motion for a two-phase flow previously derived by Kataoka. In Kataoka’s development, phase interaction terms are cast in terms of a Dirac delta distribution on the phase interface. For this reason, it is difficult to close these coupling terms in the instantaneous formulation and this difficulty is propagated into the phase-coupling terms in the FMDF transport equation. To address this point a new derivation of the local instantaneous equations for a separated two-phase flow is given. The equations are shown to be consistent with the formulation given by Kataoka, and in the development, a direct link between the conditionally surface-filtered coupling terms, arising in the FMDF formulation, and LES phase-coupling terms is established. Clarification of conditions under which conditionally filtered interphase conversion terms in the marginal FMDF transport equations may be disregarded in a separated continuum-dispersed phase flow is discussed. Modeling approaches and solutions procedures to solve the two-phase FMDF transport equation via Monte-Carlo methods are outlined.     

Incompressible flows with combustion simulated by a preconditioning method using multigrid acceleration and MUSCL reconstruction

A numerical algorithm for the steady state solution of three‐dimensional incompressible flows is presented. A preconditioned time marching scheme is applied to the conservative form of the governing equations. The preconditioning matrix multiplies the time derivatives of the system and circumvents the eigenvalue‐caused stiffness at low speed. The formulation is suitable for constant density flows and for flows where the density depends on non‐passive scalars, such as in low‐speed combustion applications. The k–ε model accounts for turbulent transport effects. A cell‐centred finite volume formulation with a Runge–Kutta time stepping scheme for the primitive variables is used. Second‐order spatial accuracy is achieved by developing for the preconditioned system an approximate Riemann solver with MUSCL reconstruction. A multi‐grid technique coupled with local time stepping and implicit residual smoothing is used to accelerate the convergence to the steady state solution. The convergence behaviour and the validation of the predicted solutions are examined for laminar and turbulent constant density flows and for a turbulent non‐premixed flame simulated by a presumed probability density function (PDF) model. Copyright © 2001 John Wiley & Sons, Ltd.