Analysis of Annular Liquid Membranes and Their Singularities

The singularities of the equations governing the fluid dynamics of steady, axisymmetric, annular liquid membranes subject to gravity are analyzed by means of two techniques based on the membranes's slope and curvature, and the membrane's mean radius, mass per unit length, and axial and radial velocity components, respectively. It is shown that no singularity is possible at or downstream from the nozzle exit for Weber numbers greater than unity because of the gravitational pull. For a Weber number equal to one, a singularity at the nozzle exit appears and the flow slope there is undetermined; however, the slope acquires a finite value if the liquid is assumed to leave the nozzle at angle different from that of the annular orifice. It is also shown that, for Weber numbers smaller than one, a singularity may occur downstream from the nozzle exit which may also be removed, and that the shapes of annular liquid membranes for Weber numbers equal to or less than one take a rounded form which is in agreement with experimental observations. An asymptotic analysis shows that, to leading order, the shapes of capillary, annular liquid membranes are arcs of circumferences, and this result is again in accord with available experimental findings.     

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

The parallelization of a fully implicit and stable finite element algorithm with relative low memory requirements for the accurate simulation of time-dependent, free-surface flows of multimode viscoelastic liquids is presented. It is an extension of our multi-stage sequential solution procedure which is based on the mixed finite element method for the velocity and pressure fields, an elliptic grid generator for the deformation of the mesh, and the discontinuous Galerkin method for the viscoelastic stresses [Dimakopoulos and Tsamopoulos [12], [14]]. Each one of the above subproblems is solved with the Newton–Rapshon technique according to its particular characteristics, while their coupling is achieved through Picard cycles. The physical domain is graphically partitioned into overlapping subdomains. In the process, two different kinds of parallel solvers are used for the solution of the distributed set of flow and mesh equations: a multifrontal, massively parallel direct one (MUMPS) and a hierarchical iterative parallel one (HIPS), while viscoelastic stress components are independently calculated within each finite element. The parallel algorithm retains all the advantages of its sequential predecessor, related with the robustness and the numerical stability for a wide range of levels of viscoelasticity. Moreover, irrespective of the deformation of the physical domain, the mesh partitioning remains invariant throughout the simulation. The solution of the constitutive equations, which constitutes the largest portion of the system of the governing, non-linear equations, is performed in a way that does not need any data exchange among the cluster's nodes. Finally, indicative results from the simulation of an extensionally thinning polymeric solution, demonstrating the efficiency of the algorithm are presented.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

A novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.     

带导管鱼雷层流与湍流绕流计算

用分块耦合求解方法和自编三维软件计算了带导管鱼雷整体流场无冲角和有冲角的绕流问题,湍流计算采用Baldwin-Lomax湍流模型.计算雷诺数从103到107.计算冲角0～20°.在各种雷诺数计算,头部滞止压力（无黏流理论值为1.5）误差在10％范围内,总流量误差在5％范围内.对于不同冲角计算表明,升力系数在冲角15°范围内基本上线性增加,超过15°后升力系数缓慢减少.而摩擦阻力系数几乎与冲角无关,但压差阻力随冲角的增加而迅速增长.     

脉冲爆震发动机进气道气动性能的数值研究

采用有限体积法计算了脉冲爆震发动机某轴对称超音速进气道在3种 不同出口条件(单个正弦扰动压力、某脉冲爆震发动机爆震室头部表压和进气道出口堵塞) 下的进气道内结尾正激波的运动情况，得出了进气道内结尾正激波运动特性和不同出口条件 的关系. 在计算中，采用了多块结构化网格，控制体积的界面无黏通量采用三阶迎风格 式插值获得，同时采用了minmod通量限制器以确保在激波处的解的物理特性；扩散通量采 用二阶中心差分格式插值获得. 定常计算采用当地时间步法，非定常计算采用双时间步法. 离散的代数方程采用交替方向迭代法求解。     

Coupled solution of the species conservation equations using unstructured finite‐volume method

A coupled solver was developed to solve the species conservation equations on an unstructured mesh with implicit spatial as well as species‐to‐species coupling. First, the computational domain was decomposed into sub‐domains comprised of geometrically contiguous cells—a process similar to additive Schwarz decomposition. This was done using the binary spatial partitioning algorithm. Following this step, for each sub‐domain, the discretized equations were developed using the finite‐volume method, and solved using an iterative solver based on Krylov sub‐space iterations, that is, the pre‐conditioned generalized minimum residual solver. Overall (outer) iterations were then performed to treat explicitness at sub‐domain interfaces and nonlinearities in the governing equations. The solver is demonstrated for both two‐dimensional and three‐dimensional geometries for laminar methane–air flame calculations with 6 species and 2 reaction steps, and for catalytic methane–air combustion with 19 species and 24 reaction steps. It was found that the best performance is manifested for sub‐domain size of 2000 cells or more, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block (coupled) Gauss–Seidel procedure. All calculations were performed on a single processor machine. The largest calculations were performed for about 355 000 cells (4.6 million unknowns) and required 900 MB of peak runtime memory and 19 h of CPU on a single processor. Copyright © 2009 John Wiley & Sons, Ltd.     

Schwarz domain decomposition for the incompressible Navier–Stokes equations in general co‐ordinates

This paper describes a domain decomposition method for the incompressible Navier–Stokes equations in general co‐ordinates. Domain decomposition techniques are needed for solving flow problems in complicated geometries while retaining structured grids on each of the subdomains. This is the so‐called block‐structured approach. It enables the use of fast vectorized iterative methods on the subdomains. The Navier–Stokes equations are discretized on a staggered grid using finite volumes. The pressure‐correction technique is used to solve the momentum equations together with incompressibility conditions. Schwarz domain decomposition is used to solve the momentum and pressure equations on the composite domain. Convergence of domain decomposition is accelerated by a GMRES Krylov subspace method. Computations are presented for a variety of flows. Copyright © 2000 John Wiley & Sons, Ltd.     

A coupled surface‐subsurface model for hydrostatic flows under saturated and variably saturated conditions

In this paper, the governing differential equations for hydrostatic surface‐subsurface flows are derived from the Richards and from the Navier‐Stokes equations. A vertically integrated continuity equation is formulated to account for both surface and subsurface flows under saturated and variable saturated conditions. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction, a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities, and a finite volume approximation for the vertically integrated continuity equation, are derived in such a fashion that, after simple manipulation, the resulting discrete pressure equation can be assembled into a single, two‐dimensional, mildly nonlinear system. This system is solved by a nested Newton‐type method, which yields simultaneously the (hydrostatic) pressure and a nonnegative fluid volume throughout the computational grid. The resulting algorithm is relatively simple, extremely efficient, and very accurate. Stability, convergence, and exact mass conservation are assured throughout also in presence of wetting and drying, in variable saturated conditions, and during flow transition through the soil interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm.     

Numerical and experimental study of free convection above a sinusoidal horizontal plate

A numerical and experimental analysis is performed to study the laminar free convection above a horizontal plate facing upward subjected to an uniform heat flux. The surface of the plate, in contact with the fluid, is described by a sinusoidal profile. The natural convection equations are discretized, using an implicit finite difference technique, based on the finite volume approach. The SIMPLE algorithm assumes the linkage between velocities and pressure fields. The top and the lateral boundaries of the space, where free convection is developing, are determined by using an iterative procedure. The temperature fields of the fluid, over the plate, are visualized by an experimental device, which can realize a simultaneous measurement of the temperature and the position. Qualitative information about the natural convection flow above the plate is obtained by using a laser tomography technique. The numerical results show that the flow and the heat transfer are strongly affected by the amplitude, the period of the sinusoidal profile and the type of fluid. Comparisons between numerical and experimental results show a good qualitative agreement.     

A method for finite element parallel viscous compressible flow calculations

This paper presents the parallelization aspects of a solution method for the fully coupled 3D compressible Navier-Stokes equations. The algorithmic thrust of the approach, embedded in a finite element code NS3D, is the linearization of the governing equations through Newton methods, followed by a fully coupled solution of velocities and pressure at each non-linear iteration by preconditioned conjugate gradient-like iterative algorithms. For the matrix assembly, as well as for the linear equation solver, efficient coarse-grain parallel schemes have been developed for shared memory machines, as well as for networks of workstations, with a moderate number of processors. The parallel iterative schemes, in particular, circumvent some of the difficulties associated with domain decomposition methods, such as geometry bookkeeping and the sometimes drastic convergence slow-down of partitioned non-linear problems.     

A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains

A time-accurate, finite volume method for solving the three-dimensional, incompressible Navier-Stokes equations on a composite grid with arbitrary subgrid overlapping is presented. The governing equations are written in a non-orthogonal curvilinear co-ordinate system and are discretized on a non-staggered grid. A semi-implicit, fractional step method with approximate factorization is employed for time advancement. Multigrid combined with intergrid iteration is used to solve the pressure Poisson equation. Inter-grid communication is facilitated by an iterative boundary velocity scheme which ensures that the governing equations are well-posed on each subdomain. Mass conservation on each subdomain is preserved by using a mass imbalance correction scheme which is secondorder-accurate. Three test cases are used to demonstrate the method's consistency, accuracy and efficiency.     

Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries

Flow and pressure waves, originating due to the contraction of the heart, propagate along the deformable vessels and reflect due to tapering, branching, and other discontinuities. The size and complexity of the cardiovascular system necessitate a “multiscale” approach, with “upstream” regions of interest (large arteries) coupled to reduced-order models of “downstream” vessels. Previous efforts to couple upstream and downstream domains have included specifying resistance and impedance outflow boundary conditions for the nonlinear one-dimensional wave propagation equations and iterative coupling between three-dimensional and one-dimensional numerical methods. We have developed a new approach to solve the one-dimensional nonlinear equations of blood flow in elastic vessels utilizing a space-time finite element method with GLS-stabilization for the upstream domain, and a boundary term to couple to the downstream domain. The outflow boundary conditions are derived following an approach analogous to the Dirichlet-to-Neumann (DtN) method. In the downstream domain, we solve simplified zero/one-dimensional equations to derive relationships between pressure and flow accommodating periodic and transient phenomena with a consistent formulation for different boundary condition types. In this paper, we also present a new boundary condition that accommodates transient phenomena based on a Green’s function solution of the linear, damped wave equation in the downstream domain.     

A discussion and comparison of numerical techniques used to solve the Navier–Stokes and Euler equations

This paper is intended to provide some background to a number of widely used methods for solving the Navier–Stokes and Euler equations. The difference between coupled and uncoupled iterative schemes is discussed together with methods for solving the equations. Methods covered include time marching (both explicit and implicit), pressure correction and a Newton–Raphson technique. The relationship between the methods is illustrated.     

The effects of fluctuating body forces on annular liquid jets

Summary The nonlinear dynamics of axisymmetric, inviscid, incompressible, thin, annular liquid jets subjected to fluctuating body forces is studied numerically by means of an adaptive finite difference method which maps the time-dependent, curvilinear geometry of the jet into a unit square. The fluctuating body forces may arise from fluctuations in the gravitational acceleration in inertial frames or from the acceleration of a non-inertial frame of reference which translates parallelly to an inertial one. It is shown that both the pressure coefficient and the axial location at which the annular jet becomes a solid one are periodic functions of time with a period equal to that of the imposed body force fluctuations, and that their magnitude increases as the amplitude of the body force fluctuations is increased. It has also been shown that, for both intermittent, sinusoidal or rectangular excitations, increases in the frequency of the excitation result in the creation of superharmonics, broad, albeit peaked, spectra, and closed phase planes with many loops.The research reported in this paper was supported by Project PB91-0767 from the D.G.I.C.Y.T. of Spain.     

Preconditioned Krylov subspace methods used in solving two‐dimensional transient two‐phase flows

This paper investigates the performance of preconditioned Krylov subspace methods used in a previously presented two‐fluid model developed for the simulation of separated and intermittent gas–liquid flows. The two‐fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two‐step semi‐implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi‐CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, τ) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite Element Analysis for Flow Around a Rotating Body using Chimera Method

In this paper, the Chimera method with the Schwarz algorithm, which is one of overlapping domain decomposition methods, is applied for a flow around a rotating body. The incompressible Navier–Stokes equations expressed in a non-inertial frame of reference are used for the governing equations. The implicit scheme with accuracy of the second order is used for the temporal discretization. The mixed finite element formulation with the iso-P2 P1/P1 elements for velocity and pressure elements is used for the spatial discretization. For numerical examples, two-dimensional analyses of flow around a circular cylinder and an ellipse cylinder which rotate uniformly in a uniform flow were performed, the validity of the present technique was verified and the characteristics of the flow were considered.     

Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation

This paper presents the derivation of a depth‐integrated wave propagation and runup model from a system of governing equations for two‐layer non‐hydrostatic flows. The governing equations are transformed into an equivalent, depth‐integrated system, which separately describes the flux‐dominated and dispersion‐dominated processes. The depth‐integrated system reproduces the linear dispersion relation within a 5 error for water depth parameter up to kd = 11, while allowing direct implementation of a momentum conservation scheme to model wave breaking and a moving‐waterline technique for runup calculation. A staggered finite‐difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non‐hydrostatic terms in the vertical direction. An semi‐implicit scheme integrates the depth‐integrated flow in time with the non‐hydrostatic pressure determined from a Poisson‐type equation. The model is verified with solitary wave propagation in a channel of uniform depth and validated with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and fringing reefs. Copyright © 2011 John Wiley & Sons, Ltd.     

Heat transfer and two-phase flow characteristics of refrigerants flowing under varied heat flux in a double-pipe evaporator

A mathematical model based on the annular flow pattern is developed to simulate the evaporation of refrigerants flowing under varied heat flux in a double tube evaporator. The finite difference form of governing equations of this present model is derived from the conservation of mass, energy and momentum. The experimental set-up is designed and constructed to provide the experimental data for verifying the simulation results. The test section is a 2.5 m long counterflow double tube heat exchanger with a refrigerant flowing in the inner tube and heating water flowing in the annulus. The inner tube is made from smooth horizontal copper tubing of 9.53 mm outer diameter and 7.1 mm inner diameter. The agreement of the model with the experimental data is satisfactory. The present model can be used to investigate the axial distributions of the temperature, heat transfer coefficient and pressure drop of various refrigerants. Moreover, the evaporation rate or the other relevant parameters that is difficult to measure in the experiment are predicted and presented here. The results from the present mathematical model show that the saturation pressure and temperature of refrigerant decrease along the tube due to the tube wall friction and the flow acceleration of refrigerant. The liquid heat transfer coefficient increases with the axial length due to reducing the thickness of the liquid refrigerant film. Due to increase of the liquid heat transfer coefficient, increasing wall heat flux is obtained.Finally, the evaporation rate of refrigerant increases with increasing wall heat flux.     

A numerical study on flow through the spiral casing of a hydraulic turbine

Flow through the spiral casing of a hydraulic turbine was analyzed. Reynolds averaged Navier–Stokes equations were solved using a finite element method. The physical domain was divided into a number of hexahedral elements which are isoparametrically mapped onto standard cubic elements. Numerical integration for the unsteady momentum equation is performed over such hexahedral elements to obtain a provisional velocity field. Compliance with the mass conservation equation and determination of the pressure correction are accomplished through an iterative procedure. The velocity distribution inside the spiral casing corroborates the results available in literature. The static pressure at the midplane generally decreases from the outside wall towards the exit of the spiral casing. © 1998 John Wiley & Sons, Ltd.