Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

A low‐dimensional spectral approach for transient axisymmetric free‐surface flow inside thin cavities of arbitrary shape

A low‐dimensional spectral method is used to solve the transient axisymmetric free surface flow inside thin cavities of arbitrary shape. The flow field is obtained on the basis of the lubrication equations, which are expanded in terms of orthonormal functions over the cavity gap. The formulation accounts for nonlinearities stemming from inertia and front location. The work is of close relevance to the filling stage during die casting, and injection molding, or the flow inside annular (extrusion) dies. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and wall geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow, when a temporal behavior of the ‘boundary‐layer’ type develops. The multiple‐scale method is applied to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple‐scale approach, including the moderately small Re range. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity field by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

A spectral approach to inertial confined thin‐film flow

A spectral approach is proposed to determine the flow field of a thin film inside narrow channels of arbitrary shape. Although the method is easily extended to transient flow, only steady flow is considered here. The flow field is represented spectrally in the depthwise direction in terms of orthonormal shape functions, which together with the Galerkin projection lead to a system of ordinary differential equations that can be solved using standard methods. The method is particularly effective for nonlinear flow, including nonlinearities of geometrical or material origins. The validity of the proposed method is demonstrated for a flow with inertia, and, unlike the depth‐averaging method, is not limited to a flow at small Reynolds number. The problem is closely related to high‐speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing it with the fully two‐dimensional finite‐element solution, and the widely used depth‐averaging method from shallow‐water theory. It is found that a low number of modes are usually sufficient to secure convergence and accuracy. The influence of inertia is examined on the velocity and pressure fields. The pressure distributions reflect excellent agreement between the low‐order spectral method and the finite‐element solution, even at moderately high Reynolds number. The depth‐averaging solution is unable to predict accurately (qualitatively and quantitatively) the high‐inertia flow. Comparison of the velocity field reflects the expected discrepancy in a boundary layer formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of axisymmetric cavity flows using the boundary element method

This paper presents a formulation of the boundary element method (BEM) for solution of axisymmetric cavity flow problems. The governing equation is written in terms of Stokes' stream function, requiring a new fundamental solution to be found. The iterative procedure for adjusting the free-surface position is similar to that used for planar cavity flows. Numerical results are compared with finite difference and finite element solutions, showing the robustness of the BEM model.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A test problem for outflow boundary conditions—flow over a backward-facing step

A numerical solution for steady incompressible flow over a two-dimensional backward-facing step is developed using a Galerkin-based finite element method. The Reynolds number for the simulations is 800. Computations are performed on an extended channel length to minimize the effect of the outflow boundary on the upstream recirculation zones. A thorough mesh refinement study is performed to validate the results. Extensive profile data at several channel locations are provided to allow future testing and evaluation of outflow boundary conditions.     

The DRBEM solution of incompressible MHD flow equations

This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

谱元法和高阶时间分裂法求解方腔顶盖驱动流

详细推导了谱元方法的具体计算公式和时间分裂法的具体计算过程 ;对一般的时间分裂法进行了改进 ,即对非线性步分别用 3阶 Adams-Bashforth方法和 4阶显式 Runge-Kutta法 ,粘性步采用 3阶隐式 Adams-Moulton形式 ,提高了时间方向的离散精度 ,同时还改进了压力边界条件 ,采用 3阶的压力边界条件 ;利用改进的时间分裂方法分解不可压缩 Navier-Stokes方程 ,并结合谱元法计算了移动顶盖方腔驱动流 ,提高了方法可以计算的 Re数 ,缩短了达到收敛的时间 ,并将结果与基准解进行比较 ;分析了移动顶盖方腔驱动流中 Re数对流场分布的影响。     

Computational technique for flow in blood vessels with porous effects

IntroductionThehemodynamicsofflowsthroughbloodvesselsisofgreatinterest,becausethesevesselspresentasubstantialhealthriskandareamajorcauseofmortalityandmorbidityintheindustrializedworld .Researchpapersonthebloodflowhaveappearedbutmostofthemhaveneglectedtheporosityeffectsduetovesselwalls.Inthisstudyweareinterestedintheflowthroughabloodvesseltakingintoaccounttheporosityeffectsofthevessels.Fluidflowthroughaporousmediumisoffundamentalimportancetowiderangeofdisciplinesinthevariousbranchesofnaturalsci…     

Finite element Galerkin Approach for a Computational Study of Arterial Flow

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Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

The final approach to steady state in a nonsteady axisymmetric stagnation point heat transfer

An analysis is presented for the transient thermal response of a laminar boundary layer in the vicinity of an axisymmetric stagnation flow on an infinite circular cylinder. The final approach to steady state temperature field is shown to have exponential decay with time. The characteristic factors appearing in the exponents result in the solution of an eigenvalue problem in ordinary linear differential equations. Numerical results are presented for a range of values of the Reynolds number and Prandtl number.     

VIBRO-ACOUSTIC BEHAVIOR OF SUBMERGED CYLINDRICAL SHELLS: ANALYTICAL FORMULATION AND NUMERICAL MODEL

We propose both an analytical formulation and a numerical model to study the hydroelastic or vibroacoustic behaviour of cylindrical thin shells immersed in an unbounded, inviscid and heavy fluid. The analytical solution allows us to calculate the dynamic response and the pressure radiated in the far field by a baffled cylinder. This formulation uses the truncated modal basis of the dry structure to expand the displacements of the submerged shell. The analytical model is used as a reference in order to validate a numerical model which couples the finite element method (FEM) to the boundary element method (BEM). As opposed to the analytical formulation which is dedicated to baffled circular cylinders only, the numerical model allows us to treat any axisymmetric shell, such as cylindrical and spherical shells, or more complex shells of revolution. The structure is idealized by two-node ring finite elements and the boundary equation is solved using the method of singularities.     

A computationally efficient solution of the wave equation for the transient response of infinite reservoirs

In this paper, the transient response of an infinite reservoir is analyzed using the dual-reciprocity boundary element method. A vertical and an inclined-face rigid dam are analyzed under a transient loading. Sharan-type boundary-condition transmission is implemented in the formulation. The results are compared with the exact solution and those obtained by using the finite element method. It is seen that the application of the dual-reciprocity boundary element method is simpler and the results are in very good agreement with the exact solution and those obtained by using the finite element method.     

A short note on the general boundary element method for viscous flows with high Reynolds number

In this paper, the general boundary element method and the parallel computation are employed to solve laminar viscous flows in a driven square cavity, governed by the exact Navier–Stokes equations. Using the solution at Re=0 as the initial approximation, the convergent numerical results for high Reynolds number at Re=7500 are obtained, for the first time, by the boundary element method. This verifies the validity and great potential of the general boundary element method for highly non‐linear problems, which may greatly enlarge application regions of the boundary element method in science and engineering. Copyright © 2003 John Wiley & Sons, Ltd.     

Supersonic nonuniform viscous gas flow past a blunt body with supply of gas from the surface

The problem of axisymmetric nonuniform gas flow past smooth blunt bodies at high Mach numbers is investigated. The approach stream is a parallel axisymmetric flow in which the velocity and temperature depend on the radial distance from the axis of symmetry and the pressure is constant. On the axis of symmetry the velocity has a minimum and the temperature a maximum. A characteristic feature of this flow is the existence of two qualitatively different flow regimes: separated [1-4], when in the shock layer on the front of the body there is a closed region of reverse-circulating flow, and unseparated [5, 6], when there is no such zone. In this study the case of unseparated flow is investigated. The equations of a thin viscous shock layer with generalized Rankine-Hugoniot conditions at the shock and boundary conditions on the body that take into account the supply of gas from the surface are solved numerically. The effect of the gas supply on the conditions of unseparated flow is analyzed in relation to the Reynolds number, and the critical values of the nonuniformity parameter a = a_k [5] are obtained. It is shown that at high Reynolds numbers the supply of gas from the surface has practically no effect on a_k, while at low and intermediate Reynolds numbers it reduces the region of unseparated flow. For high Reynolds numbers and an intense supply of gas from the surface an asymptotic solution of the problem is obtained for the neighborhood of the stagnation point. This is compared with the numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–129, July–August, 1988.The authors wish to thank G. A. Tirskii for useful discussions of the results.