Modelling the time dependent flow over riblets in the viscous wall region

The flow over riblets is examined computationally using a time dependent model of the viscous wall region. This 2 1/2 D model, developed by Hatziavramidis and Hanratty (1979) and modified by Nikolaides (1984) and Chapman and Kuhn (1981, 1986) assumes homogeneity in the streamwise direction so that the flow is solved only in the cross-sectional plane. The flow at the upper boundary of the computational domain (y ⁺ 40) is described using a streamwise eddy model consisting of two scales, one of the streak spacing (⁺ 100), which dominates vertical momentum transport, and a larger scale that accounts for the influence of large outer flow eddies.The protrusion height concept (Bechert and Bartenwerfer, 1989) is used to define ay ⁺=0 location for surfaces with riblets. A control volume finite element method utilizing triangular meshes is used to exactly fit the riblet cross-sectional geometry. Results obtained using fairly large riblets compare well with the limited experimental evidence available. Observations of the transient flow suggest that the riblets interact with the near-wall streamwise vortices, weakening them by the generation of intermittent secondary vortices within the riblet valleys. The riblets also appear to limit the lateral spread of inrushes towards the wall and retain low momentum fluid in the riblet valleys effectively isolating much of the wall from such inrushes.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

A decoupling numerical method for fluid flow

A first-order non-conforming numerical methodology, Separation method, for fluid flow problems with a 3-point exponential interpolation scheme has been developed. The flow problem is decoupled into multiple one-dimensional subproblems and assembled to form the solutions. A fully staggered grid and a conservational domain centred at the node of interest make the decoupling scheme first-order-accurate. The discretization of each one-dimensional subproblem is based on a 3-point interpolation function and a conservational domain centred at the node of interest. The proposed scheme gives a guaranteed first-order accuracy. It is shown that the traditional upwind (or exponentially weighted upstream) scheme is less than first-order-accurate. The pressure is decoupled from the velocity field using the pressure correction method of SIMPLE. Thomas algorithm (tri-diagonal solver) is used to solve the algebraic equations iteratively. The numerical advantage of the proposed scheme is tested for laminar fluid flows in a torus and in a square-driven cavity. The convergence rates are compared with the traditional schemes for the square-driven cavity problem. Good behaviour of the proposed scheme is ascertained.     

Group theoretic analysis and similarity solution for a nonlinear viscous rod subjected to time dependent velocity impact

Unidirectional wave motion in a nonlinear viscous rod obeying Norton's law in creep, subjected to time dependent velocity impact is considered. From the basic equations of the problem and the four parameter dimensional group of transformations, absolute invariants of the group are constructed to obtain similarity transformations. Similarity representation of the original system of partial differentiation equations is formulated as a system of nonlinear ordinary differential equations with auxiliary conditions. Closed form solutions are obtained for a linear rod, for a nonlinear rod subjected to constant velocity impact and a weekly nonlinear rod. Nonlinear case is solved by a numerical approach based on the quasilinearization method.     

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 numerical method for 3D viscous incompressible flows using non-orthogonal grids

This paper presents a numerical method for fluid flow in complex three-dimensional geometries using a body-fitted co-ordinate system. A new second-order-accurate scheme for the cross-derivative terms is proposed to describe the non-orthogonal components, allowing parts of these terms to be treated implicitly without increasing the number of computational molecules. The physical tangential velocity components resulting from the velocity expansion in the unit tangent vector basis are used as dependent variables in the momentum equations. A coupled equation solver is used in place of the complicated pressure correction equation associated with grid non-orthogonality. The co-ordinate-invariant conservation equations and the physical geometric quantities of control cells are used directly to formulate the numerical scheme, without reference to the co-ordinate derivatives of transformation. Several two- and three-dimensional laminar flows are computed and compared with other numerical, experimental and analytical results to validate the solution method. Good agreement is obtained in all cases.     

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.     

A Petrov-Galerkin method for the numerical solution of the Bradshaw-Ferriss-Atwell turbulence model

The Bradshaw-Ferriss-Atwell model for 2D constant property turbulent boundary layers is shown to be ill-posed with respect to numerical solution. It is shown that a simple modification to the model equations results in a well-posed system which is hyperbolic in nature. For this modified system a numerical algorithm is constructed by discretizing in space using the Petrov-Galerkin technique (of which the standard Galerkin method is a special case) and stepping in the timelike direction with the trapezoidal (Crank-Nicolson) rule. The algorithm is applied to a selection of test problems. It is found that the solutions produced by the standard Galerkin method exhibit oscillations. It is further shown that these oscillations may be eliminated by employing the Petrov-Galerkin method with the free parameters set to simple functions of the eigenvalues of the modified system.     

A discontinuous Galerkin scheme for the numerical solution of flow problems with discontinuities

In this paper, a new discontinuous Galerkin finite element method for the numerical solution of flow problems with discontinuities is presented. The method is based on the limitation in every cell of the difference between the extrema values and the mean value of the numerical solution. The algorithm and technical details for the implementation of the method are presented in one‐and two‐dimensional problems. Numerical experiments for classical test problems are solved on unstructured triangulations to demonstrate the performance of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

A new formulation for fictitious mass of viscous dynamic relaxation method

Abstract

In this article, a new relationship is proposed for the fictitious mass of viscous dynamic relaxation (DR) method. First, incremental equations are derived for DR steps. Using transformed Gershgörin theory, a new relationship is achieved for fictitious mass of viscous DR by formulating modified time step ratio. This procedure presents a new algorithm for the viscous DR method. To evaluate the numerical efficiency of the proposed method, some 2D and 3D truss and frame structures are analyzed with elastic linear and geometrically nonlinear behaviors. Results show that by using the proposed algorithm for fictitious mass, the convergence rate of the viscous DR method is improved so that the proposed algorithm presents the structural response with lower iterations in comparison with other common DR techniques.

Communicated by Joerg Fehr.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

粘性流基于特征线的四阶Runge-Kutta有限元法

对于二维不可压缩粘性流,通过沿流线方向的坐标变换,推导了无对流项的二维N-S（Navier-Stokes）方程。采用四阶Runge-Kutta法对N-S方程进行时间离散,并沿流线进行Taylor展开,得到显式的时间离散格式,然后利用Galerkin法对其进行空间离散,得到了高精度的有限元算法。利用本文算法对方腔驱动流和圆柱绕流进行了数值计算,通过对时间步长、网格尺寸和流场区域的计算分析,进一步验证了本文算法相比经典CBS法在时间步长、收敛性、耗散性和计算精度方面更具有优势。     

A net inflow method for incompressible viscous flow with moving free surface

A new finite element procedure called the net inflow method has been developed to simulate time-dependent incompressible viscous flow including moving free surfaces and inertial effects. As a fixed mesh approach with triangular element, the net inflow method can be used to analyse the free surface flow in both regular and irregular domains. Most of the empty elements are excluded from the computational domain, which is adjusted successively to cover the entire region occupied by the liquid. The volume of liquid in a control volume is updated by integrating the net inflow of liquid during each iteration. No additional kinetic equation or material marker needs to be considered. The pressure on the free surface and in the liquid region can be solved explicitly with the continuity equation or implicitly by using the penalty function method. The radial planar free surface flow near a 2D point source and the dam-breaking problem on either a dry bed or a still liquid have been analysed and presented in this paper. The predictions agree very well with available analytical solutions, experimental measurements and/or other numerical results.     

Three-dimensional coupled numerical model of creeping flow of viscous fluid

A three-dimensional coupled numerical model is developed to describe creeping flow in a computational domain that consists of a thick viscous layer overlaid with a thin multilayered viscous sheet. The density of the sheet is assumed to be lower than that of the layer. The model couples the Stokes equations describing the flow in the layer and the Reynolds equations describing the flow in the sheet. We investigate the long-time behavior of the flow in the sheet by using an asymptotic method and derive an ordinary differential equation for the sheet boundary displacements and the velocities at the interface between the sheet and the layer. The Stokes and Reynolds equations are coupled by applying the resulting equation as an internal boundary condition. Numerical implementation is based on a modified finite element method combined with the projection gradient method. The computational domain is discretized into rectangular hexahedra. Piecewise square basis functions are used. The model proposed enables different-type hydrodynamic equations to be coupled without any iterative improvements. As a result, the computational costs are reduced significantly in comparison with available coupled models. Numerical experiments confirm that the three-dimensional coupled model developed is of good accuracy.     

A new method for measuring time constants of a thermocouple wire in varying flow states

A new measuring method is suggested for determining the time constant of a thermocouple wire to be applied for the measurement of the true fluid temperatures in varying flow states. Based on the techniques of internal heating which are commonly used to measure mean time constants, we extend the existing method to measure instantaneous time constants continuously. A method of measurement and analysis is presented and verified experimentally.List of Symbols A _s surface area [m²] - c specific heat [J/kg K] - D diameter [m] - h heat transfer coefficient [W/m² K] - I current [A] - k thermal conductivity [W/m K] - L length [m] - r resistance per unit length [/m] - T temperature [°C] - t time [s] - t _c characteristic time to reach uniform state [s] - u velocity of stream [m/s] - V volume [m³] - x axial coordinate [m] - thermal diffusivity [m²/s] - normalized temperature (TT )/(T _RT )) - density [kg/m³] - time constant [s] - angular velocity [rad/s] - a amplitude - i initial condition - j junction of thermocouple - R reference point - surrounding The work was supported by Turbo and Power Machinery Research Center at Seoul National University and the authors are grateful to Mr. M. H. Yang for his assistance in the experiment.     

A combined experimental and numerical method for the solution of generalized elasticity problems

This work presents the method for the investigation of three-dimensionally stressed bodies with arbitrary shape which are under the action of an outside system of arbitrary forces. The combined method is based on syntheses of photoelastic experimental methods (other experimental methods may also be used) and digital methods of discrete analysis. Experimental procedures are used for defining superfluous boundary conditions. The boundary-value problem with such boundary conditions is solved by numerical methods. This approach qualitatively changes the very essence of experimental methods and essentially widens their range. It reduces the amount of measurements required and, at the same time, allows one to obtain complete stress fields throughout a body in a short time. In comparison with numerical methods, the combined method increases the accuracy of problem solutions and, at the same time, reduces the time required for complete investigations.