Numerical solution of laminar recirculating flow between shrouded rotating disks

In this paper we develop a method to obtain the numerical solution of the problem of recirculating flow between shrouded rotating disks. The major difficulty of this type of problem is in obtaining convergence at high Reynolds numbers. With the technique developed in this paper we have obtained convergence for Reynolds numbers up to 10,000. The procedure can be extended to higher Reynolds numbers if desired. The contours of the stream function, vorticity function and angular velocity are presented for Reynolds numbers of 500, 2000, 5000 and 10000. The method is applicable to any problem which has similar equations of motion.     

Calculation of flow between two rotating coaxial disks by differentiation with respect to an actual parameter

A method which allows to find the dependence of the solution $\text{x(ξ, s)}$ on the parameter $\text{s}$ for a nonlinear boundary value problem will be presented. The calculation of the dependence of $\text{x(ξ, s)}$ on $\text{s}$ is performed in a non-iterative way. The technique is employed to solve a difficult two-point boundary value problem: steady state flow of an incompressible viscous fluid between two rotating coaxial disks.     

Efficient numerical approximation of compressible multi-material flow for unstructured meshes

We consider the numerical approximation of multi-dimensional multi-material flows. This is a difficult topic related to the numerical smearing of contact discontinuities (or material interfaces or slip lines). Any Eulerian scheme will produce an artificial mixing zone. In this artificial mixture, the computation of thermodynamical variables (pressure, sound speed, temperature, …) is difficult to achieve correctly. For the stiff cases considered in this paper, with solid-liquid-gas interfaces, small errors on the thermodynamical variables lead to the blow up of the computation in many cases. In this paper, we review and explain this problem, then we propose solutions for 1D and 2D flows. Contrarily to front-tracking techniques, our schemes use the same formulation everywhere on the mesh. We provide several examples with the stiffened gas equation of state (EOS): inert materials (water-air), and chemically reactive materials (solid explosive-Plexiglass-air).     

A hybrid numerical method and its application to inviscid compressible flow problems

An improved version of the artificially upstream flux vector scheme, is developed to efficiently compute inviscid compressible flow problems. This numerical scheme, named AUFSR (Tchuen et al. 2011), is obtained by hybridizing the AUFS scheme with Roe’s solver. This approach handles difficulties encountered by the AUFS scheme, in the case where the flux vector does not check the homogeneous property. The present scheme for multi-dimensional flows introduces a certain amount of numerical dissipation to shear waves, as Roe’s splitting. The AUFSR scheme is not only robust for shock-capturing, but also accurate for resolving shear layers. Numerical results for 1D Riemann problems and several 2D problems are investigated to show the capability of the method to accurately compute inviscid compressible flow when compared to AUFS, and Roe solvers.     

A revised model to study the MHD nanofluid flow and heat transfer due to rotating disk: numerical solutions

Here our main interest is to present numerical simulations for magneto-nanofluid flow and heat transfer near a rotating disk. Buongiorno model, featuring the novel aspects of Brownian motion and thermophoresis, is accounted. Heat dissipation effect is preserved in the energy balance equation. We take into account more realistic wall condition which requires passive control of nanoparticle concentration at the disk. The traditional Von Karman relations have been invoked to attain self-similar differential system. Keller–Box method has been implemented to compute similarity solutions of the problem. Streamlines are prepared in both two and three dimensions for adequate flow visualization. The behavior of involved parameters on the flow fields is examined graphically. It is predicted that the torque required to maintain disk in steady rotation increases when magnetic field effects are enhanced. Fluid flow in the radial, azimuthal and vertical directions is opposed by the magnetic field strength. Thermophoresis effect enhances temperature and reduces heat flux from the disk. However, Brownian diffusion has a marginal influence on temperature distribution. Heat transfer coefficient is reduced due to the inclusion of heat dissipation terms. Present results are consistent with those of the available studies in a limiting situation.

     

A flux splitting method for the numerical simulation of one-dimensional compressible flow

Using the technique of flux vector splitting, it is shown that one-dimensional, inviscid, compressible-flow equations possess a split conservation form. Some attractive features of this form for the design of finite-difference solution schemes are discussed. Based on the split form, two solution chemes are designed. One is a first-order accurate ‘upwind’ scheme and the other is similar to the Lax-Wendroff scheme. A hybrid scheme, based on a nonlinear weighting of these two schemes, is demonstrated to yield results superior to either of the two in the solution of an ideal shock-tube problem.     

A numerical investigation of fluid flow over a rotating cylinder with cross flow oscillation

This paper studies a two-dimensional incompressible viscous flow past a rotating cylinder with cross flow oscillation using a finite element method based on the characteristic based split (CBS) algorithm to solve governing equations including full Navier–Stokes and continuity equations. Dynamic unstructured triangular grid is used employing lineal and torsional spring analogy which is coupled with the solver by an Arbitrary Lagrangian–Eulerian (ALE) formulation. After verifying the accuracy of the numerical code, simulations are conducted for the flow past a rotating cylinder with cross flow oscillation at moderate Reynolds numbers of 50, 100, and 200 considering different non-dimensional rotational speeds based on the free-stream velocity in the range 0–2.5, and various oscillating amplitudes and frequencies. Effects of the oscillation and rotation of the cylinder on the vortex shedding both in lock-on and non-lock-on regions, the mean drag and lift coefficients, and the Strouhal number are investigated in detail. It is found that similar to the fixed cylinder beyond a critical non-dimensional rotational speed the vortex shedding is highly suppressed. In addition, by increasing the rotational speed of the cylinder, the lift coefficient increases while decreasing the drag coefficient. However, in the vortex lock-on region both the lift and the drag coefficients increase significantly.     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

On the flow between two rotating shrouded discs

Numerical solutions to the full Navier Stokes equations for the flow enclosed between two rotating discs and a sidewall are presented. The sidewall remains fixed throughout, while we consider examples in which: (i) the bottom disc is rotating and the top disc is fixed; (ii) the two discs rotate in the same sense; (iii) the two discs rotate in opposite senses. We present solutions up to a Reynolds number of 1200, using second order central finite differencing.     

Mechanical vibration and critical speeds of rotating composite laminate disks

Since fiber-reinforced composite materials have high-specific strength and stiffness values, their application to rotating disks can enhance machinery performance by increasing the dynamic stability and reducing driving energy. There have been few works on the vibration characteristics of rotating multidirectional laminated disks made of fiber-reinforced composite materials. Most of the previous studies have been confined to single lamina disks. When a disk rotates, the centrifugal force causes the in-plane loads to affect the vibration characteristics of the rotating disks. In this paper, the exact expressions for the in-plane loads acting on rotating cross-ply laminate disks are presented. The vibration equation of rotating cross-ply laminate disks was solved by using Galerkin’s method. Using numerical examples, the natural frequencies and critical speeds of the rotating disks are discussed for various cross-ply ratios.     

On the numerical analysis of compressible flow problems by the “modified FLIC method”

This paper presents a new computational technique for transonic flow problem analysis. This method, named Modified FLIC Method, is based on a time-marching technique of FLIC (fluid in cell) method and employs triangular elements conventionally used in finite element method. This technique can be applied to transonic flows with any complicated boundary shapes. Three problems were solved in this paper, the first was a supersonic flow around a circular cylinder, the second was a transonic flow between tubrine blade cascades and the last was an unsteady flow in a duct with a junction. The calculated results showed a good agreement with the experimental data.     

On the rotating beam: Some theoretical and numerical results

A bifurcation analysis for a thin beam rotating about an axis is carried out by the Golubitsky-Schaeffer theory of singularities. The rotation velocity is the bifurcation parameter, while the constants describing the position of the beam are considered as small perturbation parameters. A finite difference approximation of the problem is also introduced, and an error analysis is given.     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

Aerodynamic effect on natural frequency and flutter instability in rotating optical disks

 Experimental studies on the aerodynamic coupling effect on natural frequencies and flutter instability of rotating disks are investigated in this paper. The experiments performed using a vacuum chamber and optical disks give two main results. One is that the aerodynamic effect by surrounding air reduces the natural frequencies and critical speeds of the vibration modes in pre-flutter regions. The other is that the natural frequency of the disk rotating at ambient atmospheric pressure is equal to that in vacuum at the flutter onset speed where the disk experiences aero-induced flutter. In post-flutter regions, the aerodynamic coupling between the disk and surrounding air increases the natural frequencies of the disk. Received: 17 June 2002/Accepted: 7 October 2002 The work was supported by Grant No. R11-1997-042-090001-0 of the Center for Information Storage Devices designated by the Korea Science & Engineering Foundation. Paper presented at the 13th Annual Symposium on Information Storage and Processing Systems, Santa Clara, CA, USA, 17–18 June, 2002     

Local heat transfer characteristics in a single rotating disk and co-rotating disks

The present study investigates convective heat/mass transfer and flow characteristics inside rotating disks. The rotating disks are simulated on the commonly used 3.5 hard disk drives (HDD). The experiments are conducted for the various hub heights of 5, 10 and 15 mm in a single rotating disk and 4, 6 and 8 mm in co-rotating disks and for the various rotating Reynolds numbers of 5.53 × 10⁴, 8.53 × 10⁴ and 1.13 × 10⁵. To accommodate the general operating conditions of HDD, the experiments are also conducted with an obstruction of rectangular cross-section in the space, which simulates a read-write head arm. A naphthalene sublimation technique is employed to determine the detailed local heat transfer coefficients on the rotating disks using the heat and mass transfer analogy. Flow field measurements are conducted using laser Doppler anemometry (LDA) and numerical calculations are performed simultaneously to analyze the flow patterns induced by disk rotation. The results of a single rotating disk show that the heat transfer on the rotating disk is enhanced considerably according to the reduction of the hub height and the increase of the rotating Reynolds number. The head arm inserted in the cavity between the rotating disk and the cover enhances uniformity of the heat/mass transfer on the disk due to the deficit of the momentum in the average flow despite the enhancement of the tangential component of fluctuation velocity. The heat/mass transfer rates on the co-rotating disks have very low values near the hub in the inner region of the solid-body rotation and increase rapidly toward the outer region. The change of heat/mass transfer for various hub heights is negligible.The authors wish to acknowledge support for this study by the Ministry of Science and Technology through their National Research Laboratory program and by the KOSEF through the Center of Information Storage Device.     

Gas-kinetic numerical study of complex flow problems covering various flow regimes

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the Boltzmann–Shakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation.     

A numerical study of flow past a cylinder with cross flow and inline oscillation

Two-dimensional fluid flow around an oscillating circular cylinder is studied numerically at different values of oscillation frequency and amplitude. A novel finite element method which uses discretization along the characteristic line is used for simulation. The solver is coupled to a mesh movement scheme using the Arbitrary Lagrangian-Eulerian (ALE) formulation to account for body motion in the flow field. Two cases of cylinder motion have been studied, cross flow and inline oscillation. In both cases, occurrence of lock on is investigated and the bounds of the lock on region are determined. A comparison of the numerical results with the experimental data indicates that 2D simulation is valid up to Re = 300. Beyond that, 3D effects appear. By using flow visualization, effect of a cylinder oscillation on the flow field and wake pattern has been studied. Also, variation of the mean drag coefficient against the oscillation parameters is discussed. The numerical results are in good agreement with the experimental data available in the literature.     

Differential evolution in constrained numerical optimization: An empirical study

Motivated by the recent success of diverse approaches based on differential evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important information about the behavior of DE in constrained search spaces and (2) the role of this knowledge in the correct combination of variants, based on their capabilities, to generate simple but competitive approaches.     

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ? multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.     

An improved numerical scheme for characterizing dynamic behavior of high-speed rotating elastic beam structures

The demand for designing high-speed turbomachinery has led to intensive research in dynamic modeling of rotating elastic mechanisms in recent years. Such a demand in design setting can be addressed more effectively with the development of a more efficient computational scheme. In this paper we present an improved numerical method with three new features. First of all, the time separation concept is introduced to allow time independent terms being computed separately and assembled with time dependent terms in each time marching cycle to form global system equations. Second, the Timoshenko beam with nonlinear geometric stiffness is modeled with exact tangent matrix as opposed to conventional pseudo-tangent matrix approximation. Third, the computational scheme is implemented in homogeneous coordinates that provide a more natural and efficient vector representation. Kane's classic rotating beam problem is used to test for accuracy and computer time. The result matches very well with Kane's solution. However, the computer time needed for the present approach is reduced by more than 70%.