应用数学和力学

2012 Vol. 33, No. 6

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Applications of PSE to Predict the Transition Position in Boundary Layers

LI Jia, LUO Ji-sheng

2012, 33(6): 643-650. doi: 10.3879/j.issn.1000-0887.2012.06.001

Abstract(1676) PDF(680)

Abstract:
The phenomenon of laminarturbulent transition exists universally in nature and various engineering practices. The prediction of transition position was one of crucial theories and practical problems in fluid mechanics due to different natures of laminar flow and turbulent flow. Two types of disturbances imposed at the entrance were identical amplitude and wavepacket disturbances along the spanwise in incompressible boundary layers. The disturbances of identical amplitude consist of a two-dimensional (2-D) wave and two three-dimensional (3-D) waves. The parabolized stability equation (PSE) was used to research the evolution of disturbances and predict the transition position, and the results were compared with those obtained by numerical simulation. It’s revealed that the PSE method could investigate the evolution of disturbances and predict the transition position. At the same time, the speed of calculation was much faster than numerical simulation.

Iterative and Adjusting Method for Computing Stream Function and Velocity Potential in Limited Domains and Its Convergence Analysis

LI Ai-bing, ZHANG Li-feng, ZANG Zeng-liang, ZHANG Yun

2012, 33(6): 651-662. doi: 10.3879/j.issn.1000-0887.2012.06.002

Abstract(1530) PDF(946)

Abstract:
Stream function and velocity potential can be easily computed by solving Poisson equations in a unique way for the global domain. Because of various assumptions for handling boundary conditions, the solution is not unique when a limited domain is concerned. So, it is very important to reduce or eliminate the effects caused by uncertain boundary condition. An iterative and adjusting method based on the Endlich iteration method was presented to compute stream function and velocity potential for limited domains. This method did not need an explicitly specifying the boundary condition, while it could obtain the effective solution and was proved to be successful in decomposing and reconstructing the horizontal wind field with very small errors. The convergence of the method depended on relative value between the distances of grid in two different directions and was related to the value of the adjusting factor. Applying the method in Arakawa grids and irregular domains, the results showed that it could not only obtain accurate vorticity and divergence, but also accurately decompose and reconstruct the original wind field. Hence, the iterative and adjusting method was accurate and reliable.

Numerical Simulation of Trajectory and Deformation of a Bubble in a Tip Vortex

NI Bao-yu, ZHANG A-man, YAO Xiong-liang, WANG Bin

2012, 33(6): 663-677. doi: 10.3879/j.issn.1000-0887.2012.06.003

Abstract(1407) PDF(872)

Abstract:
According to the behavior of a bubble in the ship wake flow, the numerical simulation was divided into two stages: quasi-spherical motion and non-spherical motion, based on whether the bubble was captured by the vortex or not. Oneway coupled particle tracking method (PTM) and boundary element method (BEM) were adopted to simulate these two stages respectively. Meanwhile, the initial condition of the second stage was taken as the output of the first one and the whole simulation was connected and completed therefore. Based on the numerical results and experimental data published, the cavitation inception was studied and the wake bubble was tracked. Besides, the split of the bubble captured by the vortex and the following sub-bubbles were simulated, including the motion, deformation and collapse as well. It is aimed that the results would provide some insight into the control on wake bubbles and optimization of the wake flow.

Flow on Oscillating Rectangular Duct for Maxwell Fluid

M.Nazar, Fatima Shahid, M.Saeed Akram, Q.Sultan

2012, 33(6): 678-691. doi: 10.3879/j.issn.1000-0887.2012.06.004

Abstract(1197) PDF(1086)

Abstract:
An analysis for the unsteady flow of an incompressible Maxwell fluid in an oscillating rectangular cross section was presented. Using the Fourier and Laplace transforms as mathematical tool, the solutions were presented as sum of steady-state and transient solutions. For large times, when the transients disappear, the solution was represented by the steady-state solution. Solutions for Newtonian fluids appear as limiting cases of the solutions obtained here. In the absence of frequency of oscillation, the problem for flow of Maxwell fluid in a duct of rectangular cross-section moving parallel to its length was obtained. Finally, the required time to reach the steady-state for sine oscillations of the rectangular duct is obtained by graphical illustrations for different parameters. Moreover, the graphs are sketched for velocity for the variations of x and y.

Computational Study of the Combined Effects of Conduction-Radiation and Hydromagnetic on Natural Convection Flow Past a Magnetized Permeable Plate

Muhammad Ashraf, S.Asghar, M.A.Hossain

2012, 33(6): 692-709. doi: 10.3879/j.issn.1000-0887.2012.06.005

Abstract(1321) PDF(639)

Abstract:
The computational study of the combined effects of radiation and hydromagnetics on natural convection flow of, viscous, incompressible, electrically conducting fluid past a magnetized permeable vertical plate was presented. The governing non similar equations were solved by using (i) finite difference method for entire value of suction parameter ξ and asymptotic solution for small and large value of ξ numerically. The effects of varying the Prandtl number Pr,magnetic Prandtl number P_m, magnetic force parameter S, radiation parameter R_d ,and surface temperature θ_w on coefficients of skin friction, rate of heat transfer and current density were shown in graphical form and as well as in tables. Finally, an attempt was made to examine the effects of above mentioned physical parameters on velocity profile, temperature distribution and transverse component of magnetic field.

MHD Axisymmetric Flow of a Third-Grade Fluid Between Porous Disks With Heat Transfer

T.Hayata, Anum Shafiq, M.Nawaz, A.Alsaedi

2012, 33(6): 710-725. doi: 10.3879/j.issn.1000-0887.2012.06.006

Abstract(1444) PDF(717)

Abstract:
The magnetohydrodynamic (MHD) flow of third-grade fluid between two permeable disks with heat transfer was investigated. The governing partial differential equations were converted into the ordinary differential equations by using suitable transformations. Transformed equations were solved by using homotopy analysis method (HAM). The expressions for square residual errors were defined and optimal values of convergencecontrol parameters were selected. The dimensionless velocity and temperature fields were examined for various dimensionless parameters. Skin friction coefficient and Nusselt number were tabulated to analyze the effects of dimensionless parameters.

Thermophoresis and Brownian Motion Effects on Boundary-Layer Flow of a Nanofluid in the Presence of Thermal Stratification Due to Solar Energy

N.Anbuchezhian, K.Srinivasan, K.Chandrasekaran, R.Kandasamy

2012, 33(6): 726-739. doi: 10.3879/j.issn.1000-0887.2012.06.007

Abstract(1567) PDF(792)

Abstract:
The problem of laminar fluid flow resulted from the stretching of a vertical surface with variable stream conditions in a nanofluid due to solar energ was investigated numerically. The model used for the nanofluid incorporated the effects of Brownian motion and thermophoresis in the presence of thermal stratification. The symmetry groups admitted by the corresponding boundary value problem were obtained by using a special form of Lie group transformations viz. scaling group of transformations. An exact solution was obtained for translation symmetry and numerical solutions for scaling symmetry. This solution depended on a Lewis number, Brownian motion parameter, thermal stratification parameter and thermophoretic parameter. The conclusion was drawn that the flow field and temperature and nanoparticle volume fraction profiles were significantly influenced by these parameters. Nanofluids were shown to increase the thermal conductivity and convective heat transfer performance of the base liquids. Nanoparticles in the base fluid also offered the potential of improving the radiative properties of the liquids, leading to an increase in the efficiency of direct absorption solar collectors.

Engineering Measures for Preventing Upheaval Buckling of Buried Submarine Pipelines

LIU Run, WANG Wu-gang, YAN Shu-wang, WU Xin-li<

2012, 33(6): 740-753. doi: 10.3879/j.issn.1000-0887.2012.06.008

Abstract(1546) PDF(1039)

Abstract:
Inservice hydrocarbons must be transported at high temperature and pressure to ease the flow and prevented solidification of the wax fraction. High temperature and pressure induced addition stress in the pipeline, and its accumulation will cause upheaval buckling of the pipeline. If such expansion is resisted, for example by frictional affects of the foundation soil over a kilometer or so of pipeline, compressive axial stress will be set up in the pipe-wall. When the value exceeds the constraint of foundation soil on the pipeline, sudden deformation will occur to release internal stress, which is similar to the sudden deformation of strut due to stability problems. The upheaval buckling may jeopardize the structural integrity of the pipeline. Therefore, the effective engineering measure against this phenomenon plays an important role in submarine pipeline design. In terms of the pipeline installation and protection measures commonly used in Bohai Gulf, three engineering measures was investigated in great details. The analytical method was introduced and developed to consider the protection effect of anti-upheaval buckling of the pipeline. The analyzing results showed that the amplitude of initial imperfection had a great effect on the pipeline thermal upheaval buckling. Both trenching and burial and discrete dumping were effective on preventing the pipeline from buckling. The initial imperfection and operation conditions of the pipelines determined the covered depth and the number of layers of protection measures.

Reflection and Refraction of Attenuated Waves at the Boundary of an Elastic Solid With a Porous Solid Saturated With Two Immiscible Viscous Fluids

M.Kumar, R.Saini

2012, 33(6): 754-770. doi: 10.3879/j.issn.1000-0887.2012.06.009

Abstract(1425) PDF(714)

Abstract:
The propagation of elastic waves was studied in a porous solid saturated with two immiscible viscous fluids. The propagation of three longitudinal waves was represented through three scalar potential functions and a vector potential function represents the lone transverse wave. Displacements of particles in different phases of the aggregate were defined in terms of these potentials functions. It was shown that there could exist three longitudinal waves and one transverse wave. The phenomenon of reflection and refraction due to longitudinal and transverse wave at a plane interface between an elastic solid half-space and a porous solid halfspace saturated with two immiscible viscous fluids were investigated. For the presence of viscosity in porefluids, the waves refracted to porous medium attenuated in the direction normal to the interface. The ratios of amplitudes of reflected and refracted waves with that of the incident wave were calculated as a non-singular system of linear algebraic equations. These amplitudes ratios were used further to calculate the shares of different scattered waves in the energy of incident wave. Variations of modulus of amplitude and energy ratios with the angle of incidence were computed for particular numerical model. For postcritical incidence of SV wave, the reflected P wave became evanescent. The conservation of energy across the interface was verified. The effects of variations in non-wet saturation of pores and frequency on the energy partition were depicted graphically and discussed.

Asymptotic Analysis on Weakly Forced Vibration of an Axially Moving Viscoelastic Beam Constituted by Standard Linear Solid Model

WANG Bo

2012, 33(6): 771-780. doi: 10.3879/j.issn.1000-0887.2012.06.010

Abstract(1453) PDF(862)

Abstract:
The weakly forced vibration of an axially moving viscoelastic beam was investigated. The viscoelastic material of beams was constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration were derived from dynamical, constitutive, and geometrical relations. The method of multiple scales was applied to determine the steady-state response. The modulation equation was derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response were derived from the modulation equation. The stability of nontrivial steadystate response was examined via RouthHurwitz criterion.