Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid

In problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrix A ₀ representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrix A , the RHS is perturbed by a Taylor expansion of A ^?1 about A ₀. Each term in the resulting series requires one ‘backsolve’ using the original LU . Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy. As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy. Envisioned applications other than the computation of unsteady incompressible flow include: three-dimensional parabolic problems in tubes of varying cross-section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poisson's equation.     

Numerical solution of the Navier-Stokes equations using a finite element method

A finite element algorithm for solving the Navier-Stokes equations is presented for the analysis of high-speed viscous flows. The algorithm uses triangular elements. The unsteady equations are integrated to steady state with a Runge-Kutta time-marching scheme. A postprocessing artificial dissipation term is introduced to stabilize the computations and to dampen dissipation errors. Numerical results are compared with the calculation of uniform flow on a rectangular region which encounters an embedded oblique shock. A shock/turbulent boundary layer problem is also solved and results are compared with experimental data. It is shown that the postprocessing smoothing term and boundary conditions similar to the finite difference method work well in the present numerical studies.     

Numerical investigations of drop solidification on a cold plate in the presence of volume change

We present a front-tracking/finite difference method for simulation of drop solidification on a cold plate. The problem includes temporal evolution of three interfaces, i.e. solid–liquid, solid–gas, and liquid–gas, that are explicitly tracked under the assumption of axisymmetry. Method validation is carried out by comparing computational results with exact solutions for a two-dimensional Stefan problem, and with related experiments. We then use the method to investigate a drop solidifying on a cold plate in which there exists volume change due to density difference between the solid and liquid phases. Numerical results show that the shape of the solidified drop is profoundly different from the initial liquid one due to the effects of volume change and the tri-junction in terms of growth angles ϕ_gr on the solidification process. A decrease in the density ratio of solid to liquid ρ_sl or an increase in the growth angle results in an increase in the height of the solidified drop. The solidification process is also affected by the Stefan number St, the Bond number Bo, the Prandtl number Pr, the Weber number We, the ratios of the thermal properties of the solid to liquid phases k_sl and C_psl. Increasing St, Bo, Pr, We, or k_sl decreases the solidified drop height and the time to complete solidification. Moreover, the solidification growth rate is strongly affected by St, k_sl and C_psl. An increase in any of these parameters hastens the growth rate of the solidification interface. Contrarily, increasing ρ_sl decreases the growth rate. However, other parameters such as ϕ_gr, Bo, Pr and We have minor effects on the solidification growth rate.     

Numerical solution of the oxygen diffusion problem in cylindrically shaped sections of tissue

A mathematical model is presented which describes the diffusion of oxygen in absorbing tissue, and numerical solution of its partial differential equation is obtained by the finite difference equations. The diffusion with absorption model is associated with the process of a moving boundary which marks the furthest penetration of oxygen in the absorbing cylindrically shaped sections of tissue and also allows for an initial distribution of oxygen through the absorbing tissue. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation of polyhedral crystal growth based on a mathematical model arising from non–local thermomechanics

In this work we present some results of the numerical simulation of the growth of a crystal from its melt, taking into account faceting. The simulation is based on a numerical solution of a three–dimensional generalized Stefan problem. That problem arises from a non–local thermomechanical theory applied to a continuous system with an interface and embodies ideas from the dislocation theory of crystal growth. In the model, the crystal surface is an isotherm and the growth velocity of a crystal face depends on the velocities of the other faces and on the whole crystal configuration as well as on the temperature gradient. A front fixing formulation of the model is considered. This is a conservative form of the Isotherm Migration Method [6, 7, 8, 9, 10, 11] in spherical coordinates. The numerical solution is based on an explicit finite difference discretization of the resulting non–linear equations. We develop a theoretical analysis of the interface equations that drive the crystal face motion. Numerical results, showing evolution of complex crystals with configuration changing during the growth, are in accord with experimental results. Furthermore, numerical experiments offer useful information on the influence of certain parameters in the model on the growth process. Received: March 21, 1996     

Numerical simulation on an interaction of a vortex street with an elliptical leading edge using an unstructured grid

An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical analysis of developing turbulent flow in a 180° bend tube by an algebraic Reynolds stress model

A numerical analysis has been performed for three‐dimensional developing turbulent flow in a 180° bend tube with straight inlet and outlet section used by an algebraic Reynolds stress model. To our knowledge, numerical investigations, which show the detailed comparison between calculated results and experimental data including distributions of Reynolds stresses, are few and far between. From this point of view, an algebraic Reynolds stress model in conjunction with boundary‐fitted co‐ordinate system is applied to a 180° bend tube in order to predict the anisotropic turbulent structure precisely. Calculated results are compared with the experimental data including distributions of Reynolds stresses. As a result of this analysis, it has been found that the calculated results show a comparatively good agreement with the experimental data of the time‐averaged velocity and the secondary vectors in both the bent tube and straight outlet sections. For example, the location of the maximum streamwise velocity, which appears near the top or bottom wall in the bent tube, is predicted correctly by the present method. As for the comparison of Reynolds stresses, the present method has been found to simulate many characteristic features of streamwise normal stress and shear stresses in the bent tube qualitatively and has a tendency to under‐predict its value quantitatively. Judging from the comparison between the calculated and the experimental results, the algebraic Reynolds stress model is applicable to the developing turbulent flow in a bent tube that is known as a flow with a strong convective effect. Copyright © 2005 John Wiley & Sons, Ltd.     

Computation of free surface waves around an arbitrary body by a Navier-Stokes solver using the psuedocompressibility technique

A Navier-Stokes equation solver is developed for computing free surface wave and viscous flow around an arbitrary body, in which a free surface model is introduced into the pseudocompressibility solution. The governing equations are classified in a vectorial form, with primitive variables, and a block diagonal system is generated by the discretization of an implicit factorization method. A moving grid system fitted to both the free surface and body surface is generated by an effective cubic spline fitting technique. Two zero-equation turbulence models, namely the Cebeci-Smith model and the Baldwin-Lomax model, are used for turbulent calculations. Numerical simulations are carried out for the free surface viscous flows generated by a submerged hydrofoil and a ship model. Computed results are in reasonable agreement with measurements.     

The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.     

Analysis of an organized turbulent structure using a pattern recognition technique in a drag-reducing surfactant solution flow

This work presents the investigation for an organized turbulent structure in a drag-reducing flow of dilute surfactant solution by utilizing a particle image velocimetry system to perform the pattern recognition technique on a trajectory in four quadrants of streamwise and wall-normal velocity fluctuations. The pattern recognition is added to a new algorithm in order to directly capture the spatial rotation motion. The Reynolds number based on the channel height and bulk mean velocity was set to 1.5 × 10⁴. Surfactant solution with a weight concentration of 150 ppm was employed and the drag reduction rate was 65%. In the drag-reducing flow, we observe increased frequencies of occurrence of the flow events that correspond to a meandering motion in the wall-normal direction of the high-and low-speed regions. Three findings from investigation of the ensemble-averaged Reynolds shear stress and vortex structure are as follows: (i) the Reynolds shear stress in the large fluctuation range occurs in the narrow region; (ii) Size, strength, arrangement and inclination in the spatial vortex structure in the drag-reducing flow differ from those of the water; and (iii) all trajectory contributions for the wall friction coefficient decrease. Finally, we interpreted that the viscoelasticity characterizing the viscoelastic stress and relaxation time in rheological properties of the flow changes specific elementary vortex for the drag-reducing flow, and the trajectories of each flow pattern change drastically.     

Moving‐boundary analysis of net vapor generation point in a steam generator

This paper deals with the study of the dynamics of net vapor generation point in the boiling channel of the steam generator of Kaiga‐1 nuclear power plant. The dynamics has been studied by perturbing liquid velocity at the inlet of boiling channel with a step function and heating rate with a ramp function. Both finite volume method (FVM) and finite difference method (FDM) have been applied to solve the model equations that have been developed to predict boiling boundary. The effect of thermal non‐equilibrium conditions on subcooled boiling has been taken into consideration. A comparative study of the two methods has been carried out based on the analytical solution of the equations. The study shows that at higher system frequency, increasing number of computational grids increase the accuracy of numerical solutions using FVM while FDM fails to achieve the same. The superiority of FVM over FDM for the problem has also been confirmed by grid convergence analysis. An attempt has also been made to find the analytical solution of the effect of change of heat input on boiling boundary, which is an essential part of computations for the simulation of startup and shutdown of the steam generator. Copyright © 2008 John Wiley & Sons, Ltd.     

Homotopy Coiflets wavelet solution of electrohydrodynamic flows in a circular cylindrical conduit

In previous studies, the nonlinear problem of electrohydrodynamic(EHD)ion drag flows in a circular cylindrical conduit has been studied by several authors. However, those studies seldom involve the computation for large physical parameters such as the electrical Hartmann number and the magnitude parameter for the strength of the nonlinearity due to the existence of strong nonlinearity in these extreme cases. To overcome this faultiness, the newly-developed homotopy Coiflets wavelet method is ext...     

Direct numerical simulation of decaying two‐dimensional turbulence in a no‐slip square box using smoothed particle hydrodynamics

This paper explores the application of SPH to a DNS of decaying turbulence in a two‐dimensional no‐slip wall‐bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, results in a spontaneous spin‐up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high‐accuracy pseudo‐spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo‐spectral results, as did the evolution of the total angular momentum of the fluid. However, although the pseudo‐spectral results emphasised the importance of the no‐slip boundaries as generators of long‐lived coherent vortices in the flow, no such generation was seen in the SPH results. Vorticity filaments produced at the boundary were always dissipated by the flow shortly after separating from the boundary layer. The kinetic energy spectrum of the SPH results was calculated using an SPH Fourier transform that operates directly on the disordered particles. The ensemble kinetic energy spectrum showed the expected k^?3 scaling over most of the inertial range. However, the spectrum flattened at smaller length scales (initially less than 7.5 particle spacings and growing in size over time), indicating an excess of small‐scale kinetic energy.Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical analysis of turbulent flow separation in a rectangular duct with a sharp 180‐degree turn by algebraic Reynolds stress model

Turbulent flow in a rectangular duct with a sharp 180‐degree turn is difficult to predict numerically because the flow behavior is influenced by several types of forces, including centrifugal force, pressure‐driven force, and shear stress generated by anisotropic turbulence. In particular, this type of flow is characterized by a large‐scale separated flow, and it is difficult to predict the reattachment point of a separated flow. Numerical analysis has been performed for a turbulent flow in a rectangular duct with a sharp 180‐degree turn using the algebraic Reynolds stress model. A boundary‐fitted coordinate system is introduced as a method for coordinate transformation to set the boundary conditions next to complicated shapes. The calculated results are compared with the experimental data, as measured by a laser‐Doppler anemometer, in order to examine the validity of the proposed numerical method and turbulent model. In addition, the possibility of improving the wall function method in the separated flow region is examined by replacing the log‐law velocity profile for a smooth wall with that for a rough wall. The analysis results indicated that the proposed algebraic Reynolds stress model can be used to reasonably predict the turbulent flow in a rectangular duct with a sharp 180‐degree turn. In particular, the calculated reattachment point of a separated flow, which is difficult to predict in a turbulent flow, agrees well with the experimental results. In addition, the calculation results suggest that the wall function method using the log‐law velocity profile for a rough wall over a separated flow region has some potential for improving the prediction accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical investigation of particle lateral migration in straight channel flows using a direct-forcing immersed boundary method

Inertia-induced cross-stream migration has been recently exploited for precise position of particles in confined channel flows. In this work, a three-dimensional finite volume based immersed boundary method has been developed to study the lateral migration and hydrodynamic self-assembly of neutrally-buoyant particles in pressure-driven flows. Simulation results show that, in 2D channel flows, the equilibrium position for a circular particle is closer to the centreline for larger particle Reynolds number due to the increasing flow rate, while in 3D square duct flow, the equilibrium position for a spherical particle is near a face centre and is closer to the wall for larger particle Reynolds number. Self-assembly of a pair of particles is observed in 3D square duct flows but not in 2D channel flows. Mechanisms for the self-assembly are discussed.     

Numerical simulation of heat,mass and momentum transport behaviours in directionally solidifying alloy castings under electromagnetic fields using an extended Direct‐SIMPLE scheme

A Direct‐SIMPLE scheme is further extended for numerical simulations of heat and species mass transfer, and liquid flow behaviours in solidification processes of shaped alloy castings under various electromagnetic (EM) fields, based on a binary continuum model (a solidification transport phenomena/processes (STP)‐based dendrite solidification model). Unlike in a SIMPLE scheme, no iterative computations are needed to achieve the final pressure and velocity corrections in the extended Direct‐SIMPLE procedure, therefore extremely high computational efforts can be avoided. Using three different types of model alloys, pseudo‐binary In718 base‐4.85 wt% Nb, γ(TiAl)‐55 at% Al and Al‐4.5 wt% Cu systems, sample computations for solving strongly coupled solidification transport phenomena in directionally solidifying shaped castings under static and harmonic EM‐fields of different strengths are carried out, to demonstrate the feasibility and efficient calculation performance of the present model and numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical modelling of advection–dispersion equation in a stretched curvilinear grid using the QUICKEST scheme

A stretched version of the QUICKEST scheme for solutions of the advection–dispersion equation is presented. The scheme is accurate for large degrees of stretching, so that it can be used where large gradients are present, e.g. for the calculation of sediment in suspension close to the bed. The scheme is tested for various cases of sediment advection and dispersion in one and two dimensions. © 1998 John Wiley & Sons, Ltd.     

Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique

This paper presents a composite multigrid method and its application to a geometrically complex flow. The treatment of the interior boundary conditions within a composite multigrid strategy is described in detail for a 1D model equation. For the Navier-Stokes equations a staggered grid technique is adopted for spatial discretization and a fractional step method is used for the time advance. Lid-driven cavity flows are used to demonstrate the effectiveness of the method.     

A prediction of the pressure field in a plane turbulent jet using an algebraic stress model

A numerical prediction is obtained for the mean pressure field in the similarity region of a plane turbulent jet. An algebraic stress model, which introduces non-isotropic relations for the Reynolds stress components, is used to close the mean momentum equation. The full two-dimensional form of the transport equations is retained and the resultant equation set solved elliptically. The numerical prediction simulates many of the characteristics of the pressure field measured by experimental studies. However, the overall level of the predicted field is lower than the experimental values. The level obtained for the mean pressure field depends strongly on the prediction for the transverse normal Reynolds stress component 〈u₂u₂〉. The pressure field is shown to represent a small negative contribution to the net strearnwise momentum balance.