Numerical solution of the three-dimensional boundary layer on a spinning sharp body at angle of attack

Numerical solutions for the laminar, compressible, three-dimensional boundary layer on general, sharp bodies of revolution at angle of attack with moderate axial spin are described. Unique solutions to the boundary layer equations are obtained by a finite-difference technique which properly treats the elliptic nature of the flow in a crossflow plane. Results are presented and compared with experiment for a tangent ogive-cylinder at four degrees angle of attack with and without spin. The results indicate that viscous flow solutions for spinning bodies at angle of attack can be obtained within the framework of conventional boundary layer theory. These solutions are free from the anomalies observed in a previous investigation.     

A study on pressure-driven gas transport in porous media: from nanoscale to microscale

Gas flow in porous media can be seen in various engineering devices such as catalytic converters and fuel cells. It is important to understand transport phenomena in porous media for improvement of the performance of such devices. Porous media with pores as small as the mean free path of gas molecules are used in such devices as proton exchange membrane fuel cells. It is difficult to measure molecular transport through such small pores in the experimental approach. In addition, even when using theoretical or numerical approaches, gas flow through nanoscale pores must be treated by the Boltzmann equation rather than the Navier–Stokes equations because it cannot be considered as a continuum. Thus, conventional analyses based on the continuum hypothesis are inadequate and the transport phenomena in porous media with nanoscale pores are not yet clearly understood. In this study, we represented porous media by randomly arranged solid spherical particles and simulated pressure-driven gas flow through the porous media by using the direct simulation Monte Carlo (DSMC) method based on the Boltzmann equation. DSMC simulations were performed for different porosities and different sizes of solid particles of porous media. It was confirmed that Darcy’s law holds even in the case of porous media with micro-/nanoscale pores. Using the obtained results, we constructed expressions to estimate the pressure-driven gas transport in porous media with micro-/nanoscale pores and porosity ranging from 0.3 to 0.5. The flow velocities estimated by using the constructed expressions agreed well with those obtained in the DSMC simulations.     

Combined Lattice Boltzmann and phase-field simulations for incompressible fluid flow in porous media

A Lattice-Boltzmann method for incompressible fluid flow is coupled with the dynamic equations of a phase-field model for multiple order parameters. The combined model approach is applied to computationally evaluate the permeability in porous media. At the boundaries between the solid and fluid phases of the porous microstructure, we employ a smooth formulation of a bounce-back condition related to the diffuse profile of the interfaces. We present simulations of fluid flow in both, static porous media with stationary non-moving interfaces and microstructures performing a dynamic evolution of the phase and grain boundaries. For the latter case, we demonstrate applications to dissolving grain structures with partial melt inclusions and computationally analyse the temporal evolution of the microporosity under wetting conditions at the melt-grain boundaries. In any development state of the material, the Darcy number and the hydraulic conductivity of the porous medium are evaluated for various types of fluid.     

Lie group analysis of stagnation-point flow of a nanofluid

This article concerns with a steady two-dimensional boundary layer flow of an electrically conducting incompressible nanofluid over a stretching sheet in a porous medium with internal heat generation/absorption. The transport model includes the effect of Brownian motion with thermophoresis in the presence of chemical reaction and magnetic field. Lie group analysis is applied to the governing equations. The transformed self similar non-linear ordinary differential equations along with the boundary conditions are solved numerically. The influences of various relevant parameters on the flow field, temperature and nanoparticle volume fraction as well as wall heat flux and wall mass flux are elucidated through graphs and tables.     

A moving-grid finite element algorithm for supersonic flow interaction between moving bodies

The present study describes a finite element scheme which provides for the computer simulation of the motion of one finite element mesh relative to another along a smooth interface through which flow occurs. The scheme is designed to simulate transient flow interaction phenomena encountered in situations in which two or more solid bodies move relative to one another through an arbitrary flow field. Particular applications include simulations of rotor-stator flow interactions in turbomachinery. Although the method is applicable to any type of flow interaction problem, the applications considered here focus on the modeling of supersonic flow interactions characterized by the time-dependent Euler equations, and on problems of pure advection of a scalar field, all in two-dimensional flow domains. Several numerical examples are given to demonstrate the robustness and the performance of the numerical scheme.     

Topology optimization study of arterial bypass configurations using the level set method

We studied the arterial bypass design problem using a level set based topology optimization method. The blood flow in the artery was considered as the non-Newtonian flow governed by the Navier–Stokes equations coupled with the modified Cross model for the shear dependent viscosity. The fluid–solid interface is immersed in the design domain by the level set method and the fictitious porous material method. The sensitivity velocity derived by the level set based continuous adjoint method was utilized to control the evolution of the level set function. In order to accommodate the irregular analysis domains, the flow equations and the level set equations were computed on two different unstructured grids respectively. Three idealized arterial bypass configurations problems with the minimum flow shear stress objective were studied in the numerical examples. The results indicated that the optimal arterial bypass designs can effectively reduce integral of the squared shear rate in the artery and have a superior performance for the arterial grafting.     

Steady flow around and through a permeable circular cylinder

The steady flow around and through a porous circular cylinder was studied numerically. The effects of the two important parameters, the Reynolds and Darcy numbers, on the flow were investigated in details. The recirculating wake existing downstream of the cylinder is found to either penetrate into or be completely detached from the cylinder. It is also found that, contrary to that of the solid cylinder, the recirculating wake develops downstream of or within the porous cylinder, but not from the surface of it. These new findings provide additional evidence to Leal’s conclusion (Leal LG. Vorticity transport and wake structure for bluff bodies at finite Reynolds number. Phys Fluids A 1989;1:124) that the appearance of recirculating wakes at finite Reynolds number is due to vorticity accumulation, but not a result of the same physical phenomena associated with separation in boundary layers in adverse pressure gradients. Also presented in the current study are the variation of the critical Reynolds number for the onset of a recirculating wake as a function of Darcy number and the variation of a newly defined parameter, the penetration depth, as a function of the Reynolds number and Darcy number.     

Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.     

Asymptotic model of a thin shock layer near a wedge/cone for the Burnett approximation

Asymptotic estimations on the order of the terms of the two-dimensional Burnett equations for the characteristic regions (layers) of a hypersonic rarefied gas flow are performed. A simplified Burnett model of the flow near nonthin bodies is formulated (in accordance with the concept of a viscous shock layer) on the basis of the asymptotic analysis founded on physical and theoretical assumptions. The simplified Burnett model is presented for the surfaces in the form of a wedge/cone.     

Forced convection of air through networks of square rods or cylinders embedded in microchannels

Forced convection of air at near-standard conditions through periodical networks of micrometer square rods or cylinders is investigated. The Navier–Stokes equations subjected to first-order velocity-slip condition, and energy equations for the fluid and solid phases are numerically solved for two-dimensional structures. The flow is created by imposing pressure gradients, and heat is volumetrically generated inside the solid rods. Assuming periodicity in the direction transverse to the pressure gradient, the computational domain consists in long open channels, partially filled with solid rods placed regularly. The various structures considered are then modeled as porous media. For the permeability calculations, both volume averaging technique and multiple scale expansion technique are employed, and the results are favorably compared. The slip effect on permeability is highlighted for Knudsen number of about 0.05. On the other hand, the use of a periodic approach in the flow direction for heat transfer calculations is demonstrated not to be based on realistic assumptions. In addition, the importance of axial heat diffusion in channels of width close to one micrometer is emphasized. The reason is found in the low Péclet numbers typically encountered when the incompressible approximation is invoked. Based on numerical solutions at the microscopic scale, a new macroscopic modeling is suggested. Comparisons between numerical solutions and analytical predictions for various networks of rods are discussed. A very good agreement is shown.     

Numerical simulation of two-phase flow in deformable porous media: Application to carbon dioxide storage in the subsurface

In this paper, conceptual modeling as well as numerical simulation of two-phase flow in deep, deformable geological formations induced by CO₂ injection are presented. The conceptual approach is based on balance equations for mass, momentum and energy completed by appropriate constitutive relations for the fluid phases as well as the solid matrix. Within the context of the primary effects under consideration, the fluid motion will be expressed by the extended Darcy's law for two phase flow. Additionally, constraint conditions for the partial saturations and the pressure fractions of carbon dioxide and brine are defined. To characterize the stress state in the solid matrix, the effective stress principle is applied. Furthermore, the interaction of fluid and solid phases is illustrated by constitutive models for capillary pressure, porosity and permeability as functions of saturation. Based on this conceptual model, a coupled system of nonlinear differential equations for two-phase flow in a deformable porous matrix (H²M model) is formulated. As the displacement vector acts as primary variable for the solid matrix, multiphase flow is simulated using both pressure/pressure or pressure/saturation formulations. An object-oriented finite element method is used to solve the multi-field problem numerically. The capabilities of the model and the numerical tools to treat complex processes during CO₂ sequestration are demonstrated on three benchmark examples: (1) a 1-D case to investigate the influence of variable fluid properties, (2) 2-D vertical axi-symmetric cross-section to study the interaction between hydraulic and deformation processes, and (3) 3-D to test the stability and computational costs of the H²M model for real applications.     

Numerical simulation of a compressible turbulent boundary layer over a conductive wall with line heat source

A conjugated problem of supersonic turbulent flow over a conductive solid wall with an embedded line heat source has been investigated as a model of a separation detector and skin friction gage. The 2-D Navier-Stokes equations for compressible fluid, including a two layer eddy viscosity model, are solved simultaneously with the heat transfer equation for the solid, written in general coordinates. The effect of the interface boundary condition on the stability of the implicit scheme of the flow field has been checked. A careful investigation of the effect of heat source strength, solid and fluid conductivity and Mach and Reynolds numbers on flow and temperature fields has been performed. The results of this investigation may be used to design an optimal gage with a minimum influence on the flow field.     

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first order ordinary differential equations. The streamwise shear stress function θ and the cross-flow velocity component V are represented as suitable functions of the streamwise velocity component U. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30° incidence. The results are compared with corresponding finite difference solutions.     

Finite element implementation of stress-jump and stress-continuity conditions at porous-medium, clear-fluid interface

The boundary conditions at the interface between clear-fluid and porous-medium domains are very important for solving flow through an open domain adjoining a porous medium. In this Galerkin finite-element (FE) based simulation of such interface flows employing Stokes and Brinkman equations, the traditional interfacial condition based on the continuity of stress in fluid and porous media is compared with the stress-jump condition proposed by Ochoa-Tapia and Whitaker using the rigorous volume averaging method. A novel FE formulation employing a second-order adjustable tensor is proposed to implement this new stress-jump condition for full three-dimensional flows. The paper also clarifies the hitherto obscure relationship between flow variables in the fluid and porous media for the conventional stress-continuity condition. In the first validation study involving numerical predictions of flow parallel to the interface, our FE implementation of the new stress-jump condition agree very well with the analytical solution for flow parallel to the interface, thereby proving the soundness of our adjustable tensor approach. Similar excellent results were obtained for FE implementation of the stress-continuity condition as well. A good match with analytical solution for a constant cross-flow superimposed on the parallel flow was also achieved while differences in velocity profiles near the interfaces were studied for the two conditions. Lastly a complex 3D flow simulation involving a fluid and porous media interface within the unit-cell of a non-crimp stitched fiber mat, used in liquid composite molding process during the manufacture of composite materials, is undertaken. The permeability of this dual-scale fibrous porous medium, estimated using the newly implemented stress-jump condition, agrees well with the experimental result thereby pointing to the accuracy of the FE implementation of the condition. Our simulations reveal that the stress-jump condition leads to a much smaller boundary layer within porous medium near the interface as compared to the stress-continuity condition, and hence to a lower, more accurate net flow-rate through the unit cell. However the two interfacial conditions yield similar results with a decrease in the porosity.     

A finite element investigation of the flow of a Newtonian fluid in dilating and squeezing porous channel under the influence of nonlinear thermal radiation

The influence of nonlinear thermal radiation on the flow of a viscous fluid between two infinite parallel plates is investigated. The lower plate is solid, fixed and heated, while the upper is porous and capable of moving toward or away from the lower plate. The effects of nonlinear thermal radiation are incorporated in the energy equation by using Rosseland approximation. The similarity transformations have been used to obtain a system of ordinary differential equations. A finite element algorithm, known as Galerkin method, has been employed to obtain the solution of the resulting system of differential equations. It is observed that the radiation parameter Rd increases the temperature of the fluid in all the cases considered. Same is the case with temperature ratio parameter θ _w . The influence of the concerned parameters on the local rate of heat transfer is also displayed with the help of graphs.

     

Particle partitioning strategies for the parallel computation of solid-liquid flows

The numerical investigation of the interaction of large, solid particles with fluids is an important area of research for many manufacturing processes. Such studies frequently lead to models that are very large and require the use of parallel solution techniques. This paper presents the results of a parallel implementation of a serial code for the direct numerical simulation of solid-liquid flows. The base code is a serial, arbitrary Lagrangian-Eulerian (ALE) formulation of the equations of motion, which views that particles as solid bodies are embedded into the flow domain. This particular model poses some interesting difficulties for domain decomposition type approaches for parallel solutions. In particular, it is not fully understood how the partitioning of the particles among the subdomains influences the performance of parallel solvers. We present several strategies for the partitioning of the solid particles, focusing on the effectiveness of these techniques in terms of parallel speedup and efficiency.     

应用于冷却系统的多孔介质电渗泵研究

对多孔介质电渗泵性能进行研究,分析电渗泵的流率和压力,为多孔介质电渗泵制作提供参考.依据电渗流控制方程,利用Debye-Huckel近似得到单根圆形管电渗流的近似解析模型,采用MEMS数值仿真软件CoventorWare对电渗泵模型进行求解分析,得到电渗泵流场的分布.基于电渗流的近似解析模型,采用截断高斯分布对多孔介质...     

Numerical calculations of the effect of moisture content and moisture flow on ionic multi-species diffusion in the pore solution of porous materials

A method to analyse and calculate concentration profiles of different types of ions in the pore solution of porous materials such as concrete subjected to external wetting and drying is described. The equations in use have a solid theoretical meaning and are derived from a porous media technique, which is a special branch of the more general mixture theory. The effect of chemical action is ignored, making the presented model suitable to be implemented into codes dealing solely with chemical equilibrium.     

On Computational Modeling in Tumor Growth

The paper addresses modeling of avascular and vascular tumor growth within the framework of continuum mechanics and the adopted numerical solution strategies. The models involve tumor cells, both viable and necrotic, healthy cells, extracellular matrix (ECM), interstitial fluid, neovasculature and co-opted blood vessels, nutrients, waste products, and their interaction and evolution. Attention is focused on the more recent models which are much richer than earlier ones, i.e. they address more aspects of this complicated problem. An important element is how the governing equations are obtained and how the many interfaces between the above listed components are dealt with. These considerations suggest the definition of different classes of models comprised of diffusion, single phase flow and multiphase flow models with or without a solid phase. A multiphase flow model in a deforming porous medium (ECM) is chosen as reference model since it appears to invoke the least number of simplifying assumptions and has the largest potential for further development. The strategies adopted in the choice of the many model dependent constitutive relationships are discussed in detail. Two applications referring to two different model classes conclude the paper.