The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.     

Calculation of flows using three-dimensional overlapping grids and multigrid methods

A computational methodology combining overlapping grid techniques with multigrid methods has been developed for three-dimensional flow calculations in or around complex geometries. The computational accuracy, efficiency and capability of the present approach are investigated in this paper. The incompressible Navier–Stokes equations are discretized using a finite volume method on a semi-staggered grid. The discrete problem is solved by a multigrid algorithm. Some numerical examples are chosen for evaluating numerical accuracy: (a) a straight pipe for which the exact solution is known; (b) curved pipes where previous experimental and numerical data are available; (c) an axisymmetric sudden expansion. The performance of the multigrid method on overlapping grids is assessed. Several cases of flows in stationary and time-dependent complex geometries are given to demonstrate the capability and the potential of the methods that we employ.     

Finite element analysis of incompressible viscous flow in a helical pipe

This paper presents an accurate finite element procedure to deal with steady state, fully developed and incompressible viscous flow in helical pipes with arbitrary curvatures and torsions. The full Navier-Stokes equations and continuity equation have been explicitly derived using a non-orthogonal helical coordinate system. To obtain the final simultaneous non-linear algebraic equations, a pressure-velocity finite element formulation is formulated based on the Galerkin Method.The combined influence of finite curvature and finite torsion on the helical flow is studied. The secondary flow patterns and contours of axial velocity of helical flows show the significant distinction with those of toroidal flows. Further, the effect of torsion on flow rates can be neglected.Several numerical examples are presented. Excellent correlations between the computed results and available referenced solutions can be drawn.     

Approximate factorization scheme for transonic potential flow computations

The implicit approximate factorization scheme known asaf2 is investigated here for the purpose of application to the solution of two-and three-dimensional transonic full potential equations in conservative form. The artificial viscosity used by different authors has been deduced, and is discussed in detail. A second-order correction to the implicit artificial viscosity is tested for transonic flow past a Korn aerofoil at both design and off-design conditions. The inviscid transonic flow past different aerofoils, wings and wing-body configurations has been computed using theaf2 scheme and the solutions are compared with experimental and other numerical results. It is shown that theaf2 scheme is fast, and is not sensitive to grid stretching. Modified version of the paper presented at CAARC (Commonwealth Advisory Aeronautical Research Council) Specialists Meeting on Computational Fluid Dynamics, held during 5–10 December, 1988, at National Aeronautical Laboratory, Bangalore.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes

When large, progressive, deformation occurs under plastic or visco-plastic conditions, elastic deformations are negligible and the material flows in a viscous manner. The flow is non-Newtonian and the viscosity is a function of the current strain rates. An effective numerical treatment of such non-Newtonian flow by the finite element method is indicated here and is illustrated by examples of an extrusion process for which approximate slip line solutions are known. The methodology is extended to the solution of transient, quasi-static situations and coupling with thermodynamic equations is discussed. A stream function formulation of the problem is used.     

A two-phase model for dry density-varying granular flows

A granular flow is normally comprised of a mixture of grain-particles (such as sand, gravel or rocks) of different sizes. In this study, dry granular flows are modeled utilizing a set of equations akin to a two-phase mixture system, in which the interstitial fluid is air. The resultant system of equations for a two-dimensional configuration includes two continuity and two momentum balance equations for the two respective constituents. The density variation is described considering the phenomenon of air entrainment/extrusion at the flow surface, where the entrainment rate is assumed to be dependent on the divergent or convergent behavior of the solid constituent. The density difference between the two constituents is extremely large, so, as a consequence scaling analysis reveals that the flow behavior is dominated by the solid species, yielding small relative velocities between the two constituents. A non-oscillatory central (NOC) scheme with total variation diminishing (TVD) limiters is implemented. Three numerical examples are investigated: the first being related to the flow behaviors on a horizontal plane with an unstable initial condition; the second example is devoted to simulating a dam-break problem with respect to different initial conditions; and in the third one investigates the behavior of a finite mass of granular material flowing down an inclined plane. The key features and the capability of the equations to model the behavior are illustrated in these numerical examples.     

Boltzmann schemes for continuum gas dynamics

Many problems arising in the aerodynamic design of aerospace vehicles require the numerical solution of the Euler equations of gas dynamics. These are nonlinear partial differential equations admitting weak solutions such as shock waves and constructing robust numerical schemes for these equations is a challenging task. A new line of research called Boltzmann or kinetic schemes discussed in the present paper exploits the connection between the Boltzmann equation of the kinetic theory of gases and the Euler equations for inviscid compressible flows. Because of this connection, a suitable moment of a numerical scheme for the Boltzmann equation yields a numerical scheme for the Euler equations. This idea called the “moment method strategy” turns out to be an extremely rich methodology for developing robust numerical schemes for the Euler equations. The richness is demonstrated by developing a variety of kinetic schemes such as kinetic numerical method, kinetic flux vector splitting method, thermal velocity based splitting, multidirectional upwind method and least squares weak upwind scheme. A 3-D time-marching Euler code calledbheema based on the kinetic flux vector splitting method and its variants involving equilibrium chemistry have been developed for computing hypersonic reentry flows. The results obtained from the codebheema demonstrate the robustness and the utility of the kinetic flux vector splitting method as a design tool in aerodynamics. The work presented in this paper is based on the research work done by several graduate students at our laboratory and collaborators from research and development organizations within the country.     

A fully explicit characteristic based split (CBS) scheme for viscoelastic flow calculations

In this paper, an explicit characteristic based split (CBS) scheme is proposed for the numerical solution of incompressible viscoelastic flow equations. The scheme proposed is free from simultaneous solution to the matrices arising from the finite element discretization of the governing equations. The experience gained from the solution of Newtonian fluid dynamics problems has been applied to the solution of viscoelastic flows. The Oldroyd‐B model has been employed to solve two benchmark problems of viscoelastic flow. They are viscoelastic flow past a circular cylinder and viscoelastic flow through planar contraction geometry. The results show that the solutions obtained are stable for the Weissenberg or Deborah number range studied in this paper. The solutions obtained at lower Weissenberg or Deborah numbers are accurate and agree excellently with the majority of available numerical data. However at higher Weissenberg or Deborah numbers, results show some sign of negative influence of the artificial dissipation added to the discrete constitutive equations. Copyright © 2004 John Wiley Sons, Ltd.     

Immersed stress method for fluid–structure interaction using anisotropic mesh adaptation

This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.     

Hydrodynamic modeling of dilute and dense granular flow

We study numerically a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity. The constitutive relations at small and intermediate densities are equivalent to those derived from kinetic theory of granular flow. The existence of an inherent instability due to the vanishing kinetic or collisional pressure for small granular temperatures requires a cross over from a collisional pressure to an a thermal yield pressure at densities close to random close packing. Contrary to a kinetic viscosity, the viscosity turns into a function diverging for small temperatures analogous to the diverging viscosities of liquids close to the glass transition. In this respect the presented model is a simplified version of a model of Savage (J Fluid Mech 377:1–26, 1998), which nevertheless recovers many aspects of dense granular flow. As examples we show simulations of sandpiles with predictable slopes, hopper simulations with mass and core flow and angle dependent critical sand heights in flows down an inclined plane. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear time stepping algorithm together with a finite volume space discretization. The numerical algorithm is implemented using a finite volume solver framework developed by the authors which allows discretization on cell-centred bricks in arbitrary domains.     

Stabilization for Equal-Order Polygonal Finite Element Method for High Fluid Velocity and Pressure Gradient

This paper presents an adapted stabilisation method for the equal-order mixed scheme of finite elements on convex polygonal meshes to analyse the high velocity and pressure gradient of incompressible fluid flows that are governed by Stokes equations system. This technique is constructed by a local pressure projection which is extremely simple, yet effective, to eliminate the poor or even non-convergence as well as the instability of equal-order mixed polygonal technique. In this research, some numerical examples of incompressible Stokes fluid flow that is coded and programmed by MATLAB will be presented to examine the effectiveness of the proposed stabilised method.     

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.     

基于特征线分离算法的大涡模拟

将基于特征线的分离算法与大涡模拟相结合,推导了不可压流大涡模拟有限元离散方程组,并将该方法应用于三维流场的层流及湍流非定常计算。将不同雷诺数下的三维顶盖驱动空腔流动计算结果与实验数据以及直接数据模拟结果进行对比,吻合较好,验证了方法的可靠性和准确性。     

Application of a sensitivity equation method to the <Emphasis Type="Italic">k</Emphasis>–<Emphasis Type="Italic">ε</Emphasis> model of turbulence

In this paper, we present some examples of sensitivity analysis for flows modeled by the standard k–ε model of turbulence with wall functions. The flow and continuous sensitivity equations are solved using an adaptive finite element method. Our examples emphasize a number of applications of sensitivity analysis: identification of the most significant parameters, analysis of the flow model, assessing the influence of closure coefficients, calculation of nearby flows, and uncertainty analysis. The sensitivity parameters considered are closure coefficients of the turbulence model and constants appearing in the wall functions.     

3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach

Finite deformation contact of flexible solids embedded in fluid flows occurs in a wide range of engineering scenarios. We propose a novel three-dimensional finite element approach in order to tackle this problem class. The proposed method consists of a dual mortar contact formulation, which is algorithmically integrated into an eXtended finite element method (XFEM) fluid–structure interaction approach. The combined XFEM fluid–structure-contact interaction method (FSCI) allows to compute contact of arbitrarily moving and deforming structures embedded in an arbitrary flow field. In this paper, the fluid is described by instationary incompressible Navier–Stokes equations. An exact fluid–structure interface representation permits to capture flow patterns around contacting structures very accurately as well as to simulate dry contact between structures. No restrictions arise for the structural and the contact formulation. We derive a linearized monolithic system of equations, which contains the fluid formulation, the structural formulation, the contact formulation as well as the coupling conditions at the fluid–structure interface. The linearized system may be solved either by partitioned or by monolithic fluid–structure coupling algorithms. Two numerical examples are presented to illustrate the capability of the proposed fluid–structure-contact interaction approach.     

Ground States of Two-Component Bose-Einstein Condensates with an Internal Atomic Josephson Junction

In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.     

Melt Flow Instability and Turbulence in Electro-Magnetically Levitated Droplets

This article addresses fluid flow instabilities and flow transition to turbulent chaotic motions through numerical analysis and turbulence in electro-magnetically levitated droplets through direct numerical simulations. Numerical implementation and computed results are presented for flow instability and turbulence flows in magnetically levitated droplets under terrestrial and microgravity conditions. The linear melt flow stability is based on the solution of the Orr-Sommerfeld linearized equations with the base flows obtained numerically using high order numerical schemes. The resulting eigenvalue problems are solved using the linear transformation or Arnold's method. Melt flow instability in a free droplet is different from that bounded by solid walls and flow transits to an unstable motion at a smaller Reynolds number and at a higher wave number in a free droplet. Also, flow instability depends strongly on the base flow structure. Numerical experiments suggest that the transition to the unstable region becomes easier or occurs at a smaller Reynolds number when the flow structures change from two loops to four loops, both of which are found in typical levitation systems used for micro-gravity applications. Direct numerical simulations (DNS) are carried out for an electro-magnetically levitated droplet in a low to mild turbulence regime. The DNS results indicate that both turbulent kinetic energy and dissipations attain finite values along the free surface, which can be used to derive necessary boundary conditions for calculations employing engineering k--ε models.