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129 2006, von Karman Institute for Fluid Dynamics, Rhode-St-Gense, Belgium

CFD METHODSCFD activities are composed of two parts:On the one hand there is the developmentof advanced physical modeling capabilities,including aerothermal and chemical model-ing for high enthalpy flow, modeling oftransport properties of ionized plasmas,

modeling of two phase flow and modelingof turbulence by LES. On the other hand,the development of advanced numericaltechniques has continued, expandingresidual distribution methods and solutionadaptive hybrid grid generation tools.A new solver for LES and DNS simulations(SFELES) has been developed, as wellas an object oriented multiphysics softwareplatform (COOLFluiD) suitable for main-taining and accommodating existing andnew software developments.


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An 11-species air mixture in local thermodynamicequilibrium at atmospheric pressure serves as abenchmark to assess the physical model and numeri-

cal methods. The shear viscosity computed by a directmethod is displayed in Figure 1. Below the ionizationrange, increases with temperature and presents somemodulations linked to changes of composition due tomolecular dissociation. Above 10000K, large ion-ioncross- sections drastically decrease the shear viscosity.

Table 1: Computational cost of

CG 1.00Direct 1.51Gupta-Yos 1.03Wilke 2.16

Superiority of the conjugate gradient method with re-

spect to the direct solver and approximate mixturerules found in the literature is demonstrated in termsof accuracy and computational cost in Figure 2. Wilkesrule leads to a relative error of about 10% in the dis-sociation range rising up to 70% in the ionizationrange. Below 9000K, the Gupta-Yos formula is accu-rate and one conjugate gradient iteration is sufficient-ly precise. Above this temperature threshold, theGupta-Yos formula fails with a maximum relative er-ror of 40%, whereas one conjugate gradient iterationyields maximum 6% of error. Two conjugate gradient(CG) iterations provide accurate results. The compu-tational cost is summarized in Table 1. Wilkes rule forthe shear viscosity requires 3N-3 square roots to be

evaluated, where Nis the number of species in themixture. The number of operations of the Gupta-Yosformula scales as N2/2. The CG is the cheapest andmost accurate method.

COOLFLUID: AN OBJECT-ORIENTED

FRAMEWORK FOR MULTI-METHOD AND

MULTI-PHYSICS SCIENTIFIC COMPUTING

This research is a long term effort towards the devel-opment of a next generation multi-physics and multi-method solver, by constructing a software frameworkwhere different numerical techniques and tools cancoexist and function together.

Nowadays the science of computational fluid dynam-ics is reaching its mature state of development.The need to incorporate more and more physics intothe simulation of fluid mechanics is increasing, to-gether with the constant rise in available computerpower. More complex numerical methods are beingdeveloped, that bring new levels of accuracy and

reliability to the computed solutions. It is only reason-able that the software techniques used to implementsuch methods and complex models should evolve as

well, so that the development process scales likewise.

Experience with the previous codes for scientific sim-ulations at the VKI shows that, although very power-ful, these academic codes have reached the limit ofcomplexity and scale that their architecture allows.Some of the developments made by external partnersof the Institute on cooperative projects are not com-patible with other developments; in addition, furtherextensions to these existing codes tend to be time-consuming and tedious.

Therefore, in the beginning of 2002, a project aimingat the design and construction of a software frame-

work was initiated. This framework should work as aplatform for the collaborative development of tools forscientific simulations within the departments of theInstitute and between the VKI and external projectpartners. These tools are aimed at the simulation ofscientific problems that involve multi-physics, whichcan be discretized through a variety of numericalmethods and solved on complex 3D geometries inmassive parallel simulations. Thus, the software fromthe qualitative point of view should strive for modu-larity, runtime and memory efficiency, implementationscalability, ease of maintainability and enduredlifetime.

The current partners involved in this project are thevon Karman Institute, which is responsible for themain contribution, two research groups at the CatholicUniversity of Leuven and the ETEC group at the VrijeUniversiteit Brussel. More partners are expected tojoin as the COOLFluiD platform is involved inEuropean research projects (like ADIGMA).

The software architecture of this framework reflectsthe need for a multi-partner collaborative environ-ment: it is designed in a layered architecture, with acentral kernel which is responsible for the main inner


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workings of the framework and the handling of thecomplex data structures, but which does not imple-ment any functionality per se; the functionality is then

provided by a set of plug-in modules that can be de-veloped separately by different partners, and whichare brought into collaboration by the kernel; and fi-nally a myriad of multi-physics applications comesinto existence by the multiple combination of thesemodules. This plug-in policy allows for external part-ners to develop new functionality, which is easier tointegrate and thus creates powerful synergies fromwhich all of the project partners can profit.

As this framework serves as a development platformfor other VKI CFD projects, the current applications be-ing developed within the project are linked to other re-search projects already presented in this manuscript,ranging from incompressible to compressible tran-sonic fluid dynamics, hypersonic aerothermodynam-ics, aeroelastic fluid-structure-interaction, magneto-hydrodynamic space-weather prediction for theEuropean Space Agency, simulation of inductive cou-pled plasmas to support the operation of the plasmatunnel facilities at the VKI, and electro chemical plat-ing and deposition within the MuTech project.

The solution of problems for industrial applicationsusually involves complex geometries. To simplify theirtreatment and speed-up of the design-mesh-simulate-diagnose cycle of CFD, the framework supports auto-matically generated unstructured meshes. This ex-tends to hybrid structured/unstructured meshes, forthe improved solution of boundary layers of turbu-lent flows. Work is in progress to support high orderele-ments, thus allowing for higher accuracy on com-pact stencils. This is all achieved within a lightweightdata structure, that enables dynamic mesh movementand adaptation, driven from error estimation or fluid-

structure interaction. A large set of different numeri-cal methods have already been implemented as plug-in modules within the project. These are as diverse as:

time-stepping algorithms, for explicit and implicitmulti-order time discretization; arbitrary interfaces tomultiple linear system solvers like PETSc, Trilinos andSAMGp; multiple mesh movement techniques, Hes-sian based error estimation and conservative couplingtechniques for multi-physics problems.

One of the main features of the COOLFluiD platform, en-abled by its flexible data structure, is that it allows the

coexistence in the same codeof different space discretiza-tion techniques. The possibili-ty to use the most appropriatemethods for each specific

problem broadens the spec-trum of applications that

can be successfullysimulated using theframework. Currently

there are modules for thefinite volume, finite element

and fluctuation splittingmethod. This concept of coex-istence of numerical methodsis called the multi-methodframework.

Another key feature of this software platform is its

ability to decouple the implementation of the numeri-cal methods from the physical model description.This allows for a the truly orthogonal existence of amulti-method, multi-physics platform, where not onlythe same simulation can be solved with differentmethods for comparison, but also the more appropri-ate methods can be selected for each class of physics,and still interoperate, in a multi-physical application.A concrete example would be the aeroelastic simula-tion of a wing flutter, where the flow is modelled withthe finite volume method, the structure with high-order finite element methods and the mesh move-ment with a spring-analogy method.

One of the main difficulties for the development ofhigh performance computing software is the imple-mentation of the parallel communication algorithms.Efficiency is of paramount importance and the sup-porting libraries have typically difficult interfaces andare only well understood by software engineers.Moreover, the parallel code is usually pervasive andintertwines with the numerical algorithms, reducingthe comprehensibility and steepening the learningcurve. The COOLFluiD framework tackles this issue byconstructing a parallel layer of abstraction, whichhides the communication steps, thus releasing the

Figure 2: Comparison of the same simulation, aFalcon at Mach 0.85, solved within COOLFluiDwith two distinct methods [FSM, FVM]


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developer of numerical algorithms from the task ofthinking in global terms of the simulation. In practice,this means that one can often develop serial algo-

rithms, with little consideration of parallel issues. Theframework will automatically ensure that everythingruns concurrently on multiple processors. This rangesfrom automatic partitioning of meshes, to parallelinput-output, to automatic syncing of data and com-mu-nication of signals between CPUs.

To provide a shorter learning curve as well as easiermaintainability, the project has set its documentationof the programming interface in-source. These are lat-ter compiled into written documents which can beused by partners to develop functionality that interop-erates within the kernel. For improving the commu-ni-cation within the project partners, a collaborative

website based on wiki technology has been set up,and it also serves as the project webpage:http://coolfluidsrv.vki.ac.be/coolfluid.

FINITE-ELEMENT SIMULATION

OF ELECTROCHEMICAL PROCESSES

IN COOLFLUID

This research is part of the IWT-SBO project MuTEch("Novel Multiscale approach to Transport phenomena

in Electrochemical processes"), started in Januar 2005,and which focuses on the numerical simulation ofelectrochemical processes.

Within MuTEch, the VKI AR department collaborateswith a large number of Flemish academic and indus-trial partners. The collaboration with the electro-tech-nical group of the Vrije Universiteit Brussel (VUB) aimsat implementing electro-chemistry models in the object-oriented multiphysics framework COOLFluiD, a large,3D unstructured flow solver capable of running onparallel supercomputers. The end re-sult should be a tool,sufficiently powerful toaccurately predict lo-cal parameters of re-alistic industrial elec-trochemical systems.

E lec t rochemica lprocesses are gov-erned by multi-com-ponent mass, heatand gas transport inlaminar and turbulentflow regimes. The mass

transfer boundary layers are typically one or severalorders of magnitude smaller than thermal and hydro-dynamic boundary layers. Other phenomena that fre-

quently occur are chemical reactions and temperatureor concentration-driven natural convection. As a resultof this complexity, quantitative knowledge of the cur-rent densities, the ion concentrations and the gas frac-tion distribution is very difficult to obtain.Nevertheless, it is these local quantities that determinethe properties and quality of the final product, for ex-ample, a common plating/etching of a metallic piece.

Contemporary understanding of electrochemistry onlyallows the determination of these properties a poste-riori and their correlation to the global process pa-rameters of a reactor. Any deeper insight based uponnumerical simulations would therefore be a consider-

able improvement of the state-of-the-art. Since we ex-pect that our numerical tools will not be able to cap-ture the full complexity of electrochemical problems,the MuTEch consortium has been subdivided into anumerical and experimental group. On the one hand,this will allow the validation of the developed numer-ical models against experimental data. On the otherhand, where models are missing, empirical correla-tions will be needed; these will probably require cali-bration through experiments.

As a first step, we have implemented the simplest de-scription of electrochemical deposition by diffusion(plating) into COOLFluiD. If we assume that ionic

concentrations are more or less homogeneous, thento determine electric current distributions inside a re-actor, it suffices to solve a Laplacian problem withnonlinear boundary conditions that represent chemi-cal reactions on electrodes. We solve this problem bymeans of the 3D finite-element discretization on an un-structured mesh consisting of tetrahedral elements.

The use of unstructured meshesand parallel computers allowsus to treat large geometries ofarbitrary complexity, ratherthan simplified configura-tions that are of academic in-

terest only. For instance, inFigure 1 we show the current

density on an 8-wheel racksubmerged in an electrolyteand subject to an electric po-tential difference between the

wheel surfaces and the tank.The current density can be

shown to be directly related to thespeed of deposition of material on the wheelrack. This case was run in parallel on five proces-sors, using a mesh containing 800000 nodes.

Figure 1: Wheel rackin a plating process


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During the past year, particular attention has been giv-en to the accuracy and efficiency of the implementa-tion within COOLFluiD. This was done by comparing

solutions and code performance obtained withCOOLFluiD against the existing light-weight industri-al software PlatingMaster. In Figure 2, we compare thecurrent density distributions predicted by COOLFluiDand PlatingMaster for the plating of an oil pump.Perfect agreement is found.

By comparing with PlatingMaster, we have also beenable to perform a thorough profiling and optimisationof memory consumption and execution speed hasbeen performed. The parallel efficiency of COOLFluiDhas also been assessed and is found to be satisfactory.

The difficulty of generating coarse unstructured grids

around complex geometries makes it virtually impos-sible to implement geometric multi-grid solvers tospeed up the calculations. A solution to this problemis to use algebraic multi-grid (AMG) solvers, whichtake the linear problem on the final fine mesh as aninput, generate a sequence of coarsened approxima-tions to this linear problems in an automatic mannerand solve the linear problem using techniques analo-gous to geometric multi-grid (smoothers, restrictionand prolongation operators).

Currently the SAMGp linear system solver, developedby Fraunhofer SCAI (Institute for Algorithms andScientific Computing) group is being incorporated intoCOOLFluiD. The SAMGp solver implements the AMGsolution method which is appropriate for current prac-tical applications. SAMGp performs very well and pre-liminary results show this advantage over currentlyimplemented linear solvers. The strongest expectedadvantages, such as linear parallel scalability, are yetto be asserted. Speed and memory overhead ofSAMGp will be compared with the characteristics ofconventional one-level solvers, such as CG precondi-tioned by an ILU-type smoother.

Once all performance issues listed above will havebeen addressed, we will start implementing the moredetailed MiTREM model (Multi-ion Transfer and

Reaction Electrochemical Model) and consider turbu-lent flow effects as well as gas formation at electrodes.

IDEAL MHD SOLVER FOR SPACE WEATHER

APPLICATIONS IN COOLFLUID

This research is part of the ESA project PRODEX-8,Solar Drivers of Space Weather II. In the course ofthe project, the VKI AR department is responsible forthe development of an application code for the spaceweather community (user friendliness, documenta-tion, interfacing with grid generation ...). This is doneby means of writing an ideal MHD solver inCOOLFluiD, the object-oriented multi physics frame-work under development at the VKI. Our aim is to com-bine state-of-the-art upwind Finite Volume andResidual Distribution numerical schemes, accompa-nied by effective solenoidal constraint satisfying nu-merical techniques, with new MHD physics studied atthe Center for Plasma Astrophysics (CPA) of theCatholic University of Leuven: solar wind initiation, re-sistive effects, earth magnetic field and other models.The COOLFluiD framework will provide other essen-tial building blocks, such as the capability to run in par-

allel, implicit time integration methods, solution adap-tive grids, etc.

As solenoidal constraint satisfying numerical tech-niques, two different approaches have already beenimplemented and tested in the solver. One of them isthe well-known Powell's source term approach andthe other is the artifi-cial compressibility approach(ACA) that has been developed as part of the researchthat has been conducted during this academic year atthe VKI.

Powell's source term approach relies on a non-con-servative form of the ideal MHD equations by means

of a source term whose elements cancel out the termsproportional to the divergence of the magnetic field inthe original ideal MHD equations. Errors in the diver-gence of the magnetic field are convected away by theflow. This approach suffers mainly from two draw-backs: (1) the stabilizing convection mechanism doesnot work well in the regions of stagnant flow and (2)non-conservative discretizations suffer from incorrectshock capturing. Moreover, the solenoidal constraintis satisfied up to only the accuracy of the truncationerror of the underlying base numerical scheme.

Figure 2: Plating process on a oilpump, resultsfrom COOLFluiD (left) and PlatingMaster (right)


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The artificial compressibility approach (ACA) is a varia-tion of the classical projection scheme in which thegradient of a scalar potential field is added to the mag-

netic field at each iteration, in order to ensure that thesolenoidal constraint be satisfied up to machine ac-curacy at each discrete time step. The disadvantage ofthis classical ap-proach is that a costly Poisson prob-lem is to be solved for the potential at each time step.

In ACA, the aforementioned Poisson problem is solvedby means of a pseudo-time stepping procedure for theadditional potential variable. The coupling betweenthe original ideal MHD equations and the additionalequation for the potential variable is made by meansof adding a diagonal tensor of the potential variableto the induction equation. The modified system of ninePDEs is now fully hyperbolic and conventional upwind

Finite Volume methodscan directly be applied(e.g. Lax-Friedrichsscheme, Roe's scheme).It is proved both analyt-ically and numericallythat the method is con-sistent, which meansthat the modified sys-tem of PDEs convergetowards the correct so-lution of the originalideal MHD systemwhether in the presence

of discontinuities or notboth at PDE and dis-crete levels.

In testing so far, ACAseems to be a remedyfor drawbacks of mainfamilies of solenoidal constraint satisfying methods inthe literature. The main properties of the method canbe summarized as follows: (1) the modified system ofPDEs is purely hyperbolic (unlike classical projectionmethods), (2) the method is stable and consistent andworks even in the vicinity of the regions of stagnantflow, which can be proved by converting the inductionequation into a wave equation for divergence of mag-netic field, which has eigenvalues that are independ-ent from the flow speed (unlike Powell's source termapproach), (3) easy to implement (unlike staggered-grid approaches such as Balsara's approach) (4)the solenoidal constraint is satisfied up to machineaccuracy.

We have implemented and tested the ACA approachwithin the finite-volume context, using scalar-dissipa-tion as well as Roe-type flux functions. Second orderaccuracy in space is achieved by using linear least

squares reconstruction together with Barth-Jespersen's and Venka-takrishnan's limiters. Explicittime integration is performed by means of forward Euler

and Runge-Kutta methods whereas implicit time inte-gration is done using backward Euler time stepping.

In terms of convergence speed and robustness, ACAhas been found to be more robust, but for explicit timeintegrations the convergence is markedly slower thanwhen using Powell's technique. When using an im-plicit time integrator, on the other hand, the perform-ance of ACA is at least as good as the performance ofPowell's method. We have tested the new scheme ona wide variety of academic test cases, involving MHDflows with strong magnetic fields, such as encoun-tered in, for instance, fast solar Coronal Mass Ejections(CMEs) and other phenomena in space plasmas.

Figure 1 presents results obtained for the situationwhen a high speed MHD flow, with a strong magnet-ic field aligned with the flow field, encounters a per-fectly conducting cylinder. Computed density con-tours are shown both for Powell's source termapproach and the AC approach. The underlying basenumerical scheme is a TVD scalar dissipation scheme.As can be seen, for large magnetic fields, the obtainedshock structure can be completely different from whatwould be obtained from a conventional compressibleflow without MHD effects.

We have also made considerable progress in imple-menting the ACA approach using residual distributionmethods, which are finite-element related methods forhigh-speed flows traditionally developed and studiedat the VKI. In Figure 2, we show axial momentum andmagnetic field contours obtained for a high-speedMHD flow inside a curved magnetic flux tube.

Figure 1: High-speed magnetically dominated and field-aligned MHD flow thatencounters a perfectly conducting cylinder. Density contours computed using afinite volume solver based on Powell's source term approach (a) and on theartificial compressibility approach (b)

a) b)


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In the following months of the research in this field,the finite volume and residual distribution solvers willbe further improved. At the same time, we will attempt

to simulate more realistic physical problems, studiedby the CPA, including challenging unsteady simula-tions linked to space weather prediction.

HYBRID UNSTRUCTURED GRID

GENERATION FOR CONVECTION-

DOMINATED FLOW SIMULATIONS

Unstructured hybrid grid generation has been an im-portant subject of research since 1998. A first mile-stone was reached with the participation of VKI to theEuropean Union ESPRIT project AMeGOS (AutomaticMesh Generation and Optimisation for industrial flowSimulations), which reached its end in May, 2002.Themain task of VKI was the development of a grid gen-erator suitable for convection dominated flow prob-lems with very thin diffusion layers near the bound-aries. The main application area's targeted in theproject were automotive industry (flow in combustionengines), plating industry (electrochemically reactingflows with mass transfer at the electrode boundaries)and aerospace (high Reynolds number flows). A firstprototype grid generation package [AJ31] was com-pleted in close collaboration with the software com-pany ElSyCa, who was responsible for Graphical UserInterfacing (GUI), CAD interfacing, visualization andinterfacing with solvers. Other partners in the project

were AVL, PSA, and Volvo. Since 2002, the partnershipbetween VKI and Elsyca has been further extendedwith the University of Brussels (VUB, department of

Computational Electrochemistry), and has now result-ed in a grid generation package fully integrated withthe CAD package Solid Works.

The hybrid grid generation approach combines thestrengths of structured grid generators to producestretched anisotropic layers of elements near theboundary with the capacities of unstructured gridmethods to cope with complex geometries. The pro-cedure is hierarchic, beginning with the constructionof the edge grids which constitute the boundaries ofthe faces. In a second step the face grids are gener-ated starting from the bounding edge grids. Finally ina third step the volume grids are generated starting

from the bounding face grids. Starting from the trian-gular surface mesh, semi-structured layers of pris-matic elements are created normal to the surface.

Figure 2: High-speed MHD flow inside a curvedmagnetic flux tube computed using a residualdistribution solver based on the artificialcompressibility approach

Figure 1: Comparison of the sharp concave ridgearea with (left) and without (right) front folding


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When the prismatic elements become isotropic thelayer generation is stopped and the remaining domainis filled with isotropic tetrahedra (eventually after the

creation of some intermediate prismatic elements).The isotropic part of the volume grid (away from vis-cous layers) is based on a Delaunay triangulationmethod allowing to generate tetrahedra from a givenpoint cloud.

Figure 2: Views of volume hybrid grid at the trail-ing edge (upper figure) and leading edge (lowerfigure) for the ONERA M6 wing

After 2002, the research focused on improving quali-ty of the grids and the efficiency and robustness of thealgorithms, especially for the boundary layer part ofthe grid. This has lead to the completion of a PhD the-sis in 2004 [TH16]. During this thesis a new methodhas been developed for the generation of semi-struc-tured layers in hybrid grids around convex and con-cave ridges of the geometry. The method produces su-perior grid quality and resolution in these areaswithout modifying the underlined semi-structuredgeneration algorithm. An example of typical im-provement is shown in Figure 1, comparing the stan-dard approach with the new front folding algorithm.

The well-know ONERA M6 wing at transonic flow con-ditions was chosen as a case to validate the grids gen-erated by the proposed algorithms. Figure 2 presents

two views of the grid along the tip and root of thewing. It also shows the hybrid surface grid generatedfor this configuration, with semi-structured quadrilat-erals generated starting from the round leading edgeand the sharp trailing edge of the wing (the quadshave been triangulated to accommodate the flowsolver which only supports tetrahedral elements). Themaximum aspect ratio of the triangles on the surfaceis 20 in the area of the leading and trailing edge of thewing. The wing surface has approximately 20000points. The hybrid volume grid was created with30 semi-structured layers. The final hybrid volume gridhas 683000 nodes and 4 x 106 tetrahedra. The front un-folding algorithm was applied to the sharp trailing

edge of the wing to further improve the grid in thatarea (shown in Figure 3).

In Figure 4, the pressure distribution is plotted at twostations along the span of the wing. In the plots, theresults of the present computation (using the VKI Thorsolver) are compared with the experimental data anda reference computation using the same solver on astandard grid with the same number of mesh points[MP53]. The comparisons show that in all cases thepresent grid outperforms the one used in the refer-ence in terms of pressure distribution. Notice also thatit does not exhibit the unphysical pressure jump at thewing trailing edge, typical of RANS computations on

unstructured grids. This substantial (qualitative) im-provement is due to the combination of the hybrid sur-face grid close to the trailing edge with the front un-folding (Figure 3).

Figure 3: Zoom at the trailing edge in the ONERAM6 wing in the root area showing the two normalsproduced by the front unfolding algorithm


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Figure 4: Pressure distribution for ONERA M6wing at 65% (upper figure) and 90% (lowerfigure) span location

ANISOTROPIC SOLUTION-ADAPTIVE

GRID ALGORITHMS FOR 3D

COMPRESSIBLE FLOW SIMULATION.

APPLICATION TO SPACE-WEATHER

Solution-adaptive grid adaptation is a powerful toolfor optimizing the efficiency of large scale calcula-tions, such as the solution of the Euler and Navier-

Stokes equations for compressible flow. It allows adramatic reduction in the number of unknowns (whichis proportional to the number of meshpoints) for afixed accuracy of the simulation.

These applications typically have to deal with strong-ly anisotropic phenomena like shocks and shear lay-ers. Unfortunately for this kind of application the usu-al isotropic mesh adaptation used for 3D flows resultsin an excessive number of elements. The reason forthis is that during refinement of the grid around e.g.the shock wave the grid cells become smaller

uniformly in all directions. However, if an anisotropicadaptation is used, the grid is refined only in the di-rection normal to the shock wave.

Therefore, the goal of this work was to develop ani-sotropic mesh adaptation algorithms coupled with theVKI-developed solvers for steady and unsteady flow.An application of particular interest is the solution ofthe equations of ideal magneto-hydrodynamics (MHD)for simulating space weather, i.e. the interaction of thesolar wind with the earth's magnetic field.

Both the steady and unsteady anisotropic adaptationalgorithms considered in this work rely on remeshingof the grid. A detailed description of the anisotropicgrid generation algorithms can be found in the PhDthesis of A. Athanasiadis [TH16] and previously in this

survey. The remeshing is done in a Riemannian spacewhere the spacing of the cells is uniform. Then the fi-nal spacing (after transforming back to physical space)is a function of the metric field tensor used for the def-inition of the Riemannian space.

For the construction of the metric tensor of the Rie-mann space it is assumed that the solution error canbe estimated from the interpolation error, which isproportional to the second derivatives of the solution.Hence, the metric tensor is based on the evaluation ofthe Hessian matrix of a tracer variable (e.g. Mach num-ber or density), calculated from the solution on an ini-tial (or previous mesh). Several approaches for

Hessian reconstruction have been verified in order tochoose the most robust and efficient one: Hessian re-construction based on Green's formula or based onleast squares fitting on a stencil of neighbours.

Both steady and unsteady mesh adaptation has beenperformed. To save computation time, the unsteadyadaptation is only made after a (fixed) number of timesteps of the CFD solver. Since it is not possible to pre-dict in advance the motion of flow features (e.g. shockwaves), it could happen that the adapted grid is nolonger valid after a few time steps of the solver. In par-ticular the shock wave leaving the refined area candrastically deteriorate the quality of the solution. Thesolution to this problem is to follow the flow featuresin time on the previous grid, which is not yet adapted;then the metric is calculated as an intersection ofmany metric fields, obtained for snapshots of the so-lution taken during the time interval between two gridadaptations.

The numerical results that will be discussed belowhave been obtained using a 2nd order Cell-CenteredFVM with WENO reconstruction. The calculation wasmade using the COOLFluiD Finite Volume solver withscalar dissipation.


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As a first test case, the steady MHD flow througha planar 3D channel with sinusoidal bump is comput-ed. The inlet is supersonic

and the wall is sup-posed to be fully con-ducting; it can beshown that in such asituation, a vanishingnormal magnetic fieldneeds to be imposed.The initial grid and thegrid after the 3rd adapta-tion step are shown onFigure 1. The correspon-ding fields of magni-tude of magnetic induc-tion vector are shown on

Figure 2. It is worth notingthat a drastic improve-ment of the solution couldbe obtained while the num-ber of points was increasedby a factor less than 1.6.

The next test case is the 2D unsteady Euler flow ob-tained by injecting a forward facing step in a uniformflow channel with Mach number of 3.0. The solution ofthis problem has been presented in numerous refer-ences as a flow with complex unsteady shock waves.The initial grid with uniform spacing h = 0.02 is pre-sented on Figure 3. The adapta-

tion time step is equal to 0.02,which means that 150 adapta-tions were performed beforereaching global time t = 3.0. Foreach time step, three levels ofadap-tations were done, grad-ually increasing the number ofnodes. The grid for the final lev-el of adaptation was createdsetting the minimum cell sizeto 0.0015. The number of nodeson the finest grid for time t = 3.0was equal to of 42000. A uni-form grid with equivalent reso-lution would contain ~ 13 x 106

nodes!

The grids used for calculationsfor time t = 1.0 are presented onFigure 4 and the corresponding density fields can beseen on Figure 5. The final grid after the 3rd adaptationstep and its corresponding density field for time t = 3.0can be seen on Figure 6. The improvement of the solu-tion done by means of adaptation is clearly visible - theshock waves are thin, the slip lines starting from triplepoints have been recovered.

Figure 1: Grids used for computing a magneto-hydrodynamic flow problem inside a curved, planar

flux-tube: initial grid (23000 nodes, upper); grid after3rd adaptation loop (36000 nodes, lower)

Figure 2: Iso-contours of magnetic field intensityin the mid-plane of the flux tube: computed

on the original grid (upper);computed after 3 adaptation steps (lower)


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Figure 3: Initial grid with uniform spacing used for calculations without adaptation (7400 nodes)

Grid after third adaptation (22000 nodes)

Grid after first adaptation (8500 nodes)

Grid after second adaptation (10200 nodes)

Figure 4: Grid used to represent the solution at t = 1.0 time units


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Figure 5: Density fields obtained at t = 1.0 time units

Figure 6: Grids and density fields obtained at t = 3.0 time units and three adaptive refinements (42200 nodes)

Grid after first adaptation (8500 nodes)

Grid after second adaptation (10200 nodes)

Grid after third adaptation (22000 nodes)


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RESIDUAL DISTRIBUTION METHODS

FOR STEADY AND UNSTEADY HYPERBOLIC

CONSERVATION LAWS

The development ofResidual Distribution(RD) methods for hyper-bolic conservation lawsis one of the long termresearch efforts at VKI.In the last decade, theRD method has beenshown to be an appeal-ing alternative to finite

volume, finite elementand discontinuousGalerkin methods onunstructured meshes.

During the last years,efforts have been madeto increase the accuracyof the method. Insteadof discretizing in spaceby means of P1 finite el-ements, P2 (or P3) ele-ments are used, yield-ing a parabolic (or

cubic) approximation ofthe solution. Followingthe approach of Abgrall,the element is subdivid-ed in sub-elements. Theresidual on each sub-element is computedand distributed to each node of the sub-element.

The work done on this topic during the last year hasbeen oriented towards three objectives. The first onehas been the extension of the method to the solutionof steady and unsteady viscous problems. This hasbeen done using a Petrov-Galerkin formulation,equivalent to the RD Method in the case of a linearpure advection problem [L33]. The issue was to keepthe accuracy constant for all Peclet numbers (repre-senting the ratio between advection and diffusion). Toachieve this, it is necessary to blend the Petrov-Galerkin formulation with the pure Galerkin method,via a blending parameter dependent on the Pecletnumber and hence on the flow regime. The method isextended to the time-dependent case through a fullspace-time formulation [AJ65] of the mathematicalproblem, which is then discretized in a (d+1)-dimensional space, where d is the number of space

dimensions and time is the additional one. On Figure1, the unsteady rotation and diffusion of a cosine hillis shown. Contour plots of the initial condition and of

the solution after one full rotation are displayed. Theresults clearly show the effect of the (physical) diffu-sion, however, the method preserves fairly well theshape of the initial profile.

On Figure 2, we com-pare the results ob-tained with P1 andP2 elements on thesame problem, butwithout any viscosi-ty. Clearly, the shapeis preserved betterby the higher order

discretization.

The second objectivehas been the con-struction of mono-tone high order RDschemes. The problemis that such schemesmust be non-linear(Godunov's theorem:a monotone schemewhich is linear is atmost 1st order accu-rate). To achieve this,

first a monotone linearscheme is built. Then,it is limited or blendedwith a high orderscheme [LS33]. The lin-ear monotone schemehas been constructed

using a general formulation of the N scheme that hasbeen extended to P2 and P3 elements. The schemes thatwe obtained in this way are not rigorously monotone,but they behave as such. On Figure 3, we report the so-lution of a steady problem with discontinuities, and

Figure 1 : Initial condition (left) and solutionof the viscous unsteady problem after one rotation (right)

Figure 2 : Solution after 3 full turns, obtainedusing P1 elements (left) and P2 elements (right)

Figure 3: Resultobtained withblending schemeon a discontinuousproblem with asource term


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with a discontinuous source term. Thesolution has been obtained with ablended scheme. We can see that

there are no oscillations around dis-continuities.

Last, the third objective of the researchhas been the application of high ordermonotone RD schemes to complexsystems of conservation laws. Up tonow all applications only involve thesecond order version of the schemes,the higher order discretization stillbeing under development. As an ex-ample, Figure 4 shows a Schlieren-like visualization of the simulation ofthe interaction of a shock with a bub-

ble of hot air. The simulation hasbeen run on an unstructured meshwith a second order scheme.Although the mesh used is not par-ticularly fine, one can see the wavystructure of the interface of the bub-ble (middle pictures), giving aglimpse of a flow instability. This is aknown to be a sign of the accuracy ofthe discretization.

Similar results have been obtainedwhen simulating a similar problemfor a two-phase mixture. The details of

the model used can be found in[AJ74]. In Figure 5, we report snapshots of the gas vol-ume concentration obtained when a Mach 3 shock trav-elling in an air-water mixture hits a bubble of almostpure air (95% volume concentration).

Lastly, a thorough study of the application of RDschemes to the simulation of shallow free surfaceflows under the action of the gravity force has beenconducted [6]. First of all, many applications benefitfrom the strong monotonicity-preserving character ofthe developed schemes. As an example of such anapplication, Figure 6 shows the formation of a waterwave due to the break of a dam separating two basinsof water. The difference in water heightbetween the two basins amounts to5 meters.

Additionally, when applied to thesimulation of shallow-water flows,RD schemes can be shown to havethe important property of preservingexactly some physically very impor-tant solutions [AJ87]. One such so-lution is given by the well known lake

at rest state: whatever the shape of the bed of the riv-er is, if there is no flow, the water on the top is per-fectly flat. This is shown on Figure 7, where we reportthe simulated interaction of a water wave due to aninitial perturbation with a hump in the bed of the riv-er. In the figures, where non-dimensional quantitiesare plotted, we see the shape of the bed (in grey) andthe water perturbation (in blue with coloured con-tours) splitting in two halves, one of which interactswith the non-flat bed. The lake-at-rest state ahead ofthe perturbation is perfectly preserved. More such re-sults can be found in [AJ87].

Figure 5: Two-phase shock bubble interaction. Contours of the gasvolume concentration at different times

Figure 6: Break of a dam. Water wave formation

Figure 4: Schlieren-like visualization of a numericalshock-bubble interaction


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CFD ALGORITHMS FOR THE SIMULATION

OF BUBBLY TWO-PHASE FLOWS

Within the IWT project MuTEch (Novel Multiscale ap-proach to Transport phenomena in Electrochemicalprocesses, IWT/SBO 040092), the VKI AR departmentis responsible for the development of efficient CFD al-

gorithms for simulating electrochemical reactors. Thedeveloped CFD methods should allow to predict localparameters of complex electrochemical processes, in-cluding gas bubble formation and transport under tur-bulent flow conditions. This requires the developmentof numerical solvers for dispersed turbulent twophase flow, which should be coupled with software forthe simulation of electrochemical phenomena.

Presently, two different approaches to two-phase flowmodeling are used to simulate dispersed bubblyflows. On the one hand an interface-resolving methodis used to track the bubble interface directly

(microscale approach), while a second approach fea-tures Lagrangian tracking of bubbles inside a contin-uous carrier flow (mesoscale approach).

In the microscale approach, the extended FiniteElement Method (XFEM) is used to perform direct nu-merical simulations of bubble growth, lift-off and co-alescence near electrodes. The main idea behind theXFEM technique is to locally use (slope) discontinu-ous elements near liquid/gas interfaces, in order tocapture, for instance, the pressure jump encounterednear a bubble interface.

The XFEM method is to be combined with level setmethods to track moving interfaces between differentphases. Level set methods have many advantages.They work in an arbitrary number of space dimensions

and can cope with complicated changes in topology.They therefore work even when velocity fields and/ordiscontinuities are very complex.

The underlying principle of the level set method is toadd a scalar unknown to the system of governingequations. The zero eigensurface of this contour coin-cides with the gas/liquid interfaces. Away from inter-faces, ideally this scalar field should correspond to thesigned distance to the nearest point on the interface.Figure 1 shows an advective test case for a full periodof motion. The impressive capabilities of the level setmethod used can be seen from the fact that the finalposition of the interface coincides with the initial po-

sition, even though the calculation was made on avery coarse mesh.

To speed up our level-set implementation, we exploitthe fact that only the position of the interfaces itself isimportant; it is not necessary to know the level set

Figure 7: Interaction of a water wave with a non-flat bed. Preservation of the lake-at-rest state

Figure 1: Advection of a circular interfaceby means of a vortical flow field,using the level set method.


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variable in the whole solution domain. We there-fore compute the level set variable only in a nar-row zone of elements close to interfaces

(the narrow band). We have developedan efficient search algorithm to find theclosest point of the interface and useit to regularly re-initialize the lev-el set unknown inside the nar-row band to the signed dis-tance to the interface.

In the mesoscaleapproach, asolver using anEulerian-Lagrangiantwo-phase flow model isused to track dispersed bubbles in

a continuous flow field. In this tech-nique, the flow field is computed in a single-phase manner, by solving the regularincompressible Navier-Stokes equations, while thegas bubbles are tracked sequentially in a Lagrangianmanner, by solving their equations of motion. Whenthe two algorithms are coupled together, we obtain anaccurate simulation of bubbly two-phase flow.

Flow turbulence is one of the main challenges whentrying to simulate electrochemical reactors of indus-trial interest. Reynolds-Averaged approaches oftenfail, due to the complexity of the flow fields. We havetherefore chosen to use Direct Numerical Simulations

(DNS) instead. In DNS, one resolves all turbulentlength scales on a fine mesh. This is feasible only forsufficiently low Reynolds numbers, which fortunatelyhappens to be the case in electrochemistry (typicalReynolds numbers ~ 10000).

As a DNS solver, we have so far used the hybrid spec-tral/finite-element code SFELES, developed togetherwith the Universit Libre de Bruxelles and BrighamYoung University. This solver is capable of performingDNS in planar and axisymmetric geometries of other-wise arbitrary complexity, with a very high efficiency.

To extend SFELES to dispersed two-phase flow, wehave developed the parallel Lagrangian particle/bub-ble tracking library PlaS, which simulates the dis-persed phase and couples it with the carrier phasesolver using standardized data-interfaces. To efficient-ly fulfill this purpose, the Lagrangian solver also fea-tures an MPI implementation to run on a distributed-memory parallel computing platform.

In this additional module, dispersed entities such assolid particles or gas bubbles are tracked sequential-ly via their equations of motion. They are coupled withthe carrier flow by exchange of momentum in terms

of drag, lift, virtualmass, pressure,gravity and buoy-

ancy forces. We candistinguish between cas-

es where the momentumtransfer from the bubbles to the

carrier fluid is neglected (one-waycoupling) or taken into account (two-

way coupling). For very dilute flows, a one-way coupling approach is suitable. Back-cou-

pling effects should be taken into account whensufficiently high concentrations of bubbles are pres-ent. It is well known that this can significantly influ-ence the turbulent flow structures. For even denserflows, interactions between bubbles, such as hydro-dynamic effects between nearby bubbles and bubblecollisions also have to be modeled.

The PlaS code is designed to model two-phase flowswith bubbles as well as particles. In the validationphase of the solver, particulate flows were used, asthere are more reference results available from the lit-erature for particulate than for bubbly flows. Figure 2shows the particle accumulation around streamwise

vortices in the boundary layer of a turbulent channelflow, seeded with tiny particles.

The development of the microscale and mesoscaletwo phase flow solvers is still ongoing. For the mi-croscale solver based upon XFEM and level set prin-ciples, a working 2D prototype solver has been im-plemented successfully. The next goals are (1) toparallellize this 2D solver and (2) to extend the solverto 3D. As far as the mesoscale solver is concerned, thecoupling between the DNS solver SFELES and theLagrangian tracking solver PlaS outlined above is al-ready fully operational. A set of numerical test cases(isotropic turbulence, channel flow, parallel-plate re-

actor) has been set up. First results for one-way as wellas two-way coupled dispersed two-phase flow havebeen obtained and will be published as soon as pos-sible. Further research may cover a variety of topics,such as the development of a bubble-bubble collisionmodel, the development of a numerical model thattakes into account the effect of coalescence and break-up of bubbles, and the study of turbulent wakes be-hind bubbles of high bubble Reynolds number. We willalso couple the currently developed solver for two-phase bubble flows with an electrochemical solver ca-pable of simulating multi-ion mass transport.

Figure 2: Particle accumulationaround streamwise vorticesin the boundary layer

of a turbulent channel flow


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DIRECT AND LARGE EDDY SIMULATION

ON UNSTRUCTURED GRIDS

Over the past four decades, three main approacheshave been developed to numerically study turbulentflows. The fastest, but least accurate approach, con-sists in solving the Reynolds-Averaged Navier-Stokes(RANS) equations. Turbulent fluctuations are averagedout in time in order to obtain a pseudo-steady Navier-Stokes formalism, in which additional Reynoldsstresses have to be modeled in order to represent tur-bulent transport. The most rigorous and costly ap-proach to turbulence consists in resolving all turbu-lent length and time scales, by solving the unsteadyNavier-Stokes equations on a very fine mesh for manysmall time steps. The total computational cost of suchDirect Numerical Simulations (DNS) can be shown toscale like the third power of the Reynolds number; forthis reason, only flows at low to moderate Reynoldsnumbers can be treated. Large Eddy Simulations (LES)can be seen as a compromise between RANS andDNS. The larger turbulent structures are fully resolvedin space and time, whereas small-scale structuressmaller than the size of the grid cells are supposed tobe sufficiently random to be amenable to statisticalmodeling.

Due to their high cost, DNS or LES of incompressible

turbulent flows are often restricted to academic con-figurations with trivial boundary conditions, for whichthe full power of pseudo-spectral methods can be har-nessed: superior accuracy combined with high com-putational speed and very low memory overhead. Theuse of higher-order finite-difference methods on mul-ti-block structured meshes allows to consideration ofmore challenging flows. Nevertheless, geometries re-main fairly simple; for instance, one couldstudy flows in cavities or around a cube.Recently, turbulence researchers have start-ed to explore the use of finite-element meth-ods (FEMs) on unstructured meshes, whichin principle can handle entirely general

geometries. For instance, using FEMs, itshould be possible to perform LES aroundcar-shaped geometries for the automotiveindustry.

Considerable work remains to be done before FEMswill become accepted by the wider turbulence com-munity. The feasibility of FEMs has not yet been suffi-

ciently demonstrated for the standard DNS and LEStest cases. Also, the dispersive and dissipative be-havior of FEM schemes requires deeper investigation.Last but not least, unlike for finite-differences, classi-cal FEMS inherently require the solution of a largesparse linear system at each time step; it is not obvi-ous to do this in acceptable computation times.

Rather than to directly create an unstructured FEMsolver for flows over fully 3D geometries, it has beenchosen to first address these open problems using amore lightweight hybrid spectral/FEM research code,which allows the simulation of fully 3D flows aroundplanar or cylindrical geometries of otherwise arbitrary

complexity. The first prototype of the SFELES(Spectral/Finite-Element Solver for LES) solver waswritten at the VKI / Utah State University in 2002[TH25]. Since then, the development of SFELES hasbeen continued by team of researchers at the VKI, theUniversit Libre de Bruxelles and Brigham YoungUniversity (Utah). Modern software managementtechniques have been an essential tool to make thisinternational collaboration a success; at present, aboutten researchers are contributing to SFELES on a reg-ular basis.

The basic principle behind SFELES is illustrated inFigure 1. The unsteady incompressible Navier-Stokes

equations are discretized in 2D cross planes using (lin-ear) triangular FEMs. The transverse (planar geome-tries) or azimuthal (cylindrical geometries) direction isrepresented by means of a truncated Fourier series,which assumes periodicity in the z direction. Complextwo-dimensional geometries can be accommodatedthrough the unstructured in-plane mesh, while stor-

age requirementsare reduced be-cause only a singletwo-dimensionalmesh and associat-ed overhead needbe stored.

To advance the solu-tion in time, at eachnew time step a fully3D linear problemneeds to be solved.From a computation-al point of view, themain advantage of

the hybrid spectral/FEM approachis that this 3D linear problem decouples into a se-

ries of much more affordable 2D linear problems,

Figure 1: Typical hybrid spectral/FEMmesh for incompressible circular cylin-der flow, where linear FEMs are em-ployed in two-dimensional cross planes,while a truncated Fourier series is usedin the transverse direction. Shown areisosurfaces of vorticity magnitude atReynolds number 8000 (DNS result)



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147

which can be solved using the fastest linear solverscurrently available. One 2D linear problem needs tobe solved for each mode in the truncated Fourierseries.

This approach has been implemented for use on dis-tributed-memory parallel computers as follows: the2D FEM mesh is partitioned in physical space. Eachprocessor then takes care of the calculations on a giv-en mesh partition. To avoid the need for parallel linearsolvers, data belonging to the different Fourier modesis gathered on individual processors before each lin-ear solve. The linear systems are then solved sequen-tially in Fourier space, after which the updated solu-tion is scattered back to the mesh partitions in physicalspace. This hybrid parallellization concept is illustrat-

ed in Figure 2. The main philosophy behind this par-allellization is that all parallel communications arenow grouped in two large collective communicationsteps. This approach is found to work very well onmodern LINUX clusters, which have large bandwidthsbut tend to suffer from latency problems.

SFELES is currently used as the vehicle for applied andfundamental research carried out by several scientists.For instance, at the VKI, SFELES is used to simulateturbulent bubbly two-phase flows and particle-ladenflows in electrochemical reactors. Also, SFELES isused to develop new unsteady solution methods,which are much faster and more robust than existing

methods for certain classes of problems (Figure 3).Finally, SFELES is used as the test-bed for newAlgebraic MultiGrid (AMG) techniques. AMG methodsare state-of-the art linear solvers, which reduce all er-ror frequencies simultaneously, by automatically gen-erating a sequence of increasingly coarse approxima-tions of the original fine grid linear problem. The gainin performance when using AMG techniques, ratherthan conventional linear solvers, is found to be im-pressive both in terms of computational times andmemory usage. For instance, the SFELES team has

recently run a 50 million node DNS calculation on 100nodes of the MaryLou supercomputer at BrighamYoung University [AJ88].

The main development efforts on SFELES for the nearfuture consist in replacing the stabilized finite-elementmethod currently used by a P1-iso-P2 method, whichdoes not require any artificial stabilization mecha-nisms. This approach can be shown to have many de-sirable properties for DNS/LES; for instance, it can beshown to preserve the discrete kinetic energy for in-viscid problems. Once this method will have been im-plemented, we will consider variational multi-scalemethods and perform large LES calculations with thecode.

Figure 2: Illustration of the parallelpartitioning schemes employed inSFELES (P0 through P3 represent

different processors).(a) Physical space partitioningused for calculation of the termsin the matrix equation.(b) Fourier space partitioningused for the solution of thelinear systems, which allowsthe use of multiple instancesof a serial linear solver

Figure 3: Instantaneous velocity magnitude andstreamlines obtained for the DNS of the incom-pressible flow over a circular cylinder at a highReynolds number. Preliminary test result,obtained using our new fast unsteady algorithmon an extremely fine 2D mesh in a much shortertime than possible using other existing algorithms

2006, von Karman Institute for Fluid Dynamics, Rhode-St-Gense, Belgium


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