Progress in Aerospace Sciences 46 (2010) 274–283

Contents lists available at ScienceDirect

Progress in Aerospace Sciences

0376-04

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/paerosci

Level sets for CFD in aerospace engineering

H. Xia a, P.G. Tucker a,�, W.N. Dawes b

a Whittle Lab, Department of Engineering, University of Cambridge, 1 JJ Thomson Avenue, Cambridge CB3 0DY, UKb CFD Lab, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

a r t i c l e i n f o

Available online 8 April 2010

21/$ - see front matter Crown Copyright & 2

016/j.paerosci.2010.03.001

esponding author. Tel: +44 1223 337582; fax

ail address: pgt23@cam.ac.uk (P.G. Tucker).

a b s t r a c t

In the past two decades, the level set concept has been extensively explored. The superiority of the

differential level set to other more ad hoc methods as a formal framework for directly/indirectly solving

‘front-propagation’ natured problems is now fully established. Nowadays, in many areas of aerospace

related computational fluid dynamics, applications of level set methods can be found. This paper gives a

brief review of these applications and how level sets can be useful in tackling challenging

computational aerospace problems. The use of level sets in premixed turbulent combustion, aero-

acoustics, geometry definition/morphing, meshing and turbulence modeling is explored in detail and

other applications discussed.

Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

2. Level set equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

2.1. Hamilton–Jacobi type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

2.2. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

3. Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

4. Aero-acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

5. Geometry description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6. Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7. Turbulence modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

8. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

1. Introduction

By definition, the set of variables x(x1, x2,y,xn) for which a real-valued function f(x) is equal to a given constant is a level set. If n¼2,the level set is a plane curve known as a level curve, and for n¼3 thelevel set is known as a level surface. However, despite beingconceptually simple, the link between level sets and moving fronts/interfaces in an n-dimensional space has only been discovered anddeveloped significantly during the last 15–20 years. As noted bySethian [1], the development of level set techniques for complexphysical or engineering problems is frequently not trivial, requiringcareful thought as well as a little artistry. Due to the simplicity andflexibility provided by an arbitrarily customizable speed function,

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level sets have become a framework for modeling the evolution ofboundaries, interfaces and advancing fronts. They are more efficientthan the traditional techniques for tracking interfaces such as themarker/string [2], volume-of-fluid [3] and gradient approximationmethods [1]. This explains why level set type methods have beenapplied to many areas of computational science and engineering insuch a short time, ranging from fundamental physics and chemistry,fluid mechanics, combustion, image processing, material science,computer vision, computer-aided design, optimal control theory andmany others. Readers are referred to [1] for a review on generalmethods and applications of level sets.

Due to the broad application areas of level sets, it is difficult tosummarize all applications in one short article. Hence, here wewill mainly focus on the vigorously developing application field ofcomputational fluid dynamics (CFD) where level sets have recentlyplayed major roles. These include, for example, combustion flamefront tracking [4–6], acoustic front/ray tracing for aero-acoustics

rights reserved.

Nomenclature

a speed of soundd wall distance~d modified wall distancedc starting distance of the mixed zonedA, DA surface elementdV , DV volume elementF speed functionf fluxf ðfÞ, gðfÞfunctions of fG the level set scalar in flame front trackingH HamiltonianHij(d) Hessian matrixn outward normal unit vectorL characteristic length scalem, n positive integersNv, Ns number of volume, surface pointssl laminar burning velocityt physical time

U pseudo-velocity, U¼rfUn normal pseudo velocity, Un ¼U � nv flow velocityx Cartesian coordinates

Greek symbols

b switch for stationary Hamilton–Jacobi equationd user specified Laplacian thresholde small numbere0 Laplacian coefficiente1 damping coefficientk curvaturel sound wave lengthf the level set variable, first arrival time and the eikonal

variablec Poisson equation auxiliary variablet pseudo-timeO control volume@O control volume boundary

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283 275

[7,8], implicit surface for geometry definition [9,10] and deforma-tion/morphing [11], level set based medial axis extraction for multi-block structured [12–16], overset/chimera and hybrid meshing [17],and distance functions for turbulence modeling [18–20].

Perhaps the time-dependent flame front in premixed turbulentcombustion is the best case to demonstrate the power of levelsets. With level set’s implicit representation, the flame front canbe easily tracked by a scalar hyperbolic partial differentialequation of the G variable, e.g. defined by Williams [21]. Basedon the time-dependent flow (burnt/unburnt gases) information,the position of the flame front can be evaluated in both time andspace. Likewise, the gradient of the level set function representsrays radiated from a boundary. The rays and the level sets are thedual properties of the level set function, which are akin to thestream and the velocity potential functions in a potential flowanalysis. This aspect is particularly compatible with ray tracingproblems, such as high frequency acoustics and aero-optics [22],avoiding computationally crude and potentially expensive La-grangian techniques [23]. By using the acoustic differentialequation based level sets, the solution procedure becomesrelatively simpler and naturally parallel. The third applicationis the so-called ‘implicit’ geometry or surface. The explicit wayof describing geometries involves complex CAD systems torepresent the boundaries, abbreviated as BREP (boundary repre-

sentation) [9], but when geometric complexity increases anddifferent parts of the geometry come from different CAD systems,such representations become conceptually and practically flawed.With implicit surfaces, the boundary of the geometry is notexplicitly described. Indeed it is defined by the 0-level set of thedistance function in a pixel/voxel space. This shift of paradigm candramatically increase the capability of CFD when dealing withcomplex geometries in a parallel computing environment. Thelast theme this survey will cover is meshing related automationissues. Several decades of CFD practice suggests automated meshgeneration will play a crucial role, leading the path to a new era ofcomplex geometry flow modeling with design-optimization. Aswill be seen in later sections, level sets can help enormously interms of automatically extracting the medial axis of a shape/domain (to assist, for example, with mesh generation), definingoverset mesh interfaces, and mesh hybridization.

Exact, nearest surface distance functions are one of the moststraightforward and fundamental applications of level sets. One canthink of the iso-values of distance function d(x) as the frontspropagating from the boundary, where the propagating speed isconstant. The distance is then equal to the first arrival time of thefront. The distance function has direct uses for many problems, forexample, accurate wall proximity is needed for turbulence modeling,to account for the effects of walls on turbulence. Before use of thelevel set differential equation to acquire d(x), inefficient methodswere generally used. For example, the crude search for the nearestboundary vertex to the vertex of interest can take up to OðNvNsÞ

operations where Nv and Ns are the number of interior and boundaryvertices, respectively. This can be much improved using differentialequations. For example, the level set equation methods [24,25,15]can achieve OðNvlogNvÞ and even O(Nv) in practice. Differentialequations (other than classic level set equations) such as the Poissonequation [26–28] have also been studied over recent years withsimilar hopes of achieving efficiency whilst also hoping to improveturbulence physics modeling. Level sets can also be applied toinvestigate more physically natural front propagation problems.Nevertheless, all applications are based on the same simple level setequations, which will be discussed next.

2. Level set equations

2.1. Hamilton–Jacobi type equations

We start with the general framework (suggested by Sethian[1]) of level set equation (1), and explain the simplified form andits physical meaning in relation to the front propagation. TheHamilton–Jacobi equation for level sets in either boundary orinitial value form can be expressed as ðe-0Þ

b@f@tþHðrf,x,bÞ ¼ er2f ð1Þ

with Hðrf,x,bÞ ¼ Fjrfj�ð1�bÞ. It is apparent that when b¼ 1Eq. (1) becomes the initial value level set formulation:

@f@tþFjrfj ¼ er2f ð2Þ

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283276

which is particularly useful for tracking time-dependent inter-faces such as the flame front of turbulent combustion. Recallingthe level set definition, that is to say tracking the interface or frontis equivalent to obtaining xf such that fðxf ,tÞ ¼f0 in time, wheref0 is a constant and xf describes the interface/front position.

Although different in form, a boundary value formulation canalso describe the wave front propagating nature. The stationary1

eikonal equation (or boundary value formulation) also withhyperbolic nature [1] can be obtained when b¼ 0:

Fjrfj ¼ 1þer2f ð3Þ

where the dependent variable f describes first arrival times ofpropagating fronts from boundaries, and F(x) is the local speedfunction of these fronts. The distance function is then simplyd¼ Ff, if F � const: With e-0, Eq. (3) can be solved by numericalschemes with just enough dissipation to gain an entropy(physically sensible) solution [1,29]. As shown by Tucker [19],the right hand side Laplacian is useful and often employed tocontrol the front propagation velocity. The meaning of Eq. (3) canbe interpreted as a multi-dimensional extension from the simpleone-dimensional front traveling relation: distance¼ speed� time,i.e. dx¼ F df or 1¼ F df=dx. In multi-dimensions, rf is orthogo-nal to the level sets f, and its magnitude jrfj is, similar to df=dx

in one-dimension, proportional to the inverse of speed. HenceFjrfj ¼ 1, which is exactly Eq. (3) with e� 0.

2.2. Numerical methods

Most applications use finite differences to numerically solveEq. (2) on Cartesian grids. One side upwinding schemes areemployed to discretize the gradient rf, followed by timeadvancement. However, to track all the level sets throughoutthe entire computational domain would cost enormous comput-ing time. The estimate by Sethian and Smereka [30] shows that ina two-dimensional calculation on an N�N grid it would requireO(N2) operations for each time step. The key point here is that totrack all level sets is clearly a waste, because the interest is alwaysconfined only to the 0-level set or the level set at f0. To overcomethis problem, Adalsteinsson and Sethian [31] introduced the ideaof narrow band approach (NBA), which limits the active region ofcalculation to a thin region around the 0-level set. Thus, thecomputation is greatly reduced to O(kN), where k is a constant.

Finite difference solutions have also been broadly explored forthe eikonal equation (3) on Cartesian grids using fast methodsbased on causalities such as the fast-marching method (FMM) [24]and fast-sweeping method (FSM) [25]. By nature, FMM is quiteefficient, requiring an intricate complex data structure to achieveO(Nv log Nv) complexity. On the other hand, the Gauss–Seideliteration based FSM is intrinsically O(Nv) but has to limit thespeed function so that the level sets of d have only simplecharacteristics. A similar approach has also been proposed in arecent study [32], called fast-iterative method (FIM).

Sometimes solutions on unstructured meshes are required.Kimmel and Sethian [33] extended FMM for triangulated meshes,while Qian et al. [34] suggested the FSM is also extendable to suchmeshes. However, the limitation is that both FMM and FSM haveto restrict triangulations to be acute in type. Another extension ofFMM to three-dimensional tetrahedral meshes and even arbitraryfinite elements is studied by Elias et al. [35]. In the recent work ofTucker [29,19] and Xia and Tucker [15], the eikonal equation issolved in a form compatible with existing CFD solver frameworks.This is achieved through the introduction of an auxiliary velocity

1 The stationary level set equation is mostly adequate when there is no

backward propagation, i.e. F40. The level sets can be projected to t¼0 plane.

variable. On computational efficiency, they find that in practice alinear complexity of O(Nv) can be achieved with multigridconvergence acceleration. This finite volume approach, expressedasZO

@f@t

dVþ

I@OfU � n dA¼

ZO

1

F2þ f ðfÞr2fþgðfÞ

� �dV ð4Þ

using pseudo-time t and a pseudo/auxiliary velocity U¼rf, doesnot limit the type of the mesh or the shape of the element and isespecially useful for customizing speed functions. For example,various forms for f ðfÞ and gðfÞ, to obtain modified distancefunctions, are suggested in Refs. [18,19]. The common iterativeform can be expressed as a modified speed function:

~F ðx,fÞ ¼FðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ½f ðfÞr2fþgðfÞ�F2ðxÞq ð5Þ

This is a more general form of that which Osher and Sethian[36,37] suggested earlier, namely the speed F can be a function ofcurvature k, such as FðkÞ ¼ 1�ek.

3. Combustion

Perhaps the most important energy source for industry anddaily lives is via combustion. For example, in aero-engines, thecombustor is one of the most critical parts. A good combustordesign will significantly improve the efficiency—the thermalefficiency of a gas turbine engine is the area where the greatestgains in performance can be made. Premixed turbulent combus-tion is a highly complex process. A good model for premixedturbulent combustion should take into account the interactionsbetween turbulence, chemical reactions and flow. Due to the factthat the chemical length scale, namely the flame thickness, ismuch smaller than the turbulent length scale—Kolmogorov, theflame is regarded as a thin layer. Hence its inner structure is notinfluenced by turbulence, and therefore can be treated as locallyone-dimensional. This is the so-called flamelet regime [38].

Based on the flamelet modeling assumptions, the G-equationmodel proposed by Williams [21] essentially uses a level setmethod to describe the evolution of the flame front as an interfacebetween the unburnt and burnt gases, illustrated in Fig. 1.Because the frame front can move both forward and backward intime, the time-dependent or the initial value level set formulationis best suited. The level set function G is a scalar field defined suchthat the flame front position is at G(x, t) ¼ G0. The implicitrepresentation of the front can be derived by differentiating G(x, t)in time, i.e.

@G

@tþ

dxf

dt� rG¼ 0 ð6Þ

where the local position of the frame front xf satisfies

dxf

dt¼ vþsln and n¼�

rG

jrGjð7Þ

Fig. 1. Schematic of instantaneous and averaged/filtered flame front position.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283 277

In the above, v is the flow velocity, sl the laminar burning velocityand n the unit normal of the flame front. Re-arranging (6) and (7),one reaches the instantaneous level set equation of G:

@G

@tþð�slÞjrGj ¼ �v � rG ð8Þ

The advantage of the level set approach here is that it allows thecomplicated physics and chemistry of the flame front to be linkedto the interface propagating at velocity sl. To reduce thecomputational costs of direct numerical simulation (DNS) coupledwith Eq. (8), both RANS [39,4] and LES [40,41,5] based numericalsimulations were performed with encouraging results. Becausethe above equation is only valid for the instantaneous flame front,for RANS the Favre-averaged level set G-equation was suggestedby Oberlack et al. [42]. Pitcsh [41] proposed an LES filter forEq. (8). Both time-averaged and spatially filtered G equations canbe solved with the efficient NBA numerical method.

4. Aero-acoustics

The eikonal equation (3) is also the high frequency limit to thefull wave equation [8]. Since it is the high frequencies that can bemost annoying to the human ear, eikonal equation solutions arepotentially useful for sound ray calculations and also aero-opticsin relation to studying guided missile systems. The eikonalequation is based on ray theory. This is valid for l5L orfL=ab1, where L is a characteristic length scale; l, f and a beingthe wavelength, frequency and speed of sound, respectively. Theacoustic eikonal equation can be derived in a variety of ways. Forexample, by performing a Fourier transform in space and time ofthe linearized Euler equations, and keeping just the highest orderterms in frequency (e.g. Colonius et al. [43]) leads to

L2a2jr ~fj ¼ 1�Lu � r ~f ð9Þ

In the above u is the velocity of the flow and ~fðxÞ is the non-dimensional eikonal where f¼ ~fL. This is proportional to thepropagation time of the sound ray. Broadly, since Eq. (9) isessentially consistent with Eq. (3), the solution approach for theacoustic eikonal equation is the same as the methods discussed inSection 2.2. Although the Cartesian grid (or pixel/voxel) basedmethods can be applied without any problem, once the main flowfield is obtained, the finite volume approach adjoint to the mainflow equations seems more convenient and straightforward,despite the Mach number restrictions of the approach (see Tuckerand Karabasov [8]). Once the solution ~f is obtained, acoustic rayscan be calculated from r ~f field.

Fig. 2. Conceptual noise shielding design, (a)

Blokhintsev [44] first studied sound propagation in turbulentatmospheric flows using the eiknoal equation and noted that rayvelocity is equal to the sum of the fluid velocity and sound speeddirected along the wave normal, making ray tracing a commonhyperbolic problem. Colonius et al. [43] solved the full Navier–Stokes comparing results with ray-theory, namely characteristicconstruction in the Lagrangian frame of reference. Freund [45],Khritov et al. [46], Secundov et al. [7] and Tucker and Karabasov[8] directly solved the eikonal equation to obtain acoustic rays.Agarwal [47], Moore and Mead [48] and Tucker and Karabasovshowed that, apart from theoretical studies the eikonal equationcan be a valuable acoustic design engineering tool. For example,with aircraft, the engines can be located so that the airframeshields observer noise, thus reducing the environmental impact.Fig. 2 shows a conceptual design (a jet engine above a wing) fornoise shielding based on an unstructured solution of eikonalequation.

Fig. 3 shows a cross section in the downstream of a large eddysimulation of a jet. Acoustic sources are placed at inside the jetmixing layer. The spread of these rays computed from the eikonalequation is clearly shown and is strongly influenced by the jetturbulence. Note, for these simulations the approach of Avila andKeller [49] is used. We note that, analogous to the work ofOberlack et al. [42], Khritov et al. also explored the use of a RANSconsistent form of the acoustic eikonal equation. This was againapplied to jets.

5. Geometry description

There are a number of current trends in CFD simulations. Oneis for ever larger model sizes. For example, in an aero-engine a fullannulus turbine stage with modest resolution can take up to100M nodes. The second is the coupling of CFD with otheranalysis tools such as finite element analysis for the solid. Anothertrend is to gather all this together in ‘process chains’ for routineindustrial exploitation—leading towards automated design andoptimization. The common and most important element herewhich links all others is the ‘geometry’.

Dawes et al. [10] outlined the process chain for the currentsimulation paradigm and the possible future pattern. These areshown in Fig. 4. Even though the final quality of the predictedfluid dynamics might be satisfactory, the current process chain isfar from ideal. There are a number of critical, serial bottlenecks,impeding data flow. Critical tasks like geometry ‘carving’ andmesh ‘stiching’ and ‘chunking’ have to be performed manually.This severely affects the design/optimization cycle since changing

domain and (b) acoustic eikonal solution.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283278

geometry is the key activity of a design engineer. A far preferableprocess chain is sketched in Fig. 4b.

To achieve this, as Dawes suggested [9], the currentorthodox CFD operating pattern has to be changed. The essenceof this change is to integrate a geometry kernel based on thelevel set approach into the meshing, flow solving and post-processing system. Traditionally, there are two major ways torepresent geometry, in other words solid modeling. Theseare termed constructive solid geometry (CSG) and BREP. CSG isbased on the definition and manipulation of simple analyticalbodies. It is simple but has difficulty in representing free-formshapes often encountered in aerospace engineering. BREP on the

Fig. 3. Eikonal equation ray tracing viewing from jet downstream towards

upstream, from Secundov et al. [7] with permission.

Fig. 4. Current and future paradigm of simulations. (a) A typical

Fig. 5. Bodies represented by clipped distance fields: (a) schematic of implicit surface d

mesh.

other hand is based on combining patches, edges and topologybindings into ‘watertight’ solids. Although conceptually reason-able, the biggest disadvantages of BREP is its impracticalrequirement of producing and manipulating ‘watertight’ solidsunder constant changes.

A third way for solid modeling is spatial occupancy (SO), whichconsists of Cartesian hexahedral cells. It is simple, generic andmore importantly naturally linked with the concept of the‘implicit’ surface [50,51]. By definition, an implicit surface is the0-level set of a signed distance field d in space. The marked solidCartesian cells are treated as the boundary. Then the level setequation is generally solved using fast approaches such as FMMand FIM to obtain the signed distance field. The set d¼0 is so-called implicit is because it does not necessarily need to berepresented in any explicit geometric form. For example, Fig. 5ashows a schematic of the implicit surface describing a boundary.A signed d field for a turbine section (with cooling holes) and thecorresponding Cartesian mesh are shown in frames (b) and (c).A more complex example, involving a 3D turbine blade with ashroud is shown in Fig. 6. Frame (a) gives the CSG representedgeometry and (b) the corresponding distance field. In frame (c)the implicit surface at d¼0 is highlighted. A last case is theundercarriage area of a Boeing-747 aircraft in its full landingconfiguration, shown in Fig. 7. The viscous type Cartesian meshcaptures well the geometrical features and refines according tothe local complexity.

Since the geometry is now represented by a field variable d,simple means of editing the d field can be employed fortopological changes. Typical solid modeling operations from CSGare Boolian sums of the form: C ¼ A [ B (standing for the union ofsolid A and B), and now replicated in distance fields via the simpleand inexpensive voxelwise operation:

dC ¼minðdA,dBÞ

process chain. (b) An end-to-end fully parallel process chain.

escribing a boundary, (b) a periodic turbine blade section, and (c) Octree Cartesian

Fig. 6. 3D turbine shroud geometry: (a) conventional geometry description, (b) distance function and (c) implicit surface.

Fig. 7. Undercarriage of a Boeing-747 in full landing configuration with the

viscous mesh generated based on the implicit surface rendered geometry.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283 279

Dawes [9] pointed out this forms the basis for geometry sculpting.This is also true for the wider topic of 3D shape metamorphosis,see for example Breen and Whitaker [11]. As noted in their work,the use of level sets for modeling surfaces as iso-values of adensely sampled scalar function over a space is of great benefit.This is especially true since surface movements can be encoded aschanges in the gray-scale values of the voxels without any explicitsurface representation.

6. Meshing

Level sets have also proved to be of great benefit for evaluatingthe medial axis transform (MAT) [15,16], automatic hole cutting foroverset meshes [17], and near wall hybrid viscous meshgeneration [10]. Wang [52] recently summarized some applica-tion of eikonal equation for offsetting geometry boundaries sothat the near boundary prismatic mesh can be automaticallygenerated. As ever, automated hex meshing is a key topic. This isdue to the demand from the new simulation requirements. Forexample, in the CFD, especially large-eddy simulation (LES), asTucker [53] summarized, it is generally accepted that high qualityhexahedral meshes are most preferable on accuracy grounds.Also, if used as part of a structured grid solution, high computa-tional efficiency can be gained relative to unstructured solvers.Manual meshing (boundary conformed) is extremely time con-suming for the increasingly complex geometries engineers arenowadays handling. In the general fields of shape analysis andsolid modeling, especially automated meshing, obtaining themedial axis for a given geometry (or shape/domain) is regarded asan essential step [54,55,12–14]. Efficient and robust techniquesare therefore needed.

Xia and Tucker [15,16] proposed a d-MAT approach hybridiz-ing the level set distance function and thinning techniques viasimple extracting criteria based on the Hessian. They found thatthe combination of using level set differential equations andthinning techniques is more robust than the pure geometricapproaches, such as the Voronoi diagram. The key step for the

MAT construction here is to detect the medial axis feature in agiven d field. Notice that the unique property of a medial axispoint is that it has equal distance to multiple boundaries. Hence,in x�d space, the medial axis represents the ‘local maxima’ ornon-smoothness of the distance function d(x). The criteria arebased on the threshold of r2d or the determinant of the HessianHijðdÞ ¼ @

2d=@xi@xj. Often, the marked medial axis features need tobe thinned. This can either be achieved by applying the a-shapemethod [56,57] recursively for arbitrary point cloud extraction, orthe pixel/voxel thinning techniques [58,59] for uniform Cartesianpoint sets. Fig. 8 demonstrates the d-MAT approach for a 2Dcompressor passage. Frame (a) shows the level set field andHessian identified medial axis zone. Frame (b) shows theextracted and thinned medial axis. Finally, frame (c) shows theresulting mesh.

Level sets are also ideal for identifying computational inter-faces. For bodies with relative movement, the use of overset gridsoften proves robust and efficient. Then the key issue becomes howto define the inter-grid boundaries. Nakahashi and Togashi [17]use a local level set d solution to determine the classification ofvertices and the hole cutting. Fig. 9a shows a wing section withslat and flap. In Fig. 9b, 2D unstructured overset meshes for theslat, wing and flap are shown, and the interface between eachdomain is defined by medial axes. An inviscid flow solution (atMa¼0.3, AoA¼51) is obtained on the overset mesh shown inframe (c). The Mach number contours and surface pressurecoefficient are shown in frames (d) and (e), respectively.

Another main issue often encountered by airplane designers[60] is to define a boundary for the prismatic advancing meshgenerator (a highly industrialized meshing tool) to extrude from asurface mesh so that it will not collide with itself. This isespecially difficult for concave regions, such as the joint area ofthe fuselage and wing of an aircraft. The level set distance solutionfor complex configurations can easily be obtained, see Fig. 10. InFig. 11, we take a cross section of the DLR-F6 wing-bodyconfiguration and extrude the meshes separately from thefuselage and wing surfaces. As shown in frame (a), the oversetgrids stop extruding once the medial axis is passed. In frame (b),this happens in a similar way, but the extrusion stops beforereaching the medial boundary to leave an unfilled region fortriangular elements. The level set based medial axis does not justneed to be ‘equal’ distanced but can also be ‘biased’. Xia andTucker [61,15] use different constant level set speed functions F tocreate biased medial axes. For example, in Fig. 11c, the medial axisis biased towards the fuselage giving control to the meshgeneration process.

Near wall advancing structured grid generation for hybridmeshes is also important for many applications. This is mainlybecause of the demand of RANS simulation which requires viscoustype meshes. Hyperbolic structured mesh generators [62] arepreferable for extruding surface meshes into near wall regionsleaving the remaining domain for unstructured tetrahedralelements. Hence, the hyperbolic natured eikonal equation (3) issuitable for this task. Sethian [37] first explored this capability.

Fig. 8. Medial axis for domain decomposition: (a) d solution and marked medial axis, (b) thinned medial axis (CAD entity), and (c) a sample structured mesh based on the

division.

X

Y

-3

-2

-1

0

1

2

3

MachNumber:

X

Cp

-1

-4

-3

-2

-1

0

1

2

30 2 4 6 0 1 2 3 4 5

0.050.325 0.6

Fig. 9. Determination of the inter-grid boundary for unstructured overset grids: (a) 3D wing section with slat and flap deployed, (b) inter-grid boundary by equal medial

axes, (c) inter-grid boundary by biased medial axes, (d) Mach number contours, and (e) Cp distribution.

Fig. 10. The DLR-F6 wing-body configuration. (a) Distance level sets contour on a cut plane through the wing-body-nacelle; (b) level set iso-surfaces at 5, 10 and 20.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283280

Fig. 11. Collision boundary detection using medial axis: (a) overset grids, (b) hybrid grids, and (c) biased medial axis.

X

Y

0.6

0.4

0.6

0.8

1

1.2

X

Y

0.6

0.4

0.6

0.8

1

1.2

0.8 1 1.2 1.4 1.6 1.8 0.8 1 1.2 1.4 1.6 1.8

Fig. 12. Near wall advancing layer meshes using level sets from curvature dependent speed functions: (a) F¼1 and (b) F ¼ 1�0:5k.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283 281

Other work, such as Gloth [63] and Dawes [9,10] also found levelsets are specially useful. To explain this, we can use the previouswing-body cross section case. In Fig. 12, the level set d solutionautomatically gives mesh lines parallel to the boundary and rd

forms the mesh lines orthogonal to the boundary and level sets.With different speed functions, dependent on various quantities(e.g. curvature k), we can produce different representations of theboundary curvature as the level set moves away from theboundary. In short, the current curvature formula (used in frame(b)) reduces the influence of the boundary curvature and smoothsout the concave feature, which is more preferable in many cases.

7. Turbulence modeling

In contrast to Eq. (3), other differential equations describing the‘distance’ function/level-set are also possible. These methods werespecifically sought to find the wall proximity for turbulencemodeling and, we note again, here, unlike searches, are readilyparallelizable. This is vital for the current trajectory of CFD, i.e.dealing with calculations of the increasing scale. One such differentialdistance function approach goes back to the work of Spalding [26], inwhich the Poisson equation of an auxiliary variable c is solved:

r2cþ1¼ 0 ð10Þ

Near walls, Eq. (10) becomes: @2c=@n2 ¼�1, where n is the surfacenormal. After integrating this relation twice (with c¼ 0 at n¼0), d isfound as the roots of the resulting quadratic equation:

d� n¼�@c@n

��������7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@c@n

� �2

þ2c

sð11Þ

where the negative solution is usually discarded unless some localestimate of system scale is needed. Eqs. (10) and (11) overestimatewall distance around sharp convex surfaces and underestimate themaround concave. This has consistent traits with the solid angle basedintegral equation for distance function equation proposed by Launderet al. [64] and Spalart [65]. Indeed Eqs. (3) and (5) with gðfÞ ¼ 0 andf ðfÞ ¼ cf where c is a constant replicates these traits which arecompatible with improved turbulence modeling physics. Tucker andLiu [20] give examples of the use of differential distance functionequations in Reynolds-averaged Navier–Stokes (RANS) turbulencemodeling.

The eikonal equation is highly detached-eddy simulation (DES)compatible [66,27]—the near wall distance field can be naturallymarched to the modeled LES length scale and when reached thepropagation naturally terminated, i.e. ~d ¼minðd, CDESDÞ, whereCDES is a coefficient and D the local grid spacing, as shown inFig. 13a. It can also be used for hybrid implicit LES-RANS. This,in its most DES reminiscent form, needs the exact near walldistance field for RANS. However, the implied modeled lengthscale in the implicit LES (ILES) zone needs to be zero. These twovery different implicit modeled length scales can again beconnected using Eqs. (3) and (5), where a modified speedfunction ~F ðx,fÞ can be devised to diminish to zero in the mixedzone between RANS and ILES.

Fig. 13b shows candidate distance functions, where the solidline is the exact wall distance and lines with symbols are modifieddistances from Eq. (4) using different f ðfÞ and gðfÞ. For instance,Tucker [18] suggested the forms of f ðfÞ ¼ e0f and gðfÞ ¼e1ðf=dcÞ

n, where e0 and e1 are constant coefficients, n an integerand dc the starting distance of the mixed zone (usuallycorresponding to y+

¼60). The readers are referred to Ref. [18]

Fig. 13. DES and hybrid RANS-LES wall distance functions. solid: exact d, dash line and symbols: modified d.

Fig. 14. Hybrid RANS-LES numerical simulation for a jet pylon configuration. (a) modified wall proximity contour and (b) instantaneous jet velocity field.

H. Xia et al. / Progress in Aerospace Sciences 46 (2010) 274–283282

for more detailed discussions of f ðfÞ and gðfÞ. Fig. 14a showsdistance function contours for a hybrid ILES-RANS by Eastwood[67] for a jet pylon configuration. The predicted instantaneousvelocity field is shown in Fig. 14b. The hybrid ILES-RANS approachis generally attractive giving encouraging agreement withbenchmark data (see [18]). We note here that the H–J equationcan also be used to set sponges to prevent numerical reflection atflow domain boundary (see Tucker et al. [68]).

8. Concluding remarks

The current status of level set type methods in aerospace CFDhas been briefly reviewed. Some important application areas arediscussed including: turbulence modeling, combustion, aero-acoustics, implicit surfaces for solid modeling and morphing,medial axis transform, computational interface identification andmesh hybridization.

In flame front capturing, as an analogy, the scalar G-equation isessentially a time-dependent initial value formulation level set atG0, which is solved with the efficient NBA method together withother equations of the system, such as the flow equation:instantaneous. Favre-averaged or LES. Also as an analogy to thestationary level set equation, the acoustic eikonal equation can bederived from the linearized Euler equation, allowing acoustic raytracing for installation design and more fundamental studies ofnoise propagation in fluid media.

The 0-level set (often regarded as an implicit surface) has beenextensively studied in many fields outside the scope of this article. InCFD, it also holds the key towards the future fully parallel era,namely the simplicity and generality of describing geometries (orsolid modeling) in an implicit way. Compared with the traditionalCSG or BREP descriptions, Boolian operations become simple and

inexpensive distance arithmetics. It also makes shape-to-shapetransform robust, creating new techniques for geometric morphing.As for mesh generation and interface identification, hybrid d-MATtype approaches make medial axis extraction more robust, espe-cially in 3D space. This in return helps define blocking for structuredmeshing. The potential for easy medial axis sculpting is also avaluable feature of differential equation based level sets, which helpidentifying computational interfaces such as overset/hybrid gridboundaries. When associated with the geometric features, such asthe curvature, the speed function of the level set can enhance nearwall advancing layer meshes in ways defined by the user.

Broadly, the eikonal equation is a specific case of the moregeneral Hamilton–Jacobi type equation. It can be directly appliedto calculate distance functions, therefore wall proximity informa-tion for turbulence modeling customized for different turbulencemodeling physics, and other generalized stationary level sets.

Acknowledgement

Funding from UK Engineering and Physical Sciences ResearchCouncil (EPSRC) under Grant no. EP/E057233/1 is gratefullyacknowledged.

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