High-order curvilinear hybrid mesh generation for CFD simulations
Julian Marcon, Michael Turner, Joaquim Peiró, David Moxey, Claire R. Pollard, Henry Bucklow, Mark Gammon
HHigh-order curvilinear hybrid mesh generationfor CFD simulations
Julian Marcon ∗ , Michael Turner † , and Joaquim Peiro ‡ Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
David Moxey § University of Exeter, Streatham Campus, Exeter EX4 4QF, United Kingdom
Claire R. Pollard ¶ , Henry Bucklow ‖ , and Mark Gammon ∗∗ ITI – International TechneGroup Limited, Cambridge, United Kingdom
We describe a semi-structured method for the generation of high-order hybrid meshessuited for the simulation of high Reynolds number flows. This is achieved through the useof highly stretched elements in the viscous boundary layers near the wall surfaces. CADfixis used to first repair any possible defects in the CAD geometry and then generate a medialobject based decomposition of the domain that wraps the wall boundaries with partitionssuitable for the generation of either prismatic or hexahedral elements. The latter is a noveldistinctive feature of the method that permits to obtain well-shaped hexahedral meshes atcorners or junctions in the boundary layer. The medial object approach allows greater controlon the “thickness” of the boundary-layer mesh than is generally achievable with advancinglayer techniques. CADfix subsequently generates a hybrid straight-sided mesh of prismaticand hexahedral elements in the near-field region modelling the boundary layer, and tetrahedralelements in the far-field region covering the rest of the domain. The mesh in the near-fieldregion provides a framework that facilitates the generation, via an isoparametric technique,of layers of highly stretched elements with a distribution of points in the direction normalto the wall tailored to efficiently and accurately capture the flow in the boundary layer. Thefinal step is the generation of a high-order mesh using NekMesh, a high-order mesh generatorwithin the Nektar++ framework. NekMesh uses the CADfix API as a geometry engine thathandles all the geometrical queries to the CAD geometry required during the high-order meshgeneration process. We will describe in some detail the methodology using a simple geometry,a NACA wing tip, for illustrative purposes. Finally, we will present two examples of applicationto reasonably complex geometries proposed by NASA as CFD validation cases: the CommonResearch Model and the Rotor 67. ∗ PhD Candidate, Department of Aeronautics. AIAA Member. † PhD, Department of Aeronautics ‡ Reader, Department of Aeronautics, South Kensington Campus § Lecturer, College of Engineering, Mathematics and Physical Sciences ¶ Software Developer and Digital Marketing Coordinator ‖ Product Manager. AIAA Member. ∗∗ Technical Director and CADfix Product Manager. AIAA Member. a r X i v : . [ c s . G R ] J a n omenclature Ω = High-order element Ω st = Reference element (cid:101) Ω st = Subelement of the reference element ξ = Parametric coordinates in the reference element χ = Mapping from reference element to high-order element f = Affine mapping from reference element to subelement J f ( ξ ) = Determinant of the Jacobian of the mapping fP = Polynomial order r = Growth ratio of boundary-layer element heights in the direction of the normal. I. Introduction O ne of the bottlenecks to the development of high-order CFD simulations of high-Reynolds number flows and theirindustrial uptake is mesh generation [1, 2]. The main challenge is to systematically and robustly generate validcurvilinear high-order boundary-conforming meshes which incorporate stretched elements in the near-wall boundary-layer regions. If the complexity of the computational domain lends itself to structured multi-block decomposition[3], then the mapping between the blocks and the unit cube provided by this approach facilitates high-order andboundary-layer meshing, but domain decomposition for general domains remains a very difficult and open problem.Current unstructured high-order mesh generators are based on a posteriori approaches that deform a coarse linearmesh to accommodate the curvature at the boundary, see for instance a brief review of these methods in reference [4].Robust mesh generators are available for generating the linear mesh, but a posteriori high-order mesh generators ofcurvilinear meshes tend to have difficulties in ensuring the validity of the mesh when highly stretched elements typicalof boundary-layer meshes are present.Here we propose a semi-structured approach that combines two complementary mesh generation procedures. Weemploy a linear mesh generator within the commercial software CADfix [5], that employs the medial object approach todecompose the domain into partitions which can be discretized into structured or unstructured meshes. A restriction ofsuch partitioning to the near-wall regions and an appropriate design of the medial object partitioning reduces significantlythe complexity of the generation process and makes it possible to obtain high-quality boundary-layer type hybrid meshesnear the wall surfaces. Further, CADfix provides powerful CAD healing and modification tools through an interface,CFI, for handling CAD geometrical operations and queries. The high-order mesh is generated by the open-source codeNekMesh which is part of the Nektar++ spectral/ hp element framework [6]. All the geometrical interrogations to theCAD definition of the boundary of the computational domain are handled by NekMesh via CFI that provides directaccess to the data structures describing the CAD geometry and the linear mesh.The generation of a high-order mesh using this semi-structured approach involves two steps. We first generate astraight-sided mesh using CADfix with a coarse boundary-layer mesh composed of a single layer within the medialobject based partitions adjacent to the wall boundaries. Additional points are then added, following essentially themethod described in reference [7], to obtain a high-order curvilinear mesh compliant with the CAD definition. Next, aboundary-layer mesh is generated using the isoparametric approach proposed in reference [8] where elements in thecoarse mesh adjacent to the wall are subdivided along the normal direction according to a user-defined resolution. In thiswork, we outline an extension to this method that leverages the medial-object decomposition available through CFI togenerate high-quality meshes in corners and junctions by performing splitting in two separate directions perpendicular toeach surface of the corner section. This approach is very flexible, modular, and permits defining a variety of resolutionsfrom a base coarse high-order mesh that remains unchanged.These procedures will be described in more detail in the following sections. Section II describes the CAD interfacethat handles the geometrical queries during the generation of both linear and high-order meshes. Section III provides an2verview of the medial object approach and discusses its application to the decomposition of the domain into near-fieldand far-field regions which are then subdivided into blocks that are discretized into a hybrid linear mesh. The generationof a high-order mesh from this linear mesh is described in Section IV. Finally, the methodology is applied to thegeneration of high-order meshes for two geometries and the meshes presented in Section V. II. CAD interface for geometrical queries
Processes for both linear and high-order meshing regularly interrogate the CAD geometry and thus a robust CADinteraction is required. NekMesh provides a lightweight wrapper that hides the complexity and size of the CAD interfacefrom users and developers. In the examples presented here we have used CFI, the CAD interface of CADfix [5], butNekMesh also provides a CAD back-end to OpenCascade [9] as its CAD engine.The use of CADfix, and its interface CFI, is motivated by the more stringent requirements on CAD quality forhigh-order meshing. CAD representations that may work very well within linear mesh generators, may not work fortheir high-order counterpart. For example, distortion levels in the surfaces, which might be perfectly acceptable forgenerating linear meshes, could induce poor quality or invalid elements in high-order meshes. Therefore access to highquality CAD and CAD repair tools for poor quality CAD, along with a robust CAD interface, is vital to the creation ofrobust quality high-order meshing tools.The flowchart of Fig. 1 depicts the integration of CFI into NekMesh. In a nutshell, a CADfix session producesa linear mesh that NekMesh will read via CFI and process through its own high-order routines. More details of themethod will be given in the following sections.
CAD geometryLinear mesh
High-order mesh
Nek M esh C A D r e p a i r CADfix/CFI
CAD queries
Fig. 1 Flowchart of the proposed semi-structured approach to high-order mesh generation. NekMesh interro-gates the CAD geometry via CFI, the CADfix API. The high-order mesh is produced by NekMesh from a linearmesh generated by CADfix.
III. Linear mesh generation via the 3D medial object
In order to produce a high-order mesh, we first need to generate a linear mesh. CADfix is a commercially availabletool with functionality covering the import, preparation and interrogation of geometry but the 3D medial object basedpartitioning and linear mesh generation uses results from active research projects [10] and are not yet commerciallyavailable. However, in order to provide appropriate geometry and prismatic meshes for upgrade to high-order, CADfixhas additional functionality which has been designed for this framework and is under active development. There areseveral automatic and semi-automatic tools which are included in the pipeline for generating linear meshes. First, thegeometry is prepared to repair any CAD defects and to define a valid domain. Second, we automatically subdivide thedomain to create our partitions for meshing. Finally, the partitions are meshed using a coarse set of divisions, designedto be balanced, well aligned and to allow periodicity at the boundary. Each part of this process has been designed to besuitable for a posteriori high-order mesh generation. In the following sections, these steps are described in more detail.To illustrate the various steps of the procedure for constructing a high-order mesh, we will employ a simplegeometrical domain that consists of an unswept wing of rectangular planform composed of NACA0012 aerofoil sections3nd a round tip, essentially a wing tip, enclosed in a rectangular box. This geometry is also of aerodynamic interest as acase study of vortex roll-up proposed and experimentally measured by Chow et al. [11] which has been used in CFDvalidation studies, see for instance [12].
A. Geometry preparation
Not all CAD models are suitable for CFD analysis. CAD geometry often lacks outer domain definitions, it may havedefects such as sliver surfaces or small gaps and the geometry may not be watertight. For our purposes we require a
CFD-ready CAD geometry : the fluid domain must be a watertight CAD solid. CADfix can import the geometry from awide range of design sources and provide automatic, manual and diagnostic driven tools for repairing poor quality CADgeometry, constructing outer domain boundaries and building a watertight and well connected CAD model. The 3Dmedial object algorithm also needs a certain level of quality from the input CAD model. Sharp corners, large vertex-faceand edge-face gaps all need to be repaired before the medial object can be generated to guide the partitioning andultimately the meshing. As the domain partitioning and meshing process respects the CAD topology, excessively shortedges and narrow sliver faces should also be removed. The requirements outlined here are not that different to thoseimposed by standard surface and volume meshing algorithms, and typically can be automatically detected and removed.
B. 3D medial object
The medial axis, first introduced by Blum [13], is a method for analysing shapes. For a fluid domain, it can bedefined as the set of all points in the domain which have more than one closest point on the boundary of the domain. Ifthese points are taken together with their distance to the domain boundary (the medial radius ), they form a completedescription of the flow domain. The medial axis is computed and returned as a non-manifold CAD object called themedial object, which contains extra information to describe the relationships between the different components of themedial object along with medial radius information. See Fig. 2 for an example 3D medial object of the fluid domainabout the NACA wing tip.The medial object can be used for structured meshing, feature recognition and mid-surfacing as well as the automaticpartitioning used here, and robust generation of the medial object has been a long standing challenge for the CAEcommunity. Our algorithm is based on a domain Delaunay triangulation [14], and recent developments [15] allow it torobustly work on a range of production CAD models or in the air volume around such models.
Fig. 2 Example of a 3D medial object for the fluid domain about the NACA wing tip.C. Using the 3D medial object for partitioning
The 3D medial object is used to guide our partition generation in complex junctions. Firstly, we must construct the3D medial object, and then use this to generate an offset surface, or shell , from the boundaries of the fluid domain.The medial object is used to locate lines where simply offsetting the CAD faces would cause the shell to self-intersect,known as medial halos (the red lines in Fig.2). The shell (Fig. 3) generated splits the fluid domain into two partitions:4ne near-field partition close to the boundary and one far-field partition. The near-field partition is subdivided intomultiple smaller partitions using feature lines on the CAD model to guide the location of the partition faces.
Fig. 3 A "shell" around the NACA wing tip geometry that divides the domain into two partitions: near-field(close to body) and far-field (away from body). The near-field partition will be used to generate a boundary-layermixed mesh of prisms and hexahedra and a tetrahedral mesh is generated in the far-field partition.
If the fluid domain contains a sharp concave corner or edge (for example, at a wing/fuselage junction), flows willoccur with potentially large velocity gradients in two or three directions. Ideally our coarse linear mesh requires elementsaligned with these principal directions. Using the medial object, there are several options available to achieve a meshsuitable for high-order upgrade.The medial halos and medial object itself can be used to guide partition construction around concavities in wingroot junctions, giving better mesh alignment when using hexahedral meshing. This gives us an H-type topology, similarto those constructed with a structured multiblock system, as shown in Fig. 4(a). This H-type topology is not ideal for thehighly stretched meshes required for high-order meshing. Instead, we have chosen a C-type topology, illustrated inFig. 4(b), which removes the need for hexahedral elements along the trailing edge, replacing them with large prismaticelements. However, this topology still requires hexahedral elements in the wing root junction partition to allow us tocomplete meshing of this partition structure. Therefore an O-type topology shown in Fig. 4(c) has been designed togenerate the highly stretched meshes needed to simulate near-wall flows. This allows a structured prism dominant linearmesh to be built in all near-field blocks, removing all hexahedral elements in junction regions. The C-type topology hasbeen adopted in the following examples to generate coarse linear boundary-layer meshes.(a) (b) (c)
Fig. 4 An illustration of the different shell structures designed to create: (a) H-type, (b) C-type, and (c) O-typetopologies. The C-type topology has been adopted here to generate a prismatic boundary-layer linear mesh.
The use of a C-type topology, together with hexahedral elements, enhances the quality of the linear mesh at junctions.This is illustrated by Fig. 5 which compares the meshes obtained with a O-type and a C-type approach and shows thatthe C-type mesh avoids the distortion of prismatic layers at the corners. The O-type mesh could potentially prevent thepropagation of the prismatic layers due to self-intersection.
D. Linear mesh generation
The medial object based partitioning of the flow domain has been designed for use with certain mesh styles.The O-type near field topology allows the use of prismatic linear elements which can be swept from the CADsurfaces through the near-field partition to interface with a tetrahedral mesh in the far field, with only one elementgenerated through the thickness of the near field partition. 5 ig. 5 C-type mesh advantages: the medial object guides partitioning in junction regions, avoiding the effectsof layer stopping at corners.
We follow a bottom-up mesh generation process to ensure the mesh is fully conformal between all partitions. Wemesh lines, then surfaces are meshed with elements conforming to the lines, and finally the volumes of the partitionsare meshed with elements which conform to the faces. The C-type topology features a structured hexahedral junctionpartition, and in this case the line meshes must be “balanced” to satisfy rules which are imposed via a structured meshstyle. This is solved as an integer programming problem [16], and solved using an open source solver [17]. To furtherensure good quality in the final mesh, a least-squares optimisation is performed to the line nodes to reduce potentialskew.The swept meshing of the partitions is performed by Delaunay triangulations of the designated template faces. Thesurface meshes have a simple 30 degree turn angle sizing applied to take into account changes in curvature to ensure themesh generated is coarse enough. This Delaunay mesh is swept into prismatic elements using the CADfix sweep mesher.Once the partition mesh and the tetrahedral far-field mesh have been completed, a mesh quality test is performed tomake sure all elements produced during the linear meshing stage are not inverted. An example of a coarse linear meshobtained with this method for the NACA wing tip geometry is shown in Fig. 6. The boundary-layer mesh in the near-fieldregion consists of 1 224 triangular prisms and 25 hexahedra and the far-field region is discretized into 12 576 tetrahedra.
Fig. 6 NACA wing tip: view of the linear mesh.E. Linear mesh periodicity
For the Rotor 67 example that follows in Section V, a rotationally periodic mesh is required. The far-field boundaryedges and surfaces need matching divisions so identical meshes can be made to ensure periodicity of the solution. This isachieved by performing an additional step in the linear mesh generation process during the balancing and alignment step.By calculating a rotational transformation which takes one side of the far field to the other, we can geometrically matchedges and copy the divisions from one side of the outer far field faces to the other. As these edges and surfaces have6lready been balanced and aligned we are safe to duplicate the divisions on the other side of the far field, maintainingthe density and quality required for our coarse meshes. Once the divisions have been duplicated the volume can bemeshed and quality checked as outlined above.
IV. High-order meshing
The a posteriori generation of a high-order mesh from a linear mesh proceeds in a bottom-up fashion followingthe ideas proposed in reference [7]. The additional points required for the high-order polynomial discretization areincorporated sequentially first along the curves, then on the surfaces of the CAD geometry and, finally, in the interior ofthe domain. The generation of points along the curves is essentially the one proposed in reference [7], the followingsections describe the improvements that we have incorporated into the methodology to achieve the type of meshessought in this work.
A. Surface optimisation
Inaccuracies in the representation of the geometry of the boundary of the computational domain due to CADdistortion, even if small, could have a significant impact on the accuracy of the flow solution. To overcome this problem,NekMesh optimises the location of the high-order nodes in the mesh to reduce distortion by modelling the mesh entitiesas spring networks and minimising their deformation energy. The optimal location of the mesh nodes is obtained in abottom-up fashion.The first step is to optimize the location of mesh nodes belonging to edges that lie on curves by minimizing thedeformation energy of a spring system in the parametric space of the curve with the vertices in the linear mesh fixed.This is followed by the processing of mesh nodes on edges that lie on the CAD surfaces. Again, we work on the2D parametric space of the surface and find the optimal position of the mesh nodes of an edge by minimizing thedeformation energy of the 3D high-order edge. As a result, the optimised high-order edge will lie approximately onthe geodesic between the two end points on the surface. The final step is the relocation of the mesh nodes in interiortriangle faces that lie on CAD surfaces. Here we follow a slightly different approach to the previous ones. The meshnodes on the edges of the triangles are fixed and the interior nodes are free to move. Each of the free interior nodesis connected to a system of six surrounding nodes by springs. The minimum deformation energy of this system ofspring is found using a bounded version of the BFGS algorithm that accounts for the limits of the parameter space in theCAD entities [18]. The gradients required by the optimization procedure can be evaluated from the CAD informationprovided by CFI. This procedure leads to an overall distribution of points which ensures the surface mesh is smooth,unless pathological distortion is present in the CAD geometry.
B. Boundary layer meshing
The generation of highly stretched elements with high aspect ratios, say 100:1, which are required to accuratelysimulate the high shear of boundary-layer flows at aeronautically-relevant Reynolds numbers, poses a significantchallenge for high-order mesh generation. If the high-order mesh is produced using a posteriori methods, then curvingthin elements in the boundary-layer mesh will almost certainly produce self-intersecting elements in regions of highcurvature. To avoid this, we generate high-order boundary-layer meshes by applying the isoparametric approach [8] tothe linear meshes produced via the medial object.Firstly, a macro boundary-layer hybrid mesh consisting of a single layer of hexahedra and triangular prisms isgenerated by the medial object method in the near-field region and the far-field partition is discretized into tetrahedra.The medial object allows us to control the thickness of the near-field region and the height of the elements to a muchgreater extent than most commercial mesh generators. By selecting a thickness of the shell that gives enough room toaccommodate the surface curvature we reduce the likelihood of generating invalid high-order elements within the macroboundary-layer mesh.The volume generation proceeds next to split these high-order elements using the isoparametric approach [8]. If theelements are valid, there exists a bijective mapping χ between a reference element Ω st and the physical space element Ω . We use the mapping to introduce subdivisions, according to a user-defined criterion, of the reference elementalong the height to generate layers along the normal in the physical space, as shown in the left-hand side of Figure 8.This way we can generate very thin boundary layer elements that are themselves valid if the mapping satisfies certainrestrictions, as shown in [8, 19]. The splitting strategy used here is to specify a number of subdivisions, or layers, alongthe parametric coordinate representing the wall normal and a growth, or progression, rate for the height of the elements.7 =1 r=1.5 Fig. 7 In the isoparametric approach a macro high-order prismatic element in the near wall region (left figure)is split along the normal to produce elements of high aspect ratio. We prescribe the number of elements andthe growth ratio of heights between adjacent boundary-layer elements, r . For example, r = (middle figure)corresponds to constant height, and r = . , increases the element height by half in the normal direction (rightfigure). This progression rate is characterized by a factor, r , that is the ratio of heights of adjacent elements. The generation ofthe boundary-layer mesh by this approach is illustrated in Fig. 7 which shows a boundary layer region of macro-prismssplit to produce highly curved valid boundary-layer elements with very high aspect ratios. C. Extension of isoparametric splitting in multiple directions
The generation of C-type meshes in junction regions allows for the generation of structured, hexahedral meshes thatavoid the effects of layer stopping in corners and produce higher-quality meshes. However, a significant downside tothis approach from the high-order perspective is that it breaks the isoparametric splitting of prismatic stacks of elementsthat has been used thus far to produce boundary layer meshes of arbitrary thickness. In this section, we show how thismethod can be adapted to deal with C-type boundary layer refinement in order to generate valid, curved meshes througha straightforward extension of the technique.The original isoparametric refinement technique first proposed in reference [8] splits a valid prismatic macro-elementinto a stack of high-order prismatic elements by using the polynomial mapping that defines the curvature of the element.Mathematically, this mapping χ : Ω st → Ω is defined between a reference element Ω st and a given element Ω . The keyobservation in the isoparametric splitting technique is that to generate the stack of elements, we may split the standardelement Ω st into reference sub-elements, and then apply χ to these to produce sub-elements in Cartesian space. Thisprocess is depicted visually in Fig. 8 for a simple quadrilateral element. When this procedure is applied to a boundarylayer mesh, the result is a curved valid mesh, regardless of the element type.To apply this technique for the present problem of junction meshing, we require an adapted version of this approachwherein the reference element is split in not one but two directions that correspond to each surface of the junction. Fromthe perspective of the mathematical justification for the validity of the method, this aspect actually makes very littledifference. As noted in reference [8], the splitting of the reference element in the original isoparametric technique can beviewed as an affine mapping f : Ω st → (cid:101) Ω st , where (cid:101) Ω st is a sub-element of Ω st . The curvature mapping of a sub-elementof the Cartesian element (cid:101) Ω can then be viewed as the composition f ◦ χ : Ω st → (cid:101) Ω . Then, as long as f is defined suchthat its Jacobian determinant J f ( ξ ) > ξ ∈ Ω st , then this new mapping is valid so long as χ is also valid.To adapt this technique for our junction meshing problem, we therefore require a slightly different refinementstrategy, as depicted in Fig. 8, where the standard element is split in each direction. A small extension to the method hasbeen applied which computes the orientation of the hexahedron and alters the distribution of the splitting points in thereference element accordingly, but the core of the method remains mostly the same.An example of the boundary layer mesh generated by this method for the wing tip geometry is shown in Fig. 9.The linear mesh in the near-field region has been subdivided into 10 layers using a growth rate, r = .
5. The resultingboundary-layer mesh is formed by 12 240 prismatic and 2 500 hexahedral elements with a maximum aspect ratio of 70.8 ! ! ! ! ! ⌦⌦ st Refine2 sides
Refine1 side
Fig. 8 Adapted version of the isoparametric splitting to work for generating high-order junction-adaptedmeshes. (a) (b)(c) (d)
Fig. 9 NACA wing tip boundary-layer mesh: (a) coarse high-order mesh, and (b) after the isoparametricsplitting; (c) close-up near the tip of the coarse high-order mesh and (d) after the isoparametric splitting. . Volume meshing The introduction of the curvature of the CAD surfaces onto the high-order surface triangulation produces high-orderelements in the interior of the volume with curved faces and edges which could become invalid. Controlling the thicknessof the near-field via the medial object permits the generation of linear boundary-layer meshes that can accommodate thedeformation induced by surface curvature without producing invalid elements. The positions of the additional nodesrequired for the polynomial representation of the high-order elements are obtained by means of a mapping between areference element and the physical element which accounts for the presence of curvature on its faces and edges lying onthe CAD definition, whilst the other edges are straight and their faces planar.A high-order mesh of the volume is shown in Fig. 10 which contains, in addition to the elements in the boundary-layermesh, 12 576 tetrahedral elements of polynomial order 4.
Fig. 10 NACA wing tip: view of the high-order mesh.
V. Examples of Application
This section presents an illustration of the proposed mesh generation methodology and the high-order meshes itproduces using two geometries proposed by NASA for CFD validation: the Common Research Model and the Rotor 67.These are described in the following sections and, for reference, all the corresponding high-order meshes have beengenerated using a polynomial order of 4.
A. NASA Common Research Model
The Common Research Model (CRM) presented here is one of the five configurations designed by NASA [20]for CFD validation. It is a wing/body alone configuration with a fuselage with a maximum radius of 0.17m, and a 35degrees backward-swept wing of aspect ratio of 9 and span of 1.60m.In the first instance, we processed the original definition of the CAD geometry in STEP format [21] through CADfixto clean it and fix a number of inconsistencies and severe distortions that might prevent the successful generation of themedial object decomposition and the high-order mesh.The medial object interface, depicted in Fig. 11(a), was designed to generate a hexahedral mesh at the wing-fuselagejunction. Fig. 11(b) shows the block partitioning of the near-field region that provides the framework for generating theboundary-layer mesh around the aircraft. Fig. 12 provides a more detailed view of the blocks in the near-field region viaa wireframe representation of the edges of the partitions in that region.The medial object decomposition was used to produce an initial coarse linear mesh with a single layer of elements inthe near-field partition. This boundary-layer mesh consisted of 33 hexahedra and 2 042 prisms. The far-field region wasdiscretized using 18 084 tetrahedra. The characteristics of the linear mesh can be observed in Fig. 13(a) that shows a cutnormal to the fuselage through the mesh and in the enlargement of that mesh in the wing-fuselage junction of Fig. 14(a).In the final step of the generation process we apply the isoparametric approach to subdivide the coarse boundary-layermesh into 10 layers with a progression ratio r = .
5. This produces a high-order mesh with 20 420 prisms and 3 30010a) (b)
Fig. 11 NASA CRM medial object: (a) interface of the medial object at the wing-fuselage junction, and (b)partitions in the near-field region. (a) (b)
Fig. 12 A wireframe representing the edges of the partitions in the near-field region: (a) global view of blocksaround wing and fuselage, and (b) close-up near the wing. (a) (b)
Fig. 13 A cut view normal to the fuselage of the CRM mesh: (a) linear mesh, and (b) high-order mesh.
Fig. 14 Close-up of the CRM mesh in the region adjacent to the wing-fuselage junction: (a) linear mesh, and(b) high-order mesh. hexahedra with a maximum element stretching ratio of 60. Views of the cut through the high-order mesh and theenlargement near the wing-fuselage junction are shown in Fig. 13(b) and Fig. 14(b), respectively.
B. NASA Rotor 67
The geometry considered here is a first stage rotor (NASA Rotor 67) of a two-stage transonic fan designed and testedat the NASA Glenn center [22]. The original rotor has 22 blades with a tip leading-edge radius of 25.7cm and a tiptrailing-edge radius of 24.25cm. The hub to tip radius ratio is 0.375 at the leading edge and 0.478 at the trailing edge.Here we consider the geometry for a single blade with periodic boundary conditions and without tip clearance. Thedistinctive feature of this geometry is the incorporation of periodic features both in the medial object decomposition andthe linear and high-order meshes.The medial object for this geometry is shown in Fig. 15(a). Fig. 15(b) depicts the block decomposition in thenear-field region. The linear mesh in that region, which consists of 53 830 prisms and 236 hexahedra, is shown inFig. 15(c).Enlarged views of the surface mesh and the boundary-layer mesh in the vicinity of the blade root are shown inFig. 16(a) and Fig. 16(b), respectively. The surface mesh contains 53 830 triangles and 472 quadrilaterals.The coarse mixed prismatic-hexahedral linear mesh is split via the isoparametric mapping into 10 layers with aprogression ratio r = . VI. Conclusions
We have proposed a semi-structured approach for generating high-order meshes with high aspect ratio elements toefficiently simulate boundary-layer flows. It combines a linear mesh generator for hybrid linear meshes of hexahedral,prismatic and tetrahedral elements, based on a medial object approach for domain decomposition, and an a posteriori high-order mesh generator.The domain is decomposed into near-field and far-field regions using a medial-object approach currently underdevelopment within CADfix. Its parameters have been tuned to produce O-type, described in a sister publication12a) (b) (c)
Fig. 15 NASA Rotor 67: (a) Medial object, (b) near-field region and (c) near-field linear mesh. (a) (b)
Fig. 16 NASA Rotor 67: (a) surface mesh, and (b) boundary-layer mesh.Fig. 17 NASA Rotor 67: Enlargement of the root surface mesh near the blade’s leading edge.
Fig. 18 NASA Rotor 67. Enlargement near the mid-chord of the blade in the near-field region: (a) linearboundary-layer mesh, and (b) high-order boundary-layer mesh. Enlargement of the hybrid mesh in the sameregion: (c) linear mesh and (d) high-order mesh. [23], and C-type topologies, described here, for the near-field region that facilitate its discretization into meshes ofprismatic, and of mixed hexahedral and prismatic elements, respectively. The medial object permits to design near-fieldregions that allow a boundary-layer mesh which is “thicker" that those achievable by most commercially available meshgenerators. The far-field region is discretized into tetrahedra. Both topologies are simpler to obtain than a generalmulti-block decomposition, and yet they are sufficiently flexible to deal with reasonably complex geometries.There are two main contributions of the paper. The first one is the design of a C-type topology for the near-fieldregion in combination with a modified isoparametric approach that produces hybrid meshes with hexahedral elements atjunctions. These meshes have a higher quality and fewer elements than those corresponding to O-type topologies. Thesecond contribution is the incorporation of periodic surfaces both in medial-object decomposition and mesh generation.This permits the numerical treatment of flow simulations incorporating periodic boundary conditions.The proposed method is robust for both linear and, to some extent, high-order meshing, but its ability to producehigh-order meshes of very good quality depends strongly on the quality of the linear mesh and of the distortions inducedby the CAD surface mappings. The major contributing factors to this problem are the coarseness of the linear meshrequired and the higher sensitivity of high-order algorithms to distortions in the mappings defining the CAD curvesand surfaces. However, through the use of the semi-structured approach proposed here and in reference [23] we haveimproved significantly our rate of success at producing high-order meshes with complex geometries.The quality of the mesh can be further enhanced by applying the variational approach proposed in reference [4] andalso by allowing elemental faces to be curved at the interface between the near-field and far-field regions. This shouldbe possible since CADfix represents these as CAD surfaces.
Acknowledgments
Mike Turner received financial support from Airbus and EPSRC under an industrial CASE studentship. This projecthas also received funding from the European Union’s Horizon 2020 research and innovation programme under theMarie Skłodowska-Curie grant agreement No 675008. 14 eferences [1] Vincent, P., and Jameson, A., “Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community ofFluid Dynamicists,”
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