Near-Wall Flow Visualization in FlattenedSurface Neighborhoods

Christoph Petz∗

petz@zib.deSteffen Prohaska†

prohaska@zib.deLeonid Goubergrits‡

leonid.goubergrits@charite.de

Ulrich Kertzscher§

ulrich.kertzscher@charite.deHans-Christian Hege¶

hege@zib.de

Abstract

We present a method that flattens a curved surface and its neighborhood to createeffective, uncluttered visualizations by applying standard flow visualization techniquesin the deformed space. Level sets of the distance from the curved surface are mappedto parallel planes. Data fields are mapped accordingly. Our method eases the visualinspection and analysis of a curved surface’s vicinity by providing a flat space thatsupports the creation of standardized views. This is particularly useful for visualizingflow fields in the boundary layer.

1 IntroductionFlow visualization is an important analysis tool in many engineering and scientific tasksand a large number of different techniques are available. Applications come from a widerange of areas, such as automotive industry, aerodynamics, meteorology, climate modeling,and medical visualization. For example, a good understanding of the flow around boundarygeometries can lead to improved designs in engineering applications and to a better under-standing of blood flow in life science research. Laramee et al. [LHD+04] and Post et al.[PVH+03] provide a good overview over the field.In particular, flow visualization and analysis in the vicinity of the boundary can help todetect and evaluate important structures in the 3D flow domain [SG04]. For example, theeffect of the viscosity of a fluid becomes apparent especially in the near-wall flow. Turbu-lence emerged at the boundary wall and boundary layer flow separation propagates into thefluid and influences the overall flow significantly. The opposite influence is true as well,vortical structures in the flow cause characteristic patterns in the wall shear stress. In CFDsimulation practice, the boundary layer is very often modeled separately with a much finer

∗Zuse-Institut Berlin, Takustr. 7, D-14195 Berlin, Germany†Zuse-Institut Berlin, Takustr. 7, D-14195 Berlin, Germany‡Biofluidmechanics Laboratory, Charite, Spandauer Damm 130, D-14050 Berlin, Germany§Biofluidmechanics Laboratory, Charite, Spandauer Damm 130, D-14050 Berlin, Germany¶Zuse-Institut Berlin, Takustr. 7, D-14195 Berlin, Germany

Figure 1: Streamline visualization in the 3D boundary layer (left) and in flattened parameterspace (right). The velocity field on the left was transformed to a space of parallel planesover the flattened surface (on the right). The visualization in the flattened space clearlyreveals a zone with streamlines parallel to the boundary surface, and two zones of flowseparation and flow attachment. This is harder to see in the left visualization.

computational grid, or modeled with a wall-function approach, to capture near wall flowphenomena accurately.The application of flow visualization methods in a thin boundary layer around a curvedsurface often yields images that are hard to understand. First, it is difficult to find goodviews onto the curved surface without self-occlusions of the region under investigation.Especially in highly curved boundary regions finding a view to grasp the flow at a glanceis often impossible. Second, some of the field quantities of a computed flow field mustbe related to the boundary distance. The increase of the velocity magnitude with respectto the boundary is related to the thickness of the boundary layer. This relation is hard todetect visually above a curved surface geometry. Third, it is often desirable to distinguishbetween tangential and perpendicular properties of a flow, e.g., a large velocity componentperpendicular to the boundary might have an important implication to the flow. The localorientation of extracted flow features is also very hard to detect clearly from visualizations.Surface flattening, also called surface parametrization, is a well established technique tomap a problem from a curved surface to a planar domain to make a solution feasible. Sur-face flattening is used to tackle fundamental computer graphical problems, such as texturemapping [BVI91], texture synthesis [Tur01], 3D painting [HH90], and digital geometryprocessing [GSS99]. In medical visualization, surface flattening is, for example, appliedin analysis of CT cross-sections along the track of a blood vessel [KFW+02], as wellas visualization of anatomical structures [SFH06], and thickness analysis of thin objects[MVG04]. Wiebel et al. [WTS+07] recently presented an application of surface flatteningto track topological flow features on the boundary geometry over a time-dependent domain.

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Eck et al. [EDD+95] was one of the first who presented a parametrization method with ap-plications to computer graphics. Today, a great number of methods for surface parametriza-tion exist. A survey can be found in [FH05].In this paper, we address the following questions: Can surface flattening be applied fornear-wall flow visualization? And how can the parametrization of a flattened surface beextended to the surrounding 3d neighborhood?We present a method that overcomes the problems described earlier by applying surfaceflattening to map the curved geometry under investigation into a two-dimensional param-eter space and extend this mapping to the surrounding three-dimensional neighborhood.The third parameter dimension maps into the field in normal direction of the curved sur-face. That yields a local mapping from a three-dimensional parameter space into the curvedneighborhood of the original surface.Visualization methods can then be applied in parameter space in a standardized coordinatesystem: The boundary wall lies in the UV-parameter plane with the flow field in the adjacenthalf space. Vector fields, e.g. velocity and shear stress, are transformed such that theirangles to the normal and tangential direction are preserved point-wise in the mapping.Visualization of the transformed flow fields in parameter space turns out to be an effectivetool to study near-wall flow in a simplified way. Navigation to a specific area at the 3Dgeometry is not necessary for an initial analysis. Instead, the region of interest can bespecified by a center point on the boundary geometry and a radius. The flattening of thesurface prevents self-occlusion of the geometry and allows an overall view onto the surfacefrom many different viewpoints.The reminder of the paper is organized as follows: In Sec. 2 a detailed technical descriptionof the proposed method is presented. Results are shown in Sec. 3 based on an applicationin the field of bio fluid mechanics. We discuss our results in Sec. 4, conclude and showdirections for future work in Sec. 5.

2 Boundary Layer ParametrizationThe flattening of the wall geometry can be done for selected regions that are homeomorphicto a disc. We assume that the region of the surface under investigation is smooth, suchthat the surface normals are continuous over the geometry. The application of the surfaceparametrization algorithm gives a mapping s2 from a two dimensional parameter spaceU2 ⊂ R2 to the surface part V2 ⊂ R3:

s2(u, v) : U2 → V2 (1)

The parametrization is necessarily distorted in comparison to the 3D surface. By the choiceof an appropriate parametrization method, this distortion can be controlled and optimizedto a certain degree. For all the examples in this paper, we applied the surface parametriza-tion method from Eck et al. [EDD+95], where the shape of the parameter domain U2 isselected automatically by the algorithm. The algorithm described by Eck et al. exploits thisfreedom to reduce the overall distortion of the mapping. The necessary distortion has to beconsidered in the evaluation of the visualizations in parameter space.

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Next, the flat mapping s2 will be extended into the nearby flow field. Therefore, a thirdparameter dimension is added, which will be related to the distance to the 3D surface ge-ometry. The visualizations shall depict the neighborhood of the curved surface in a naturalway, assuming a locally flat world. This is achieved by pointing the third parameter dimen-sion into the normal direction of the corresponding surface point. To avoid ambiguities, it isdesirable that the mapping from the 3D parameter space into a 3D region around the bound-ary geometry is injective and continuous, at least in a small vicinity around the boundary.We denote a normal vector of the 3D boundary surface with ~n(s2(u, v)). The normal func-tion is assumed to be continuous on the surface. We define the mapping s3 : U3 → V3,where U3 ⊂ R3, V3 ⊂ R3 with U2 ⊂ U3 and V2 ⊂ V3 as follows:

s3(u, v, w) = s2(u, v) + wα~n(s2(u, v)) (2)

Where α is a uniform scaling factor. This mapping s3 maps planes in parameter space,parallel to the uv-plane to surfaces with constant distances to the boundary wall. This isan important property of the mapping, since we want to be able to relate the results to theboundary distance visually. In the case of concave geometries, injectivity of the mapping isonly achievable in a small layer around the geometry. The size depends on the curvature ofthe geometry. The parametrization domain U3 must be restricted to that region.Continuity of the mapping requires continuity of the normal function ~n(x, y, z). In theanalysis of CFD data, boundary geometries are normally given as discretized triangle orquadrilateral surface meshes, not in the form of smooth representations. The surfaces arepiecewise flat, and the normal directions at the points on the surface are not continuousacross the edges of the triangles and quads. Using these normals directly would lead to afunctions s3 that would neither be continuous nor injective in concave regions.For triangulated surfaces, we establish smooth normals by interpolating normals that arecomputed at the vertices of the surface mesh over the triangles. This is a common practicein computer graphics surface rendering: interpolated normals are also used for shading.Under the assumption, that the discretized surface is an approximation of a real smoothgeometry, this is a reasonable choice.Having the mapping from 3D parameter space U3 into a 3D neighborhood V3 of the bound-ary in world coordinates, the data fields of the flow solution are now transformed into pa-rameter space, too. This allows the analysis of the near boundary flow in parameter space.For scalar fields, this mapping follows directly from the definition of the parameter spacemapping s3. We expect that a scalar value at a parameter space location is equivalent to thevalue at the corresponding location in world coordinates. Hence for a given scalar functionf in the vicinity of the boundary V3, the corresponding function fp in parameter space is:

fp(u, v, w) = f(s3(u, v, w)) (3)

Any scalar valued flow field solution such as the pressure or the velocity magnitude can bemapped into parameter space in this way.More is needed to transform vector fields into parameter space. Integral curves must beinvariant under the choice of the coordinate system and parametrization. For a point p0 ∈U3 with x0 = s3(p0), the computation of a streamline starting at x0 in world space, and

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the streamline computed at the start point p0 in parameter space and afterwards mappedinto the world coordinate system must yield the same result. To achieve this, the directionsof the vectors have to be transformed: Tangent directions of the parameter curves, e.g.∂s3/∂u, must map to the corresponding parameter axis, e.g. u. The Jacobian matrix Js3 ofthe mapping s3 is needed for that transformation:

Js3(u, v, w) =

∂s3x

∂u∂s3x

∂v∂s3x

∂w∂s3y

∂u∂s3y

∂v∂s3y

∂w∂s3z

∂u∂s3z

∂v∂s3z

∂w

(4)

For a given vector field ~v : V3 → R3, the corresponding vector field ~vp : U3 → R3 inparameter space is

~vp(u, v, w) = [Js3(u, v, w)]−1~v(s3(u, v, w)), (5)

with the Jacobian matrix Js3 evaluated at (u, v, w) ∈ U3.With those mappings, visualization and analysis of the boundary layer flow can be per-formed in the flat parameter space. All the flow field quantities are transformed into theparameter space, and also field quantities on the boundary surface, such as the wall shearstress are transferred into parameter space. Any visualization algorithms for scalar- andvector fields can then be applied in parameter space. Results can easily be interpreted withrespect to the boundary geometry: The third parameter direction corresponds directly tothe distance to the surface, tangential vector directions are parallel to the flat uv parameterspace, and directions normal to the boundary geometry are perpendicular to that plane.We have implemented this method within the visualization software Amira [SWH05].When the flattening is applied, the parametrized geometry is represented as an explicitflat surface, and the fields are accessible in the transformed space. For the images in thispaper, the visualization algorithms are applied to these fields in parameter space withoutmodification.

3 ResultsWe applied the method to a CFD data set. The data set results from a biofluid mechanicssimulation of blood flow. The geometry, representing a cerebral aneurysm, was extractedfrom a patient’s CT head scan. The shape of the aneurysm is depicted in Fig. 2, with oneinflow artery and three outflow vessels. The simulation grid consists of 770k tetrahedra and250k prisms. As shown in Fig. 2 (right), the prisms are used to resolve the boundary layeraccurately.We start with a visualization showing the Line Integral Convolution (LIC) technique andthe extraction of critical points in a flattened region of a highly curved part of the geometry.Fig. 3c depicts the location of the geometry part. Fig. 3a and 3b depict the visualization onthe curved surface, the colored spheres indicate critical points. Fig. 3d displays the wholeregion in a single flattened view, while 3a and 3b fail to do so because of the curved surface.The locations and colors of the critical points, and the border shape of the regions facilitatesthe identification of the flat with the curved geometry part.

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Figure 2: Geometry of a cerebral aneurysm (left), used for the simulation blood flow. Theaneurysm has one inflow artery (bottom of the image) and three outflow vessels (top). Onthe right, the grid structure at the inflow boundary is depicted. The grid resolution in thevicinity of the boundary layer is very fine.

The near-wall velocity field in the same region is shown with streamlines in Fig. 1. In theflattened visualization (right), two zones of flow separation can clearly be identified. In thecenter of the image, the streamlines are closely attached and parallel to the boundary ge-ometry. On the left and right of that area, the streamlines are oriented nearly perpendicularto the geometry. Here the flow separates from the boundary. The same flow phenomena atthe curved geometry is depicted in Fig. 1 (left). Only one of the separation zones is clearlyvisible on the left side of the geometry. The second zone is hard to identify, nearly hiddenat the right part of the curved surface.Fig. 4 shows an iso-surface of the velocity magnitude. The flattened view (right) allows avisual estimation of the distance to the boundary, while this is hard to grasp on the curvedsurface (left). Fig. 5 supplements the visualization with streamlines. Streamlines emergingfrom the left and right side in the flattened view (right) leave the area to the front and tothe back. The separation area, where the left/right and front/back streamlines are separated,corresponds to the area with a low near-wall velocity, i.e. the area where the iso-surface isfarther from the boundary. The same flow is visualized at the curved geometry (left), butharder to track visually.

4 DiscussionIn all examples, the flattened surface proved superior in providing a complete view of thesituation. Flattening avoids self-occlusion of the curved surface and supports visual esti-mation of distances perpendicular to the boundary. Note though, that the information aboutthe geometry of the surface gets lost. The user either needs to remember the geometry orthe geometry needs to be provided in a separate view.

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Choosing a parametrization method that preserves the triangular structure of the regionsupports identification of the flattened surface with its corresponding curved surface. In theexamples displayed in this paper, the shape of the parameter domain and the shape of theaccording curved surface geometry exhibit similarities. This can especially be seen in thetriangles and the shape of the region borders in Fig. 4. This facilitates the identification ofthe two regions for a user of the method. The shape of the parameter domain was computedby the parametrization algorithm. Not to force the parametrization domain into a particularpre-defined shape such as a disk or a square gives the algorithm the freedom to reduce thedistortions of the mappings as far as possible.Including landmarks in the visualization helps for the orientation in the flattened space.In Fig. 3, critical points of the wall shear stress are depicted as differently colored spheres.Besides their meaning in the flow field, the figure also illustrates that their role as landmarkson the flat and curved surface help to identify the positions of the parameter domain withthe corresponding positions on the boundary geometry. This suggests that the placement ofadditional landmarks, differentiated by shape, texture, or color can give an observer goodsupport for the assessment of the mapping.The factor α in Eq. (2) is a free parameter of the mapping to scale and emphasize theimmediate vicinity of the boundary. By adjusting this parameter within a visualization, theuser of the method can decide on how close the vicinity of the boundary should be resolvedin the visualization. By amplifying the region, even very thin layers in the vicinity of theboundary can be analyzed. Such an independent scaling in normal direction is not possiblearound a curved surface. A natural choice for the scaling, that preserves the proportions ofthe surface size to the normal distance does not exist, due to the non-uniform distortion ofthe surface mapping. A reasonable choice for α is to relate its value to the ratio of the areaof the parameter domain (A) and the area of the corresponding curved surface part (B). Achoice of α =

√B/A fulfils the requirement globally on average. All examples shown in

this paper are created with these values.The presented method is also applicable to other curved surfaces within the flow, not onlyto boundary geometries. For example to surfaces that are extracted from the fields of a flowsolution, e.g. iso-surfaces of the pressure field, or stream surfaces of the velocity field. Afterthe transformation of such an inner surface, the flow field can be analyzed on both sides ofthe flattened surface.Our method is limited to disk-shaped parts of smooth surfaces. Surface parametrization re-quires, that the region under investigation must be homeomorphic to a disc. As the bound-ary of a 3D volume is automatically a manifold surface, this requirement does not restrainthe applicability of our method. Non-smooth boundary geometries with sharp edges anddiscontinuous normals can not thoroughly be analyzed in a flat space: A crease in the ge-ometry has a strong impact to the surrounding flow, and should therefore not be hiddenin the analysis and visualization. Technically, the mapping of Eq. (2) would not even beone-to-one in a small neighborhood of the crease, and made any interpretation of analysisresults potentially misleading.The analysis in flattened space of concave regions with high curvatures should be restrictedto a very thin neighborhood. In such regions, the mapping gets non-injective in the spaceclose to the curved surface. Points in the vicinity of the surface are mapped several times

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into the parameter space. The mapping gets ambiguous, and the interpretability of the vi-sualization results gets questionable. Hence, we recommend to restrict the analysis to onlya thin neighborhood.The actual choice of the parametrization is not crucial for the flattening to work, the solerequirement is that the mapping must be one-to-one. We used the method of Eck et al.[EDD+95] in the examples shown here. This method does not guarantee to produce one-to-one mappings. To make the flattening method more robust in a visualization system, onecould consider to use mean value coordinates [Flo03], as they guarantee to provide a one-to-one mappings. The appearance of the visualization depends heavily on the parametrizationmethod in use. A distortion-free mapping of a curved surface into a flat parameter spaceis generally not possible. But the choice of the parametrization method influences the dis-tribution of the distortion. The extension of the mapping into the vicinity of the surfaceshould work equally well for any valid parametrization; the distortion just propagates intothe adjacent space. Note however that we did not conduct experiments to verify how thisinfluences perception of the created visualizations. Different priorities can be set: For ex-ample, by choosing a method that minimizes the distortion in a given focus-region, or theuse of a method that tries to distribute the distortion uniformly over the parametrizationdomain.The transformed vector fields in parameter space are in general non-continuous across thetriangle boundaries of the flattened space. The derivatives of the mapping s2 in Eq. (1) areconstant within each mapped triangle, and are thus non-continuous across triangles. Thisimpacts the continuity of the Jacobian matrix Js3 in Eq. (4) and the vector field transforma-tion in Eq. (5). The non-continuity influences streamline integration algorithms: Integrationstep sizes must be adapted accordingly.The streamlines computed in the flattened parameter space are identical to correspondingstreamlines in physical space. For a lookup of a vector in parameter space, we compute thecorresponding point position in physical space, the Jacobian matrix of the transformationat that point and the interpolated vector value. With that, we transform the vector back intoparameter space. Sadarjoen et al. [SvWHP94] compared streamline integration methods inparameter space and in physical space and found differences in the streamlines. In contrastto our approach, they transformed vector fields on grid vertices into parameter space andinterpolated the vector values in parameter space. That led to different streamlines.Our method supplements the analysis of CFD flow solutions in a valuable way for theanalysis of the near-wall flow behavior. A possible practice for the analysis might be tofirst visualize the wall shear stress on the boundary geometry with the LIC technique andthen select interesting regions for the analysis with our presented method. The pattern ofthe LIC visualization shows separation areas and near-wall vortical structures in an easyidentifiable and characteristic shape. These areas can then be selected via an interactivepoint-and-click interface for the application of our flattening method, to gain insight intothe boundary layer flow.It is important not to loose the context of the flattened region in the original physical space,otherwise the visualization can be misleading: Even in the case of incompressible flow,streamline densities do not indicate the mass flow of the fluid any more, incompressibilityis not preserved under the vector field transformation. Also streamline curvatures can easily

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be under- or overestimated. We recommend to always visualize the original physical spacein parallel.With our approach, visual complexity is traded for the complexity of interpretation. Incontrast to traditional methods, such as careful viewpoint selection in combination withcutting planes and clipping volumes, surface flattening solves the inherent problem of self-occlusion of the boundary geometry and requires less user interaction.

5 Conclusion and Future WorkFlattening a curved surface in space to solve a data analysis task is a method to reducethe complexity of the object of investigation. This method has important applications ina variety of different areas, amongst others, in cartography, computer graphics and medi-cal visualization. Our results show that flattening in flow visualization is useful as well, inparticular for the analysis of the boundary layer. The 3D extension of the flattened regionmapping into the near-wall flow field, including the transformation of the flow fields aspresented in this paper has some nice features for the analysis. An occlusion-free view ontothe flattened geometry is always possible, even in highly curved regions, visualizations ofthe near-wall flow can be done in a standardized coordinate system, and the interpreta-tion of the flow solution related to the geometry is simplified: Flow features parallel andperpendicular to the boundary are clearly visible as such.The extension of the mapping to 3D was straight forward and proved to be useful in thepresented applications, but further research is needed to address the question on how tobest handle world coordinates that are mapped twice to the parameter domain in highlyconcave regions, and to study the practical implications of the non-injectivity of the map-ping. Another direction for further research is to find a good way to handle non-smoothgeometries with sharp edges and corners, and to find a method to map these geometriesinto a standardized coordinate system.An interesting application to be tackled next is to use this method for the analysis of theresults of a time-dependent flow simulation. In this case, the flow fields of the time steps aredifferent, but the underlying geometry does not change. The comparison of near-wall flowstructures and their development over time can be studied and compared in the simplifiedflattened space.For improving the usability, we aim to further complement the presented method with othertechniques and implementation variants. Methods to automatically segment the bound-ary surface into disks, either fully automatic or user controlled, can ease the selectionof disk-shaped regions. The limits of the mapping, before the space in normal directionself-intersects can be predicted and should clearly be shown in the user interface of thevisualization system.

6 AcknowledgementsThis work was partly supported by German Research Foundation (DFG Grant HE 2948/5-1). We thank Dr. Andreas Spuler, Neurochirurgische Klinik, HELIOS Klinikum Berlin-

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Buch for the provision of the aneurysm data set and helpful discussions

References[BVI91] Chakib Bennis, Jean-Marc Vezien, and Gerard Iglesias. Piecewise surface

flattening for non-distorted texture mapping. In Proc. SIGGRAPH ’91, pages237–246, New York, NY, USA, 1991. ACM.

[EDD+95] Matthias Eck, Tony DeRose, Tom Duchamp, Hugues Hoppe, Michael Louns-bery, and Werner Stuetzle. Multiresolution analysis of arbitrary meshes. InSIGGRAPH ’95: Proceedings of the 22nd annual conference on Computergraphics and interactive techniques, pages 173–182, New York, NY, USA,1995. ACM.

[FH05] Michael S. Floater and Kai Hormann. Surface parameterization: a tutorial andsurvey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advancesin multiresolution for geometric modelling, pages 157–186. Springer Verlag,2005.

[Flo03] Michael S. Floater. Mean value coordinates. Comput. Aided Geom. Des.,20(1):19–27, 2003.

[GSS99] Igor Guskov, Wim Sweldens, and Peter Schroder. Multiresolution signal pro-cessing for meshes. In Proc. SIGGRAPH ’99, pages 325–334, New York,NY, USA, 1999. ACM Press/Addison-Wesley Publishing Co.

[HH90] Pat Hanrahan and Paul Haeberli. Direct WYSIWYG painting and texturingon 3d shapes. In Proc. SIGGRAPH ’90, pages 215–223, New York, NY, USA,1990. ACM.

[KFW+02] Armin Kanitsar, Dominik Fleischmann, Rainer Wegenkittl, Petr Felkel, andMeister Eduard Groller. CPR: curved planar reformation. In Proc. IEEEVisualization ’02, pages 37–44, 2002.

[LHD+04] Robert S. Laramee, Helwig Hauser, Helmut Doleisch, Benjamin Vrolijk,Frits H. Post, and Daniel Weiskopf. The state of the art in flow visualization:Dense and texture-based techniques. Comput. Graph. Forum, 23(2):203–222,2004.

[MVG04] Matej Mlejnek, Anna Vilanova, and Meister Eduard Groller. Interactive thick-ness visualization of articular cartilage. In Proc. IEEE Visualization ’04,pages 521–528, Washington, DC, USA, 2004. IEEE Computer Society.

[PVH+03] Frits H. Post, Benjamin Vrolijk, Helwig Hauser, Robert S. Laramee, and Hel-mut Doleisch. The state of the art in flow visualisation: Feature extraction andtracking. Comput. Graph. Forum, 22(4):775–792, 2003.

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[SFH06] Laurent Saroul, Oscar Figueiredo, and Roger D. Hersch. Distance preserv-ing flattening of surface sections. IEEE Trans. Visualization and ComputerGraphics, 12(1):26–35, 2006.

[SG04] Herrmann Schlichting and Klaus Gersten. Boundary-Layer Theory. Springer,2004.

[SvWHP94] I. Ari Sadarjoen, Theo van Walsum, Andrea J. S. Him, and Frits H. Post.Practicle tracing algorithms for 3d curvilinear grids. In Scientific Visualiza-tion, pages 311–335, 1994.

[SWH05] Detlev Stalling, Malte Westerhoff, and Hans-Christian Hege. Amira: A highlyinteractive system for visualdata analysis. In Charles D. Hansen and Christo-pher R. Johnson, editors, The Visualization Handbook, chapter 38, pages 749–767. Elsevier, 2005.

[Tur01] Greg Turk. Texture synthesis on surfaces. In Proc. SIGGRAPH 2001, pages347–354, New York, NY, USA, 2001. ACM.

[WTS+07] Alexander Wiebel, Xavier Tricoche, Dominic Schneider, Heike Jaenicke, andGerik Scheuermann. Generalized streak lines: Analysis and visualization ofboundary induced vortices. IEEE Transactions on Visualization and Com-puter Graphics, 13(6):1735–1742, 2007.

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(a) (b)

(c) (d)

Figure 3: Wall shear stress visualization on the 3D boundary from two different viewpoints(a) and (b). The wall shear stress vector field is visualized with the Line Integral Convo-lution (LIC) visualization technique. Critical points are extracted and shown as well. Twoviews are necessary to see all details of this highly curved boundary part. The locationof the boundary part is depicted in (c). The same visualization is applied on the flattenedsurface in (d). A single view gives an overview of the wall shear stress. The critical pointsserve as landmarks on the surfaces as well: They support the user in finding correpondingregions of the flattened space to the original curved surface.

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Figure 4: Iso-surface visualization of the velocity magnitude near the boundary indicatesthe thickness of the boundary layer flow. In the vicinity of the original geometry (left),the distance of the iso-surface to the geometry is not obvious. The same visualization inflattened parameter space (right) shows the iso-surface relative to the boundary distancemore clearly.

Figure 5: In addition to the velocity magnitude iso-surface as shown in Fig. 4, the stream-lines of the flow show the separation behavior of the flow. In the flattened flow visualization(right), the structure is better recognizable.

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