IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007
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Visualization of Vorticity and Vortices in Wall-Bounded Turbulent Flows Anders Helgeland, Member, IEEE, B. Anders Pettersson Reif, Øyvind Andreassen, and Carl Erik Wasberg
Abstract— This study was initiated by the scientifically interesting prospect of applying advanced visualization techniques to gain further insight into various spatio-temporal characteristics of turbulent flows. The ability to study complex kinematical and dynamical features of turbulence provides means of extracting the underlying physics of turbulent fluid motion. The objective is to analyze the use of a vorticity field line approach to study numerically generated incompressible turbulent flows. In order to study the vorticity field, we present a field line animation technique which uses a specialized particle advection and seeding strategy. Efficient analysis is achieved by decoupling the rendering stage from the preceding stages of the visualization method. This allows interactive exploration of multiple fields simultaneously, which sets the stage for a more complete analysis of the flow field. Multifield visualizations are obtained using a flexible volume rendering framework which is presented in this paper. Vorticity field lines have been employed as indicators to provide a means to identify “ejection” and “sweep” regions; two particularly important spatio-temporal events in wall-bounded turbulent flows. Their relation to the rate of turbulent kinetic energy production and viscous dissipation, respectively, have been identified. Index Terms— 3D vector field visualization, unsteady flow visualization, time-varying volume data, features in volume data sets, multifield visualization, fluid dynamics, turbulence
I. I NTRODUCTION Turbulent flows, bounded by impermeable surfaces, probably constitute the most frequently occurring flow configuration in practice. Examples are external boundary layers on cars, ships and aeroplanes, and internal flows in turbines, pumps and pipes, to only mention a few. The large-scale structure of turbulent flows near rigid boundaries is affected in several ways: by strong mean shear; by kinematic blocking of turbulent fluctuations; by fluctuating pressure reflections; and by moving internal shear layers as produced by the large-scale structures themselves. Elongated streamwise vortices are formed with length scales comparable to, or larger than, the boundary layer thickness. These structures vigorously mix momentum – high momentum fluid in the outer part is transported toward the surface and, conversely, low momentum fluid is transported from the near-wall region toward the outer part of the boundary layer. The complexity of these structures, and many of the characteristic phenomena associated with them, are still not well understood. Advances of our understanding of the physics of fluid turbulence is however of crucial scientific and A. Helgeland is with the University of Oslo and the University Graduate Center, Kjeller, Norway. E-mail:
[email protected] B. A. Pettersson Reif, Ø. Andreassen and C. E. Wasberg is with the Norwegian Defence Research Establishment, FFI, Kjeller, Norway. E-mail: {Bjorn.Reif,Oyvind.Andreassen,Carl-Erik.Wasberg}@ffi.no
practical importance. An excellent introduction to the theory of turbulent shear flows can be found in the text by Townsend [1]. Carefully conducted direct numerical simulations (DNS) enable a deterministic approach to study the seemingly stochastic turbulent motion – DNS provides a pointwise solution to the Navier-Stokes equations both in time and space. This study was motivated by the interesting prospect of applying advanced visualization techniques to highly accurate DNS data in order to gain further insight into various spatio-temporal characteristics of turbulent flows. The study is based on DNS of fully developed plane turbulent channel flow using an advanced high-order spectral element code. The primary objective is to employ a vorticity field line approach to study the spatio-temporal behavior of wall-bounded incompressible turbulent flows. The very nature of turbulent motion makes the utilization of flow vorticity the most convenient approach to characterize the flow field. The premise of the work is that the kinematical and dynamical evolution of an incompressible fluid can equivalently be expressed in terms of the velocity and the vorticity fields. As such, no information about the numerically simulated flow field is lost by considering the latter. The cascade of turbulent kinetic energy is characterized by vorticity processes like straining and connection/reconnection, all of which can be quantified pointwise in a DNS field. Since vorticity is linked to flow topology, the time evolution of the vorticity field expresses changes in flow topology. Such processes can be identified through visualization. Of particular interest in the present paper, is the study of spatio-temporal kinematical “ejection” and “sweep” events and their relation to the rate of production and viscous dissipation of turbulent kinetic energy. A notable feature of the vorticity field in an inviscid fluid, is that the corresponding field lines are equivalent to material lines of the fluid. Material lines, formed by injecting tracers, are used to visualize the fluid motion in physical experimental settings with the objective to reveal dominating structures/patterns of the flow field. Material lines “per se” do not reveal dynamically important features as vorticity field lines are able to. There exist, however, no flow visualization techniques in physical fluid experiments that capture the 3D vorticity field. In numerical experiments, on the other hand, the vorticity field is easy to derive. Nevertheless, viscous dissipation is always present in turbulent flows, and one of the goals of the present study is to provide a quantitative measure of the error of interpreting the vorticity field lines as material lines. In this paper, we discuss visualization and animation of
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007
z U(z)
y x
Fig. 1. The computational domain and the corresponding sub-domain used for visualization purposes. Lower wall is colored blue.
the vorticity field through field lines, and through “fuzzy objects” referred to as “vortices”. The depiction of the latter can be based on a number of existing criteria that includes the vorticity field – although no universally accepted definition of a vortex in turbulence presently exists [2]. The vorticity field line animation is done by injecting a collection of particles into the domain. These particles are then tracked along their path lines. At each time step, the particles are used as seed points to generate vorticity field lines. In this way, the animation shows the advection of particles, while each frame in the animation shows the instantaneous vorticity field. To improve rendering performance, we decouple the rendering stage from the preceding stages of the visualization method. This allows interactive exploration of multiple fields simultaneously, which sets the stage for a more complete analysis of the flow field. To facilitate the analysis of multiple data fields, we present a flexible volume rendering framework which is capable of producing effective visualizations of multiple volumetrical fields. II. T URBULENT CHANNEL FLOW AND DIRECT NUMERICAL SIMULATIONS
The present study considers fully developed turbulent flow in a plane straight channel bounded by infinite plates. Fig. 1 displays a schematic of the channel flow configuration. This particular configuration constitutes a well defined case that is suited for the purpose of this study. Although the mean flow field (i.e. the ensemble averaged flow field) is unidirectional and steady, the turbulence field is highly three-dimensional and time-varying, and as such extremely complex. The present DNS has been performed atpfrictional Reynolds number Reτ ≡ uτ h/ν = 180, where uτ ≡ τwall /ρ , h, and ν ≡ µ /ρ denote the friction velocity, channel half height, and kinematic viscosity of the fluid, respectively. µ and ρ are the dynamic viscosity and fluid density, whereas τwall ≡ µ (dU/dz)wall represents the wall shear stress. Physical parameters throughout the paper have been made nondimensional (referred to as ’plus units’ or ’viscous units’) by using the velocity scale uτ and kinematic viscosity ν . Mean flow kinetic energy is transferred to turbulent kinetic energy by the action of mean flow gradients at the largest scales of motion. The energy is on average transferred from the largest turbulent scale to the smallest scales through the turbulent cascade process, where it subsequently is dissipated into heat by the action of viscosity. Since the channel flow field is fully developed, i.e. statistically steady, it reaches a
statistical equilibrium between the rate of production and rate of viscous dissipation of turbulent kinetic energy, and is as such also independent of inflow conditions. The direct numerical simulation was conducted using a spectral element method, where the computational domain is divided into elements, and the solution is represented by high order polynomials on each element. For details on the implementation, see [3], [4]. The simulation was carried out on a computational domain of size (Lx+ , Ly+ , Lz+ ) = (1440, 720, 360) viscous units (Fig. 1). The total number of nodal points equals 128 × 128 × 129 in the streamwise (x), spanwise (y), and wall-normal (z) directions, respectively. The solution was advanced in time with a time-step corresponding to 0.18 viscous time-units (t + = ν /u2τ ), and with 50% polynomial filtering [5] on each time-step. In order to ensure sufficiently converged statistics the flow was evolved approximately 54 flow-through times along the computational box. This is important in order to achieve accurate fluctuating velocity and pressure fields used in the visualization (these are obtained by subtracting the mean flow from the instantaneous field). No-slip boundary conditions are applied at the solid walls, whereas periodicity are imposed in the streamwise (x) and spanwise (y) directions. III. VORTICITY AND VORTICES IN TURBULENT FLOWS In the early 1990’s, numerical realizations of turbulent flows at low to moderate Reynolds numbers revealed that vorticity was in part concentrated in “tubes” with characteristic thickness in the order of a few dissipation scales [6], [7]. Although these “tube-like” structures often are referred to as vortices, there has not yet been given any universally accepted definition of a “vortex”. Several authors have proposed a number of different criteria that can be used to identify and classify a vortex within complex fluid topologies, cf. e.g. [8], [9]. The concept of tube-like vortices and the recognition of their dynamical importance in turbulent flows has spawned a number of research efforts with the objective to develop simpler models of near-wall turbulent motion. In the mid 1990’s, Banks and Singer [10], [11] (e.g.) proposed a technique for vortex tube identification, representation, and reconstruction in turbulent flows through a predictor-corrector technique where vortices were visualized and used as an exploratory tool. In [12], a predictor-corrector scheme is used to study vortex structures in a wall-bounded flow during the transition phase from laminar to turbulent motion. A comprehensive comparison between various vortex identification schemes can be found for instance in [2] and [13], [14]. As alluded to above, the majority of previous studies focused on the identification of vortex topology as compact tube-like objects occupying local regions. Due to the nature of such methods, vortex identification schemes are, in general, not able to represent the kinematic behavior related to internal velocity shear. Velocity shear spawn regions of vorticity that are characterized by sheet-like structures. These structures are for some applications, such as wall-bounded flows, dynamically very important. For example in the immediate vicinity of an impermeable wall, the kinematic blocking of wall-normal
HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS
velocity fluctuations causes the flow field to be more sheet-like than tube-like, even if the flow is fully turbulent. It is within this narrow region turbulence is produced and subsequently dissipated into heat by the action of the fluid viscosity; in every respect this is a fundamentally crucial portion of wallbounded turbulent flows. It is therefore highly desirable to not only consider tube-like vortices but also to characterize the topology of the vorticity field in general. This requirement calls specifically on the ability to visualize vector fields, and in particular vorticity field lines. While vorticity field lines were visualized in [15] to reveal the flow topology in an engineering application, the present study is chiefly concerned with developing and assessing faithful methodologies that can assist our interpretation of the physics of turbulent flows. In this paper, we present an animation technique that can be used to visualize temporally evolving vector fields. We discuss the quantification of the error of interpreting vorticity field lines as material lines. This is of interest when evolving vorticity lines are used to infer the physical characteristics of temporally evolving turbulent flow fields. It should be noted, however, that irrespectively of the error, the presented field line animation technique will provide qualitative information about the flow evolution. The vorticity transport equation as derived from the incompressible Navier-Stokes equations states that the local time change of vorticity at a fixed point in space can be written
∂ω = ∇ × (u × ω ) + ν ∇2 ω , (1) ∂t where ω ≡ ∇ × u is the vorticity vector and u denotes the velocity vector. Let us now consider the flux of the vorticity field integrated over a moving contour attached to a fluid element. Let dA be an element of the surface outlined by the contour and let ds be a line element along the contour. Then the total rate of change in the vorticity flux (φ ), can be written using Stokes’ theorem as dφ dt
∂ω · dA + (u × ds) · ω ∂t ZZ ∂ω − ∇ × (u × ω ) · dA = ∂t =
ZZ
= ν
ZZ
I
∇2 ω · dA,
(2)
which vanishes if viscous effects can be neglected; the flux of vorticity through a contour that follows the fluid particles is then constant. Taking into consideration that for incompressible fluids the velocity field is solenoidal i.e. ∂ uk /∂ xk = ∂ u/∂ x + ∂ v/∂ y + ∂ w/∂ z = 0, equation (1) can be rewritten as d ωi ∂ 2 ωi = si j ω j + ν , dt ∂ xk ∂ xk
(3)
where si j ≡ 21 (∂ ui /∂ x j + ∂ u j /∂ xi ) denotes the components of the rate-of-strain tensor. Mathematically, in order for the vorticity field lines to be perfectly advected by the fluid, the dynamic influence of the viscous term (ν∂ 2 ωi /∂ xk ∂ xk ) in equation (3) must vanish. If the viscous term is small compared to the strain term, the idea of a perfectly advected field line is
3
0.04
Rate of strain term Viscous term
0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
0
20
40
60
80
z+
100 120 140 160 180
Fig. 2. Instantaneous distribution over horizontally averages planes (xy) of the terms in equation (3) across half the channel (nondimensionalized by uτ and ν ).
at best approximative. This constraint seems to be fulfilled in the present case only within a quite narrow region close to the wall (5 / z+ / 60). This is demonstrated in Fig. 2, where the norm of the averaged rate-of-strain term (hksi j ω j kixy ) is compared with the norm of the average viscous term (hkν∂ 2 ωi /∂ xk ∂ xk )kixy ) plotted against the distance from the boundary. Here, xy means the average is calculated over an xy plane parallel to the bounding wall. The no-slip boundary condition, u = 0, implies that the rate-of-strain source of vorticity si j ω j vanishes at the boundaries. Although the results shown in Fig. 2 would suggest approximative match of material lines and vorticity field lines within the narrow region close to the wall, the analysis presented in section VII shows that this picture is too simple and that the vorticity field lines and material lines exhibit significant spatial separation after a relatively short time interval. IV. VORTICITY F IELD A NIMATION A technique for animating three-dimensional timedependent vector fields was developed in [16]. The method is a hybrid solution based on both the use of path lines and field lines, similar to the ideas used in DLIC [17] and UFAC [18]. In principle, the field lines can be based on a different vector field than the velocity field, but only the velocity field was studied in [16]. In this paper, we develop the previous work further to animate the field lines of the vorticity field in combination with the underlying flow field. The vorticity field line animation is done by injecting a collection of “evenly distributed particles” throughout the physical domain (Section V-A). These particles are then tracked along the time-dependent velocity field by calculating their path lines. At each time step, the particles are used as seed points to generate field lines using the vorticity field. In this way, the animation shows the advection of particles, while each frame in the animation shows the instantaneous vorticity field. To obtain a global representation of the flow, particles are injected and removed in areas with too low or too high density, respectively. The particle placement is also dependent on a second distribution criterion, which has the effect of producing field lines that are approximately evenly spaced at each time step. Such a solution gives still images (volumes) that are very easy
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007
Initial Seed Point Placement
Next Time Step
Particle Advection
Inflow Treatment
Maintaining Particle Density
Fig. 3.
Output Texture
Flowchart of the particle advection algorithm.
to interpret and is nearly optimal for diminishing cluttering issues. The time-dependent vector field visualization algorithm is divided into three parts: 1) Particle Advection – pre-process (Section V), 2) Field Line Generation – pre-process (Section VI), 3) Volume Rendering (Section VIII). To improve rendering performance, the rendering stage is decoupled from the rest of the visualization pipeline. Hence, the final rendering of the flow data will achieve an increase in performance [16] compared to the combined advection and volume rendering algorithms. A compact description of the algorithm is given in the next few sections. More details can be found in [16]. V. T HE PARTICLE A DVECTION S TRATEGY One of the main challenges encountered when visualizing three-dimensional vector fields is to find a good visual representation of the vector field due to perceptual issues such as occlusion and cluttering. This is especially true for dense texture-based representations of 3D flows such as the Lagrangian-Eulerian Advection (LEA) [19], [20] and the Image Based Flow Visualization (IBFV) [21], [22] methods. Visual perception can be improved by replacing the completely dense input noise texture by a sparse input texture [23], [24]. The sparsely collected points will act as seed points with the result of rendering a collection of densely placed field lines instead of a more or less solid object as is the case for a dense representation. The fundamental problem with the sparse texture approach is to choose appropriate seed points for the particle tracing of the flow. Such a particle advection method needs to address a number of issues: It must be able to depict all important features of a flow, maintain a certain particle density both in time and space, properly take into account certain boundary effects such as inflow and outflow, and make sure that the field lines traced from these seed points are separated by a minimum distance. All these issues are addressed in the algorithm presented in this paper. A flowchart of the algorithm is shown in Fig. 3. A. Initial Seed Point Placement The purpose of the seed point placement algorithm is to obtain a collection of evenly distributed seed points throughout
the domain, while meeting two criteria. First, the chosen seed points should be separated in such a way that all field lines traced from the points are separated by a minimum distance. Second, the distribution of points should be somewhat random in order to avoid a completely uniform distribution, which produces visual artifacts. The initial seeding algorithm consists of inserting one seed point at a time randomly into the domain. Then, a field line is computed in positive and negative direction for a user-defined length. If any sample point along the computed field line is closer to any already inserted field line than a prescribed minimum separating distance, the particle is rejected. If not, the particle is inserted into the domain and all cells or voxels covered by the field line are marked. A single sample point is checked for validation by checking neighboring voxels for marked values. The sparse particle texture is defined by setting the voxels at the chosen seed points to 255 (which is the maximum value for a byte texture of type unsigned char), while the remaining voxels are set to zero. This texture will at a later stage of the algorithm be treated as input to a texture-based method for representing directional information via patterns of correlation in a texture. This is covered in Section VI. B. Particle Advection After the initial placement of the seed points, these points are treated as particles and are advected along the velocity field for a short time period. Both path lines and field lines are computed using a fourth-order Runge-Kutta method. To achieve correct discrete Runge-Kutta integration along path lines, intermediate steps are calculated from linearly interpolated vector field values. To avoid cluttering of field lines, all particles that come too close to any other particle are removed after each time step. When particles are inserted into the domain, the minimum separating distance between adjacent particles apply for all sample points along the field lines traced from these particles. However when removing particles, the separating distance is just computed between the particles. The reason for this weaker constraint during removal of particles, is that we wish to preserve the life span of each individual particle as long as possible. This is motivated both for physical reasons as well as for producing smooth and coherent animations. For instance, the formation process of vortex structures leads to a concentration of vorticity field lines. Such a flow property could be suppressed if it was not for this weaker constraint when removing particles. As the proposed field line animation technique allows individual field lines to fall closer to each other locally over time, significant flow information such as concentration of vorticity field lines can still be seen from the animation even though we suppress some flow information by allowing particles to be removed. C. Inflow Treatment While all particles leaving the physical domain are removed naturally in the particle advection step, special attention has to be given to particles entering the domain. After each advection step, every cell or voxel on the boundary is checked in
HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS
random order for inflow. New particles are then inserted at the boundary using the same criteria as were used for the initial seed point placement. D. Maintaining Particle Density in Time and Space As time evolves and particles start to cluster, some areas of the domain will have lower particle density than the initial distribution. To maintain an approximately even distribution in space, particles are injected in areas with low density. This ensures that all parts of the flow are represented at all times. By doing this, we are in fact emulating a field description rather than a true particle description of the flow. However, since efforts are made not to distort the interpretation of the physics, some clustering of particles is allowed to happen. As a result, the separating distance between adjacent field lines will vary during the animation. The injected points are chosen according to the same criteria as was used for the initial seed point placement (see Section V-A). In order to maintain the avarage density of particles in time, a maximum number of seed points allowed in the domain is computed by the initial seed point placement algorithm as the number of starting particles. Only when particles are removed, either due to outflow or clustering, new particles can be injected into the domain. VI. F IELD L INE G ENERATION After each time step in the particle advection algorithm, the resulting sparse particle texture can be used as input to a texture-based algorithm for visualizing field lines. Several different approaches for generating field lines were discussed in [16], including Seed LIC [23] and anisotropic diffusion [25]. Due to significantly shorter running times, we here only employ the direct approach described in [16]. In this field line generation method, all voxel values are set directly during the field line integration step. This means that the final ’particle and field line’ (PFL) volume can be generated directly during the particle advection algorithm presented in Section V. To incorporate orientational as well as directional information in each output 3D texture, each voxel value along the field line in the negative direction is set with decreasing intensity values. This will have the same effect as the OLIC method proposed by Wegenkittl et al. [26]. While all voxels along field lines in the positive direction are set to 255, the voxel intensity, I, in the negative direction at xi = σ (si ) is computed by the formula I(xi ) = (M − i)/M · 255. Here, M denotes the max number of samples along the field line in each direction. The curve σ (s) is parameterized by the arc-length s. To convey the 3D shape and depth relations among the field lines we employ the limb darkening technique used in [23]. Limb darkening creates a halo effect around each field line and is obtained by manipulating transfer functions (TFs). Since all TFs are handled by the graphics card and only require a 1D texture stored in memory, this is an efficient method for shading. Limb darkening is achieved by assigning darker
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values and decreasing opacity near the edges of a volume feature. To emphasize the halo effect, the output textures are convolved with an isotropic 3 × 3 × 3 filter [23]. This leads to a smearing of the field lines, resulting in a smoother representation. VII. P HYSICAL I NTERPRETATION The animation technique used in this work employs a hybrid solution based on both the use of path and field lines. The physical interpretation of the animation is not straightforward and therefore deserves additional comments. Path lines are used to track individual particles over time, while field lines are used to convey the instantaneous topology of a vector field at each instant of time. The field lines in our solution are to be regarded as advanced glyphs attached to each individual particle. During the animation, these instantaneous curves or glyphs are tracked along the time-dependent velocity field. This is done by tracking the center point of the curve along its path line. This means that particle motion is only observed by watching a sequence of images. The main objective with this particular visualization technique was to clearly convey the instantaneous topology of time-dependent vector fields and their time evolution. However, when used to visualize inviscid flows, the animation of vorticity field lines has additional physical meaning. The reason is that in such cases they can be considered to be material lines (Section III). For such flows, the presented technique can also be used to track individual vorticity field lines over time. As the results shown in Fig. 2 suggest a reasonably good match of material lines and vorticity field lines, it is of interest to quantify this difference. A. Analysis - Vorticity Field Lines VS Material Lines The spatial separation (i.e. the “error”) between material lines and vorticity field lines is calculated in the following way. First, a section of a given length of a vorticity field line is computed in both directions from the seed point. Then a set of particles is inserted along the vorticity field line at evenly distributed sample points. All particles, including the seed point, are subsequently advected along the velocity field for a given time period. A new vorticity field line is then computed from the new seed point position, and finally the error at each sample point is computed by calculating the distance between the sample points along the new vorticity field line and the advected particles (Fig. 4). For simplicity, the local errors are calculated from points that constitute original neighbors, and not from the closest point along the vorticity field line. To obtain sufficient statistics, vorticity field lines are inserted densely into the domain using the initial seed point placement algorithm presented in Section V-A. In order to obtain error information at different z+ positions, all local errors found at each sample point are averaged in xy-planes. From Figs. 5 and 6 it is clear that the vorticity field lines and material lines disperse after a relatively short time interval. The errors shown are given in percentage of the distance traveled (i.e. advected length). For instance, an error of 5% means that when a particle has traveled 20 units, the error between the advected
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007
Vorticity field line
New vorticity field line
Advected seed point
Seed point Error Particles
Advected particles
Fig. 4. A sketch of how error measurements between the vorticity field line and the material line are obtained. +
t =5.4 t+=10.8 t+=16.2
14 12 L+=112
error (%)
10 8 6
L+=56
4
L =28
+
2 0
0
20
40
60
80
z+
100 120 140 160 180
Fig. 5. Error measurements between the vorticity field lines and material lines as a function of wall distance. L+ denotes the length of the vorticity field line.
vorticity field line and the material line could be as much as one unit. A visualization of the error is shown in Fig. 7 in a plane parallel to the wall for a few field lines inserted at z+ = 40 and advected for t + = 9.72 viscous time units. This implies, according to the mean flow, that each field line in average has been advected about 142 x+ units. The length of each vorticity field line is L+ = 56 units. VIII. VOLUME R ENDERING During the stages of particle advection and field line generation, all vector data are converted to a series of byte scalar data sets carrying information of all the advected particles and their resulting field lines. This leads to a reduction in storage requirements by a factor of 12, assuming the original three component vector data was represented as floats, and the resulting particle and field line volume is of the same 10
x y z
9 8 error (%)
7 6 5 L+=56
4 3 2 1 0
0
20
40
60
80
z+
100 120 140 160 180
Fig. 6. Error measurements (shown in the three coordinate directions) between the vorticity field lines and material lines as a function of wall distance. L+ denotes the length of the vorticity field line.
Fig. 7. Visualization of the error between vorticity field lines (red) and material lines (blue) in a horizontal plane parallel to the wall at z + = 40. The seed points are shown in yellow.
resolution as the original data. (i.e. 3 floats = 12 bytes → 1 byte). Once these first two stages of the visualization pipeline are finished, the output, which is a time series of 3D textures, is sent to our rendering framework. Since the rendering stage is completely decoupled from the first two stages, it will ensure a faster rendering compared to algorithms that have to compute texture advection in addition to volume rendering on the graphics card [16]. These texture advection algorithms also have to handle much more data including the vector data, all of which need to be stored in texture memory. The presented algorithm is therefore capable of handling larger volumes interactively than the combined advection and volume rendering algorithms. The volumetric data sets are rendered using a standard 3D texture-based direct rendering approach [27], [28]. A stack of view-aligned slicing polygons, serving as proxy geometries, are used to sample the volume and blended together in a backto-front order to create the final image. A. Multifield Visualization Even though most CFD simulations involve the computation of a multiple set of related data fields, much of the previous visualization research have focused on methods and techniques for visualizing a single field variable only [29]. While singlevariable visualizations can satisfy the needs of the user in many applications, it is clear that for some areas, such as in fluid mechanics research, it would be extremely useful to be able to effectively visualize multiple fields simultaneously and to visualize interactions between them. However, due to perceptual issues such as clutter and occlusion it can be very challenging to produce an effective visualization of multiple volumetrical fields. Despite the less received attention, there are several techniques available that are capable of visualizing multiple volume data sets simultaneously. The frameworks presented in [23], [30] use a similar slice-based multifield approach. Here, each volume in a scene is rendered using a separate set of slice planes. To achieve correct blending, the slices are intermixed and rendered in the correct geometrical order. Another approach is presented by Grimm et al. [31]. Here, scenes containing multiple volumetric data objects are represented using a flexible data structure called V-Objects. Other
HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS
methods involving the representation of multiple data sets are approaches using multi-dimensional transfer functions [32], [33] and flexible focus+context visualization techniques based on interactive feature specification [34], [35]. Our Multifield Approach: Here, we present a flexible volume rendering framework for the rendering and analysis of multiple data fields. The framework was loosely presented in [16], however, in this paper we present a more formal and detailed description. The framework basically consists of data objects, volume objects, and a scene. First, the selected data fields are stored as data objects together with metainformation including data range and time step. Then, volume objects are created from the data sets and put in the scene. Each volume object is defined by an uniform 3D scalar field and stored as a separate 3D byte texture on the graphics card. For data objects stored as floats or doubles, normalized byte data are generated from the original data. The actual rendering of the scene depends on the precise order and type of the volume objects. The volume rendering framework currently supports three kinds of volume objects, which are luminance (L), alpha (or opacity) (A) and mask (M). Luminance This is the standard type of volume object. Each value in the data set is used to define both the color and opacity in each voxel. Alpha The data set values will replace the alpha values of the preceding luminance object. Mask The alpha values of all preceding luminance objects are multiplied by the values associated with the mask object. Using these three types of volume objects, we can create a number of different visualization scenes. To facilitate interactive displays, all supported scenes are written as various shader programs. This could either be as a fragment program or via high-level shading languages such as Cg [36] and GLSL [37]. We have chosen to write all of the shaders in a high-level shading language, which is more human-readable and easier to maintain. In order to identify the different scenes, we use a system where each type of volume object is identified by a letter (see above). So, if we want to visualize the ’particle and field line’ (PFL) volume together with another property field as two separate luminance objects, the shader program used would be identified as LL. A scene where both of these fields are masked by a third volume, would be identified as LLM. Whenever multiple luminance objects are used to create a visualization scene they are blended together. This is done in the following way. First, every scalar value associated with a specific volume object is mapped to a color and opacity value through a transfer function. Then, the blended color (C) and opacity value (A) at each voxel or fragment are derived using the following formulas n
A = 1 − ∏(1 − Ai ) i=1
n
n
i=1
i=1
C = ( ∑ AiCi )/ ∑ Ai ,
(4) (5)
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where n denotes the number of luminance objects in the scene and a single luminance object L after lookup, at each voxel, is given as Li = (Ci , Ai ). Whenever a masking object is used, all alpha values of the “left hand side” volume objects of a mask field are multiplied by the mask values. This means that for instance in an LMLL scene, the two rightmost luminance objects will remain unaffected by the mask field. Both operations associated with the alpha and mask objects are evaluated before the blending of the individual luminance objects. All volume objects used in a scene are associated with a separate transfer function, enabling the user to create a various number of visualization effects. Fig. 9 shows an example of a complex visualization scene generated using the above rendering framework. Here, three different fields are visualized using an LLAL-scene. In this scene, the first luminance object holds the vorticity field through the PFL volume. This volume is then blended with the luminance/alpha object, where the vorticity field has been used to define the opacity and the z component of the velocity field has been used to define the color. While the first volume object is used together with a transfer function to create shading effects through limb darkening, the latter volume object has the effect of coloring the individual vorticity field lines. The last luminance object is used to visualize the rate of production of turbulent kinetic energy together with the other two volume objects. Fig. 12(b) shows another example of a complex visualization scene. Here, three different fields are used to create an LMLL-scene. Similarly to the above scene, the first luminance object holds the vorticity field through the PFL volume. This volume is then blended with the two fields enstrophy (kω k) and λ2 [9], both revealing vorticity structures of the flow. In order to show the direction of the vorticity field only inside the structures associated with the other two fields, a masking field has been applied to the PFL volume. In Fig. 12(b), enstrophy has been used as a masking field. All three luminance fields have been used together with transfer functions to create shading effects through limb darkening. The blending procedure of multiple volume objects presented here has similarities both with the slicing techniques presented in [23], [30] and the V-Object approach [31]. However, there are some notable differences. For instance, the number of slices used in our approach is independent of the number of volume objects. This is not the case for the other slicing methods [23], [30], which use a separate set of slice planes for each volume object. Instead we consider, just as in the V-Object approach, the volumes to be like clouds of particles and take into account all volumes simultaneously at each sample position. Another difference lies in the compositing scheme. Similar to the other three techniques, our method employs the standard over operator [38] when blending the individual slices. However, for the blending of multiple volumes at each voxel, we present an alternative formula which opposed to the over operator is independent of the precise order of the volume objects to be blended (equations (4) and (5)). This merge operator is identical to the plus operator [38], with the exception that we use the standard compositing scheme when calculating the
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(a)
(b)
(c)
Fig. 8. Rendering results for different blending schemes used to visualize a torus together with a sphere. The torus and sphere data are white everywhere except for a small region in red and blue colors, respectively. The alpha value for each volume is A = 0.5. The rendering shows a single slice polygon inside the volume. (a) Blending formula presented in equations (4) and (5). (b) Sphere over torus. (c) Torus over sphere.
blended alpha values. This ensures that both the final color and the alpha values in each voxel are in the range [0,1]. Hence, a multifield visualization of several transparent data fields will continue to be transparent when using the merge operator. This is not the case when using the plus operator which very quickly accumulates opacity values higher than one. In many applications, the merge operator is preferable compared to the over operator since this expression holds no precedence in any area covered by multiple data fields (see Fig 8). For other visualization purposes, other compositing operators could be the natural choice. The presented rendering framework can also easily be extended to support a wider set of useful operators such as the compositing operators presented by Porter and Duff [38]. Our rendering framework also differs from the two other slicing methods and the V-object approach, in that our framework supports more than a single type of volume object. By using luminance objects in combination with mask and alpha objects, we can create far more advanced scenes such as the ones in Figs. 9 and 12(b). Since each volume object is associated with a separate transfer function, our framework can also be used to create focus+context visualizations [34], [35], [39]. In focus+context visualizations, some objects or parts of the data are shown in detail, while other objects or parts act as a context. While the data “in focus” often are displayed rather opaque, the rest of the data can be shown rather transparent. Since too much blending of different colors at a single voxel could lead to visualizations which can be difficult to interpret, the presented rendering framework should not be used uncritically. Careful use of the framework, however, enables in-depth analysis of topologically very complex data. To facilitate the analysis, each volume object is accompanied by a check box that can be used to quickly turn on and off individual objects from the scene. This way, the user can interactively investigate the effect and contribution made by the individual data fields. B. Animation - Interactive Analysis Once the desired data is selected and an appropriate visualization scene is created, our rendering framework handles two types of navigations through the time-varying data set.
The data set can be explored by either dragging a time slider or by using the animation utility. The time slider is very useful for investigating the data at different time steps. Once a new time step is selected, the visualization scene is automatically updated using the same transfer functions. The time slider feature also simplifies the process of finding color and opacity tables that are well-suited for the whole time-series. Finding good transfer functions is often a tedious process that sometimes involves clipping of data value ranges. The animation utility allows a more continuous visualization of the time-dependent data. Here, the user can choose the order of the data sets to be loaded, the step size, as well as pregenerated user interactions such as rotation and zooming. In addition to manipulating the time slider, our rendering framework supports an additional set of tools to facilitate the analysis of a three-dimensional time-dependent flow field. This includes manipulation of transfer functions, clip planes, datasubset selections, and other user functions at interactive rates. These are all tools that can be used to diminish the occlusion effects, by for instance creating transparent visualizations and reducing the complexity of the scene by focusing on a region of interest. C. Visual Perception While geometric objects have reflective properties so that the use of light sources emphasizes the 3D shape perception, a volume could be thought of as emitting light, where the emitted light expresses the data value of a particular voxel. Without the use of light sources, one could then anticipate that volume rendering would result in “flat” images with little information about the depth relations in the volume. However, “simple” nonphotorealistic rendering (NPR) techniques can be used to enhance spatial structures and to give the necessary three-dimensional appearance. In our framework, we use the emission-absorption [40] model in combination with limb darkening [23] to enhance interesting features. Even though several other NPR techniques exist, including warm to cool shading, gradient enhancement, and depth enhancement [41], [42], we feel that our rendering strategy in combination with interactivity provides enough spatial cues for the purpose of data analysis. In this context, it is important to emphasize that interactivity is far more important than shading. Of course when producing illustrations of volume data, there is much to gain on using more advanced NPR techniques than the ones presented in our framework. Our framework can easily be extended to support a broader amount of various NPR techniques as well as volume shading, as all visualization scenes are written as separate shader programs. One benefit with the presented rendering framework, is its flexibility when rendering multiple data features in a single scene. In order to communicate data field correlations with several data fields, it can sometimes be useful to focus on some specific features using limb darkening while visualizing other properties more transparent without further feature enhancements. Examples of such visualizations can be seen in Figs. 9 and 11(b). In Fig. 9, the field “in focus” is the vorticity field. Here, correlation with high rate of turbulent production is
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shown through a more transparent visualization and through the use of a clip plane. In Fig. 11(b), the field in focus is enstrophy, whereas the regions of high energy dissipation are shown without any further feature enhancement. IX. R ESULTS We have visualized various fields and combinations of fields derived from the turbulent channel flow simulation. In order to reduce the amount of data, only 1/8 of the entire computational domain has been considered, cf. Fig. 1. We employ vorticity field lines as indicators to provide a means to identify “ejection” and “sweep” regions; two particularly important spatio-temporal events in wall-bounded turbulent flows. In sweep regions, high momentum fluid elements far from the walls are brought closer to the surface, thereby increasing the local momentum density in the vicinity of the wall. This results in increased internal shear close to the walls, and subsequently also increased vorticity, and increased kinetic energy dissipation rate. In ejection regions, on the other hand, low momentum fluid close to the wall is brought outwards whereby the resulting momentum density is reduced. This results in reduced internal shear, and consequently also a locally reduced vorticity and reduced kinetic energy dissipation rate. The outward motion in ejection regions creates vorticity loops with associated vortex stretching and subsequent vortex intensification. The rate of energy dissipation is also increased in these loops. In Fig. 9, the fields are seen through a clip plane perpendicular to the streamwise direction. The vorticity field lines are colored according to the vertical component of the velocity (w). They are colored green in the sweep regions where the fluid elements are approaching the wall, i.e. w < 0, and yellow in the ejection regions where w > 0. The field lines in the figure are distorted according to the fluid motion. The vorticity field line topology shows that the ejection regions are associated with a low vorticity field line density, whereas the field line density is increased in the sweep regions. In the same scene the rate of instantaneous turbulent production P(x,t) = −uw ∂ U(z)/∂ z is visualized. Positive values are assigned a red color. As expected from kinematical considerations, positive production occurs in regions where the streamwise (u) and wall-normal (w) fluctuating velocity components have different signs. It can clearly be seen that both ejection and sweep regions are associated with P > 0; ejections and sweeps are thus characterized by streamwise velocity fluctuations u < 0 and u > 0, respectively. Positive turbulent energy production is associated with a transfer of mean flow energy to the turbulent motion. Mean flow energy is provided by the imposed streamwise pressure gradient. Fig. 10 shows the instantaneous rate of turbulence production. Positive production is colored in red while negative production is colored in green. In regions with negative turbulence production, kinetic energy is extracted from the turbulence and transfered to the mean flow field. Positive energy transfer dominates in an ensemble averaged sense, however, and this is also indicated in the instantaneous distribution displayed in Fig. 10. The area and saturation of the
Fig. 9. Vorticity field lines colored to identify sweep (green) and ejection (yellow) regions. Regions with high rate of turbulent production are colored red.
Fig. 10. Regions of positive (red) and negative (green) rate of turbulent production.
red patches are larger compared with the green ones, indicating a net positive turbulent production rate. In Fig. 11(a), the instantaneous turbulent energy dissipation rate ε (x,t) = 2ν (∂ ui /∂ x j )(∂ ui /∂ x j ) is visualized together with “vortices” through the λ2 criterion [9]. The rate of dissipation is strongest close to the wall, where the velocity shear has its maximum. Here, the shear is manifested as strong vorticity sheets consisting mainly of spanwise vorticity; vortices per se do not exist in this thin layer immediately above the wall. The λ2 structures are shown in green and are visualized as tubes with transparent cores in order to make the dissipation rate visible inside them. The turbulent energy dissipation rate is shown in red to yellow where the red color indicates maximum values. Apparently the energy dissipation inside the core of the vortices is weak compared with the levels observed within the sheet-like structures. The latter thus dominates. In order to capture vorticity sheets in addition to “tube-
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like” structures, enstrophy (kω k) (colored in white to gray) is visualized together with turbulent energy dissipation rate, and the result is shown in Fig. 11(b). The highest values of enstrophy are found close to the wall, through intense vorticity sheets. Further into the channel, the enstrophy is weaker and forms structures that are a mix of tubes and sheets. The opacity corresponding to the interiors of the enstrophy structures are given low values so they are made transparent. Similarly to Fig. 11(a), the energy dissipation is represented in colors varying from red to yellow with decreasing values. Apparently there is a close spatial correspondence between enstrophy and turbulent kinetic energy dissipation, as shown in Fig. 11(b), which in fact is loosely manifested through the exact relation ε = 2ν (∂ ui /∂ x j )(∂ ui /∂ x j ) = 2ν [ωi ωi + (∂ ui /∂ x j )(∂ u j /∂ xi )], where the last term has a secondary effect. Visualization of enstrophy together with λ2 structures show that the vortices often have vorticity sheets wrapped around them (Fig. 12(a)). To make the vortices more visible, the interiors of the enstrophy structures are made completely transparent. Vorticity sheets are formed in the strong shear region close to the wall. They are distorted and advected around in the fluid and can be wrapped up around the vortices. The orientation of the vorticity field within the vorticity sheets is generally inclined relative to the direction of the nearby vortex cores, whereas the field lines inside the vortex cores are approximately oriented along the cores (Fig. 12(b)). Contrary to isotropic turbulence, the turbulence in the channel is highly anisotropic, and the degree of anisotropy is higher toward the walls of the channel than in the center. The vorticity is created close to the wall by internal shear and is advected away from the wall in the ejection regions. This leads to local intensification of the wall-normal vorticity through straining. In sweep regions, the flow is directed toward the wall with a diverging motion close to the wall, again leading to intensification of vorticity through straining. Although the channel flow is stationary from a statistical point of view, it is surprising that also the instantaneous vorticity structures develop relatively slowly in time. The animations show that during the time it takes for the structures to traverse the computational domain (we visualize only half of that), the structures show a relatively stationary behavior. However, on the smallest scales, the temporal changes are substantial. Animation of the vorticity field together with the vorticity structures provides qualitative information about the details of the evolution of the flow field. X. C ONCLUSION We have visualized and animated vorticity fields in a turbulent channel flow by letting the seed points for the vorticity field be material particles following particle paths. This gives qualitative information of the spatial and temporal evolution of the vorticity field. The resulting vorticity fields show the evolution of the flow. Usually scalar fields like enstrophy or fields obtained by the λ2 criteria is used to visualize vorticity fields. We have demonstrated that these approaches give a limited view of the vorticity field. For example, the λ2 method is only of limited interest close to the wall since there are
few “tube-like” structures there. The direction of the vorticity field contains important information, which is also lost when visualizing the vorticity as a scalar field. In the case of turbulent channel flow, due to the presence of the wall, the turbulence is highly anisotropic. This is seen in the structure of the vorticity field. In regions close to the wall, vortices (tube-like structures of vorticity) do not dominate as in isotropic turbulence. In the channel flow there is an intricate mix of vortices and vorticity sheets. The latter dominates close to the wall, where they are produced at a constant rate by the strong velocity shear. The production of vorticity by straining also peaks in a region close to the wall. The vorticity sheets are advected away from the wall by ejection events and distorted through straining. In the sweep regions, the vorticity magnitude is increased by straining, leading to preferential spanwise vorticity close to the wall. The energy dissipation is strongest in these regions. The error involved by assuming that vorticity field lines and material lines are equivalent in wall-bounded turbulent flows has been quantified in the present study. It has been demonstrated that the error is largest (and approximately equal) in the directions parallel to the wall, and that the error grows approximately linearly in time as the fluid elements are advected downstream. Even though the vorticity field lines, in general, do not evolve as material lines, the proposed animation technique still enable a trustworthy qualitative interpretation of the spatio-temporal behavior of the vorticity field in turbulent flows. The technique cannot be used to track individual vorticity field lines over time, but is still able to show the instantaneous topology of the vorticity field and its time evolution. However, the time evolution of the vorticity field is an indicator of the motion of the fluid particles. We animated the vorticity field according to the above mentioned seeding and tracking strategy. The vorticity field is rendered in regions of non-neglectable enstrophy in the same scene as the λ2 structures. The animation is carried out over the time interval it takes for the structures to traverse the whole channel. To the best of our knowledge, this is the first time an animation like this has been done in wall-bounded turbulent flows. A notable outcome of the animation is the observed stationarity of the vorticity structures (in a reference frame moving with the flow). The rendering stage of the field line animation technique is decoupled from the preceding stages of the visualization method, in order to increase the efficiency. In combination with the presented volume rendering framework, this allows interactive exploration of multiple fields simultaneously, and thus sets the stage for a more complete analysis of the flow field. ACKNOWLEDGMENT The authors would like to thank Dr. Xing Cai and the anonymous reviewers for their valuable comments and suggestions. R EFERENCES [1] A. A. Townsend, The Cambr. Univ. Press, 1976.
Structure
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Fig. 11. (a) Vortices defined by λ2 < 0 are colored green, while regions of energy dissipation are colored from yellow for moderate dissipation to red for maximum dissipation. Evidently the dissipation is weak within the vortices. (b) Energy dissipation shown in red to yellow is visualized together with enstrophy in white to gray. Areas with high energy dissipation is spatially well correlated with enstrophy.
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Fig. 12. (a) λ2 structures visualized together with high enstrophy (yellow) showing vorticity sheets surrounding the vortices (red). (b) Vorticity field lines rendered in regions of high enstrophy together with vorticity sheets and vortex cores. The direction of the vorticity field within these sheets is rarely oriented along the direction of the underlying vortices, as opposed to the field lines rendered inside the vortices.
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Anders Helgeland received BS and MS degrees in computer science from the University of Oslo, Norway, in 2000 and 2002. He is a PhD candidate in computer science at the University Graduate Center, Norway. His current research interests include flow visualization and volume visualization. He was a recipient of the best poster award at the Proceedings of the IEEE Visualization Conference 2002. He is a member of the IEEE, the IEEE Computer Society and ACM SIGGRAPH.
B. Anders Pettersson Reif received MS and Dr.Ing. degrees in Applied Mechanics at Lule˚a Technical University in 1992, and at the Norwegian University of Science and Technology in 1997, respectively. Pettersson Reif is working as a principal scientist at the Norwegian Defence Research Establishment (FFI) and is appointed adjunct professor in turbulence modeling at Chalmers University of Technology. Research interests include turbulence physics and modeling, and computational fluid dynamics.
Øyvind Andreassen studied astrophysics at the University of Oslo, where he received a MS in 1981 and a PhD in physics in 1994. Andreassen is currently chief scientist at the Norwegian Defence Research Establishment (FFI) and adjunct professor in applied mathematics at University Graduate Center at Kjeller and University of Oslo. His research interests include scientific computing and visualization, wave physics, computational fluid dynamics. turbulence and flow noise.
Carl Erik Wasberg received a MS degree in Industrial Mathematics from the Norwegian Institute of Technology in 1986, and a PhD degree in Applied Mathematics from the University of Bergen in 1995. He is now a senior scientist at the Norwegian Defence Research Establishment (FFI), where he works in computational fluid dynamics. His research interests include numerical solution of partial differential equations, large-scale computing, and fluid mechanics.
