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Visualization-Driven Structural and Statistical

Analysis of Turbulent Flows

Kenny Gruchalla1, Mark Rast2, Elizabeth Bradley1, John Clyne3, and Pablo

Mininni4

1Department of Computer Science, University of Colorado, Boulder, Colorado

2Laboratory for Atmospheric and Space Physics, Department of Astrophysical and

Planetary Sciences, University of Colorado, Boulder, Colorado

3Computational and Information Systems Laboratory, National Center for

Atmospheric Research, Boulder, Colorado

4Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de

Buenos Aires, Argentina and Geophysical Turbulence Program, National Center for

Atmospheric Research, Boulder, Colorado

Abstract. Knowledge extraction from data volumes of ever increasing

size requires ever more ﬂexible tools to facilitate interactive query. In-

teractivity enables real-time hypothesis testing and scientiﬁc discovery,

but can generally not be achieved without some level of data reduction.

The approach described in this paper combines multi-resolution access,

region-of-interest extraction, and structure identiﬁcation in order to pro-

vide interactive spatial and statistical analysis of a terascale data volume.

Unique aspects of our approach include the incorporation of both local

and global statistics of the ﬂow structures, and iterative reﬁnement fa-

cilities, which combine geometry, topology, and statistics to allow the

user to eﬀectively tailor the analysis and visualization to the science.

Working together, these facilities allow a user to focus the spatial scale

and domain of the analysis and perform an appropriately tailored mul-

tivariate visualization of the corresponding data. All of these ideas and

algorithms are instantiated in a deployed visualization and analysis tool

called VAPOR, which is in routine use by scientists internationally. In

data from a 10243simulation of a forced turbulent ﬂow, VAPOR allowed

us to perform a visual data exploration of the ﬂow properties at interac-

tive speeds, leading to the discovery of novel scientiﬁc properties of the

ﬂow, in the form of two distinct vortical structure populations. These

structures would have been very diﬃcult (if not impossible) to ﬁnd with

statistical overviews or other existing visualization-driven analysis ap-

proaches. This kind of intelligent, focused analysis/reﬁnement approach

will become even more important as computational science moves to-

wards petascale applications.

1 Challenges to data analysis

A critical disparity is growing in the ﬁeld of computational science: our abil-

ity to generate numerical data from scientiﬁc computations has in many cases

exceeded our ability to analyze those data eﬀectively. Supercomputing systems

have now reached petaﬂop performance [1], supporting numerical models of ex-

traordinary complexity, ﬁdelity, and scale. In supercomputing centers, terabyte

data sets are now commonplace and petabyte data sets are anticipated within a

few years. However, analysis tools and the computational machinery that sup-

ports them have not been able to scale to meet the demands of these data.

For many computational scientists, this lack of analysis capability is the largest

barrier to scientiﬁc discovery.

The imbalance of scale between numerical simulation and data analysis is

largely due to their contrasting demands on computational resources. Large-

scale numerical simulation is typically a batch processing operation that proceeds

without human interaction on parallel supercomputers. Data analysis, in con-

trast, is fundamentally an interactive process with a human investigator in the

loop, posing questions about the data and using the responses to progressively

reﬁne those questions[2]. While some data analyses certainly can be performed

in batch mode, this is only practical for completely predetermined investiga-

tions. Exploratory analysis depends on hypothesis generation and testing, which

requires an interactive environment that can provide timely and meaningful feed-

back to the investigator. Unfortunately, this kind of interactive workﬂow is not

well-suited to batch access on a parallel supercomputer. Another key bottleneck

in the analysis process is data storage. If the data exceeds the size of the available

random access media, one must manage its storage and exchange across diﬀer-

ent media. Disk transfer rates are generally inadequate to support interactive

processing of large-scale computational data sets.

These dilemma can be addressed by data reduction, and a variety of schemes

have been developed to reduce data volumes while maintaining essential proper-

ties. Their methods and results depend on the scientiﬁc goals of the simulation

and analysis. For example, in the investigation of turbulent ﬂows, analysis of

strictly statistical or spectral properties can enable signiﬁcant reduction in data

dimensionality, while analysis of local ﬂow dynamics, thermodynamics, or stabil-

ity does not. In the later cases, the only solution is to reduce the physical volume

under analysis. There are two general classes of methods for this. First, one can

isolate and extract local sub-regions from the global domain. The success of this

strategy depends on locating those regions in the solution that are of particu-

lar scientiﬁc importance or interest, which is a real challenge to intelligent data

analysis. There has been some very interesting work in the IDA community on

dimensional reduction for this purpose[3, 4]. The visualization-driven approach

described in this paper extracts regions of interest using an iterative interactive

ﬁltering technique that employs a combination of global and local ﬂow statistics.

The second class of data-volume reduction techniques uses a coarsened global

approximation of the discrete solution to subsample the data over the entire do-

main. The obvious challenge here is selecting an appropriate coarsening (method

and scale) to maintain accuracy — and conﬁdence in the results.

Both of these data-reduction techniques have been implemented in VAPOR,

an open-source desktop visualization-driven analysis application (available at

http://www.vapor.ucar.edu). It closely couples advanced visualization with quan-

titative analysis capabilities, and it handles the complexities of large datasets

using a hierarchical data model. It is designed to support a multi-phase analysis

process, allowing the investigator to control a speed-and-memory versus a locus-

and-quality trade oﬀ. This is an ideal context within which to explore intelligent

data reduction. Using the ideas described in the previous paragraph, we have

extended VAPOR’s capabilities to include deﬁnition, manipulation, and reﬁne-

ment of feature sub-spaces based on multi-scale statistics of turbulent structures

contained within the data. The user can explore the data volume at a coarsened

resolution to gain a qualitative understanding and identify regions/structures of

interest. Those regions/stuctures can then be investigated at progressively higher

resolutions with computationally intensive visualization and analysis performed

on progressively smaller sub-domains or structure populations. The base func-

tionality of VAPOR provides data coarsening in the form of multi-resolution

access via wavelet decomposition and reconstruction [5]. VAPOR’s data coars-

ening approach coupled with a simple a simple sub-domain selection capabil-

ity and has a successful track record in the analysis of large-scale simulation

data using only modest computing resources [6]. With the addition of intelligent

region-of-interest extraction, VAPOR can now provide interactive, scientiﬁcally

meaningful explorations of tera-scale data volumes.

The following Section describes VAPOR’s visualization-driven analysis ca-

pabilities; Section 3 demonstrates their power using data from a 10243forced

incompressible hydrodynamic simulation.

2 VAPOR: A desktop analysis and visualization

application

Many applications have been developed speciﬁcally for the visualization and

analysis of large-scale, time-varying numerical data, but all of them have signiﬁ-

cant limitations in terms of visualization, analysis, and/or scalability. In many of

these tools, the emphasis is on the algorithms and the generation of aesthetic im-

ages, rather than on scientiﬁc discovery [7]. Visualization-oriented applications,

like Paraview [8], Visit [9], and Ensight, lack quantitative analysis capabilities,

and many of them demand specialized parallel computing resources. High-level,

fourth-generation data languages such as ITT’s IDL and Mathworks’s Matlab

are on the opposite end of the analysis-visualization spectrum. They provide a

rich set of mathematical utilities for the quantitative analysis of scientiﬁc data

but only limited visualization capabilities, and they do not scale well to very

large data sets.

The goal of the VAPOR project was to address these shortcomings. It pro-

vides an integrated suite of advanced visualization capabilities that are specif-

ically tailored to volumetric time-varying, multivariate numerical data. These

capabilities, coupled with the intelligent data reduction strategies introduced in

the previous sections, allow investigators to rapidly identify scientiﬁcally mean-

ingful spatial-temporal regions in large-scale multivariate data. VAPOR’s design

— both its functionality and its user interface — is guided by a steering com-

mittee of computational physicists to ensure that it truly meets the needs of

the end-user community. Scalability is addressed through a multi-phase analysis

process that is based on a combination of region-of-interest isolation, feature

extraction (with our extensions), and a hierarchical data model. Finally, VA-

POR interfaces seamlessly with high-level analysis languages like IDL, allowing

its user to perform rigorous quantitative analyses of regions of interest.

2.1 Visualization

VAPOR incorporates a variety of state-of-the-art volume rendering and ﬂow-

visualization techniques, including both direct and indirect volume rendering [10].

Direct volume rendering describes a class of techniques, which generate images

directly from volumetric data without any intermediate geometric constructions,

while indirect volume rendering constructs geometric isosurfaces. To support the

visualization of vector ﬁelds, VAPOR provides both sparse and dense particle-

tracing methods [11]. The former render the geometry of individual trajectories

of particles seeded in a ﬂow ﬁeld, and can support both steady (time-invariant)

and unsteady (time-varying) trajectory integration. Dense particle-tracing meth-

ods synthesize textures that represent how the ﬂow convolves input noise.

VAPOR’s integrated design allows these volume rendering and ﬂow-visualization

techniques to be used in diﬀerent combinations over a single analysis run, in con-

cert with the intelligent data reduction strategies described in Section 2.3, as the

investigator progressively isolates and reﬁnes scientiﬁcally meaningful regions of

the data. The hierarchical data model that supports this is described in the next

Section. By utilizing the optimized data-parallel streaming processors of modern

graphics processing units (GPUs), VAPOR can eﬀectively work with volumes of

the order of 15363[6].

2.2 Hierarchical data model

The VAPOR data storage model is based on wavelet decomposition [5, 12]. Data

are stored as a hierarchy of successively coarser wavelet coeﬃcients; each level

in this hierarchy represents a halving of the data resolution along each spatial

axis, corresponding to an eight-fold reduction in data volume. In this manner,

VAPOR maintains a series of useful coarsened approximations of the data, any

of which can be accessed on demand during an analysis run, without an undue

increase in storage requirements. Wavelet data are organized into a collection

of multiple ﬁles: one binary ﬁle containing the wavelet coeﬃcients for each time

step, each variable, and each wavelet transformation level, and a single metadata

ﬁle that describes the attributes of the ﬁeld data (e.g., the grid type, the time

steps, the spatial resolution, the ﬁeld names, etc.).

This storage model naturally supports intelligent, interactive data decom-

position. It allows VAPOR to operate on any subset of time steps, variables,

and wavelet transformation levels, which has a variety of important advantages,

including iterative focus and reﬁnement of the analysis eﬀort. An investigator

can control the level of interactivity by regulating the ﬁdelity of the data, ﬁrst

browsing a coarsened representation across the global spatial-temporal domain

to identify regions or features of interest and then examining those reduced do-

mains in greater detail. The hierarchical data/metadata combination also allows

VAPOR to work with very large data collections, as data components can be

stored oﬀ-line until required, handling incomplete data sets smoothly.

2.3 Multivariate feature extraction

The VAPOR volume rendering capability forms the basis for the multivariate

feature extraction technique we have implemented to isolate structures of inter-

est in large data sets. A multidimensional transfer function is used to deﬁne a

mapping from data values to the color and opacity values in the volume render-

ing. The opacity component of this function visually separates the volume into

opaque features. VAPOR users can construct and reﬁne these functions itera-

tively, and use them in diﬀerent thresholding schemes to visually separate the

volume into opaque regions.

Once these regions have been visually identiﬁed using the transfer function,

the individual structures are extracted and tagged using a connected-component

labeling algorithm [13], an image-processing technique that assigns groups of ǫ-

connected data points5a unique feature label. Once the features have been iden-

tiﬁed in this manner, they can be visualized and analyzed as individual features

oppose to a set of opaque voxels. The individual features can be visualized in

isolation or as members of sub-groups, and the data points and geometry of each

can exported to external analysis packages for further analysis, as described in

Section 2.4. This allows any user-deﬁned physical property to be computed on

the associated data points contained in each feature, over any ﬁeld or combina-

tion of ﬁelds in the data. VAPOR presents the resulting values and distributions

to the user as a table of feature-local histograms and property values, as shown

in Figure 1. Using this table, the set of features can be culled based on the central

moments of their distributions to further focus the study. The entire reduction

process—including the transfer function design and feature deﬁnition—can be it-

erated to progressively reﬁne the analysis, providing insight into the multivariate

properties of structures across multiple scales.

2.4 Coupled visual, quantitative, and statistical analysis

Understanding large-scale simulation data is an exploratory process that can be

greatly facilitated by combining highly interactive, qualitative visual examina-

tion with quantitative numerical analysis. Visualization can be used to motivate

analysis through the identiﬁcation of structures in the data, giving rise to hy-

potheses that can be validated or rejected through numerical study. Likewise,

the analysis can be used to drive the visualization, identifying salient quanti-

tative characteristics of the data through numerical study, and then visualiz-

ing their associated geometric shapes and physical properties. VAPOR’s design

5i.e., those that are connected by an ǫchain

Fig. 1. The VAPOR structure analysis dialog, which displays feature-local histograms

of user-selected ﬁeld distributions.

seamlessly combines qualitative visual and quantitative numerical investigation,

enabling its users to interactively transition between the two. Its multi-resolution

visualization and region-of-interest isolation capabilities, in conjunction with its

hierarchical data representation, allow its users to cull data intelligently and pass

appropriate subsets to an external quantitative analysis package.

Smooth integration of all of these capabilities required some interesting de-

sign decisions. VAPOR performs GPU-accelerated visualization natively, as de-

scribed in Section 2.1, and hands numerical analysis oﬀ to IDL. VAPOR and

IDL sessions are run simultaneously; after regions of interest are identiﬁed in

the former, the associated data volumes are exported via metadata descriptors

to the latter for further study. The tight coupling between IDL and VAPOR is

accomplished by a library of data-access routines, which allow IDL access to the

wavelet-encoded data representation. (This approach is readily generalizable to

other analysis packages, complementing and enhancing existing user capabili-

ties.) The qualitative/quantitative tandem is very eﬀective: IDL, as mentioned

at the beginning of Section 2, does not scale well to large data sets [12], but VA-

POR’s ability to focus the study neatly bypasses that problem, and the results

of IDL analysis on focused regions can be seamlessly imported back into the

VAPOR session for visual investigation. By repeating this process, very large

data sets can be interactively explored, visualized, and analyzed without the

overhead of reading, writing, and operating on the full data volume.

3 Application to vortical structures in Taylor-Green ﬂow

As an illustration of the power of the ideas described in the previous sections,

we use VAPOR to explore data from an incompressible Taylor-Green forced tur-

bulence simulation with a microscale Reynolds number of Rλ∼1300 [14]. The

particular structures in this data that are of scientiﬁc interest involve vortic-

ity, but the volume contains so many of these structures, of diﬀerent sizes and

strengths, as to pose a truly daunting analysis problem. The small-scale struc-

tures are particularly hard to isolate, so that is what we set out to analyze with

VAPOR.

Fig. 2. A volume rendering of areas of strong vorticity in Taylor-Green turbulence

isolates tens of thousands of vortical structures.

3.1 Global vorticity and structure identiﬁcation

Vortices play important roles in the dynamics and transport properties of ﬂuid

ﬂows, but they are surprisingly hard to deﬁne, which complicates the task of de-

signing a vortex extraction method. Jiang et al. [15] provide an extensive survey

of current techniques. As a working deﬁnition, we treat a vortex ﬁlament, tube,

or sheet as a connected region with a higher relative amplitude of vorticity than

its surrounding [16]. Many vortex detection and visualization methods use the

same deﬁnition, and most of them operationalize it by thresholding the magni-

tude of the vorticity. This is the starting point for our analysis, but VAPOR’s

capabilities allowed us to add other scientiﬁcally meaningful analysis steps and

iteratively focus the process. In this particular case, it allowed us to investigate

the correlation between vorticity and helicity across multiple scales and discover

important structural properties that were previously unknown.

The ﬁrst step in the process is to threshold the vorticity of the Taylor-Green

data using the opacity contribution of the multidimensional tranfer function.

The ﬁelds in the data include the simulated velocity vector ﬁeld and two derived

ﬁelds: a vorticity vector ﬁeld and a normalized helicity ﬁeld. Vorticity is deﬁned

as the curl of a velocity ﬁeld, ω=∇ × v, characterizing the pointwise rotation

of ﬂuid elements. Helicity is a scalar value, Hn=v·ω

|v||ω|, the cosine of the angle

between velocity and vorticity. An initial vorticity threshold was chosen to begin

separating the tube-like vortical structures in the data volume. This step isolates

tens of thousands of vortical structures, as shown in Figure 2. Using VAPOR’s

iterative reﬁnement capabilities, we focus the study by further considering the

helicity within these structures. A global analysis across the entire data volume,

Figure 3, shows that both helicity and its pointwise correlation with vorticity

are distributed in a nearly uniform fashion—i.e., that all angles between velocity

and vorticity vectors occur with similar frequencies across all values of vorticity.

While this is a useful result, it lumps the whole data volume together, possibly

obscuring important local diﬀerences. Using VAPOR to generate feature-local

histograms, we ﬁnd that diﬀerent high-vorticity regions do indeed have distinct

helicity distributions, Figure 3(c). Three populations of structures are conspic-

uously evident: those whose helicity distributions span the full range with no

distinct peak, those with a peak at high absolute values of helicity (i.e., domi-

nated by nearly aligned or anti-aligned velocity and vorticity vectors), and those

whose helicity distributions peak near zero (i.e., dominated by nearly orthogonal

velocity and vorticity).

Fig. 3. The relationship between vorticity and helicity in Taylor-Green Turbulence a)

the histogram of global normalized helicity, indicating helicity, measured point wise in

the domain, has a nearly uniform distribution; b) the scatter plot of vorticity magni-

tude versus normalized helicity, showing that helicity has a nearly uniform distribution

across all values of vorticity; c) a selected subset of the feature-local helicity histograms

from features deﬁned by high vorticity that show individual regions of strong vorticity

have distinct helicity distributions.

Using VAPOR’s intrinsic capabilities, we have thus eﬀectively diﬀerentiated

the regions of strong vorticity into three structure populations, based on their

helicity distributions. In order to investigate the statistics and local dynamics

of these structures, we next extend the analysis through a combination of vi-

sualization in VAPOR and focused study in the coupled quantitative analysis

package. By visualizing individual features in isolation, we ﬁnd that the wide

noisy distributions belong to composite structures that were not well separated

into individual components by the original vorticity thresholding, while the other

two populations are those of individual tube-like structures. This result allows

us to further cull the dataset and focus on the tube-like structures with either

high or low helicity magnitude. Both populations have similar geometries, but

streamlines seeded in these regions, as shown in Figure 5, reveal that their ﬂow

properties are quite diﬀerent. In the low-helicity tubes, the streamlines twist

around the core; in the high-helicity tubes the streamlines more closely follow

the writhe of the tube.

Further interactive analysis of these distinct vortex structures can proceed

either by examining the statistical properties of the population or the detailed

dynamics of any one of them. Looking ﬁrst at statistics of the population of

vortical structures as a whole, we note that, while structures with all values of

helicity exist, there seems to be a small deﬁcit of those with high absolute mean

value compared to the point-wise helicity distribution (Figure 4a). Moreover,

the helicty of any given structure is well deﬁned and symetrically distributed

about its mean value (Figure 4b and 4c). The helicity distribution within a

great majority of the structures has both small variance and skewness.

Fig. 4. Distributions of the ﬁrst three central moments of the feature-local helicity

distributions.

The detailed dynamics underlying any single vortex structure is also acces-

sible. By exporting planar cross sections through tubes using VAPOR’s cross-

section capability, average radial proﬁles of the helicity and vorticity can be

constructed (Figure 5c & 5d). Distinct diﬀerences between the maximally and

minimally helical structures are apparent. The maximally helical structure has

one sign helicity throughout, while the minimally helical twisted structure shows

a change in the sign of helicity near its border (Figure 5c). This appears to be

associated with inward (toward the pinched section midway along the tube )

axial ﬂow surrounding the outside of the vortex tube and outward (diverging

from a pinched section midway along the tube) axial ﬂow in its core, (Figure 6).

A temporal history of these ﬂows would be critical in conﬁrming what looks

to be a signiﬁcant vorticity ampliﬁcation mechanism in this minimally helical

vortex ﬁlament. Also critical in future analysis would be the ability to combine

the statistical and dynamical analyses presented here to determine how common

this mechanism is and whether particular dynamical processes are statistically

linked to speciﬁc structure populations.

Fig. 5. Local dynamics of two strucutures with diﬀerent helicity distributions showing:

a) streamlines seeded within segmented region; b) the feature-local helicity histogram;

c) an average radial helicity proﬁle; d) an average radial vorticity proﬁle. The shaded

region of the radial proﬁles represents the inside of the visualized structure.

The primary advantage of coupling visual data investigation with a data

analysis language is the ability to defer expensive calculations of derived quan-

tities until they are needed and then perform them only over sub-domains of

interest. The computational requirements for computing such variables in ad-

vance, across the entire domain, is often impratical, overwhelming the available

analysis resources. Furthermore, some quantities, as was shown by our analysis

of the Taylor-Green ﬂow, can only be computed with reference to the location

of a ﬂow structure and are therefore not in principle a priori computable. The

coupling between VAPOR and IDL facilitates the calculation of derived quanti-

ties as needed over sub-regions of the domain, realizing considerable savings in

storage space and processing time.

Fig. 6. Top: Streamlines colored by y-component velocity, which approximates the

axial velocity. Bottom: vorticity magnitude and Y-component velocity cross-sections

taken at positions a, b, & c (extents of the structure are bounded by the dotted line)

4 Conclusions

VAPOR’s tight integration of visualization with traditional techniques like statis-

tics, fourth-generation data languages, and eﬀective information-management

strategies meets the challenges that are inherent in visual exploration of com-

plex turbulence data. This will only become more important as data volumes

increase. The Taylor-Green ﬂow simulation described in the previous section,

which has 10243degrees of freedom, can be readily computed on today’s ter-

aﬂop supercomputing platforms. The emergence of petaﬂop-capable machines

will enable simulations at vastly greater scales, resulting in substantially larger

data volumes. 40963simulations have already been conducted on existing su-

percomputers [17] and the recent NSF Track 1 Petascale computing solicitation

calls for a system capable of executing a homogeneous turbulence simulation

with 12,2883degrees of freedom [18]. The interactive analysis model in this pa-

per, with its reliance on progressive data reﬁnement, visual data browsing, and

region/structure-of-interest isolation, is intrinsically highly scalable. We have

described our experiences with this analysis model in the context of investigat-

ing numerically simulated turbulence. However, we believe that these techniques

have applicability across a broad spectrum of data-intensive sciences.

5 Acknowledgements

We wish to thank the National Science Foundation and the National Center for

Atmospheric Research for their computational support.

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