Fluid dynamics, describing the flow of liquids and gases, is one of the most important of all areas in the fields of science and technology. Life would not exist without fluids, particularly air and water. The movement of air, for example, keeps us warm and enables us to breathe oxygen, while water makes up most of our body mass.
The study of the motion of fluids underpins many aspects of physics, biology, astronomy and engineering. Fluid dynamics can be subdivided into disciplines such as aerodynamics, describing the flow of air and other gases, which plays a significant role in areas such as aerospace development, and wind turbine generators, and hydrodynamics, describing the flow of water and other liquids, modelling concepts such as ocean and cardiovascular flow.
Modelling fluid flow
There are numerous applications of fluid dynamics. Mathematical modelling of the movement of fluids plays a significant role in the developments of modern infrastructure such as the flow of traffic on our road networks. The recent improvements in observation and measurement technologies allows the extraction of information from ultrasonic images of cardiovascular flows and satellite images of coastal flows enables the numerical simulation of various scenarios.
The volume of data obtained from numerical simulations, together with observation and measurement data necessitates an efficient way of describing flow properties in order to spot trends and make predictions from big data.
Dr Takashi Sakajo and Dr Tomoo Yokoyama have developed a novel approach to classifying patterns that describe the flow of fluids using topology (the study of geometrical properties and spatial relations which are unaltered by elastic deformations such as a stretching or a twisting) and dynamical systems theory (used to describe the behaviour of complex dynamical systems where a function describes the time dependence of a point e.g. a clock pendulum swinging or the flow of water in a pipe).
In order to analyse fluid flow, it is necessary to be able to visualise the key features of the ﬂow in the situation being studied. It is also important to remember that it is the flow and not the fluid that is of interest. While all physical fluid flows are three dimensional, it can be convenient and quite accurate to visualise them as two dimensional (2D) in order to model flow phenomena. Dr Sakajo’s research is underpinned with his classifying the global streamline patterns of 2D incompressible flows in vortex structures.
Definitions and assumptions
The streamline is the path taken by an imaginary particle suspended in the fluid as it is carried along. While the fluid moves, these streamlines are fixed and can be interpreted in a similar way to contours on a map. In streamline flow, velocity, pressure and other flow properties remain constant. When the streamlines are close together, the fluid is moving relatively quickly. Where the streamlines are more spread out, the fluid is moving slowly and may be relatively still. In streamline flow, velocity, pressure and other flow properties remain constant.
While no fluid can really be incompressible, when modelling flow, a fluid can be assumed incompressible when the effects of pressure on the fluid density are zero or negligible. The density and the specific volume of the fluid, therefore, do not change when the pressure changes. Similarly, ﬂuids possess varying degrees of viscosity, but in some situations, the forces on ﬂuid elements that arise from viscosity are small compared with other forces and can be considered negligible, so we can treat the ﬂow as inviscid and ignore the effects of viscosity.
When analysing 2D incompressible flow, a stream function is employed to make both the associated calculations and understanding the flow easier. A stream function’s derivatives give the velocity components of a particular flow situation. The stream function can be used to plot streamlines, the lines with a constant value of the stream function, representing the trajectories of imaginary particles in a steady flow, i.e. a flow with constant velocity.
A ﬂow of 2D incompressible and inviscid ﬂuid is an example of a Hamiltonian vector ﬁeld, a geometric representation of Hamilton’s equations in classical mechanics, where its Hamiltonian, or energy function, corresponds to the stream function whose curves are streamlines.
Dr Sakajo uses genus elements, depicted as circles, to represent the physical obstacles in the flow. These are also the elliptic centres, and singularities of the Hamiltonian vector ﬁelds i.e. the points where the values of the Hamiltonian, or stream function, diverge.
Hamiltonian vector ﬁelds with uniform ﬂow and solid boundaries are employed to model engineering and geophysical ﬂow problems such as aerodynamic ﬂows around multiple wings and coastal ﬂows around various islands. Initially, Dr Sakajo and Dr Yokoyama developed a classification process that involves assigning a sequence of letters, which they refer to as a maximal word, to a structurally stable Hamiltonian vector ﬁeld, whose streamline patterns are robustly unchanged subject to general perturbations and noise. Each letter of the maximal word symbolises a speciﬁc local streamline structure, and the maximal word denotes the global streamline structure.
This innovative word representation proved useful, particularly as a maximal word can be assigned to every structurally stable streamline topology, so large quantities of streamline plots, obtained from experiments and simulations, can be converted into small sets of simple symbolic data ready for analysis. Further work, however, revealed that the word representation has a uniqueness problem. While a unique maximal word can be assigned to every structurally stable streamline topology, a given maximal word only represents an equivalence class of streamline patterns. This means the correspondence between streamline patterns and maximal words is one-to-many.
In order to resolve the uniqueness problem, the researchers at the Kyoto University and the Kyoto University of Education decided to model the streamline topology for each Hamiltonian vector ﬁeld is with a discrete structure: a rooted, labelled and directed tree. Referring to graph theory, a tree is a graph, or network of vertices and edges, where any two distinct vertices are connected by a single path of one or more edges. Where a tree has one vertex identified as the root, the tree is a rooted tree. Where the edges have a direction associated with them, the tree is directed. A tree is labelled when a particular symbol or label is specified for each vertex.
Employing these rooted, labelled and directed trees, which we will refer to as just ‘trees’, enabled the researchers to develop an alternative combinatorial classiﬁcation theory with a one-to-one correspondence between streamline topologies of Hamiltonian vector ﬁelds and their new symbolic sequence called regular expressions.
Establishing uniqueness means that these specific sequences of characters form codes that represent particular flow functionalities and their significant properties and can be considered as ‘DNA’ for flow patterns. The streamline topology of every structurally stable Hamiltonian vector ﬁeld is now in one-to-one correspondence with a unique tree and its associated exclusive sequence of symbols defined by a distinct regular expression. This uniqueness means that every streamline topology is identiﬁed globally by the regular expression. Hence users can carry out a simple symbolic comparison of regular expressions to differentiate between topological streamline patterns regardless of the structures’ complexity.
Application of the methodology
While the underpinning mathematics is complex, the conversion of a specific streamline pattern into its unique regular expression is performed in a straightforward manner and a user’s guide is available to aid those who are not mathematicians. Once familiar with the expression components, the conversion from a streamlined pattern into its unique regular expression becomes intuitively recognisable.
The researchers have developed software that converts large quantities of streamline plots, obtained from laboratory experiments and numerical simulations, into small sets of simple symbolic data made up of regular expressions. This is amenable to a big data analysis for streamline patterns and facilitates the analysis of massive flow pattern data obtained via numerical simulations and physical measurements recorded from engineering and medical studies. The software also employs an automatic conversion that can convert regular expressions into their corresponding maximal words s-o the established analysis techniques can be employed. This has also facilitated the successful interpretation of latent knowledge hidden behind flow data.
The advanced algorithmic conversion to tree structures and their associated regular expressions is easily performed.
Transition between streamline patterns
Now that word representations can be generated for all structurally stable topologies as well as all possible transitions between them, we can visualise the global network of transitions as a graph, the transition graph, provided the number of genus elements is fixed.
Work has also been carried out to develop a mathematical theory that describes all generic transitions of global 2D flow patterns in terms of the changes to the sequence of letters. This has led to the discovery that any transition between two structurally stable Hamiltonian vector ﬁelds is topologically equivalent to one of the 34 possible transitions through marginal structurally unstable Hamiltonian vector ﬁelds. The one-to-one correspondence between streamline topologies and regular expressions means that the transitions between streamline topologies can now be described as conversion rules of regular expressions. By simply comparing the sequences, it is now possible to predict the possible changes to global flow patterns that could happen in future.
Modelling viscous fluids
The original modelling assumptions of the word representation method treated all fluids as inviscid, ignoring the effects of viscosity on their flow, has already been discussed. The tree representation theory, however, can be extended beyond the original assumptions and can be used to model viscous fluids as well. The streamline topologies are established locally and globally without any ambiguity and a specific sequence of characters contained within the regular expressions can characterise a variation of a physical quantity.
This innovative characterisation of streamline topologies in terms of their regular expression is of practical signiﬁcance in a variety of ways. It provides the classiﬁcation of Hamiltonian ﬂows in the presence of uniform ﬂow and multiple solid boundaries. These particular ﬂow models appear in many engineering and geophysical problems. The tree representation also provides a complementary topological characterisation for Hamiltonian vector fields and can be utilised in the study of generic vector fields.
The technique can be applied to categorise instantaneous snapshots of streamline patterns observed in laboratory experiments and numerical simulations that are usually structurally stable. Only a snapshot of level curves needs to be recorded from experiments and simulations. Consequently, the methodology provides efﬁcient data compression whereby a vast amount of streamline plots of ﬂow evolutions can be converted into a small set of regular expressions. This speeds up the statistical analysis thus enabling the user to spot trends and predict future flow patterns from big data. The regular expressions are also utilised as new global identifiers for machine learning – topological machine learning.
Unique DNA-like sequences can be generated for all structurally stable topologies and all possible transitions between them, so possible changes to global flow patterns can be envisaged through the comparison of sequences and used to predict future flow patterns. This can be employed in situations such as forecasting the diffusion of contaminants affected by the ﬂow in rivers and ocean ﬂows, containing obstacles such as water breaks, sandbanks and islands.
Although the tree representation was initially devised for incompressible and inviscid ﬂows with vortex structures, Dr Sakajo has shown that its application can now be extended. In addition to modelling incompressible and viscous ﬂows, the methodology can be applied to any physical phenomena described by Hamiltonian vector ﬁelds, such as electromagnetism.
Dr Sakajo and his colleagues are keen to show that their methodology can be applied to real-world situations. They have already employed their software to analyse numerical flow data used in the development of hydraulic machines for an established engineering business, and the investigation of medical image data of cardiovascular flows.
- Sakajo, T. and Yokoyama, T. (2018). ‘Tree representation of topological streamline patterns of structurally stable 2D Hamiltonian vector fields in multiply connected domains’. The IMA Journal of Applied Mathematics, 83, 380-411.
- Sakajo, T. and Yokoyama, T. (2014). ‘Transitions between streamline topologies of structurally stable Hamiltonian flows in multiply connected domains’. Physica D, 307, 22-41.
- Sakajo, T., Sawamura, Y and Yokoyama, T. (2014). ‘Unique encoding for streamline topologies of incompressible and inviscid flows in multiply connected domains’. Fluid Dynamics Research, 46 (3), 031411.
- Yokoyama, T. and Sakajo, T. (2013). ‘Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains’. Proc. Roy. Soc. A, 469.
Professor Sakajo is researching the dynamical phenomena observed in the evolutions of incompressible fluid motion and aims to develop a new way to describe fluid flow properties. This underpins his development of innovative software, which compresses vast amounts of ﬂow data into small sets of regular expressions; thus accelerating the statistical analysis of large data sets to predict future flow patterns from big data.
- Japan Science and Technology Agency (JST)
- Japan Society of Promotion of Science (JSPS)
Dr Tomoo Yokoyama, Associate Professor at the Kyoto University of Education.
Takashi Sakajo holds a BSc from Kyoto University and a PhD from the Graduate School of Science, Kyoto University, Japan. He was awarded the Takebe Awards 2000 from the Mathematical Society of Japan and Paper of the Year 2000 from the Japan Society of Industrial and Applied Mathematics. He is currently a Professor in the Department of Mathematics, and Chief Coordinator of the Math Clinic at Kyoto University.
Dr Takashi Sakajo
Kitashirakawa Oiwake-cho, Sakyo-ku,
Kyoto 606-8502, Japan
Japan Science and Technology (CREST project page) www.jst.go.jp/kisoken/crest/en/index.html
JST CREST project mathematics field that support this research www.jst.go.jp/crest/math/en/index.html