Flow visualization is concerned with displaying features and behaviors of fluid flow. The features may be phenomenological (that is, empirical, observable, perceptual) manifestations of the underlying equations governing the flow, and their definitions are not based on simple principles like "conservation of mass". For example, it is difficult to define a turbulent region mathematically (or even algorithmically) in a way that would specify a region experts agree is turbulent.

The unifying theme in visualizing fluid flow is to show curves that illustrate particle motion. The velocity vectors guide the motion of the particles.

Textbooks on fluid dynamics generally follow a standard organization. Stationary fluid is described. Moving fluid is described. At low velocities, simple equations are obtained by neglecting terms. At higher velocities these terms must be retained or improved. The equations lead to phenomena like vortex stretching, turbulence, shock formation, generation of acoustic waves. An advanced textbook will consider additional conditions within a fluid, such as the multiple species that constitute it (nitrogen, oxygen, hydrogen) and their interaction (such as through combustion).

Textbooks on applications of fluid dynamics (such as meterology, oceanography, hydrology, and geodynamics) specialize the governing equations of fluids to a particular area of interest. For example, flow may be studied on a rotating sphere (modeling the Earth).

Computational visualization of fluid flow requires some knowledge of computer graphics, flow physics, and differential geometry. The computer-science students who write visualization software eventually become consultants and specialists in visualization at research labs and in industry. But they may never have been exposed to the physics and math that lie at the foundations. This reading list is targeted at them. They took 15 computer science classes (programming, theory, hardware, electives), 4 math courses (calculus I-II, discrete math, statistics), introductory chemistry and physics. Their other 20 classes covered the gamut of core requirements.

That curriculum leaves out some important math and physics. The math builds on algebra, topology, and advanced calculus, which restricts its customer base to graduate students. There's really not enough room in a curriculum for the student to become acquainted with physics up through turbulent flows, become acquainted with math up through differential geometry, and become a proficient computer programmer. The interested person can skim this reading list in a day and become acquainted with essential elements underlying flow visualization.

1. Physics: Fluid Dynamics

   1. James Wilkes, Fluid Mechanics for Chemical Engineers
      1. 248-250 The convective derivative D/Dt.
   2. G. Batchelor, An Introduction to Fluid Dynamics
      1. 71-73 Eulerian and Lagrangian specifications. Streamline, stream-tube, path, streakline. Material derivative.
   3. John Douglas et al., Fluid Mechanics

      This book is unusual for containing FORTRAN code examples. For the object-oriented programmer who wants to work with physics code to produce 3D images, this gives a flavor of what the difference in culture can be like. The book also contains a few experimental images (from real life) and hand-drawn images, which are suggestive of what kinds of images an author would like to get from a computed flow visualization.

      1. 86-97 Obligatory definition of pathline, streakline, streamline, streamtube. Steady and unsteady flow. Substantive derivative. Turbulent flow.
      2. 226-228 FORTRAN program ROTCYL for calculating stagnation points, lift, and pressure on a cylinder.
      3. 321-323 FORTRAN program CBW for calculating friction factor of duct flow.
      4. 377-382 Images: pathlines over an airfoil, hand-drawn illustrations of turbulence behind an airfoil amd vortices behind an airfoil. The images suggest the type of feature extraction and simplified rendering that make a flow visualization effective for explanation (like non-photorealistic rendering).
   4. Hunter Rouse, Elementary Mechanics of Fluids

      An old book with some good images.

      1. 238-248, 280-283 The von Kármán vortex trail. The author includes both a streamline image and a streakline image of vortex trails past a cylinder. The empirical images are similar to images produced by spot noise and line integral convolution.
   5. Robert Granger, Fluid Mechanics
      1. 482 A photograph showing transition to turbulence, and a hand-drawn illustration of the same.
      2. 548-552 (Essential reading) Satellite photograph of large atmospheric eddies and a hand-drawn illustration of the phenomenon. page 489 gives a descriptive overview of vortex-stretching as a precursor to turbulence, and as a mechanism for transfering engergy to small spatial scales. Pages 550-551 show hand-drawn illustrations: inspirational for flow visualization.
      3. 757-768 More examples of photographs and hand-drawn illustrations of turbulence in a boundary layer.
   6. Frank White, Viscous Fluid Flow
      1. 4-10 This section does 3 things. First, it gives descriptive examples of different flow configurations (airfloi, cylinder, pipe, air filter). Second, it gives photographic and hand-drawn illustrations. Third, it gives an obligatory mention of streamlines and timelines.
      2. 59-69 This section serves as a typical example of how a fluids text introduces the Navier-Stokes equations. First, there is a particle derivative. Then conservation of mass, leading to the continuity equation. Second, conservation of momentum leads to a discussion of the stress tensor, the linear (Newtononian) rate of change of this tensor, and viscosity. Third, conservation of energy leads to an equation involving temperature.
      3. 367-377 (Essential reading) This includes the end of a section on the involvement of vortices in transition-to-turbulence; there are a couple of nice hand-drawn illustrations that are inspirational for flow visualization. Next comes a narrative describing how the minor fluctuations in a laminar flow transition to unsteady flow with large instabilities. The discussion includes photographs and hand-drawn images as illustrations of the process.
      4. 394-396, 458-461 Turbulence. With pictures and narrative.
   7. James Holton, An Introduction to Dynamic Meterology

      Fluid dynamics on a rotating sphere.

      1. 27-38 The total derivative is motivated by the Taylor series of a function measured by a balloon carried by the wind. The author derives the momentum equation in rotating coordinates by applying "simple physical reasoning." It would be instructive to see the very same derivation from the differential geometry point of view, starting with the Navier-Stokes equations and then applying them to a manifold.
      2. 70 The author calls pathlines "trajectories."
      3. 300-308 This section offers several hand-drawn illustrations of atmospheric circulation. These are great examples of stylized, minimalist flow visualization.
   8. G. Tokaty, A History and Philosophy of Fluid Mechanics

      The book, first published in 1971, is available in paperback. It traces the history of the people and ideas involved in developing the field of fluid mechanics.

      1. 88-90 Claude Louis M. H. Navier. Navier thought about the flow in a cylindrical pipe and the effect it has on a volume element. That led him to his famous differential equation with a viscous term.
      2. 114-119 Osborne Reynolds. Reynolds observed that turbulence resulted from the shear forces in the boundary of a viscous flow over a long distance. Turbulence developed when density*pipeDiameter*velocity/viscosity exceeded 1400, even when the individual parameters are allowed to vary.
      3. 134-138 This section contains five hand-drawn 3D illustrations of vortices on wing tips.

2. Math: Differential Geometry

   The surface of a solid body is a submanifold of R³. Streamlines on the surface are integral curves associated with the vector field on the surface. The curves can be produced in the "parameter space" of the surface, but the Jacobian of the mapping must be consulted. Differential geometry makes these ideas precise, and allows them to generalized far beyond the scope of ordinary flow visualization.

   These sections of textbooks introduce the technical vocabulary that mathematicians use for discussing the integration of streamlines in a vector field.

   1. Alfred Gray, Modern Differential Geometry of Curves and Surfaces.

      The main distinctive about this book is that it uses Mathematica and includes lots of 3D images of surfaces. There is an ftp site with the examples: ftp://bianchi.umd.edu/pub/CandS/ and a gallery of surfaces at http://bianchi.umd.edu/ on the Web.

      1. 163-178 Definition of tangent vector, tangent space, directional derivative, tangent maps, vector field, derivative of a vector field. This is essentially a compressed presentation of the material in O'Neill.
      2. 524-531 Definition of the Lie or Jacobi bracket (for vector fields; not the Dirac bracket for quantum mechanics). Definition of tangent bundle, section, tensor field, differential 1-form. The presentation is intended as a review, not a tutorial.
   2. Barrett O'Neill, Elementary Differential Geometry.

      This is a typical textbook on differential geometry. The new (second) edition uses Mathematica. The author maintains a Web page for the book at http://www.math.ucla.edu/~bon/.

      1. 6-10 Definition of tangent vector, tangent space, vector field.
      2. 26-31 Definition of differential form, wedge product, exterior derivative. Problem 8 on page 31 relates differential forms on E³ to the more conventional engineering calculus (for example, the divergence theorem). The idea is that differential forms permit the fundamental theorem of calculus to be stated in a single uniform way.

   3. Richard Sharpe, Differential Geometry

      This book includes generous stretches of explanation, not just proofs and definitions. If flow visualization is to be considered as sampling the foliation induced by a section of a vector bundle on a manifold, this book is a good place to get a sense of what all that means.

      1. 65-83 Definitions of integral curve, distribution (not the Dirac delta "function" and not a distribution in the sense of random variables), integrability, foliation.
   4. Michael Spivak, Differential Geometry, Volume I.

      Spivak wrote five volumes in this set. Unfortunately, they were written just a little before everyone had a word processor, so it's all typed, which makes difficult material more difficult to parse. This volume can be thought of as a long slow answer to the question: what does it mean to visualize a flow on the surface of an object in space? Spivak takes many opportunities to express himself informally and intuitively, and he includes lots of illustrations. These expository passages really help. But it can be a little demoralizing to read difficult passages that Spivak casually tosses out as being trivially simple to understand.

      1. 86-101 Definitions of tangent bundle, fiber, tangent vector, n-plane bundle, section.
      2. 153-158 (Essential reading) Spivak explains how the old-fashioned "differential" became the modern "differential form." How covariant and contravariant got their names.
      3. 186-191 Definition of integral curve. This is what a streamline or a pathlines is.
      4. 244-254 Definitions of 1-dimensional distribution, integral manifold. This section addresses the issue of whether a vector field can be integrated to produce flow lines.
      5. 325-331 Line integrals.

   5. Louis Auslander, Differential Geometry

      This book serves as a reminder of the way textbooks used to always be written. Definition definition proof. It's covers some of the same material as the other more recent texts, but the purpose of differential forms and integral curves seems to be a closely guarded secret.

      47-61 Tangent spaces: In the first paragraph, the author says the tangent plane is a "subset of Eⁿ". Cotangent spaces and vector fields: without actually introducing fiber bundles, the author defines vector fields and 1-forms as if they were sections of the tangent bundle and the cotangent bundle. Integral curves.
   6. Serge Lang, Fundamentals of Differential Geometry.

   This a very abstract treatment of differential geometry. If you have ever coded twenty variations on a basic idea, say in graphics or visualization, and wished you could generalize all your functions into just three or four really generic abstractions, you share the spirit of the author.

   1. 3-6 As if it were not already difficult enough to formulate visualizing the physics of fluid flow in terms of foliations of manifolds by sections of the cotangent bundle, Lang has found a way to be even more abstract. He considers the category of topological vector spaces that are Banach spaces, and deals with functors of morphisms. While this book may be completely out of reach, skimming it after paging through more elementary differential geometry texts at least gives a sense of the very general setting in which the flow-visualization problem lies.
   2. 43-52 Vector bundle morphisms. Exact sequences of bundles.
   3. 66-72 "We do not go into these theories" [properties which are invariant under a given group of automorphisms of X]. But Lang does go into time-dependent vector fields, local flow, uniqueness.
   4. 88-96 A vector field is a morphism, and a curve is a morphism. A neighborhood of a non-critical point can be re-parametrized as a constant (morphism). This may have been in an episode of Dilbert.

3. Visualization: Flow Lines

   The readings listed below provide analytic solutions for creating flow lines.

   1. Robert Kirchhoff, Potential Flows: Computer Graphic Solutions

      Computational scientists are often reluctant to share their data, which puts a burden on the programmer developing a better technique for displaying flows. This book gives equations for several potential flows, allowing the programmer to create 2D and 3D datasets without relying on a flow physicist in the neighborhood to cooperate.

   1. 35-40 Cylinder and doublet in 2D.
   2. 86-91 Flow over a step in 2D.
   3. 71-76 A row of vortices (including Karman vortex street) in 2D.
   4. 121-125 Point dipole and flow over a sphere in 3D.
   5. 130-131 Flow over a Rankine solid in 3D.
   2. Gregory Nielson et al., Scientific Visualization: Overview, Methodologies, and Techniques

      This book is designed as a textbook for a course on scientific visualizaiton.

   Chapter 21 explains how to compute the analytic solution for a streamline through a triangle, given the vector field evaluated at the three vertices. The linearly varying vector field is described by a matrix A. The eigenvalues of A fall into five cases, with an explicit solution given for each case.