Playing Poseidon: A Lattice Boltzmann Approach to Simulating Generalised Newtonian Fluids

Nitish Tripathi (homepage)

Abstract

Imitating the behaviour and characteristics of fluids with the help of a computer is called fluid simulation. Real world fluids are fickle. They are subtle and gentle at times; at times they are ravenous and tumultuous. Needless to say, complex equations are behind even the tiniest of ripple, so much so that often fluid mechanics has been described as ”the physicists nightmare”. Yet there are few, if any, substances which are so beautiful and graceful in motion to observe. To an ardent student of hydrodynamics, everything in the discernible world is fluid. Solids may just be classified as fluids which flow extremely slowly! Given time, every substance has the tendency to flow under the influence of an external force.

The history of fluid simulations, thus, rightly, begins with the formulation of the Navier Stokes’ equations. These were a set of partial differential equations originally developed in the 1840s on the basis of conservation laws and first order approximations. What followed was the Conventional study of fluid flows for more than a century. Arriving at computational models to solve fluid equations has been a subject of research since the early 1950s. Finding solutions to the partial differentials of Navier Stokes’ equations using discrete algorithms was area of focus. Many modern day techniques, of which some will be skimmed through in the succeeding chapters, came up during that time. Staggered marker-and-cell (MAC grid structure), Particle in Cell (PIC method) etc. are two of those.

However, most of the models and techniques developed by the CFD community then was complex and unscalable for visual effects oriented computer graphics. In the succeeding years, fluid effects was generated using non-physics based methods, such as using hand drawn animation (key frame animation) or displacement mapping.

The development of fluid simulations has traditionally been in two concurrent streams, viz., Eulerian and Lagrangian Simulations. Eulerian Method involves modelling the fluid as a collection of scalar fields (density, pressure etc.) and vector fields (velocity etc.). Each field is calculated using Navier Stokes’ equations and fluid is visualised as crossing the volume at fixed grid points, where the value of each field is known. Lagrangian simulations on the other hand take a more intuitive approach. They treat fluid particles as carriers of the field values in accordance with the Navier Stokes’ equations. The dependence of both the methods on Navier Stokes’ makes them essentially top down simulations methods - these methods look at what the perceptible fluid properties are without concerning themselves about the kind of particular interactions which give rise to the said properties.

Lattice Boltzmann Method developed around the same time but did not come into widespread usage until much later. Unlike the conventional methods, it is a statistical method based on Kinetic Theory. It treats fluids as a collection of logical mesoscopic particles. These are constrained to move in a discrete set of directions across a cartesian grid. They follow continuous alternating iterations of colliding at each grid centre and redistribution around it, and, progressing to the neighbouring centre. It was shown that for a particular kind of collisions these particular interactions give rise to Navier Stokes’ properties at the macro level. However, we can tweak certain parameters during the development so that quantities, taken to essentially be constants in the final Navier Stokes’ equations, can be varied to simulate fluids outside their scope. As we will see in the succeeding chapters, such fluids are in abundance around us and are called non-Newtonian fluids. The Navier Stokes’ equation, as will also be seen in succeeding chapters, are essentially Newton’s second law of motion. It is therefore unfit to deal with fluids which are non-Newtonian in nature and requires regular tweaking to model them.

In this work, we combine physical models of non-Newtonian fluids with Lattice Boltzmann Method. We give the CPU implementation, showing how easy it is to understand and code. We show how the method, inspite of its ease of implementation, doesn’t compromise on physical realism or accuracy. We also give a model for GPU implementation for increased efficiency and interactive frame rates.

Related Publications

• Tripathi, N. and Narayanan, P.J. Generalized newtonian fluid simulations. 2013 Fourth National Conference on Computer Vision, Pattern Recognition, Image Processing and Graphics (NCVPRIPG-2013). Jodhpur, India. [PDF]
• Jain, Somay and Tripathi, Nitish and Narayanan, P. J. Interactive Simulation of Generalised Newtonian Fluids Using GPUs. Proceedings of the 2014 Indian Conference on Computer Vision Graphics and Image Processing (ICVGIP-2014). Bangalore, India. [PDF]

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