Introduction

This is a web app that simulates the fluid flow around a falling sphere, taking into account the dynamic effects of fluid viscosity, sphere radius and density, along with static parameters such as gravity, buoyancy and drag.

The simulation measures the time taken for the sphere to travel 15cm at terminal velocity. It is meant as a lab auxiliary for the Chemical Engineering course at the University of Birmingham.

Usage

The simulation parameters can be changed from the menus at the top of the screen. The time can be altered from real time to 4 times slower or 4 times faster. The viscosity is measured in centistokes (cSt).

Any sphere can be selected by clicking on it. The radius is provided in imperial measurements, like the laboratory resources.

At the bottom of the screen is the table for physical, dynamic parameters. On the right is the data log that outputs the measurements.

The Jacobi iterations parameter represents the number of calculations for the fluid movement. A higher number of iterations provides higher fidelity in the fluid flow, but requires more power from the GPU.

More in-depth information is given in the next section.

Technicalities

The graphical fluid flow is modelled using the Navier-Stokes equations: $$\frac{\partial \vec{u}}{\partial t} = -\vec{u} \cdot \nabla \vec{u} - \frac{1}{\rho}\nabla \rho + \nu \nabla^2\vec{u} + \vec{F}$$ $$\nabla \cdot \vec{u} = 0$$ However, for purely graphical purposes, leaving the viscosity term out provides a very good trade-off in terms of computational efficiency and graphical fidelity. The equations are solved numerically for pressure using Jacobi iterations.

The graphical sphere movement is modelled using gravitational acceleration, drag force and buoyancy. The sphere moves in discrete time-steps of around 0.065ms. Therefore, at terminal velocity, the following is true: $$ m g = C_d A \frac {\rho u^2}{2} + \rho V g $$ The drag coefficient C_d is defined in terms of the Reynolds Number: $$ Re = \frac {\rho u d}{\mu} $$

• For Stokes' flow (Re < 0.2) $$C_d = \frac{24}{Re}$$
• For Allen flow (0.2 < Re < 1000) $$C_d = \frac{24}{Re} (1 + 0.15 Re^{0.687})$$
• For Newton flow (1000 < Re < 2x10⁵) $$C_d = 0.44$$
• For Re > 2x10⁵ $$C_d= 0.11$$

The simulation screen is comprised of two superimposed HTML canvases:

• The fluid simulation is in the background and is rendered on the GPU.
• The sphere movement is in the foreground and is rendered on the CPU.

The programming technicalities can be accessed in the well-commented javascript file fluid-sim.js. I have written it to be as modular and intuitive as possible. It is available in this website's sources in the developer menu, as well as on my github account

References and Further Reading

To make this, I had to draw from many different sources. The main ones are:

• Jamie-Wong's WebGL Fluid Simulation: This is by far the most important resource I used. It describes the mathematical-computational concepts used for fluid simulation in a very user-friendly format. I used most of the WebGL code from here, updating all the libraries and enhancing numerical stability.
• GPU Gems Chapter 38. Fast Fluid Dynamics Simulation on the GPU: It walks through specific implementation ideas, and gave me a much better intuition for advection. It is written in a much more technical language, being more of a scientific paper.
• Jonas Wagner’s fluid simulation on canvas and particularly the source for it (fluid.js) were helpful for understanding what a full solution actually looks like.
• Amanda Ghassaei vortex shedding simulation: I used this to analyse vortex formation and actual coding of the simulation.
• Burak Kanber's modelling physics in javascript: I used this as a tutorial for modelling forces in javascript and rendering them in an HTML canvas.
• Evan Wallace's LightGL library: This javascript library provides a very light abstraction layer on top of WebGL. Very helpful for prototyping the simulation on the GPU.
• Dr. Phillip Robbins, Senior Lecturer of the School of Chemical Engineering of the University of Birmingham: I got the lab description and the physical properties of the lab resources from the exercise paper written by Dr. Robbins.
• Many thanks for also having Ciara Albas-Martin MEng and Marten Reichow MEng conduct this experiment in the lab and taking lots of measurements.

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