---
title: "Pretty Convincing?"
subtitle: "My Opinionated Introduction to CFD"
kicker: "MEC Lecture · Computational Fluid Dynamics"
author: "Professor Andrew Garrard MEng(hons) PhD CEng FIMechE PFHEA"
details: "MAC118"
date: "2026-08-08"
theme: "white"
base: "../../../"
---
## What we will learn today
::: {.learning-outcomes}
By the end of this lecture you will be able to:
1. Outline how CFD developed into its current state of the art
2. Explain why and how CFD is used alongside experiment and theory
3. Describe the steps involved in simulating a fluid flow numerically
4. Relate broad CFD concepts to the applications in the tutorials
:::
# Introduction
## CFD stands for
* C - Computational
* F - Fluid
* D - Dynamics
> Computational fluid dynamics (CFD) is a branch of both fluid mechanics and computational physics that uses numerical analysis and data structures to analyze and solve problems that involve flows.
>
> — [Wikipedia, "Computational fluid dynamics"](https://en.wikipedia.org/wiki/Computational_fluid_dynamics)
. . .
::: {.fragment}
_Alternatively_
> Colours for Directors
:::
::: {.fragment}
_a.k.a_
> "The Beautiful Lie"
:::
## Contemporary uses of CFD
A simulation to determine safety for Urban Air Mobility.
: Hovering quadcopter rotor in ground effect. Image: Patricia Ventura Diaz and D-G Caprace, Tim Sandstrom. NASA/Ames.
Watch the simulations on the NASA page
# A history of CFD
## Fluid mechanics before computers
: From the 17th to the 19th century, fluid mechanics was split into two disciplines.
## Early CFD
: CFD arrives, and it is not well defined as either theory or experiment.
## A third way
: Three fields of fluid mechanics, of equal importance, each informing the others.
## Driving and bottlenecking CFD progress
: Moore's law. Via Wikimedia Commons.
## Commercial computers
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: A Cray Y-MP. ~1988: ~2.5 GFLOPS, ~20 million USD, ~55 million now USD. NASA Goddard, via Wikimedia Commons.
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: A Dell Latitude D630. Core 2 Duo PC, ~2008: ~25 GFLOPS, ~1,000 USD, ~1,500 USD now. Via Wikimedia Commons.
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## Pushing boundaries in fluid mechanics
: A Lockheed Martin hypersonic vehicle.
## Pushing boundaries in fluid mechanics
: A flow visualization of NASA’s X‑59 supersonic, low-boom aircraft concept. This computational fluid dynamics simulation, produced on the Pleiades supercomputer using the NAS-developed Launch Ascent and Vehicle Aerodynamics (LAVA) code, will be used to help determine which components contribute to the pressure signatures below the aircraft that create the sonic boom in flight. Red and blue represent high and low pressure, respectively; the symmetry plane—which bisects the vehicle along its center line—is colored by Mach number. James Jensen, NASA/Ames.
[https://nas.nasa.gov/pubs/stories/2021/feature_X-59.html](https://nas.nasa.gov/pubs/stories/2021/feature_X-59.html)
## Wind tunnels
The most powerful wind tunnel in the world.
## A couple of issues with wind tunnels
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### Power
The JF-22 uses 15 GW of power.
. . .
: 180 × Trent XWB engines.
: 5 × Iron Men.
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. . .
### Scale
The JF-22 has a working section of 4 m.
Real planes are bigger than 4 m across.
$$Re = \frac{\rho V L}{\mu}$$
. . .
The run time is less than 120 ms.
:::
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## Why use CFD?
* Cheaper
* Safer
* Unphysical problems
* Complete dataset
* Data after experiment
* Increasingly automated
. . .
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_But..._
* No exact solution
* No idea if the answer is correct – "black box engineering"
* Good for some situations, not for others - this requires some wisdom
:::
. . .
::: {.fragment}
_Why I like CFD_
* I can work independently
* Easy "what if" scenarios
* Work from anywhere
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## Numerical experiment
CFD can be a portable, cheap wind tunnel.
> Numerical experiment
* Used to test a model, theory or hypothesis.
* Fewer factors are outside your control than in a physical experiment, but not none.
* Error will occur and has to be understood and accounted for.
* Good experimental design needs to be performed before starting.
* Experiments should adhere to the scientific method.
* Focus should always be maintained on the reasons for conducting the experiment.
## Something for everyone
: The CFD stack: every answer rests on six layers the engineer never sees.
## Some of my work
* Simulation of a "personal cloud"
* Prediction of smoke ingress and movement in offshore installations
* GRP spraying of boat hulls
* Electric motor cooling
* Fuel cell electrochemical modelling
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: Cooling and windage loss in a switched reluctance motor.
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: Swirl in a fuel cell stack manifold.
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## Commercial software
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### In the recent past
Dedicated CFD platforms
* Fluent
* CFX
* STAR-CD
* COMSOL (general)
* OpenFOAM (open source)
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. . .
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### Now
Large multiphysics packages
* Ansys (Synopsys)
* Dassault Systèmes
* CAD bolt-ons (not very good)
* COMSOL (general)
* OpenFOAM (open source)
:::
::::
. . .
**At Sheffield we use Ansys Fluent.**
## Fluent started in Sheffield
: Joe Scarborough's painting of Fluent Europe's Sheffield, commissioned in 2000.
# How CFD works
From equations to numbers.
## The Navier–Stokes equations
Conservation of mass
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$
Conservation of momentum, in $x$, $y$ and $z$
$$\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \mu\left(\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}\right) + \rho g_{x}$$
$$\rho\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \mu\left(\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}\right) + \rho g_{y}$$
$$\rho\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu\left(\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}\right) + \rho g_{z}$$
$u$, $v$, $w$ velocity components · $p$ pressure · $\rho$ density · $\mu$ dynamic viscosity
## Solving the Navier–Stokes equations
The Navier–Stokes equations cannot be solved analytically.
. . .
::: {.fragment}
_Despite what Sam Altman says_
: OpenAI didn't solve the Navier–Stokes equations. From BBC News, https://www.bbc.co.uk/news/articles/cy7zygy3rl2o, 8 September 2026.
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## Basic conservation equation
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: Mass is conserved. The mass entering across the boundary minus the mass leaving across it is the mass accumulated inside.
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: Momentum is also conserved in the same way, but has a third route in: forces on the boundary.
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## Discretising
::: {.demo src="/teaching-demos/cfd/cfd_cell_refinement.html" height="580" title="Mesh refinement demo"}
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## Typical CFD simulation workflow
: The steps of running a CFD simulation.
## The more holistic CFD simulation workflow
: Running the simulation is one step of six in a CFD study.
## The complete CFD simulation workflow
: The whole simulation pipeline is one box in the study.
## Geometry creation
This is basically CAD. But the _fluid_ needs to be created, not (necessarily) the solid.
: To predict drag, mesh the air surrounding the rocket. The rocket’s surface forms a boundary of the fluid domain.
## Meshing
Meshing is the task of dividing the geometry into many small "elements".
: A tetrahedral mesh around a sphere, with inflation layers at its surface.
* The quality of the mesh has a dramatic impact on the results produced by the simulation.
* Meshing is becoming an increasingly automated process.
## Pre-processing
The mesh divides the fluid into cells. We now set up the calculation:
* **Which physics?** Turbulence, heat transfer, chemical reactions.
* **Which fluid?** Air, water, and its material properties.
* **What happens at the boundaries?** Flow entering, flow leaving, solid walls.
* **How should the calculation run?** Solution methods and solver settings.
Each of these decisions will have an impact on the result of the simulation.
The same mesh can be used for different calculations by changing these choices.
## Solved by numerical methods
* Some equations can be solved analytically.
* Any question where the function of the unknown variable produces a known result can be solved _numerically_.
* For example
$$x = \cos x$$
$$\cos x-x = 0$$
* Use a "predictor-corrector" method.
$$r_n = \cos x_n - x_n$$
* where $r_n$ is the _residual_.
## Predictor-corrector
Start between $0$ and $1$; keep the half with a sign change. Angles are in radians.
| Guess | Calculation | Residual | New guess |
|------:|:------------|---------:|----------:|
| $0.5$ | $\cos(0.5) - 0.5 =$ | $+0.3776$ | $0.75$ |
| $0.75$ | $\cos(0.75) - 0.75 =$ | $-0.0183$ | $0.625$ |
| $0.625$ | $\cos(0.625) - 0.625 =$ | $+0.1860$ | $0.6875$ |
| $0.6875$ | $\cos(0.6875) - 0.6875 =$ | $+0.0853$ | $0.71875$ |
| $0.71875$ | $\cos(0.71875) - 0.71875 =$ | $+0.0339$ | $0.734375$ |
| $0.734375$ | $\cos(0.734375) - 0.734375 =$ | $+0.0079$ | $0.742188$ |
| $0.742188$ | $\cos(0.742188) - 0.742188 =$ | $-0.0052$ | $0.738281$ |
| $0.738281$ | $\cos(0.738281) - 0.738281 =$ | $+0.0013$ | $0.740234$ |
* For most practical flows, the Navier–Stokes equations have no known analytical solution, so we solve them approximately using numerical methods.
## Residuals
: Residuals fall as the solver iterates, then level off at different magnitudes.
## Post-processing
. . .
What you _think_ a valuable result from CFD is
: Pictures: streamlines and surface colouring for flow past a sphere.
What an _actually valuable_ result from CFD is
: Numbers: Fluent's force report, with the total drag on the sphere circled.
## Validation
Validation checks whether the simulation gives the _right_ answer.
**But what does “right” mean?**
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::: {.keypoint}
The answer predicts what would happen if the simulated conditions existed in the real world.
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* Simulations consist of assumptions, estimations, decisions (Mesh, models, physics and numerics) which all influence the answer.
* The simulation will **merrily** and **convincingly** generate you _an_ answer.
* Just like analytical calculations, web searches, AI output, you need to check what is being produced is correct.
::: {.keypoint}
A convincing answer still needs validation.
:::
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## You are liable.
> THE ENTIRE RISK ARISING OUT OF THEIR USE REMAINS WITH THE RECIPIENT. IN NO EVENT SHALL ANSYS AND/OR ITS RESPECTIVE SUPPLIERS BE LIABLE FOR ANY DIRECT, CONSEQUENTIAL, INCIDENTAL, SPECIAL, PUNITIVE OR OTHER DAMAGES WHATSOEVER (INCLUDING WITHOUT LIMITATION, DAMAGES FOR LOSS OF BUSINESS PROFITS, BUSINESS INTERRUPTION, OR LOSS OF BUSINESS INFORMATION), EVEN IF ANSYS HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
https://ansys.synopsys.com/legal/legal-notices
## How to validate CFD results.
Compare the simulation result to
* Experimental results (not necessarily yours)
* Theoretical result (that has been validated)
* Another CFD result (that has been validated)
If the result of the simulation is already known to validate against, why do the CFD simulation in the first place?
## Summary
* CFD complements theory and experiment; all three inform each other.
* Computing advances have made CFD more accessible for numerical experiments.
* Start with an engineering question and plan the study before simulating.
* The steps in the workflow for a CFD simulation:
* **Geometry creation:** define the fluid domain.
* **Meshing:** divide it into cells; mesh quality affects the results.
* **Pre-processing:** choose physics, fluid properties, boundaries and solver settings.
* **Solving:** solve conservation equations numerically and monitor residuals.
* **Post-processing:** extract useful engineering quantities, such as drag.
* **Validation:** check against real-world evidence before trusting a convincing answer.