Maths in a Minute: Computational Fluid Dynamics
Source:Plus Magazine
Computational fluid dynamics (CFD) is the use of numerical methods and computers to approximate the behaviour of moving fluids such as air, water and blood. Fluid motion is governed by the Navier–Stokes equations, which relate quantities such as velocity, pressure and viscosity. In a common incompressible form, the momentum equation can be written as
$\rho\left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right)=-\nabla p+\mu\nabla^2\mathbf{u}+\mathbf{f}$,
together with the incompressibility condition $\nabla\cdot\mathbf{u}=0$. For realistic objects such as cars, heart valves or buildings, obtaining exact analytical solutions is generally impossible, so the equations must instead be approximated computationally.
CFD tackles this by dividing the physical region into a huge collection of small elements forming a mesh. The equations are approximated locally on each element, while neighbouring elements interact so that a numerical approximation to the overall flow gradually emerges. The result can provide estimates of air velocity, density, temperature and pressure throughout the region. Once the pressure distribution around an object is known, integration can be used to calculate quantities such as total aerodynamic forces and drag—conceptually similar to estimating an area using the trapezium rule. The article highlights applications ranging from designing the supersonic Bloodhound car to biomedical engineering and modelling airflow and aerosols in buildings.
Key takeaways
ARTICLE / PDF
Source:Plus Magazine
Computational fluid dynamics (CFD) is the use of numerical methods and computers to approximate the behaviour of moving fluids such as air, water and blood. Fluid motion is governed by the Navier–Stokes equations, which relate quantities such as velocity, pressure and viscosity. In a common incompressible form, the momentum equation can be written as
$\rho\left(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right)=-\nabla p+\mu\nabla^2\mathbf{u}+\mathbf{f}$,
together with the incompressibility condition $\nabla\cdot\mathbf{u}=0$. For realistic objects such as cars, heart valves or buildings, obtaining exact analytical solutions is generally impossible, so the equations must instead be approximated computationally.
CFD tackles this by dividing the physical region into a huge collection of small elements forming a mesh. The equations are approximated locally on each element, while neighbouring elements interact so that a numerical approximation to the overall flow gradually emerges. The result can provide estimates of air velocity, density, temperature and pressure throughout the region. Once the pressure distribution around an object is known, integration can be used to calculate quantities such as total aerodynamic forces and drag—conceptually similar to estimating an area using the trapezium rule. The article highlights applications ranging from designing the supersonic Bloodhound car to biomedical engineering and modelling airflow and aerosols in buildings.
Key takeaways
- CFD turns difficult differential equations into large numerical problems that computers can solve approximately.
- The physical domain is split into a mesh of many small regions, where simplified local calculations are performed.
- Integration of the resulting pressure and velocity fields allows engineers to predict forces such as drag and lift.
- CFD is important far beyond aerodynamics, with applications in medicine, architecture, ventilation and many other areas involving fluid flow.
ARTICLE / PDF
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