by Ben Evans, CFD Engineer
Most engineering fluid flow problems are now solved, at least in part, by means of computational fluid dynamics (CFD); the use of computers to aid in the solution of the governing equations of fluid dynamics by numerical methods. The governing equations for the majority of practical fluid flow problems are partial differential equations (PDEs), as is the case for many naturally occurring phenomena. It is likely that if you have studied mathematics at A-level, you will have come across some simple differential equations and solved them analytically (using pen and paper). The set of equations that are most relevant for describing the aerodynamic flows around Bloodhound are the so-called ‘Navier-Stokes’ equations. These are a set of 5 PDEs describing quantities such as the density, velocity and pressure of the airflow (plus a 6th if you want to model turbulence in the flowfield). To give some idea of the degree of complexity of this set of equations, they are shown in the figure below:-
The Navier-Stokes equations for viscous, compressible fluid flow
There is no hope of solving such a complex and coupled set of equations such as this by hand, and in fact, even large supercomputers have to work hard to obtain solutions!
Right: Swansea University School of Engineering Supercomputing Cluster
The development of CFD techniques has followed closely the development of the mathematical tools of numerical methods for solving partial differential equations. Numerical methods have been known since the time of Newton in the 1700s, but without the aid of the computer, the full exploitation of these techniques was impossible.
Modern CFD has its roots in the 1950s with the advent of the digital computer. At the heart of all CFD numerical schemes is the fundamental question of how one should represent a continuous function in discretised form. In other words, how should one store a function defined for an infinite number of points (i.e. every possible position in space and time) in some finite way, and as accurately as possible?
Left: Example of the discretisation / approximation of a pressure function
In the example above, a pressure function as been discretised so that the value of the function at a finite number of ‘nodes’ is stored and a linear interpolation of the solution is assumed between the nodes. The jump between each node is referred to as an element, and hence the solution method is often referred to as ‘the finite element method’.
The most popular method of achieving a 3D finite element solution is to discretise the solution domain into a finite number of small cells / elements forming a mesh or grid, and to then apply a suitable algorithm to values stored at the intersections of the mesh (the nodes) to solve the governing equations, in our case, the Navier-Stokes equations. The computational mesh can consist of elements of a whole variety of 3D shapes. For the CFD work carried out on BLOODHOUND, a mesh of hexahedra, prisms and tetrahedra numbering into the tens of millions was used!
The procedure for performing the CFD analysis on BLOODHOUND can be described in the following stages:
1. The definition of the vehicle surface to be examined is provided to the team at Swansea University as a CAD (computer-aided-design) output from the design engineers based in the design office in Bristol. This geometry definition is then analysed via the FLITE computer system, developed at the School of Engineering.
2. This CAD output is processed and a mesh generation computer program is used to construct the computational mesh.
3. A ‘pre-processing’ computer program is used to format the mesh in such a way that the Swansea University supercomputing cluster can be used to run the solver program.
4. The ‘equation solver’ program containing the Navier-Stokes solution algorithm is run on the supercomputing cluster.
5. A ‘post-processing’ software package is used to convert the solutions coming from the solver into meaningful flow visualisation plots and force distributions.
A selection of outputs from final visualisation stage is shown below. These visualisations help the design team understand the behaviour of the car’s aerodynamics in terms of flow phenomena such as shock waves, boundary layers and pressure distributions. These pictures and force distributions are analysed and decisions made that evolve the vehicle design and steps 1 – 5 are repeated.
Left: Streamribbons and pressure contours over an intermediate BLOODHOUND configuration
Left: Visualisation of a vortex being shed at the rear of BLOODHOUND impinging on the rear wheel struts
The applications for the computational modelling technologies being developed at Swansea are wide ranging and span from areas in medicine through to lightning strike modelling, and from building structural analysis to interstellar plasma flows. Essentially, any
Biomedical modelling of stresses in a human femur phenomenon that can be described by a set of partial differential equations can be modelling using the techniques of finite elements shown here for the BLOODHOUND aerodynamics.
Below left: Fluid-structure interaction modelling of a typical jet airliner. Right: Biomedical modelling of stresses in a human femur