All geometries are constructed using the CAD engine CATIA and imported into a mesh- ing tool. Both meshing and flow simulations are executed using the commercial CFD package GambitTM(FluentTM, 2003a) and Fluent. To ensure that solutions yield suffi- cient accuracy within Fluent, a mesh dependency study has been performed on a straight walled diffuser prior to the optimization studies. An inlet mass flow rate of 10.8kgs−1 (Reynolds number, Re = 6×106) is fixed, determined for a car travelling at approxi- mately 70ms−1 or approximately 150mph, and a paved quadrilateral/triangle structure is used for the mesh. Structured quadrilateral cells of a fixed start size, growth rate and depth were grown from the wall to capture the boundary layer. A ‘standard’ wall function (the default setting in Fluent (FluentTM, 2003b)) was used which is based upon the proposal of Launder and Spalding (1974). With this wall function, the logarithmic
law for mean velocity is know to be valid for y+ > 30, where y+ is the wall-normal
coordinate normalised by the local viscous length scale ν/uτ. For this study y+ ≈50. Various mesh sizes were tested and solved with the same CFD model with a standard k-ǫturbulence model at each attempt. Starting from a cell count of approximately 2000 cells, the mesh was progressively refined testing approximately 3, 6, 12, 24, 39, 75 and 120 thousand cell meshes. The profile of velocity U was set as uniform across the inlet and the dependent variable of turbulent dissipation rate was calculated from a turbulent intensity value of 1%, and the dependent variable of turbulent kinetic energy calculated from a hydraulic diameter of 0.2m. Using these values, the static pressure recovery value, Cp, was calculated for each of the different mesh sizes and the results shown in Figure 4.1. The percentage difference in Cp value between the meshes comprising of approximately 1500 cells and the finest mesh was 437%, whereas the percentage differ- ence in Cp value between the meshes comprising of approximately 39000 cells and the finest mesh was 1.7%; this is considered to be a sufficiently accurate result. Hence a mesh with approximately 39000 cells was chosen for the optimization studies presented in this chapter. It is difficult to control the exact cell count when using an automated process to mesh each design discovered by the optimizer. However, the meshes used in this initial dependency study were created with a size function capability within the Gambit meshing tool which allows the same growth rate from the boundary layer to be developed regardless of the shape of the diffuser. Using this functionality, finer meshes are produced by decreasing the distance between the mesh nodes on the diffuser walls while maintaining a consistent boundary layer depth across meshes. The fixed boundary layer depth is such that it will capture all of the boundary layer as it develops over the streamwise distance x. The fixed boundary layer depth, fixed distance between wall nodes and a fixed cell growth rate mean that the meshes produced during the optimiza- tion processes later in this chapter will only vary by a small amount in terms of total cell count given varying wall shapes.
The CFD analysis for all the studies carried out in this chapter involves solving the in- compressible two-dimensional steady-state Reynolds-averaged Navier-Stokes equations. Thek-ǫturbulence model (Launder and Spalding, 1974) is used for the straight diffuser study which is in keeping with the references studied, and the Spalart-Allmaras tur- bulence model (Spalart and Allmaras, 1992) is applied in the constant width turning
0 2 4 6 8 10 12 x 104 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of cells Cp
Figure 4.1: Graph illustrating the dependency ofCp value with mesh density for a
2D straight walled diffuser
Figure 4.2: An example of a∼39000 cell mesh with inlet, filter and outflow positions elbow and the final study in section 4.6. The change to using a Spalart-Allmaras model is due to the fact that this model is more economic than the standard k-ǫ model and more accurate for wall-bounded flows and flows with mild separation and recirculation (FluentTM, 2003b). The boundary conditions for studies conducted in sections 4.4 and 4.5 comprise a fixed mass flow rate of 10.8kgs−1 with a uniform profile at the inlet and
a pressure outflow positioned at the exit of the duct. The exit of the duct is situated downstream of the engine filter. The position of the engine filter is chosen to be at the end of the diffuser expansion in the straight diffuser study, and at the end of the bend in the elbow study. This is to ensure that any separation arising within the diffuser or elbow does not pass through the outflow boundary. Mass-averaged static pressure values are taken at the inlet and at the position of the filter. Figure 4.2 illustrates a mesh of approximately 39000 cells and depicts the positions of the inlet, filter and outflow for the straight diffuser.
For the study carried out in section 4.6, the large expansion required over such a short distance coupled with the 90◦ bend means that a pressure outflow boundary condition
at the filter is insufficient due to the unstabilized flow at this point caused by separation. The diffuser could be extended downstream of the filter creating a long constant width outflow duct as carried out in sections 4.4 and 4.5, but this is unrealistic in terms of the nature of the airbox setup within an F1 race car. To ensure that an accurate converged solution is obtained, the airbox model is extended to include the engine filter, represented in Fluent by a one-dimensional porous jump, and the trumpet tray, an area of fixed width situated between the filter and the engine trumpets which are positioned over the cylinders. The area represented by the engine trumpets is classified as a velocity inlet boundary condition.
A breathing engine sucks the air out of the airbox through the trumpets and so the velocity inlet condition at the trumpets is given a negative velocity value. It is assumed that the 4-stroke engine is at wide open throttle running at 18000 revolutions per minute (rpm), i.e., 9000 intake strokes per minute or 150 intake strokes of the pistons per second. Assuming 100% volumetric efficiency, the 3-litre engine requires 0.003 m3of air and hence the engine requires a volume flow rate ˙V = 0.45m3s−1. This flow rate is also equal to
the product of the total cylinder area and the average piston velocity required by the engine. From this the piston velocity can be calculated given the total cylinder area. For the two-dimensional model to be representative of the real 3D case, the total trumpet length is calculated by matching the 3D total cylinder area to filter area ratio. However, within the 3D airbox there is a second expansion ratio of total cylinder area to total inlet area to consider. For this 2D case, both cannot be considered. Thus, the length of the diffuser inlet and length of the filter are fixed. The first expansion ratio of cylinder area to filter area is considered more appropriate as the complete shape represents the centre plane of a 3D airbox and the geometry parameterization techniques for defining the walls can potentially be carried over into an airbox design strategy in three dimensions. Since the 2D diffuser is effectively doing half the work needed to expand the flow through a 3D airbox for the required breathing engine velocity, the velocity of the flow being sucked out of the 2D trumpets needs to be increased to represent sensible inlet speeds akin to those seen in the 3D case. The velocity through the engine trumpets is such that Re = 2×104 giving a velocity of 58ms−1. A pressure inlet is imposed at the diffuser inlet. The porous jump values used here are given by the thickness and permeability of the engine
filters typically used by F1 teams and have the following values: face permeability = 1.3×10−8m2, filter thickness = 15mm and inertial resistance factor = 1000m−1. Table 4.1 provides the boundary conditions and turbulence model setup values for each of the three two-dimensional intake cases: the straight diffuser, the elbow, and the diffuser turning through 90◦. The under-relaxation values shown in Table 4.1 are used
to control the change of calculated nodal value during each CFD iteration. This occurs due to the non-linearity of the equation set solved by Fluent. So, for a variable ψ, the new variable value depends on the old value through the relationψ=ψold+q∆ψ. The higher the under-relaxation values (q), the faster the convergence rate. Each simulation requires 1000 iterations to converge satisfactorily.