Benzing BE 122-125 Airfoil Three-Dimensional Analyses

within recommended values), and for an unusual growth rate between the layers of the inflation (usually no more than 1.2-1.3), the comparison with the experimental values is exceptional considering that it was the analysis with the fewer number of elements.

To conclude the CFD model validation with two-dimensional NACA 4415 airfoil, the same CFD model from test case ’XXIV’ was applied to the two-equation turbulence models realizablek-(without inflation and non-equilibrium wall functions activated) and k-ω SST to check the importance of choosing an appropriate turbulence model for the prediction of transition behaviour in two-dimensional analyses of cambered airfoils as it follows in Table 4.6. It is notable that these models require much more refined mesh, although from the present studies have shown that they will not perform as good as theγ-Reθt

model for the CFD two-dimensional analysis of the flow around airfoils.

Table 4.6: Comparison of turbulence modelling influence for test case ’XXIV’ against experimental data.

Model y⁺min/max ∆Cl% ∆Cd% ∆Cm%

γ-Reθt(Transition SST) 0.14/1.81 −0.40 −1.52 −2.53 k-ωSST 0.18/1.87 −9.88 +71.89 −14.69 Realizablek- 19.10/312.76 −11.89 +45.79 −39.56

(a) Numerical wind tunnel geometry. (b) Test case ’I’ mesh model of Table 4.8.

Figure 4.3: Three dimensional virtual wind tunnel geometry and mesh of test case ’I’ (withγ-Reθttran- sition model).

The blockage ratio computed for the wing surface areaS in relation to the test section area (7.2×7.5 m²), gave a value of1%. Reference values for these investigations are summarised in Table 4.7.

Table 4.7: Three-dimensional fluid flow reference values for ANSYS^R Fluent and wing dimensions, using Benzing BE 122-125 airfoil. [bis the wing span.]

Re ρ(kg m⁻³) µ(kg m⁻¹s⁻¹) U (m s⁻¹) c(m)×b(m) α(deg) 1.0269×10⁶ 1.225 1.7894×10⁻⁵ 50 0.3×1.8 1

The first test case performed for this wing (test case ’I’ in Table 4.8) was the application of two- dimensional ’XXIV’ test case settings with the addition of the span-wise number of divisions, set to400 divisions, based on the inspection of chord-wise elements size (Figure 4.3(b)). The same approach was used for the endplates and later vertical supports full-size geometries of the designed rear wings.

The three-dimensional meshing strategy resulted in a structured mesh for the wing surface and inflation layers using volumetric tetrahedronWed6type elements, andTet4type elements for the remaining flow elements with Patch Conforming Method. Despite not having experimental data to corroborate these results, as previously mentioned, this was considered to be the role model for three-dimensional CFD analysis using Fluent in this thesis. However, considering the number of analyses to be performed in the parametric studies, optimisation and rear wings surface area increase (which increases the total number of elements and consequent computation time), the mesh density and computational time required were not reasonable to accomplish the proposed objectives.

The first simplification made was to reduce the mesh density. The aftermath test case ’IV Opt.’, in Table 4.8, reduced it in more than70%(Figure 4.4(a)) with small change in the force values. Nonetheless, when performing rear wings parametric studies, it was found that for minimum chord dimensions, in particular for the proposed rear wing lower wings (0.20m),Fluentstruggled to get reliable results (force coefficients did not converge and positive order of magnitude for the residuals statistics) and displayed errors such as”turbulent viscosity ratio limited to1×10⁵and reversed flow in the outlet”. Two reasons could explain such behaviour. The first related to the meshing strategy of inflation combined with low

number of divisions (60) and bias factor (10), that resulted in poor quality mesh with extremely high skewness and very low minimum orthogonal quality elements for the first layers of the inflation and in particular at the trailing edge of the wing. The second one, associated with the specificγ-Reθttransition model application for low Reynolds numbers (as a result of reducing the chord dimension). Such model delays the flow transition and is also extremely dependent on the inlet conditions for low Reynolds numbers. This can result in a non-converged solution relative to the predominant laminar regime and consequent problematic transition prediction.

(a) Test case ’IV Opt.’ with60(BF:10) number of division for the airfoil and200spanwise. Inflation options are the same of two-dimensional test case ’XXIV’.

(b) Test case ’VI Opt.’ with80number of division for the airfoil and200spanwise. Inflation suppressed.

Figure 4.4: Closer look at the airfoil mesh for optimisation test cases ’IV Opt.’ usingγ-Reθtmodel and

’VI Opt.’ using realizablek-model with non-equilibrium wall functions in ANSYS^R Meshing.

In order to overcome this and improve significantly poor quality elements from test case ’IV Opt.’

mesh, the decision was to use a turbulence model that doesn’t necessarily need prismatic layers to guarantee the correct fluid flow computation at the near-wall region through the use of wall functions (subsection 3.1). The two-equation turbulence model realizable k- satisfies such requirements. This turbulence model used in combination with non-equilibrium wall functions is widely used in academic and industrial applications, and in particular for the automotive road and racing industry [19, 34, 35, 63]

which showed that it is’possible to achieve good results in terms of integral values (e.g.,CD, which are within 2−5%’ [63]. Non-equilibrium wall functions are sensitised to the effects of pressure gradients and distortion of the velocity profiles [31, 63] and is strongly recommended to be selected for external aerodynamic simulations by [63].

The third test case presented in Table 4.8 (test case ’II’) is the application of mesh from the first one (test case I with suppressed inflation) for realizablek-model in combination with non-equilibrium wall functions. The resulting mesh, with only Tet4type elements was significantly reduced in density as well as computational time required. It performed well against test case ’I’ values using γ-Reθt

transition model with inflation method (within the expected from [63]). The values ofy⁺where of10.44<

y⁺ < 385.48, although99% of the wing surface where within 30.00 < y⁺ < 385.48. Comparing with two-dimensional numerical analyses, the reason why the realizable k- model performed a lot better when compared to the γ-Reθttransition model is related to the type of drag predominance. For two- dimensional flow around an airfoil, the predominant drag was the friction drag as for three-dimensional flow around a wing, it was the resulting induced drag from the pressure distribution and vortices at the wing tip (and endplates).

Despite considerable reduction on the computational time, resources and mesh density, with respect

to the rear wings parametric studies and optimisation, it was advised by the supervisor to run these analyses with a coarser mesh and maximum number of elements about two million. To fulfil such rec- ommendations, further studies were done and the resulting test case ’VI Opt.’ performed good when compared to test case ’I’. The y⁺ values for this test case where of 39.49 < y⁺ < 479.18and cross section view of mesh (and characteristics) can be seen in figure 4.4(b).

Table 4.8: Three-dimensional fluid flow studies withFluentfor different turbulence modelling and mesh- ing strategies. [NE is the total number of elements;∆tis the duration of the analysis, for 1000, 1000, 500 and 500 iterations respectively;∆is the relative percentage variation calculated in relation to the first ’I (γ-Reθt)’ test case.]

Test case (Model) NE ∆t −L(N) ∆ % D(N) ∆ % −L/D ∆ %

I (γ-Reθt) 6 699 559 8h 938.75 — 59.09 — 15.89 —

IV Opt. (γ-Reθt) 1 805 605 3h 913.45 −2.70 58.23 −1.47 15.69 −1.25 II (realizablek-) 2 887 477 45min 929.06 −1.03 61.92 +4.77 15.00 −5.55 VI Opt. (realizablek-) 1 445 577 20min 953.89 +1.61 64.38 +8.95 14.82 −6.74

Based on the discussion and results obtained from these studies, succeeding rear wings aerody- namic parametric studies (Section 5.3) are presented for both model application of test cases ’IV Opt.’

with γ-Reθt transition model and ’VI Opt.’ with realizable k- turbulence model combined with non- equilibrium wall functions. Optimisation analyses were performed only with application of test case ’VI Opt.’ considering the minimisation of the computational effort. Final results of rear wings aerodynamic performance were conducted using the test case ’II’ application (with realizable k-turbulence model) taking into attention the evidence of small values change when compared to theγ-Reθttransition model test case ’I’ and the significant computational time advantages.