Entre aspiraciones, problemáticas y fracasos

In Chapter 2, an array of computational techniques currently employed in the field of aerodynamic design was outlined, and a distinction was made between two design approaches: direct design optimization and inverse design. The EMFID process combines these two methods, with the aim of improving the efficiency of performing design optimization using high-fidelity CFD analyses. The underlying prediction is that a low-dimensional parameterization of flow features can, following inverse design, result in a larger range of geometrical variation and higher quality designs than a geometry-based parameterization of the same dimensionality. This reduces the number of design parameters required for optimization, leading to a more efficient search process. The EMFID method has been demonstrated in this thesis using four case studies in airfoil design and two further case studies considering wing design. Table 7-1 briefly describes these case studies, augmenting Table 4-7 with the 3-D cases from Chapter 6. The conclusions drawn from the six case studies are described next.

Table 7-1 Summary of the case studies reported in this thesis.

Case studies Flow equations Flow speed Mach number Case description

1 RANS 0.15 Comparison of EMFID and benchmark methods 2 RANS 0.15 Benchmark run using a low dimensional model 2-D

airfoil

design 3 RANS 0.15 VGK, 0.73 FLUENT Comparison using a transonic drag calculation with a subsonic Cp model in EMFID

4 RANS 0.73 EMFID run using a transonic Cp model

5 Euler 0.7 Comparison of EMFID and benchmark methods – minimizing induced drag

3-D wing-tip

design 6 RANS 0.7 Comparison of EMFID and benchmark methods – minimizing total (induced and viscous) drag

In Chapter 3, the concept of flow feature parameterization was introduced and the practicalities in implementing the method for 2-D airfoil design were detailed. Following this, in Chapter 4, the EMFID process was applied to 2-D airfoil design in case studies 1 to 4. The conclusions from this work are fourfold.

• First, it has been shown that a B-spline representation of the airfoil surface pressure distribution can be an effective parameterization technique and is able to generate high-performing designs.

The low-dimensionality of this method results in rapid convergence of the design search. In addition, the smooth nature of the B-spline representation means that the resulting airfoils are inherently sensible

shapes, and this further accelerates the design search towards promising designs.

• Second, it has been demonstrated that a geometrical parameterization, using the same number of variables as the EMFID process, is unable to generate such high-quality and detailed shapes, and the corresponding design search finds poorer designs than the EMFID method.

Thus, it is apparent that a parameterization of the pressure profile can result in a more efficient design search than a geometrical parameterization of the same dimensionality. However, it is noted that the EMFID method benefits from the relatively simple shape of the pressure distribution corresponding to the optimum (or at least, very low drag) airfoil design. This simplicity means that a low-dimensional parameterization can perhaps more effectively represent the pressure distribution than the geometry.

• Hence, the third conclusion states that, to be successful, the choice of flow feature to parameterize in EMFID should take into account the simplicity or complexity of the flow features corresponding to the optimum design.

• Fourth, and finally, the EMFID concept has been applied to the design of transonic airfoils, and it was found that the method benefits from using the same flow conditions for the parameterization, inverse design, and the final drag calculation.

In particular, it is important that the parameterization of the pressure profile includes the specification of the shock. Incorporating the shock strength and position into the design variable set provides the optimizer with a direct means of controlling the wave drag, and gives a simpler objective function landscape. Conversely, if the target pressure distribution is specified for subsonic flow conditions, the relationship between the design variables and (transonic) drag is likely to be complex. In addition, the use of a transonic inverse design procedure means that the resulting airfoils are inherently tailored for the transonic regime.

A parameterization of the surface pressure distribution has been shown to be an effective approach for subsonic and transonic airfoil design. Although, ultimately, higher-dimensional schemes may be able to represent finer detail, a design search using the six-variable EMFID parameterization is a highly efficient practice, producing higher performing designs than a 13-variable geometry-based method for a given

computational expense. Following the success of the EMFID concept in the latter guise, a logical progression was to attempt to apply the method to 3-D wing design. Chapter 5 introduces the application of wing-tip device design, and reports a study examining an appropriate flow feature to parameterize in EMFID and a suitable geometric representation for the wing-tip device. The details associated with the wing-tip vortex are a novel choice of flow feature, but the vorticity predictions of the panel code and high-fidelity CFD code do not agree sufficiently to provide a meaningful design search. Consequently, the spanwise loading distribution was chosen as the design flow feature. Concerning the geometry description, a set of five distinct geometrical wing-tip device parameters are inappropriate because these are mapped non-uniquely to the spanwise loading over the wing-tip. Thus, the spanwise variation of chord over the wing-tip is used, since each design corresponds to a unique lift distribution.

Chapter 6 presents two case studies in which the EMFID and benchmark methods are applied to the design of the chord profile over a wing-tip device. The use of a quadratic or cubic polynomial to represent the lift distribution over the wing-tip device allows the EMFID process to converge more quickly than the six-variable benchmark method. As in the 2-D airfoil case studies, this is due to both a lower dimensionality in EMFID and an inability to generate nonsensical shapes. When the objective function, drag, was calculated using Euler simulations, a large range of final geometries were generated using the design searches. The induced drag is relatively insensitive to the chord function inboard of the tip, but is more dependant on the chord at the tip station, and hence all final designs feature highly aft-swept tips. However, the benchmark method produced marginally better performing designs, and it was found that the lift distribution corresponding to the best benchmark designs is too complex to be represented using the EMFID quadratic or cubic curve. When the drag objective is calculated using RANS simulations, there is a clear trend to minimize the wetted area while maintaining the tip sweep. Although the EMFID design search only reached the optimum design because of the geometry repair process, it converged significantly more quickly to this optimum than the benchmark method. Again, the EMFID parameterization is not sufficiently detailed to be able to represent the optimum lift distribution. However, when the method is not forced to conform to the benchmark constraints, the EMFID scheme can produce geometries closely matching those that are repaired.

There are four key conclusions resulting from the 3-D application of EMFID.

• First, it is important that the flow feature to be parameterized in EMFID must map uniquely to geometry, i.e., each wing-tip lift profile should correspond with a single geometry.

• Second, the parameterization of the lift profile has been shown to be effective in the sense that it facilitates rapid convergence of the design search through a reduction in the number of design variables.

• Third, the loading distribution corresponding to the optimum design must be sufficiently simple that it can be represented using a low-dimensional parameterization. In the 3-D design scenario presented in this work, the optimum flow feature is rather too complex. If the optimum lift profile were simpler then perhaps a quadratic or cubic would be sufficient to reproduce it. However, because the optimum geometry is relatively complex, it remains the case that the EMFID approach has generated better performing designs than would be possible with a benchmark search using the same number of variables.

• Thus, fourth, it can be concluded that a parameterization of the lift profile is able to produce finer detailed designs than a geometrical-based scheme of the same dimensionality.

The significance of the EMFID concept as a design tool for aerospace design should be considered. Fundamentally, the method can be used as a low-dimensional means of representing any surface subjected to fluid flow, provided that there exists a suitable flow feature to parameterize. Crucially, due to the expense of performing inverse design at every iteration, the method is only likely to be computationally efficient when high-fidelity CFD analyses are used to calculate the design objective. From a practical point of view, implementation of the EMFID method requires more computational setup time than the benchmark method; in particular the method requires an inverse design procedure which calls upon a low-fidelity CFD code.

Referring to the initial discussions in Chapter 1 surrounding parameterization and Figure 1-2, a need has been identified for a design approach which uses a small number of design variables but which can generate high-performing designs. The concept of flow feature parameterization has been shown to address this need. However, to be successful there must exist a prominent flow feature which exhibits a simple variation. Further, the EMFID process is only advantageous when the flow feature variation is

simple but the geometrical shape is complex. Thus, a parameterization of the surface pressure distribution for 2-D airfoil design is very effective, while a parameterization of the lift profile for 3-D wings has been shown to be promising in terms of efficiency but unproven in ultimate performance. Nonetheless, in all of the case studies demonstrated here, the EMFID strategy has provided gains in computational efficiency; for this reason, it can be a useful tool in the arsenal of an engineer.

Finally, the major contributions of this thesis are listed below.

• A low-dimensional parameterization of flow features can produce high quality geometries following inverse design. Further, such a parameterization can produce finer detail and local control of the shape than a geometry-based parameterization of the same dimensionality.

• For a given level of design improvement, a parameterization of flow features uses fewer design variables, and, combined with an inherent ability to generate smooth designs, this leads to a highly efficient optimization process.

• The flow feature parameterization concept is most advantageous when a flow feature can be represented simply while the corresponding geometry is relatively complex.

• The parameterized flow features, the inverse design operation, and the design objective should all be calculated for the same flow conditions, such as Mach number.