# Euler Method Technique

Euler equations are obtained when the viscous terms are omitted from Navier – Stokes equations, allowing faster predictions of pressure distributions. They can be usefully employed at the preliminary design stage. Viscous effects can be included by integrating boundary-layer methods and by displacing the surface of the aero­foil, wing, or aircraft by an amount equal to the local boundary-layer displacement thickness.

14.6.4 Full-Potential Flow Equations

The full-potential flow equations assume that the flow is irrotational. Compressible flows can be modeled but the “shocks” that are predicted are isentropic. The method is now quite dated but it can provide rapid information about pressure distributions and – like the Euler method – it can be integrated with a boundary-layer method.

14.6.5 Panel Method

This is simplest of all numerical methods for predicting flow around an aircraft. The surface of an aircraft is covered with panels, each one a source of sink, and some (e. g., those on lifting surfaces) are assigned a bound vortex (with its associated trailing vortex system). The strengths of the sources and bound vortices are initially unknown but can be determined through application of the boundary conditions (e. g., flow tangency at solid surfaces).

Descending through the hierarchy, the methods provide less physical fidelity but also require less computational effort. It is conceivable that the panel method, full-potential flow equations, Euler equations, and RANS method can be used in an undergraduate aircraft-design project (as a separate task), although not at the conceptual design stage. These methods provide a qualitative pressure-distribution pattern to help shape the geometrical details. Whichever method is used, the issue of grid generation must be addressed: More time is spent on the generation of a suitable mesh than on the prediction of flow.

A 3D model created in CAD is useful at this stage. The planning to prepare the 3D model in CAD should be done in such a way that Boolean operations can build it from isolated components, while still retaining the isolated components for a separate analysis. The wing-fuselage analysis provides the tail-less pitching moment data, which are useful in designing the aircraft H-tail and its setting relative to the fuselage to minimize trim drag.

CFD results can be compared with results obtained through use of the semi­empirical relationship (e. g., drag) (see Chapter 9). Generally, semi-empirical drag results are considered to provide good accuracy, validated on many aircraft consis­tent over a long period of use.

Figure 9.8 presents the wave drag, CDw, for the Mach number. CFD provides an opportunity to generate a more accurate viscous-independent wave drag versus the Mach number. When the CFD results are available, the data in Figure 9.9 may be replaced, thereby obtaining a further iteration on the drag polar of the aircraft. CFD is also a good place to generate ACDp values to be used for comparison. In general, CFD-generated ACDp values should provide good values if the CFD is set up properly.

If the CFD results are within 10% of the results obtained using semi-empirical relations, then they may be considered good. Some adjusting of the CFD runs should

improve the results – this is an area where experience is beneficial. Once the CFD is set up to yield good results, it is useful to improve and/or modify an aircraft configu­ration through extensive sensitivity studies. The spectrum plots in color show the hot spots that contribute to drag (e. g., local shocks and separation). These details can­not be seen as easily by any other means. Designers can follow through by repairing the hot spots to reduce drag. These opportunities are unique to CFD, making it an indispensable tool for optimizing a configuration for minimum drag. Any signifi­cant difference between the CFD and semi-empirical results should be investigated properly.

14.7 Summary

CFD simulation is a digital-numerical approach to design incorporated in the virtual-design process using computers. The current status is adequate for com­parative analyses at low cost and time; therefore, it must be applied early in the conceptual design phase as soon as a CAD 3D model drawing is available. The development of CFD is not necessarily driven solely by aerodynamic considerations but rather by the requirement to have a tool to design a better product at a low cost and in less time.

CFD continues to develop with greater computing power at lower cost and in less time along with advances made in the algorithms for resolving solutions, provid­ing considerable ease and automation to benefit users. Although researchers have achieved a degree of accuracy in drag prediction for a clean aircraft configuration, the generalized application by engineers is yet to achieve consistency in results. Ver­ification and validation of results from CFD analysis are essential for substantiation, and the state of the art is still being scrutinized and continuing to develop. Verifi­cation of new CFD software comes before validation; together, they involve a pro­tracted process in which research continues.

As a conservative user, the industry must ensure fidelity in a design. However, CFD is capable of comparison to recognize the better designs, even if the absolute values remain under scrutiny. This capability produces the best compromise in an early phase of the project at low cost and in less time, thereby avoiding subsequent costly modifications of an aircraft configuration – that is, it provides the opportunity to make the design right the first time. The design is subsequently tested in a wind tunnel for substantiation. Today, this approach requires few changes to the design after the final flight-test results.

The industrial effort in CFD is extensive and is not suitable for an undergrad­uate course. However, coursework can follow the industrial approach by solving smaller problems, such as those described in this chapter.