Correlación entre la motivación de los estudiantes y el post-test de álgebra

In the previous sections, some of the existing techniques available to solve the aerodynamic inverse problem have been reviewed. Although by no means comprehensive, this review shows ongoing efforts to develop or improve inverse methodology for aerodynamic design. Here the main observations are summarized and figure 2.5 depicts the main points.

Different inverse design methods based on potential flow were described in section 2.3.1. In general, these methods are highly efficient and only a few iterations are required to compute the desired blade shape. However, this is only possible because assumptions are applied to simplify the flow equations. At the simplest level, the flow is assumed to be isentropic, irrotational and incompressible (for example, Lighthill's conformai mapping technique). Most of the methods in this category are also not fully

three-dimensional except those of Yu (1980) and Fung (1980) and those o f Tan et al.

(1984), Borges (1986), Zangeneh (1991) and Soulis (1985).

The methods by Yu and Fung are based on the Fictitious gas model which do not allow

the user to have full control over the flow specification or the geometry. Their methods thus fall short of a systematic design methodology. The second group of methods by Tan, Borges, Zangeneh and Soulis based on the specification of the swirl velocity is systematic and effective but the applications are limited to the design of thin blades.

In addition to the above, a common limitation in all the surveyed methods in this category is that they cannot cope with high-speed flow where there is shock. The methods may therefore be applied only in cases when the flow is entirely subsonic. Although the two-dimensional method proposed by Schmidt (1980, 1986) is adequate when the shock waves are weak, the rotational effects are neglected. The assumption of irrotationality is not acceptable in high-speed turbomachinery since the flow is rotational downstream of any shock (including weak shocks). Not only is this important to the description of the flow field around shock waves, but rotational effects are also important for accurately predicting real fluid behaviour and quantifying the actual force

on the blade during the working of the fluid machine. The methods in this category are thus not applicable for the design of high-speed transonic turbomachinery blades.

Section 2.3.2 described some Euler-based methods. The primary advantage o f Euler- based inverse methods over those based on potential flow is that they are applicable for all flow regimes including high transonic flow cases where there is strong shock. In addition, the flow model behind the design is more realistic since the flow is not assumed to be isentropic or irrotatonal.

Substantial effort in the use of the surface pressure distribution as the design specification has been reported. All such procedures including that using a

Transpiration model suffer from the difficulty of obtaining realistic blades if the

pressure distributions are specified on both the pressure and suction sides of the blade. As a result, a compromise is usually made to yield an acceptable result. Common approaches are to design only one side of the blade or to relax the design specification. More seriously, the prescription of this design distribution in three dimensions is extremely difficult since the distribution at the hub and shroud cannot be specified independently and it is not possible to know their relationship in the absence of a known blade geometry. This is a major limitation of the method.

An alternative to the surface pressure transpiration model method is to impose the blade pressure difference distribution together with a chosen thickness distribution. As this approach is in the early stages of development, only details in two dimensions have been reported and improvements are possible. The applicability of the design specification in three dimensions and its ability to design blades with finite thickness also make it attractive for further development. This is therefore pursued as one part of the current work.

The other part of the current work involves implementing another Euler-based method. From the successes reported in the use of the swirl velocity in sections 2.3.1.6 and 2.3.1.7, this is chosen to be design parameter for the method.

In recent years, there has been interest in developing design methods based on viscous flow solutions and a few methods employing Navier-Stokes equations have emerged. Due to the high computational requirement involved, however, these methods are still too expensive for routine use. In contrast, in the current work, viscous modelling is implemented to improve the Euler solution. The viscous coupling is efficient and allows designs to be carried out while taking into consideration the viscous effects, without incurring a severe increase in computational cost. Details of the inviscid/viscous solver code and the design methods are given in chapters 3 and 5 respectively.

Potential Equations Euler Equations Navier-Stokes Equations

Can be com pressible or incompressible Inviscid, isentropic and irrotational

Applicable for subsonic flow

Complicated and inaccurate for high-speed flow

M ainly com pressible Inviscid and rotational Entropy generation properly accounted for A pplicable for high­ speed flow Shock capturing ability

Mainly com pressible V iscous and rotational Entropy generation properly accounted for A pplicable for high­ speed flow Shock capturing ability

Singularities m ethod C onfom al m apping Potentiai- Stream T aylor Series H odograph F ictitious

Im perm eable and m ovable surface Clebsh representation o f How circulation - M ethod has no control on flow or geom etry, o r - M ethod is approxim ate Applicable in Mainly T ranspiration Transpiration or m ovable surface Specified tangentially averaged Swirl v elocity, rV'e A pplicable m Different Approaches Difficult to use in 3-D Specified surface pressure (or velocity)

Applicable only to thin blades (zero o r negligible thickness) Flow is inviscid y v Applicability in 2-D/3-D Specified surface pressure difference (API Different Design Parameters

• N o control over blade • A rbitrary specification o f thickness AP does not give required • Problem o f ensuring

blade closure open o r • Flow is inviscid cro ss-over ends • D etails is given only in • N o definitive rules to ensure a solution 2-D This design param eter is chosen for the current work C o n s tr a in e d B lade D esign - D esign on only one side

o f the profile - Relax the design

specification during the process

- Vary the blade solidity during the process

This method is developed and improved in the current work Shortcomings

CHAPTER 3

DEVELOPMENT OF THE TIME-MARCHING EULER