Shock Fitting a Transonic Cascade Solution into an Inverse Design Technique

Abstract

We present a new Lagrangian-based shock-e tting technique for inversely designing transonic turbomachinery cascade geometries. Thismethod, which consistsofa two-dimensional e owe eld integrator, a camberline generator, and a passage-averaged momentum/pressure boundary condition, generates a cascade geometry to match a prescribed e ow turning distribution. A complex-lamellar e ow decomposition is used, e rst to show how discontinuous geometries are created when one’ s total turning distribution is specie ed to be continuous and shock-generated entropy gradients are present and then to construct a shock-e tting treatment that actively modie es the specie ed turning distribution to counter this effect. Finally, numerical results are presented to illustrate that, with this new shock-e tting approach, our transonic cascades are both geometrically continuous and faithful to the prescribed e ow turning distribution. I. Introduction W ITH the increase in their accuracy and efe ciency, numerical simulations are now being implemented into every aspect of the aerodynamic design process. In fact, whereas they were once used mainly for generating postmortem analyses of intermediate designs, numerical methods are now being used for both design optimization 1i3 and inverse design. 4i6 Although inverse methods are often most efe cient, they require one to specify loading or pressure distributions, which, without proper judgement, can lead to poorly performing designs. Target distributions are often specie ed without any prior knowledge of their appropriateness or ability to be actively modie ed throughout the inverse design procedure. Thus, design-optimization schemes have begun to attract a tremendous amount of attention. In these schemes, one examines a large design space in hopes of identifying the optimal solution to a given number of constraints and objective functions. Unfortunately, global optimums are not easily obtained without careful construction of the appropriate geometric constraints, adjoint equations, and objective functions. 7;8 In fact, Drela 9 has shown that, whereas a multipoint optimization is needed to control both design andoff-design performances,geometries that havebeenoptimized overmultiple objectivefunctions areoftensusceptible to small-scale irregularities of signie cant consequence in viscous and transonic e ows. Thus, mixed inverse design/design optimization scheme are being developed to exploit the strengths of each of these approaches. 1;10;11 With this work we present a new Lagrangian-based shock-e tting techniqueforinverselydesigningtransonicturbomachinerycascade geometries. This technique, which is based on the inverse-design theoriesofHawthorneetal. 4 andTanetal. 12 ingeneral,andDang 5 in