This dissertation presents a technique for the solution of incompressible equivalents to planar steady subsonic potential flows. Riemannian geometric formalism is used to develop a gauge transformation of the length measure followed by a curvilinear coordinate transformation to map a subsonic flow into a canonical Laplacian flow with the same boundary conditions. The method represents the generalization of the methods of Prandtl-Glauert and Karman-Tsien and gives exact results in the sense that the inverse mapping produces the subsonic full potential solution over the original airfoil, up to numerical accuracy. The motivation for this research was provided by the analogy between linear potential flow and the special theory of relativity that emerges from the invariance of the wave equation under Lorentz transformations. Whereas elements of the special theory can be invoked for linear and global compressibility effects, the question posed in this work is whether other techniques from relativity theory could be used for effects that are nonlinear and local. This line of thought leads to a transformation leveraging Riemannian geometric methods common to the general theory of relativity. The dissertation presents the theory and a numerical method for practical solutions of equivalent incompressible flows over arbitrary profiles. The numerical method employs an iterative approach involving the solution of the incompressible flow with a panel method and the solution of the coordinate mapping to the canonical flow with a finite difference approach. This method is demonstrated for flow over a circular cylinder and over a NACA 0012 profile. Results are validated with subcritical full potential test cases available in the literature. Two areas of applicability of the method have been identified. The first is airfoil inverse design leveraging incompressible flow knowledge and empirical data for the potential field effects on boundary layer transition and separation. The second is aerodynamic testing using distorted models.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-12T13:25:21Z No. of bitstreams: 1 arquivototal.pdf: 1212419 bytes, checksum: 6d3ca7625ab07b583f3f68045187a552 (MD5) / Made available in DSpace on 2017-09-12T13:25:21Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1212419 bytes, checksum: 6d3ca7625ab07b583f3f68045187a552 (MD5) Previous issue date: 2016-02-25 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In 1918 the German Hermann Weyl developed a unified theory of gravitation and electro­magnetism becoming geometrical both interactions, that is, he associated the potential elec­tromagnetic a 1-form a, after considering that the length of a vector is not preserved under parallel transport as well as with the direction, this also meant that the covariant derivatives of the metric tensor ceased to be null becoming Vag" = a, The gravitational and electromagnetic field equations are obtained from the action I = f (R2 ± v Ft" ) —gd4x in a gauge any and "natural gauge"R = A = constant taking into account that they, as well as the action, should be both invariant under coordinate transformations as invariant under the gauge transformations introduced, namely, gliv = of gin, and di, = cp.+ hi, actually, the first person to speak in scale invariance in physics was the Weyl himself in his article. It is also found that the solutions to the emptiness of Einstein's field equations are also solutions of the corresponding Weyl's field equations. Finally it is shown that the Weyl affine geo­desic may not come from a variational principle by analysing the Helmholtz conditions for the inverse problem of the calculus of variations and discusses about Einstein's criticism of the theory, on which it is concluded that the even seized an inadequate definition of proper time to give his opinion on the work of Weyl, thus, a problem to be solved was to find a good definition of proper time, which leaves open a final version of the Weyl theory. / Em 1918 o alem-do Hermann Weyl desenvolveu sua teoria de unificndo entre gravitacao e eletromagnetismo geometrizando ambas as internOes, isto é, ele associou o potencial eletromagnetico a uma 1-forma a, depois de ter considerado que o comprimento de um vetor nao preserva-se sob transporte paralelo assim como acontece com a direcao, isso tambem fez com que a derivada covariante do tensor metrico deixasse de ser nula tornando­se V agii, = glivaa. As equagOes de campo gravitacionais e eletromagneticas sdo obti­das da nao I = f (R2 ±AF,,,,Filv)\/—gd4x em um calibre qualquer e no "calibre natural" R = A = const ante levando em conta que elas, assim como a nao, devem ser tanto invari­antes por transformnOes de coordenadas como invariantes sob as transformnOes de calibre introduzidas, a saber, gliv = of gin, e di, = al, + hi, na verdade, a primeira pessoa a falar em invarifincia de escala na fisica foi o prOprio Weyl em seu artigo. Tambem é verificado que as solucOes para o vazio das equagOes de campo de Einstein tambem sac) solucOes das equnOes de campo de Weyl correspondentes. Por fim mostra-se que as geodesicas afins de Weyl nao podem advir de um princfpio variacional atraves da analise das condicOes de Helmholtz para o problema inverso do calculo de varinOes e discute-se sobre a critica de Einstein a teoria, onde conclui-se que o mesmo se apoderou de uma definicao inadequada de tempo proprio para dar seu parecer sobre o trabalho de Weyl, assim, um problema a ser resolvido seria encontrar uma boa definicao de tempo pr6prio, o que deixa em aberto uma versdo final da teoria de Weyl.

When calculating scattering amplitudes in gauge and gravitational theories one encounters infrared (IR) divergences associated with massless fields. These are known to be artifacts of constructing a quantum field theory starting with free fields, and the assumption that in the asymptotic limit (i.e. well before and after a scattering event) the incoming and outgoing states are non-interacting. In 1937, Bloch and Nordsieck provided a technical procedure eliminating the IR divergences in the cross-sections. However, this did not address the source of the problem: A detailed analysis reveals that, in quantum electrodynamics (QED) and in perturbative quantum gravity (PQG), the interactions cannot be ignored even in the asymptotic limit. This is due to the infinite range of the massless force-carrying bosons. By taking these asymptotic interactions into account, one can find a picture changing operator that transforms the free Fock states into asymptotically interacting Faddeev- Kulish (FK) states. These FK states are charged (massive) particles surrounded by a “cloud” of soft photons (gravitons) and will render all scattering processes infrared finite already at an S-matrix level. Recently it has been found that the FK states are closely related to asymptotic symmetries. In the case of QED the FK states are eigenstates of the large gauge transformations – U(1) transformations with a non-vanishing transformation parameter at infinity. For PQG the FK states are eigenstates of the Bondi-Metzner-Sachs (BMS) transformations – the asymptotic symmetry group of an asymptotically flat spacetime. It also appears that the FK states are related the Wilson lines in the Mandelstam quantization scheme. This would allow one to obtain the physical FK states through geometrical or symmetry arguments. We attempt to clarify this relation and present a derivation of the FK states in PQG from the gravitational Wilson line in the eikonal approximation, a result that is novel to this thesis.