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The work addresses an important question of whether a discontinuity in wall curvature can cause boundary layer separation at transonic speeds. Firstly an inviscid transonic flow in the vicinity of a curvature break is analysed. Depending on the ratio of the curvatures, several physically different regimes can exist, including a special type of supersonic flows which decelerate to subsonic speeds without a shock wave, transonic Prandtl-Meyer flow and supersonic flows with a weak shock. It is shown that if the flow can be extended beyond the limiting characteristic, it subsequently develops a shock wave. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem. From an asymptotic analysis of the Karman-Guderley equation it follows that the curvature discontinuity leads to singular pressure gradients upstream and downstream of the break point. To find these gradients, we perform computations and employ both the hodograph method and the phase portrait technique. The focus is then turned to analysing how the given pressure distribution affects the boundary layer. It is demonstrated that the singular pressure gradient, which appears to be proportional to the inverse cubic root of the distance form the curvature break, corresponds to a special resonant case for the boundary layer upstream of the singularity. Consequently, the boundary layer approaches the interaction region in a pre-separated form. This changes the background on which the viscous-inviscid interaction develops, allowing to construct an asymptotic theory of the incipient viscous-inviscid interaction for our particular problem. The analysis of the interaction which takes place near a weak curvature discontinuity leads to a typical three-tier structure. It appears to be possible to obtain analytical solutions in all the tiers of the triple deck when the curvature break is small. As a result, the interaction equation may be derived in a closed form. The analytical solution of the interaction equation reveals a local minimum in the skin friction distribution, suggesting that a local recirculation zone can develop near the curvature break. In fact, the recirculation zone is formed when the ratio of the curvatures is represented as a series based on negative powers of the logarithm of the Reynolds number. This proves that a discontinuity in wall curvature does evoke boundary layer separation at transonic speeds. The result is fundamentally different from the effect of a curvature break at subsonic and supersonic speeds, as no separation takes place in these two regimes (Messiter & Hu 1975).

The correlation of cascade experimental data is one method for obtaining compressor stage characteristics. These correlations specify pressure loss and flow turning caused by the blades. Current open literature correlations used in streamline curvature codes are inadequate for general application to high-speed transonic axial-flow compressors. The objective of this research was to investigate and evaluate the available correlations and ultimately discover sets of correlations which best fit the empirical data to be used in streamline curvature codes. Correlations were evaluated against experimental data from NASA Rotor 1-B and NASA Stage 35. It was found that no universal set of correlations was valid for minimum-loss point predictions. The Bloch shock loss model showed promising results in the stall regime for supersonic relative inlet Mach numbers. The Hearsey and Creveling off-minimum-loss deviation angle prediction performed consistently better than all other correlations tested. / Master of Science

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In chapter 1, we focus on the full potential equation in an infinite nozzle with some decay cross-sections and prove the existence and stability of the transonic shock wave; which is a solution to a free boundary value problem for a quasi-linear mix-typed partial differential equation with the position of shock as a free boundary. To achieve this conclusion, we reduce it to a free boundary value problem for a quasi-linear elliptic equation in an unbounded domain. The crucial step in our analysis is to derive some uniform a priori estimates in such a domain. Then we apply the fixed point theorem to establish the existence of solutions to the full potential equation. / In chapter 2, we study the short time existence of discontinuous shock front solutions of the pressure gradient system which is the Euler system without inertial terms, where the initial data can have shock discontinuities of arbitrary strength which lie on a given smooth initial surface with arbitrary geometry. These shock solutions are constructed via a classical iteration scheme. The key step is to obtain the uniform stability for the related linearized equation by calculating the Lopatinski's determinant, which enables us to modify the technique of Majda and establish the local existence of solutions to the pressure gradient system without the structural constraints as for the full Euler system. / In this thesis we study two kinds of multi-dimensional shock phenomena for the compressible fluid dynamics. / Xie Feng. / "December 2005." / Adviser: Zhou Ping Xin. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6446. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 71-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

With the present drive in turbomachine engine developmenttowards thinner and lighter bladings, closer spaced blade rowsand higher aerodynamic loads per blade row and blade, advanceddesign criteria and accurate prediction methods for vibrationalproblems such as forced response become increasingly importantin order to be able to address and avoid fatigue failures ofthe machine early in the design process. The present worksupports both the search for applicable design criteria and thedevelopment of advanced prediction methods for forced responsein transonic turbine stages. It is aimed at a betterunderstanding of the unsteady aerodynamic mechanisms thatgovern forced response in transonic turbine stages and furtherdevelopment of numerical methods for rotor stator interactionpredictions. The investigation of the unsteady aerodynamic excitationmechanisms is based on numerical predictions of thethree-dimensional unsteady flow field in representative testturbine stages. It is conducted in three successive steps. Thefirst step is a documentation of the pressure perturbations onthe blade surface and the distortion sources in the bladepassage. This is performed in a phenomenological manner so thatthe observed pressure perturbations are related to thedistortion phenomena that are present in the blade passage. Thesecond step is the definition of applicable measures toquantify the pressure perturbation strength on the bladesurface. In the third step, the pressure perturbations areintegrated along the blade arc to obtain the dynamic bladeforce. The study comprises an investigation of operationvariations and addresses radial forcing variations. With thehelp of this bottom-up approach the basic forcing mechanisms oftransonic turbine stages are established and potential routesto control the aerodynamic forcing are presented. For the computation of rotor stator interaction aerodynamicsfor stages with arbitrary pitch ratios a new numerical methodhas been developed, validated and demonstrated on a transonicturbine test stage. The method, which solves the unsteadythree-dimensional Euler equations, is formulated in thefour-dimensional time-space domain and the derivation of themethod is general such that both phase lagged boundaryconditions and moving grids are considered. Time-inclination isutilised to account for unequal pitchwise periodicity bydistributing time co-ordinates at grid nodes such that thephase lagged boundary conditions can be employed. The method isdemonstrated in a comparative study on a transonic turbinestage with a nominal non integer blade count ratio and anadjusted blade count ratio with a scaled rotor geometry. Thepredictions show significant differences in the blade pressureperturbation signal of the second vane passing frequency, whichwould motivate the application of the new method for rotorstator predictions with non-integer blade count ratios.