Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 113-115). Also available via the Internet.

Ph. D.

An inverse method for the design of exhaust nozzles with a specified transonic pressure distribution is presented. A problem of mixed Neumann and Dirichlet boundary condition is solved. A successive line relaxation process is used to solve the array of velocity potentials in the entire flow field. The streamlines are then displaced to produce boundaries which match a desired pressure distribution. Various cases are tested to verify the reliability of the method. The design calculation proves to be efficient and accurate. / M.S.

Numerical optimization procedures have been employed for the design of airfoils in transonic flow based on the transonic small-disturbance (TSD) and Euler equations. A sequential approximation optimization technique was implemented for solving the design problem of lift maximization with wave drag and area constraints. A simple linear approximation was utilized for the approximation of the lift. Accurate approximations for sensitivity derivatives of the wave drag were obtained through the utilization of Nixon's coordinate straining approach. A modification of the Euler surface boundary conditions was implemented in order to efficiently compute design sensitivities without recreating the grid. Our design procedures experienced convergence problems for some TSD solutions, where the wave drag was found not to vary smoothly with the design parameters and consequently create local optimum problems. A procedure interchanging the role of the objective function and constraint, initially minimizing drag with a constraint on the lift was found to be effective in producing converged designs, usually in approximately 10 global iterations. This procedure was also shown to be robust and efficient for cases where the drag varied smoothly, such as with the Euler solutions. The direct lift maximization with move limits which were fixed absolute values rather than fractions of the design variables, was also found to be a reliable and efficient procedure for designs based upon the Euler equations. / Ph. D.

Transonic flow of dense gases for two-dimensional, steady state, flow over a NACA 0012 airfoil was predicted analytically. The computer code used to model the dense gas behavior was a modified version of Jameson's FLOS2 airfoil code. The modifications to the code enabled modeling the dense gas behavior near the saturated vapor curve and critical pressure region where the fundamental derivative, Γ, is negative. This negative Γ region is of interest because the nonclassical gas behavior such as formation and propagation of expansion shocks, and the disintegration of inadmissible compression shocks. The results of this study indicated that dense gases with undisturbed thermodynamic states in the negative Γ region show a significant reduction in the extent of the transonic regime as compared to that predicted by the perfect gas theory. The results of the thesis support existing theories and predictions of the nonclassical, dense gas behavior from previous investigations. / Master of Science

This thesis documents an experimental investigation of the base (trailing edge) pressure and its approximate distribution on a transonic turbine blade. Since the base pressure plays an important role in determining the profile loss on blades with thick trailing edges, both the base pressure and the blade losses are presented for a range of transonic exit Mach numbers. The overall objective of this work is to provide experimental data for improving current computer-based models used in designing turbine blades. The two-dimensional cascade was tested in the VPI&SU Transonic Cascade Wind Tunnel, a blow-down type of tunnel facility. The blade design for the cascade was based on the pitchline profile of the high-pressure turbine in a commercial jet engine with a design exit Mach number of approximately 1.2. In order to carefully instrument the thin trailing edge, the blades used in the experiment were made five times the size of the actual engine blade. With this large-scale blade, five static pressure taps were placed around the trailing edge. In addition to these taps, the rearward portion of the suction surface was also instrumented with five static pressure taps. The aerodynamic losses were quantified by a loss coefficient: the mass-averaged total pressure drop divided by the total pressure upstream of the blade row. These measured pressures were taken with a fixed total pressure probe upstream of the cascade and a pitchwise traversing probe in the downstream position. The cascade was tested for an exit Mach number ranging from 0.70 to 1.40. The results of the experiments indicate a decreasing normalized base pressure (p_B/p_t1) with increasing downstream Mach number (M₂) until the minimum value of p_B/p_t1 = 0.30 at M₂ = 1.30. The approximate base pressure distributions for all transonic downstream Mach numbers indicate nearly uniform pressure around the central 90° of the trailing edge. Results for the profile loss are displayed for exit Mach numbers between 0.70 and 1.35; the trend of increasing loss with decreasing base pressure is shown. The shadowgraph pictures taken reveal the trailing edge region of the flow for several downstream transonic Mach numbers. / Master of Science

The capability for experimental assessment of unsteady shock wave effects on turbine blade performance has been developed. A novel shock-generation system utilizing a shotgun blast has been implemented into the Virginia Tech Wind Tunnel Transonic Cascade Facility. Specialized optical systems and high-performance pressure instrumentation were utilized to obtain both qualitative and quantitative information. Shadowgraph photos of the unsteady shock wave propagation through the cascade indicate presence of a vortical region and its movement from the blade surface into the passage flow. A previously unseen distortion of the trailing edge shock wave is also identified. High-frequency blade surface pressure measurements reveal large fluctuations in surface pressure during shock passage. An estimate of unsteady blade lift is made which reveals a 120 percent peak-peak variation. Furthermore, an approximated loss coefficient was determined to fluctuate as much as 40 percent near the blade passage center. Comparisons are made with previously-published experimental and analytical results. / Master of Science

This thesis presents the results from measurements taken during the transient unstable operation of an axial-flow transonic core-compressor rotor. The measurements were taken to better understand the unstable flow physics of transonic rotors. The rotor, commonly referred to as Rotor 37, was designed by NASA Lewis to be the first stage of an advanced, eight-stage, core-compressor having a high pressure ratio (about 20:1), good efficiency and sufficient stall margin. The rotor was tested without the presence of a stator (or any of the following seven stages) at the NASA Lewis single-stage, high-speed, core-compressor test-rig. The measurements were obtained with a single circumferential, high-response, total pressure and total temperature probe. The measurements were taken immediately after the machine was ’tripped’ into unstable operation by slowly closing the downstream throttle valve. Measurements were obtained at several different span-wise locations and at two different operating speeds. The rotor was shown to exhibit many of the same characteristics typical of low-speed axial-flow machines. Both rotating stall cells and surge cycles were present during unstable operation. The surge cycles present immediately after the inception of unstable operation involved a large-extent single-cell type rotating stall that was present only during the first half of the surge cycles (the second half of these surge cycles involved operation in the stable operating region). However, as the unstable operation progressed (approximately three to five surge cycles later), surge cycles were present that contained a multiple-cell smaller-extent type rotating stall that existed throughout the entire surge cycle with no partial operation in the stable operating region. Thus, compressor system recovery from single-cell large-extent rotating stall (partial operation in stable operating range during the surge cycle) is more probable than recovery from multiple-cell small-extent rotating stall (no operation in stable operating range during the surge cycle). Rotor wheel speed was shown to be an important variable in influencing the form of unstable operation. Surge and rotating stall were shown to be coupled during the unstable operation. Furthermore, the surge/stall coupling was shown to be related more by pressure interactions than by temperature or efficiency interactions. Also, this high hub-tip ratio transonic rotor was shown to exhibit instantaneous stalling across the entire blade span (typical of low-speed, high hub-tip ratio machines). Attempts to fit the data to Greitzer’s one-dimensional lumped-parameter model are presented and the reasons for poor agreement are discussed. / Master of Science

Various sources of error can cause discrepancies among flight test results, experimental measurements and numerical predictions in the transonic regime. For unsteady flow, the effects of wind tunnel walls or a finite computational domain are the least understood and perhaps the most important. Although various techniques can be used in steady wind tunnel testing to minimize wall reflections, e.g., using slotted walls with ventilation, wind tunnel wall effects remain in unsteady wind tunnel testing even when they have been essentially eliminated from the steady flow. Even when the walls are ten chord lengths or more from the airfoil being tested, they can have a substantial effect on the unsteady aerodynamic response of the airfoil. In this study we compare numerical computations of two- and three-dimensional unsteady transonic flow with one another, and with experimental measurements, to isolate and examine the effects of tunnel walls. An extension of the time-linearized code developed by Fung, Yu and Seebass (1978) is used to obtain numerical results in two dimensions for comparison with one another and with the experimental measurments of Davis and Malcolm (1980). The steady flow which is perturbed by small unsteady airfoil motions is found numerically by specifying the pressure distribution rather than the airfoil coordinates using the procedure provided by Fung and Chung (1982). This provides results that are nearly free from effects caused by the small perturbation approximation; it also simulates the viscous effects present in the experimental measurements. A similar algorithm, developed especially for this study, is used for the related investigations in three dimensions. Different wall conditions are simulated numerically. Aside from a shift of frequency due to nonlinear effects, our numerical predictions of resonance conditions in two dimensions agree very well with those of linear acoustic theory. A substantial discrepancy between unconfined computations and wind tunnel experiments is observed in the low frequency range. This discrepancy highlights the importance of wall interference and wind tunnel measurements of unsteady transonic flows and delineates the conditions required to suppress them satisfactorily.

A numerical scheme has been developed to solve the quasilinear form of the transonic stream function equation. The method is applied to compute steady two-dimensional axisymmetric solar wind-type problems. A single, perfect, non-dissipative, homentropic and polytropic gas-dynamics is assumed. The four equations governing mass and momentum conservation are reduced to a single nonlinear second order partial differential equation for the stream function. Bernoulli's equation is used to obtain a nonlinear algebraic relation for the density in terms of stream function derivatives. The vorticity includes the effects of azimuthal rotation and Bernoulli's function and is determined from quantities specified on boundaries. The approach is efficient. The number of equations and independent variables has been reduced and a rapid relaxation technique developed for the transonic full potential equation is used. Second order accurate central differences are used in elliptic regions. In hyperbolic regions a dissipation term motivated by the rotated differencing scheme of Jameson is added for stability. A successive-line-overrelaxation technique also introduced by Jameson is used to solve the equations. The nonlinear equationfor the density is a double valued function of the stream function derivatives. The velocities are extrapolated from upwind points to determine the proper branch and Newton's method is used to iteratively compute the density. This allows accurate solutions with few grid points. The applications first illustrate solutins to solar wind models. The equations predict that the effects of vorticity must be confined near the surface and far away the streamlines must resemble the spherically symmetric solution. Irrotational and rotational flows show this behavior. The streamlines bend toward the rotation axis for rapidly rotating models because the coriolis force is much larger than the centrifugal force. Models of galactic winds are computed by considering the flow exterior to a surface which surrounds a uniform density oblate spheroid. Irrotational results with uniform outward mass flux show streamlines bent toward the equator and nearly spherical sonic surfaces. Rotating models for which Bernoulli's function is not constant show the sonic surface is deformed consistent with the one-dimensional theory.