Njuki W. Mureithi

On the bifurcations of a rigid rotor response in squeeze-film dampers

Abstract The effectiveness of squeeze-film dampers in controlling vibrations in rotating machinery may be limited by the nonlinear interactions between large rotor imbalance forces with fluid-film forces induced by dampers operating in cavitated conditions. From a practical point of view, the occurrence of nonsynchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The bifurcations in the response of a rigid rotor supported by cavitated squeeze-film dampers resulting from such interactions are presented in this paper. The effects of design and operating parameters, namely the bearing parameter (B), gravity parameter (W), spring parameter (S) and unbalance parameter (U), on the bifurcations of the rotor response are investigated. Spring parameter (S) values between 0 and 1 are considered. A spring parameter value of S=0 represents the special case of dampers without centering springs. With the exception of the case S=1, jump phenomena appeared to be a common bifurcation that occurred at certain combinations of B, W and U irrespective of the value of S. Period-doubling and secondary Hopf bifurcations which occurred for low values of S (⩽0.3) were not observed for the higher values S⩾0.5. For very low stiffness values (S

Blade Tip Clearance Flow and Compressor Nonsynchronous Vibrations: The Jet Core Feedback Theory as the Coupling Mechanism

This paper investigates the role of tip clearance flow in the occurrence of nonsynchronous vibrations (NSVs) observed in the first axial rotor of a high-speed high-pressure compressor in an aeroengine. NSV is an aeroelastic phenomenon where the rotor blades vibrate at nonintegral multiples of the shaft rotational frequencies in operating regimes where classical flutter is not known to occur. A physical mechanism to explain the NSV phenomenon is proposed based on the blade tip trailing edge impinging jetlike flow, and a novel theory based on the acoustic feedback in the jet potential core. The theory suggests that the critical jet velocity, which brings a jet impinging on a rigid structure to resonance, is reduced to the velocities observed in the blade tip secondary flow when the jet impinges on a flexible structure. The feedback mechanism is then an acoustic wave traveling backward in the jet potential core, and this is experimentally demonstrated. A model is proposed to predict the critical tip speed at which NSV can occur. The model also addresses several unexplained phenomena, or missing links, which are essential to connect tip clearance flow unsteadiness to NSV. These are the pressure level, the pitch-based reduced frequency, and the observed step changes in blade vibration and mode shape. The model is verified using two different rotors that exhibited NSV.

Stability and bifurcation of a rigid rotor in cavitated squeeze-film dampers without centering springs

Abstract The continuation method is employed to investigate the non-linear response of a rigid rotor supported by squeeze-film dampers without centering springs. The Floquet transition matrix method, which was used to examine the stability of the periodic response, indicated that the rotor may lose stability via period-doubling and saddle-node bifurcations at certain parameter values. The continuation method, which allows for switching from one solution branch to another, enabled the solution branches of period-two, four, eight and sixteen to be mapped out. By simultaneously varying the unbalance and bearing parameters, the combination of these two parameters that avoids the occurrence of non-synchronous vibration or jump phenomena in this system was determined.

Vibration excitation force measurements in a rotated triangular tube bundle subjected to two-phase cross flow

Two-phase cross flow exists in many shell-and-tube heat exchangers. Flow-induced vibration excitation forces can cause tube motion that will result in long-term fretting-wear or fatigue. Detailed vibration excitation force measurements in tube bundles subjected to two-phase cross flow are required to understand the underlying vibration excitation mechanisms. An experimental program was undertaken with a rotated-triangular array of cylinders subjected to air/water flow to simulate two-phase mixtures over a broad range of void fraction and mass fluxes. Both the dynamic lift and drag forces were measured with strain gage instrumented cylinders. The experiments revealed somewhat unexpected but significant quasi-periodic forces in both the drag and lift directions. The periodic forces appeared well correlated along the cylinder with the drag force somewhat better correlated than the lift forces. The periodic forces are also dependent on the position of the cylinder within the bundle.

Fluidelastic Instability of an Array of Tubes Preferentially Flexible in the Flow Direction Subjected to Two-Phase Cross Flow

Almost all the available data about fluidelastic instability of heat exchanger tube bundles concerns tubes that are axisymmetrically flexible. In those cases, the instability is found to be mostly in the direction transverse to the flow. Thus, the direction parallel to the flow has raised less concern in terms of bundle stability. However, the flat bar supports used in steam generator for preventing U-tube vibration may not be as effective in the in-plane direction than in the out-of-plane direction. The possibility that fluidelastic instability can develop in the flow direction must then be assessed. In the present work, tests were done to study the fluidelastic instability of a cluster of seven tubes much more flexible in the flow direction than in the lift direction. The array configuration is rotated triangular with a pitch to diameter ratio of 1.5. The array was subjected to two-phase (air-water) cross flow. Fluidelastic instability was observed when the flexible tubes were located at the center of the test section and also when the seven flexible tubes were placed over two adjacent columns. No instability was found for a single flexible tube in a rigid array nor for the case where the seven flexible tubes were placed in a single column. Tests were also done with tubes that are axisymmetrically flexible for comparison purposes. It was found that fluidelastic instability occurs at higher velocities when the tubes are flexible only in the flow direction. These results and additional wind tunnel results are compared to existing data on fluidelastic instability. Two-phase flow damping results are also presented in this paper.

Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Abstract Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.

The Tip Clearance Flow Resonance Behind Axial Compressor Nonsynchronous Vibration

Nonsynchronous vibration (NSV) is a particular type of aero-elastic phenomenon, where the rotor blades vibrate at nonintegral multiples of the shaft rotational frequencies. NSV behavior appears similar to off-design stall flutter but with a particular blade tip flow evolution. This paper demonstrates the link between NSV and the resonance induced by the tip clearance flow based on a proposed hypothesis and experimental confirmation. At off-design operating conditions, the rotor blade tip clearance shear layer flow can evolve tangentially. It is proposed that this tangential flow becomes a support for an acoustic feedback wave that settles between rotor blades. The feedback wave is driven by the blade vibratory motion and synchronizes the shear layer vortical structures with the blade vibration frequency. Depending on the blade tip local temperature, and when the feedback wavelength matches within one or two blade pitches, the system becomes resonant and very high vibrations can occur on the blade. An axial stage compressor test rig is set-up to look into the underlying mechanism behind NSV through targeted measurements using both static and rotating instrumentation. The experimental apparatus consists of the first stage of a high pressure compressor driven by an electric motor. The test-section is built to minimize the effects of the adjacent stator blade rows in order to isolate the role of rotor blade tip clearance flow on NSV. Sensitivity studies are carried out to assess and demonstrate the effects of the rotor blade tip clearance and inlet temperature on NSV and validate the predicted resonance for NSV occurrence under various conditions. Vibrations and surface pressure data from adjacent blades are collected to demonstrate the predicted interactions between neighboring rotor blades. Finally, evidence of the staging phenomenon, inherent to the proposed NSV mechanism, is experimentally obtained. All the data obtained are consistent with and thus in support of the proposed mechanism for NSV.

Correlation Between Vibration Excitation Forces and the Dynamic Characteristics of Two-Phase Cross Flow in a Rotated-Triangular Tube Bundle

Two-phase cross flow exists in many shell-and-tube heat exchangers. Flow-induced vibration excitation forces can cause tube motion that will result in long-term fretting wear or fatigue. Detailed flow and vibration excitation force measurements in tube bundles subjected to two-phase cross flow are required to understand the underlying vibration excitation mechanisms. Some of this work has already been done. The distributions of both void fraction and bubble velocity in rotated-triangular tube bundles were obtained. Somewhat unexpected but significant quasiperiodic forces in both the drag and lift directions were measured. The present work aims at understanding the nature of such unexpected drag and lift quasiperiodic forces. An experimental program was undertaken with a rotated-triangular array of cylinders subjected to air/water flow to simulate two-phase mixtures. Fiber-optic probes were developed to measure local void fraction. Both the dynamic lift and drag forces were measured with a strain gage instrumented cylinder. The investigation showed that the quasiperiodic drag and lift forces are generated by different mechanisms that have not been previously observed. The quasiperiodic drag forces appear related to the momentum flux fluctuations in the main flow path between the cylinders. The quasiperiodic lift forces, on the other hand, are mostly correlated to oscillations in the wake of the cylinders. The quasiperiodic lift forces are related to local void fraction measurements in the unsteady wake area between upstream and downstream cylinders. The quasiperiodic drag forces correlate well with similar measurements in the main flow stream between cylinders.

Quasi-Static Forces and Stability Analysis in a Triangular Tube Bundle Subjected to Two-Phase Cross-Flow

Although almost half of the process heat exchangers operate in two-phase flow, the complex nature of the flow makes the prediction of fluidelastic instability a challenging problem yet to be solved. In the work reported here, the quasi-static fluid force-field is measured in a rotated-triangle tube bundle for a series of void fractions and flow velocities. The forces are strongly dependent on void fraction, flow rates and relative tube positions. The fluid force field is employed along with quasi-steady models [1, 2], originally developed for single phase flows, to model the two-phase flow problem. Stability analysis is performed using the single flexible tube model [1] as well as constrained mode analysis [2]. The results are compared with dynamic stability tests [3] and show good agreement. The results of single flexible tube analysis and multiple flexible tubes tend to coincide at low structural damping as expected. The present work uncovers some of the complexities of the fluid force field in two-phase flows. The data are valuable since they are the necessary inputs to the class of quasi-static, quasi-steady and quasi-unsteady fluidelastic instability theoretical models. This database opens a new research avenue on the feasibility of applying quasi-steady models to two-phase flow.Copyright

CYLINDER WAKE DYNAMICS IN THE PRESENCE OF STREAM-WISE HARMONIC FORCING

The vortex wake flow dynamics downstream of a cylinder undergoing streamwise harmonic (fe /fs =1) forced oscillations has been investigated numerically using a CFD code for Re=1000. The steady-state of the wake flow has been analysed considering the amplitude of oscillations as a perturbation parameter. The resulting dynamics of the fluid lift and drag forces acting on the cylinder have been linked to the different vortex wake modes observed downstream of the cylinder. Forced oscillations lead to periodic, quasi-periodic and chaotic responses depending on the amplitude of oscillation of the cylinder. Different vortex wake patterns or modes (including 2S, P+S and S modes) have also been identified and described. Symmetry related bifurcations both in the computed fluid force dynamics as well as in the vortex wake patterns were identified. The key role played by spatio-temporal symmetry in the interaction between the wake flow and the oscillating cylinder has been elucidated by a Proper Orthogonal Decomposition (POD) of the wake velocity field. Symmetric and antisymmetric spatio-temporal modes were identified and bifurcations in the wake flow were explained in terms of mode interactions in the wake.Copyright