the geometries involved, either of these may be the primary driver in a given application. When the gap is large and the blade loading is high, the tip-gap flows are generally dominated by pressure effects, while very narrow gaps or low blade loading lead to gap flows that are more dependent on the relative movement between the blade tip and the casing. In general, however, inertial forces in this narrow gap region are sufficient to generate a tip-leakage vortex (TLV) on the suction side of the blade that not only produces substantial losses, but may also initiate blade surface cavitation . Over the life span of a turbomachine, the machine efficiency and operating margin deteriorate due to the increase in tip gap spacing.
The engineering importance of tip gap flows, and the complexity of the physics they contain, has stimulated numerous previous studies of these phenomena. Recent studies of tip-leakage flow have involved various turbomachinery configurations, including linear cascades, compressors, turbines and pumps. Experimental studies of tip-gap flows in cascades have been reported by Kang and Hirsch [2,3], while Lakshminarayana et al. , Inoue et al. [5,6], Suder and Celestina , and Foley and Ivey , have conducted measurements in compressors. Tip-gap flows in turbines have been studied experimentally by Chan et al. , Yamamoto et al. , and DeCecco et al.  while pump flow experiments have been reported by Graham  and Zierke et al. [13, 14].
In addition to these experimental investigations, many analytical studies and numerical simulations of tip-gap flows have also been reported. Because our emphasis in the present paper is computational in nature we emphasize the latter here. Previous computational studies have included the simulations reported by Hah , Dawes , Crook , Adamczyk et al. , Copenhaver et al. , and Kang and Hirsch . Of the numerous analytical studies, we here cite only the work of Chen et al.  who have developed a similarity scaling for the crossflow in the clearance region and a generalized description of the tip-leakage vortex based on inviscid-flow modeling. A comparison of our tip-leakage vortex trajectories with the model of Chen et al.  is presented later.
Although the experimental investigations and numerical simulations cited above have revealed many features of tip-leakage flows, many unsettled issues still remain. In general, the results to date have shown that several characteristic features tend to dominate the tip gap region in most cases. These include: (i) the formation, transport and trajectory of the tip-leakage vortex; (ii) the interaction of the tip-leakage flow with vortices generated by other mechanisms; and (iii) the size of the tip clearance and the blade loading. Despite these similarities, the detailed characteristics of the tip-leakage flow vary from one type of machine to another. For example, the relative motion between the blade tip and the shroud acts in an opposite direction in compressors and turbines. In compressor rotors, the relative movement between the blade tip and the casing or shroud tends to enhance tip leakage by dragging fluid through the gap region in the same direction as the pressure drop. By contrast, the relative motion in turbine rotors is in the opposite direction and tends to negate the effects of pressure drop. In fact, observations in turbines have been reported in which the tip gaps were so small that the leakage flow was reduced to zero by the relative movement .
The present study investigates the physical processes involved in the generation and transport of the tip-leakage vortex by means of numerical solutions of the three-dimensional Navier-Stokes equations. Results are presented for both a stationary cascade and a high Reynolds number axial rotor. The cascade geometry and operating conditions are chosen to match the low-speed compressor cascade experiments of Kang and Hirsch [2,3] who have provided a detailed map of the passage and tip-leakage flow at three different tip clearance conditions. The axial-flow rotor computations are chosen to match the conditions tested by Zierke et al. [13,14] who have furnished a reasonably complete outline of the wake of a pump rotor in the rotating frame including information on the transport of the tip-leakage vortex. The passage flow for this rotor has been investigated numerically by [22, 23, 24]. Dreyer and Zierke  quantitatively compared their predicted tip flow with the measured data at the design tip gap size. The cascade experiments were obtained with a stationary wall so that the gap flows are driven by pressure drop alone, while the pump rotor experiment involves the effects (in the relative frame) of a moving shroud. Thus, these two cases not only provide us with detailed experimental data against which we can check our computations, but they also represent an interesting comparison of the effects of the moving wall. Our emphasis therefore is on comparing the details of the tip gap flow in a cascade with a stationary wall with those in a rotating pump stage where there is relative motion between the blade and the casing.