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A Physical Interpretation H. P. Hodson of Stagnation Pressure T. P. Hynes and Enthalpy Changes Whittle Laboratory, University of Cambridge, in Unsteady Flow 1 JJ Thomson Avenue, Cambridge CB3 0DY, UK This paper provides a physical interpretation of the mechanism of stagnation enthalpy and stagnation pressure changes in turbomachines due to unsteady flow, the agency for E. M. Greitzer all work transfer between a turbomachine and an inviscid fluid. Examples are first given e-mail: [email protected] to illustrate the direct link between the time variation of static pressure seen by a given fluid particle and the rate of change of stagnation enthalpy for that particle. These C. S. Tan include absolute stagnation temperature rises in turbine rotor tip leakage flow, wake transport through downstream blade rows, and effects of wake phasing on compressor e-mail: [email protected] work input. Fluid dynamic situations are then constructed to explain the effect of unstead- iness, including a physical interpretation of how stagnation pressure variations are cre- Gas Turbine Laboratory, ated by temporal variations in static pressure; in this it is shown that the unsteady static Massachusetts Institute of Technology, pressure plays the role of a time-dependent body force potential. It is further shown that Cambridge, MA 02139 when the unsteadiness is due to a spatial nonuniformity translating at constant speed, as in a turbomachine, the unsteady pressure variation can be viewed as a local power input per unit mass from this body force to the fluid particle instantaneously at that point. [DOI: 10.1115/1.4007208] 1 Introduction and Scope of the Paper tion exists in the y-direction. Because the blade moves at a speed Xr, a temporal variation in static pressure is seen in the absolute, Unsteady flows are fundamentally different than steady flows. or stationary, frame of reference, As stated succinctly by Kerrebrock [1]: “…in an unsteady flow there is a mechanism for moving energy around in the gas which 1 @p 1 @p is quite distinct and qualitatively different than steady flow. In ¼Xr (3) steady flow, by and large, energy is carried along stream tubes.… q @t abs q @y rel This is not the case in unsteady flows….” This paper describes concepts associated with, and applications of, the mechanisms by In the stationary frame, for fluid particles passing through the which unsteady flows create this energy exchange. rotor blade row, @p/@t < 0, and, from Eq. (1), the stagnation Many turbomachinery texts include a demonstration that, for an enthalpy of a fluid particle falls. This is (as it must be) consistent ideal fluid, the flow must be unsteady for a turbine to produce with the energy extraction calculated from combining the expres- shaft work or for a compressor to absorb shaft work [2–4]. The sion for conservation of angular momentum and the steady flow demonstration has two conceptual parts. The first is the relation energy equation, both applied in the stationary frame. between changes in the stagnation enthalpy of a fluid particle and the local time variation of static pressure, Dh 1 @p t ¼ (1) Dt q @t or the incompressible form [5–8] Dp @p t ¼ (2) Dt @t The second is the illustration that fluid particles passing through a moving turbine or compressor blade row see a nonzero value of @p/@t. The connection between unsteady flow and work exchange was made explicit by Dean [5], using arguments similar to those sketched in Fig. 1, which gives a representation of the static pres- sure in an axial flow turbine rotor passage. For a flow that is steady in the relative (blade fixed) frame of reference the pressure falls from pressure side to suction side and a static pressure varia- Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received April 29, 2011; final manuscript received July 29, 2011; published online September 14, 2012. Fig. 1 Relative frame view of the static pressure distribution in Editor: David Wisler. an axial flow turbine rotor (after Dean [4]) Journal of Turbomachinery Copyright VC 2012 by ASME NOVEMBER 2012, Vol. 134 / 060902-1 Downloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 03/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use Turbomachinery aerodynamicists have not generally approached in accord with the unsteady pressure field associated with the blade design from the perspective of unsteady flow. The methodol- blade forces. Within the passage (along the dashed line in Fig. 1, ogy has relied on transforming the entering flow to a frame of for example) blade forces generally point from pressure to suction reference in which the flow is (assumed) steady, stagnation quanti- surface in accord with the decrease in stagnation pressure. ties are conserved, and analysis is on more familiar ground, with On another level, however, the pressure upstream of the blade subsequent transformation at the rotor exit, allowing a return to the row varies about the average in the pitchwise direction so @p/@t stationary system. This approach has led to a long history of has both negative and positive values. Some particles thus have successful and sophisticated turbomachines, but two drivers instantaneous increases in stagnation enthalpy, and their change increasingly push towards explicit inclusion of unsteady flows in along a path line is not monotonic. The association of increases in the design process. One is the availability of computations that stagnation enthalpy with the predominant direction of the blade resolve unsteady features. A second, and more important, trend is forces is less evident for these particles, as is the concept of blade that current machines have high levels of efficiency and there is forces in the upstream and downstream regions. The route to pro- incentive to grapple with unsteady flow issues as a route to possible viding an appropriate interpretation of @p/@t for these regions, as performance increases. well as more generally throughout unsteady turbomachinery flow, The arguments concerning unsteady flow that were presented is therefore not through direction considerations of blade forces, above are well recognized, but directly related aspects appear to as is often implied with reference to Eq. (1). In this context the be (at least from our observations at IGTI meetings) much less aim of the paper is to provide two items: (i) clear illustrations of appreciated. First is that thinking in terms of Eqs. (1) or (2) pro- the effects of flow unsteadiness on time-mean turbomachinery vides insight into a number of manifestations of unsteady flow in performance, and (ii) a description of the physical mechanism that turbomachines; for example the capability to define how unsteady underpins this alteration. effects scale with different parameters. Second is that the concepts presented provide a route to enhanced interpretation, and thus increased capability to make use of, computational simulations of 2 Stagnation Enthalpy and Stagnation Pressure unsteady flow. Third is the lack of a physical explanation for why Changes Due to Unsteady Flow the stagnation pressure changes if the flow is unsteady. While one In the next sections we examine four examples of turbomachi- can simply state that the equations lead to this consequence, the nery flow to illustrate the effects of unsteadiness in determining many discussions on the topic the authors have had with others the time mean flow: (i) work input in tip clearance flow, (ii) wake (and between themselves) strongly suggest that such explanations interactions with blade surfaces, (iii) the effect on losses of would be useful to workers in the field. wake behavior in downstream blade rows, and (iv) the effect of An initial document in which the two threads—turbomachinery wake phasing on compressor work input. The phenomena are design and unsteady flow phenomena—were brought together in a different, but it will be seen that the underlying ideas about clear and explanatory manner is a note by Roy Smith [9] on wake stagnation enthalpy change provide a powerful framework for attenuation (see Sec. 2.3). With this as context, and perhaps as understanding and estimation of the physical mechanisms and model, for the present discussion it is a pleasure to have the paper magnitudes of the effects. appear in a session dedicated to Roy and the insight he In the discussions below, we emphasize that the features has brought to many different aspects of turbomachinery fluid encountered can be described in terms of inviscid fluid mechanics, dynamics. although they are modified in practice (mainly damped) by vis- To frame the issues, we illustrate some additional implications of cous effects and heat transfer. Further, the essential behavior can Eq. (1) in the turbomachinery example of Fig. 1. Figure 2(a) gives a be seen from the incompressible, constant density, inviscid flow path line, as seen in the stationary system, for a “typical” fluid parti- arguments that are presented. Viscous stresses, heat transfer, and cle (i.e., defined by a velocity and pressure averaged across the pas- compressibility change the quantitative magnitudes, but not the sage) in the turbine. If the flow in the relative system is taken as central physical features. steady, relative system streamlines and path lines coincide, with the average streamline closely following the blade passage. Path lines (particle trajectories) in the stationary system, however, can be 2.1 Turbine Casing Stagnation Temperature Variation. almost normal to the blades. Figure 2(b) shows the link between the The first example is given in Fig. 3, which shows contours of time derivative of the static pressure, and the variation in stagnation enthalpy, respectively, for the typical particle. On one level, the behavior of the typical particle gives a useful picture of turbomachinery stagnation enthalpy and pressure rises; these quantities decrease in a turbine and increase in a compressor Fig.

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