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Special Section: Honoring Dr. Leroy H. Smith

A Physical Interpretation of Stagnation Pressure and Enthalpy Changes in Unsteady Flow

[+] Author and Article Information
H. P. Hodson, T. P. Hynes

 Whittle Laboratory, University of Cambridge, 1 JJ Thomson Avenue, Cambridge CB3 0DY, UK

E. M. Greitzer

 Gas Turbine Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139greitzer@mit.edu

C. S. Tan

 Gas Turbine Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139choon@mit.edu

The existence of a stagnation pressure nonuniformity can be inferred directly from Fig. 9. As seen in the stationary system, the velocity of convection of the row of vortices is the fluid velocity on the x-axis, which is uvort . The velocity associated with the row of vortices, adds to this velocity for locations above the row and subtracts from it for locations below. The fluid velocity, and the stagnation pressure, is thus higher above the row than below it.

For a nozzle of length L and a characteristic velocity in the nozzle of u, the fractional change in velocity during the time Δt is proportional to the quantity [Δt Δp/ρuL]. If the latter is small the fractional change in velocity during Δt can be neglected.

They are not force terms, because pressure forces and body forces are represented by spatial gradients.

“Succinctly, invariance under time translation implies energy conservation.” [23] (italics due to the original author).

J. Turbomach 134(6), 060902 (Sep 19, 2012) (8 pages) doi:10.1115/1.4007208 History: Received April 29, 2011; Revised July 29, 2011; Published September 14, 2012; Online September 19, 2012

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 all work transfer between a turbomachine and an inviscid fluid. Examples are first given 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 include absolute stagnation temperature rises in turbine rotor tip leakage flow, wake transport through downstream blade rows, and effects of wake phasing on compressor work input. Fluid dynamic situations are then constructed to explain the effect of unsteadiness, including a physical interpretation of how stagnation pressure variations are created by temporal variations in static pressure; in this it is shown that the unsteady static pressure plays the role of a time-dependent body force potential. It is further shown that 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.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Relative frame view of the static pressure distribution in an axial flow turbine rotor (after Dean [4])

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Figure 2

(a) “Typical” (average) fluid particle motions in absolute and relative (rotor) frames; (b) time rate of change of static pressure and stagnation enthalpy for the typical particle

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Figure 3

Computed casing stagnation temperature in an HP turbine (Thorpe [10])

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Figure 4

Tip leakage flow trajectories in an HP turbine in the relative frame (adapted from Thorpe [10])

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Figure 5

Measured time-mean absolute stagnation pressure [1/4] axial chord downstream of LP turbine blades [11]

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Figure 6

Schematic of wake-blade interactions [11]

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Figure 7

Wake growth in a pressure gradient in steady flow

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Figure 8

Moving row of (point) vortices moving along the x-axis at velocity = uvort . Time-mean velocity seen by stationary observer is u¯+ for y > 0 and u¯- for y < 0; u¯+ > u¯-.

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Figure 9

Vorticity contours in a two-dimensional IGV-rotor configuration at two different interblade-row spacings. The pink line shows the path of the counter-rotating vortices in the rotor frame [19].

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Figure 10

Unsteady vortex flow field and time-mean representation. The pink line shows the path of the counterclockwise rotating vortices in the rotor frame. Label “1” denotes fluid with high relative stagnation pressure, while label “2” denotes fluid with low relative stagnation pressure.

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Figure 11

Flow of an incompressible liquid out of a reservoir subjected to a time-dependent reservoir pressure

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Figure 12

Energy input to a fluid particle along a path line between times t and t + dt.

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