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The core axis velocity, denoted by the subscript s, is the velocity on the vortex centerline. The tip vortex core axis is not aligned with the axis of the turbomachine, and “core axis velocity” is used here to distinguish the velocity in the direction of the vortex core axis from the “axial velocity” or velocity along the turbomachine axis of rotation.

The region of reversed flow occurs behind the trailing edge, in x/c coordinates, in Figs. 1(b) and 2. In Fig. 2 even though the intersections of the last two planes with the blade are at values of x/c ≤ 1, the vortex center on these planes is actually at a value of x/c larger than unity.

The equations that describe changes in vorticity do not have a term involving pressure, and it is not consistent to use features of the pressure field in motivating the vorticity behavior [11]. In a turbomachinery design context, however, much of the thinking about blade performance is in terms of static pressure. It, thus, seems simpler to use the implied connection (through Bernoulli) of pressure and velocity and refer to changes in pressure, rather than to keep having to put the point in terms of velocity variations and then make the connection to the pressure distribution.

A Rankine vortex has a core of uniform vorticity (uθ = Ωr, where Ω is independent of radius) surrounded by irrotational flow. The velocity in the core aligned with the core axis direction us is uniform but may be different from the velocity along this direction in the surrounding irrotational flow Us.

A Burger vortex has swirl velocity uθ = $Γ/2πre{1-exp[1.26(r/re)2]}$. The velocity in the core axis direction is us for r < re and Us for r > re.

This can be regarded as a “loss derivative,” Δ(Normalized mixed-out loss)/ΔCp, as the small change in normalized pressure ΔCp approaches zero.

The volumetric viscous entropy generation is given by $ρs·=Φ/T$, where Φ is the dissipation function [19] and is nondimensionalized as $TexitρVs·/m·mainΔht$, where V is the domain volume.

e-mail: arthuang@mit.edu

e-mail: greitzer@mit.edu

e-mail: choon@mit.eduGas Turbine Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139

e-mail: edward.r.turner@rolls-royce.comTurbine Aerodynamics, Rolls-Royce Corporation, Indianapolis, IN 46241

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the Journal of Turbomachinery. Manuscript received June 28, 2012; final manuscript received September 4, 2012; published online June 26, 2013. Editor: David Wisler.

J. Turbomach 135(5), 051012 (Jun 26, 2013) (11 pages) Paper No: TURBO-12-1092; doi: 10.1115/1.4007832 History: Received June 28, 2012; Revised September 04, 2012

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## References

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## Figures

Fig. 1

(a) Suction surface isentropic Mach number at 80% span as a function of axial position. (b) Vortex core swirl number (Γ/2πreus) and vortex centerline velocity normalized by exit velocity (us,centerline/Uexit) as a function of axial position.

Fig. 2

Color contours of plane normal velocity/turbine exit velocity at different planes normal to the blade suction surface. The regions in which flow is reversed (negative core axis velocity) are indicated in white.

Fig. 3

Geometry of rectangular nozzle with area ratio 0.8. Color contours of static pressure coefficient Cp = (p-pexit)/12ρUinlet2.

Fig. 4

(a) Normalized mixed-out loss at duct exit, TS·/0.5 m·injUexit2, as a function of area ratio. Numerical simulation (solid line) and control volume approach (dotted line). (b) Vortex swirl number at x/H = 2 for ducts with AR = 0.5 to 1.7 as a function of area ratio. Dotted line shows approximate vortex breakdown limit [14,15-14,15].

Fig. 5

Velocity defect, swirl velocity, and mixed-out loss (normalized by their inlet values) as a function of farfield pressure rise Cp,∞ for inlet wake of velocity ratio 0.6 (black), and vortex of core axis velocity ratio us/Us = 0.6 and swirl number Γ/2πreus = 0.83 (gray)

Fig. 6

Flow through expanding axisymmetric ducts. Δ (mixed-out loss)/(inlet mixed-out loss) for a wake and two vortices of differing swirl numbers (0.42 and 0.83) as a function of pressure rise on the outer wall ΔCp=Δp/12ρUs2. The initial core velocity ratio in each case is 0.6. Velocity contours for cases A–E are shown in Fig. 7.

Fig. 7

(a) Contours of uθ/Us,inlet for cases A, B, and C (see Fig. 6; ΔCp = 0.05,0.19,0.31, VR = 0.6, S = 0.83). (b) Contours of us/Us,inlet for cases C, D, and E (ΔCp = 0.31,0.37,0.42, VR = 0.6, S = 0.83). The regions in which flow is reversed (negative core axis velocity) are indicated in white.

Fig. 8

Derivative of normalized mixed-out loss TS·/0.5 m·core Us2, with respect to Cp,∞ for vortices with specified initial swirl number and velocity ratio. Points represent low pressure turbine (⊙, computed), high pressure turbine (+, computed), and compressor experiments (*, [18]).

Fig. 9

(a) Redesigned turbine tips (A, F) and baseline (B). (b) Isentropic suction surface Mach number at 80% span for A, B, F blades; clearance equal to 0.5% blade height.

Fig. 10

(a) Computed clearance loss coefficient (TΔsleak/Δht) as a function of clearance for A, B, F blades. (b) Leakage mass fraction (m·leak/m·main) as a function of clearance for A, B, F blades.

Fig. 11

Loss per unit leakage flow for A, B, F blades, from numerical simulations and from control volume estimate for mixing loss, normalized with respect to computed baseline loss per unit leakage flow; clearance equal to 2% blade height

Fig. 12

(a) Vortex centerline velocity normalized by exit velocity as a function of axial position for A, B, F blades; clearance equal to 2% of blade height. (b) Tip leakage mixing loss per unit axial distance (/d(x/c)) as a function of axial position for A, B, F blades; clearance equal to 2% of blade height.

Fig. 13

Color contours of plane normal velocity/turbine exit velocity at several crossflow planes for A, F blades. Reversed flow regions are indicated in white.

Fig. 14

Color contours of volumetric viscous dissipation, normalized by domain volume, turbine enthalpy rise, passage massflow, and exit temperature, at x/c = 1.15 (location of maximum loss per unit leakage flow) for A, F blades

Fig. 15

(a) Isentropic suction surface Mach number at 80% span for A2, B2, F2 blades. (b) Loss per unit leakage flow for A2, B2, F2 blades, from numerical simulations and from control volume estimate for mixing loss, normalized with respect to computed baseline loss per unit leakage flow; clearance equal to 2% blade height.

Fig. 16

(a) Isentropic suction surface Mach number at 80% span for B2 at incidences of nominal, +10, and −10 deg. (b) Loss per unit leakage flow as a function of incidence for B2 blade, from numerical simulations and from control volume estimate for mixing loss, normalized with respect to computed baseline loss per unit leakage flow; clearance equal to 2% blade height.

Fig. 17

(a) Isentropic suction surface Mach number at 80% span for blades with solidity 0.88, 0.97, and 1.07. (b) Loss per unit leakage flow as a function of solidity, from numerical simulations and from control volume estimate for mixing loss, normalized with respect to computed baseline loss per unit leakage flow; clearance equal to 2% blade height.

Fig. 18

(a) Vortex core centerline velocity/cascade exit velocity as a function of axial position for blades with solidity 0.88, 0.97, and 1.07. (b) Normalized tip leakage loss per unit axial distance as a function of axial position.

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