22nd TURBOMACHINERY WORKSHOP 2008
SECONDARY FLOWS IN TURBINE BLADING SYSTEMS – THEORY AND COMPUTATION Piotr Lampart IMP PAN, Gdańsk
17-19.09.2008 Gliwice, Poland
Secondary flows in pipes and channels
pressure-driven
ρv 2 ∂p = R ∂n
secondary flows
∂p/∂n=const, v 0 R 0
stress-driven secondary flows
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Secondary flows in turbine cascades
Formation of horse-shoe vortex, Marchal & Sieverding (1984)
A - model of Hawthorne (1955), B – model of Langstona (1980), C – model of Sharma & Butler (1987), D – model of Goldstein & Spores (1988).
Secondary flows modify boundary layers at the endwalls
Secondary flows in cascades with a tip clearance
Endwall boundary layer, Harrison [17]
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SECONDARY FLOWS IN OTHER FIELDS
Tip vortex behind aircraft TORNADO
Tip vortex behind marine propeller
River bend flow
Endwall / secondary flow losses
Formation of the inlet boundary layer upstream of the blade leading edge;
Formation of the boundary layer downstream of the horse-shoe vortex lift-off lines;
Shear effects along the horse-shoe vortex lift-off lines, separation lines, between the secondary vortices, main flow and blade surfaces, especially at the suction surface;
Dissipation of the passage vortex, trailing shed vortex, corner vortices and other vortex flows in the process of their mixing with the main flow;
Exit non-uniformities may lead to local separations and upstream relocation of the laminarturbulent transition at the downstream blade in the secondary flow dominated region.
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Evolution of vorticity from the endwall boundary layer Lakshminarayana, Horlock
(u ⋅ ∇ )ω = (ω ⋅ ∇ )u − ω(∇ ⋅ u) − ∇ × ∇p + ∇ × ∇ ⋅ τ ρ ρ
u - velocity, ω - vorticity, p – pressure, ρ - density, τ - viscous stress tensor
ω=sωs + nωn + bω
ρq
∂ ωs 2ωn 1 ∂p ∂ρ ∂ρ ∂p = − 2 − ∂s ρq R qρ ∂n ∂b ∂n ∂b
ω ω ∂a 1 ∂ (ωn q ) = b − n b + 12 ab ∂s qρ q ∂s τ
+ viscous terms
∂p ∂ρ ∂ρ ∂p ∂s ∂b − ∂s ∂b
+ viscous terms
inviscid incompressible flow
ρq
∂ ω s 2ω n = ∂s ρq R
ωs − ωs 0 = 2ωn0 (α − α0 )
inviscid perfect gas flow
ρq
∂ ωs 2 ∂p* = ∂s ρq Rρq ∂b
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ωs ωs 2 ∇p* cos φds ρq − ρq = ∫ 2 ρ 2 1 1 Rρq
Calculation of secondary flow losses
∆ψ = −ωs1 ∂ψ ∂r
vn1 = ,.
vr1 = −
(
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∂ψ ∂n
)
hp ∫ ∫ ρ1v1 v r21 + v n21 cosα1d xdy ξ sec =
00 11
hp ∫ ∫ ρ1v13 cosα1d xdy 00
(it is assumed that the secondary kinetic energy of the relative motion in the exit section is lost during mixing)
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EFFECTS OF SECONDARY FLOWS (A) INCREASED CASCADE LOSSES, (B) REDISTRIBUTION OF EXIT VELOCITY, (C) INCREASED NON-UNIFORMITY AT EXIT, (D) REDISTRIBUTION OF STEADY AND UNSTEADY LOADS.
Durham cascade –distribution of loss coefficient and exit swirl angle at slot 10; experimental, computed by FlowER with Menter k-ω SST model and computed by Fluent with RSM – LRR model.
The effect of blade height
Entropy function contours in a rotor cascade h=20mm, 60mm i 100mm
Span-wise distribution of enthalpy losses and exit angle in a rotor cascade; 1 – h=20mm; 2 – h=60mm, 3 – h=100mm
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Profile type
Pressure distribution at the stator (left) and rotor (right) profile; blade height h=60mm Velocity vectors at the mid-span section (left) and at the root (right) in the stator (left) and rotor (right) cascades; blade height h=60mm.
Entropy function contours in the stator and rotor at the trailing edge; h=60mm
Flow turning in the cascade
Static pressure contours at the mid-span of the rotor cascade for three inlet angles
Velocity vectors at the endwalls of the rotor cascade for three inlet angles
Secondary flow vectors at the trailing edge and total pressure contours 15% axial chord downstream of the trailing edge in the rotor cascade for three inlet angles
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Flow turning in the cascade with tip clearance
Secondary flow vectors and total pressure contours in the HP rotor cascade in selected sections located 15% axial chord upstream of the trailing edge, at the trailing edge and 15% axial chord downstream of it; tip gap size – 2%, Ma=0.2, α0 = 63o, α1 = -63o.
Secondary flow vectors and total pressure contours in the HP rotor cascade in selected sections located 60% and 15% axial chord upstream of the trailing edge and at the trailing edge; tip gap size – 2%, Ma=0.4, α0 = 75o, α1 = -72o.
HIGHLY LOADED CASCADES
UHL cascade of Yamamoto Profiles used in gas turbines for a low weight-to-power ratio
Total pressure contours in normal sections Velocity vectors in sections from hub to tip
Enthalpy losses
Total pressure contours and secondary flow vectors in exit section – experiment, Yamamoto [4]
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HL cascade of NRC
Enthalpy losses
Total pressure contours in normal sections
Total pressure contours and secondary flow vectors at the exit – experiment, Moustapha et al. [4]
The case of non-nominal inflow onto the suction side of the blade
Static pressure contours and velocity vectors at the endwall of the rotor cascade for the case of non-nominal inflow onto the suction side of the blade, α0 = 0o.
Loss contours and distribution
Secondary flow vectors 85%, 55% and 5% axial chord upstream of the trailing edge of the rotor cascade for the case of non-nominal inflow onto the suction side of the blade for α0 = 0o and 30o;
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The effect of span-wise distribution of static pressure and cascade load
Straight and compound leaned stator blade
Spanwise distribution of static pressure, relative velocity and swirl angle in the stator and rotor 5% axial chord upstream of the trailing edge; stage with straight stator blades (1), stage with compound leaned stator blades (2)
Redistribution of loss in the stator and rotor; straight blades (left, 1), compound leaned blades (right, 2)
Velocity vectors at the rotor suction surface; stage with straight stator blades (left), stage with compound leaned stator blades (right)
The effect of leakage flow (the case of shrouded blades)
R1
Computational domain with source/sink-type boundaries
Velocity vectors in the rotor (upper part) at the suction surface and entropy function contours at the rotor trailing edge - computed without sources and sinks (left), computed with tip leakage (right)
S2
Static pressure contours and velocity vectors at the suction surface of the second stator blade
S2
Secondary flow vectors 35% and 75% axial chord downstream of the leading edge (left) and entropy function contours 15% axial chord downstream of the trailing edge in the second stator (right).
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S2 Total pressure contours in the second stator in subsequent sections located 90%, 75% and 25% axial chord upstream of the trailing edge and 15% axial chord downstream of the trailing edge
S2 Entropy function contours in the second stator in subsequent sections located 75%, 50% and 25% axial chord upstream of the trailing edge and 15% axial chord downstream of the trailing edge (L). Also entropy function contours behind the second stator computed without leakage (NL)
Redistribution of secondary flows due to unsteady effects unsteady effects
=
upstream interaction of the moving blade row downstream transport of 2D and 3D wakes
Aachen turbine S1/R1/S2 R1
Instantaneous entropy function contours in the rotor at the mid-span in unsteady flow
R1
Instantaneous total pressure contours at the rotor trailing edge in unsteady flow
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S2
Instantaneous secondary flow vectors in the second stator 40% axial chord downstream of the leading edge in unsteady flow
S2
Instantaneous total pressure and entropy function contours at the second stator trailing edge in unsteady flow
The effect of thickness and skewness of the inlet boundary layer
Skewed inlet boundary layer
Durham cascade: Total pressure contours for different skew configurations of the inlet boundary layer; experiment - Walsh & Gregory-Smith [16]
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Conclusions Losses in the endwall boundary layers can be found from an analytical expression. Secondary flow losses need to be evaluated numerically. RANS calculations typically overpredict losses in the secondary flow region. The predictions are improved with the Reynolds Stress Model. A decrease of the relative blade height and/or increase of flow turning in the cascade increases the intensity of passage vortices and the level of secondary flow losses. Spanwise gradients of pressure and profile load cause a redistribution of secondary loss centres along the blade span and endwall boundary layer losses. The development of secondary flows for the case of non-nominal incidence angles onto the suction surface of the blade looks different than for the classical case of nominal incidence. In front part of the blade the role of the convex and concave surface is reversed. Two passage vortices appear – a reverse (counter-rotating) and a regular passage vortex. The shroud leakage helps to remove the low-energy endwall boundary layer into the leakage slots, which retards the development of secondary flows in the current blade row. Local span-wise pressure gradients at the second stator blades induce a strong recirculating flow in the stator. It rolls up both the low-energy endwall boundary layer fluid and high-energy mixing layer of the shroud leakage and gives rise to an intensive tip passage vortex in the stator The transport of two-dimensional stator wakes leads to significant oscillations in size of the secondary flow zones. Segments of the tip leakage vortex from unshrouded rotors are found periodically within the recirculating flow in the downstream stator and a strong pulsating passage vortex is formed.
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