Variable Definitions
valid for TRACE SUITE 9.1
AT-NUM
October 24, 2018
Document Version 9.1.09
1
1
Nomenclature
Name
A
C
DF
DH
E
H
K
L
M
M V DR
Nu
P
R
Re
S
T
Tu
W
Zw
a
c
cf
cp
cv
d
e
h
k
ṁ
nblades
~n
p
q
r
s
v
v
vabs
vmer
vx , vr , vθ
w
x
y
z
Description
Units
Area
m2
Coefficient
−
Diffusion factor
−
De Haller number
−
Modal energy
Nm
Enthalpy
J
Blade span, channel height
m
Boundary layer parameter
−
Swirl of fluid
m2 /s
Length
m
Mach number
−
Meridional velocity density ratio −
Nusselt number
−
Power
W
Gas constant
m2 /s2
Reynolds number
−
Entropy
Nm/K
Overall arc length
m
Temperature
K
Turbulence level
Work
Zweifel number
Speed of sound
Blade chord length
Skin friction coefficient
Specific heat at constant pressure
Pressure coefficient
Specific heat at constant volume
Distance
Specific internal energy
Specific enthalpy
Blade height counting from hub
Turbulent kinetic energy
Mass flux
Number of blades
Outward pointing unit normal
Pressure
Dynamic Pressure
Radius, radial coordinate r (cylindrical coordinates)
Specific entropy
Blade pitch
Arc length
Velocity (vector)
Velocity (magnitude)
Absolute velocity (magnitude)
Meridional velocity component
Cylindrical velocity components
Specific work
Cartesian coordinate x
Cartesian coordinate y
Cartesian coordinate z
2
−
Nm
−
m/s
m
−
m2 /(Ks2 )
−
m2 /(Ks2 )
m
m2 /s2
m2 /s2
m
m2 /s2
kg/s
−
−
Pa
Pa
m
m2 /(Ks2 )
m
m
m/s
m/s
m/s
m/s
m/s
J/kg
m
m
m
θ
y+
αθ , αz , αθ∗
αr
ε, εcyl,abs
δ
δ1
δ2
δ3
Γ
κ
ε
ω
Ω
Φ
Ψ
ζ
µ
ν
ξ
η
π
ρ
τ
1.1
Angular coordinate (Cylindrical coordinates)
Dimensionless wall distance
Circumferential (azimuthal, pitchwise) flow angles
Radial or spanwise flow angle
Radial or spanwise flow angle
Boundary layer thickness
Boundary layer displacement thickness
Boundary layer momentum thickness
Boundary layer energy thickness
Blade surface
Ratio of specific heats
Radial or spanwise flow angle
Total pressure loss coefficient
Angular frequency
Rotational speed (radian per time)
Flow coefficient
Work coefficient
Eigenmode
Mass flow defect
Loss coefficient
Dynamic viscosity of fluid
Molecular viscosity
Normalised streamwise coordinate
Normalised spanwise coordinate
Efficiency
Pressure ratio
Fluid density
Temperature ratio
Shear stress
Subscripts
Name
LE
TE
T
Zw
abs
ax
blades
c
cool
diffuser
ht
in
is
leak
mainbleed
maincool
mainleak
mean
mer
noleak
out
Description
Blade leading edge
Blade trailing edge
Turbulent
Zweifel
Absolute frame of reference
Axial component of the corresponding quantity
Blade
Chordwise oriented coordinate system
Cooling flows
Diffuser
Defined via total enthalpy
Inflow location
Isentropic property
Leakage flow
Main bleeds
Main cooling flows
Main leaks
Mean value
Meridional
Without leakage flow
Outflow location
3
−
−
◦
◦
◦
m
m
m
m
m2
−
◦
−
s−1
s−1
−
−
m
−
−
kg/(ms)
m2 /s
−
−
−
−
kg/m3
−
Pa
pol
r
red
ref
rel
rot
rotor
s
steady
swirl
tan
t
wall
θ
12
32
*
Polytropic property
Radial direction
Reduced value
Reference value
Quantity related to a reference value
Defined via rothalpy of fluid
Value for the rotor (of a domain)
Static property
Value of steady flow field
Based on swirl
Tangential to solid body
Total property
Wall location
Pitchwise
Boundary layer relation - δ1 : δ2
Boundary layer relation - δ3 : δ2
Normalised quantity
4
2
Definitions
Name:
AreaRatio
Symbol:
−
Definition:
A2
A1
Unit:
-
Description: Area ratio outlet:inlet.
Name:
AeroDamping
Symbol:
δ
Definition:
−
Unit:
-
Re Wcyc
2E
Description: Aerodynamic damping for a structural mode.
Name:
AeroExcitation
Symbol:
ε
Definition:
|Wcyc |
2E
Unit:
-
Description: Aerodynamic excitation (in the context of forced response analysis).
Name:
AeroExcitationScaled
Symbol:
εscaled
Definition:
ε · AmplificationFactor
Unit:
-
Description: Aerodynamic excitation (in the context of forced response analysis) scaled by the AmplificationFactor used during the mapping process of the eigenmode displacements. The scaled
excitation is the aerodynamic excitation with respect to the unscaled eigenmode displacements.
Name:
AeroStiffness
Symbol:
κ
Definition:
Im Wcyc
2E
Unit:
-
Description: Aerodynamic stiffness for a structural mode.
Name:
AeroStimulus
Symbol:
S
Definition:
Wcyc /
Unit:
-
R
Γ
d
|ΨH (p~
n)0 |dS
Description: Aerodynamic stimulus.
5
Name:
AeroStimulusSigned
Symbol:
Ssigned
Definition:
Wcyc /|
Unit:
-
R
Γ
d
ΨH (p~
n)0 dS|
Description: Aerodynamic stimulus with respect to the signed mean aerodynamic forces.
Name:
AeroWork
Symbol:
Definition:
Wcyc
R
d
−iπ Γ ΨH (p~
n)ω dS
Unit:
Nm
Description: Work per cycle exerted by the aerodynamic forces for an eigenmode Ψ.
Name:
AeroWorkL1Norm
Symbol:
Definition:
kdWcyc /dSkL1
R
d
−iπ Γ |ΨH (p~
n)ω |dS
Unit:
Nm
Description: L1 -norm of the work per cycle and area.
Name:
AeroWorkModulus
Symbol:
|Wcyc |
Unit:
Nm
Description: Absolute value of (complex) work per cycle exerted by the aerodynamic forces for an eigenmode Ψ.
Name:
AeroWorkMeanPressurePart
Symbol:
Definition:
Wcyc,mp
R
bω dS
−iπ Γ pb0 ΨH ~n
Unit:
Nm
Description: Part of the modal work per cycle which is due to the mean pressure.
Name:
AeroWorkPerArea
Symbol:
dWcyc /dS
Definition:
d
n)ω
−iπΨH (p~
Unit:
kg/s2
Description: Work per cycle and area exerted by the aerodynamic forces for an eigenmode Ψ.
6
Name:
AngularMomentumFlux
Symbol:
−
Definition:
−ṁKabs
Unit:
-
Description: kg m2 s−2
Name:
AreaRatio
Definition:
Aout /Ain
Unit:
-
Description: Relation between outflow and inflow throughflow surface area.
Name:
Chord
Symbol:
c
Unit:
m
Name:
ChordAxial
Symbol:
cx
Unit:
m
Name:
CoefAreaRotationSpeedSquare
Symbol:
Cars
Definition:
Aout Ω2 /(4π 2 )
Unit:
m2 /s2
Remark:
A refers to the area of the full annulus.
Name:
CoefDiffusion
Symbol:
DF
Definition:
1 − vout /vin + |vθ,out − vθ,in |/(2 vin ) ∗ s/c
Unit:
-
Description: Diffusion coefficient - loading criterion for compressors. s/c is the mean pitch/chord ratio
(hub, tip) of the current blade row.
Remark:
Has to be checked.
Name:
CoefEntropyRise
Symbol:
C∆s
Definition:
log(pt,abs,out,is /pt,abs,in ) − log(pt,abs,out /pt,abs,in )
Unit:
-
7
Name:
CoefFlowAxial
Symbol:
Φx
Definition:
(vx,in + vx,out )/|2Ωrmean |
Unit:
-
Description: Axial component of flow coefficient.
Remark:
For axial turbomachines only! rmean is the arithmetic mean of the mass averaged radius at
in- and outlet.
Name:
CoefLossScholzV1
Symbol:
ωScholz1
Definition:
log(pout,is /pout )/ log(pt,out /pin )
Unit:
-
Description: Loss coefficient after Scholz (1).
Name:
CoefLossScholzV2
Symbol:
ωScholz2
Definition:
2
)
(hout − hout,is )/( 12 vout
Unit:
-
Description: Loss coefficient after Scholz (2). hout,is is computed from the absolute total pressure and
enthalpy at inlet and the static pressure at outlet.
Name:
CoefPressure
Symbol:
cp
Definition:
(p − pref )/(pt,ref − pref ) =
Unit:
-
Remark:
For airfoil cuts, reference location is inflow at same position as the airfoil cut is located.
For 2D surfaces the relative mass flow is used for the reference location.
Name:
CoefPressureStagDiffDyn
Symbol:
ω
Definition:
Unit:
∆p
qref
pt,in − pt,out
pt,ref − pref
General formulation:
Compressors (stages, components
or analysis volumes):
=
∆pt
qref
pt,out − pt,in
pt,ref − pref
-
Description: Total pressure loss coefficient in relative frame of reference based on dynamic pressure.
Compressors: reference is inlet, turbines: reference is outlet
Remark:
Not valid for radial compressor
8
Name:
CoefPressureStagDiffDynAbs
Symbol:
ωabs
Definition:
pt,abs,in − pt,abs,out
pt,abs,ref − pref
General formulation:
Compressors (stages, components
or analysis volumes):
Unit:
=
∆pt
qref
pt,abs,out − pt,abs,in
pt,abs,ref − pref
-
Description: Total pressure loss coefficient in absolute frame of reference based on dynamic pressure.
Compressors: reference is inlet, turbines: reference is outlet
Remark:
Stators and cascades only. Not valid for radial compressor.
Name:
CoefPressureStagDiffDynTh
Symbol:
ωtheoretic
pt,in −pt,loc
pt,in −pout
Definition:
Unit:
-
Description: Total pressure loss coefficient in relative frame of reference normalized using the theoretic
dynamic pressure.
Name:
CoefPressureStagDiffStag
Symbol:
ωt
Definition:
pt,in − pt,out
pt,ref
General formulation:
Compressors (stages, components
or analysis volumes):
Unit:
=
∆pt
pt,ref
pt,out − pt,in
pt,ref
-
Description: Total pressure loss coefficient in relative frame of reference based on dynamic pressure.
Compressors: reference is outlet, turbines: reference is inlet
Name:
CoefPressureStagDiffStagAbs
Symbol:
ωt,abs
Definition:
pt,abs,in − pt,abs,out
pt,abs,ref
General formulation:
Compressors (stages, components
or analysis volumes):
Unit:
=
∆pt
pt,abs,ref
pt,abs,out − pt,abs,in
pt,abs,ref
-
Description: Total pressure loss coefficient in absolute frame of reference based on total pressure.
Compressors: reference is outlet, turbines: reference is inlet
Name:
CoefPressureStagIsLossDyn
Symbol:
ζt,is
Definition:
pt,out,is − pt,out
pt,ref − pref
Unit:
-
=
∆pt,is
qref
Description: Isentropic total pressure loss coefficient in relative frame of reference based on dynamic
pressure.
Compressors: Reference is inlet, turbines: Reference is outlet
9
Name:
CoefPressureStagIsLossDynAbs
Symbol:
ζt,abs,is
Definition:
pt,abs,out,is − pt,abs,out
pt,abs,ref − pref
Unit:
-
=
∆pt,abs,is
qabs,ref
Description: Isentropic total pressure loss coefficient in absolute frame of reference based on dynamic
pressure.
Compressors: reference is inlet, turbines: reference is outlet
Name:
CoefPressureStagLossDyn
Symbol:
ζt
Definition:
pt,in − pt,out
pt,ref − pref
Unit:
-
=
∆pt
qref
Description: Total pressure loss coefficient in relative frame of reference based on dynamic pressure.
Compressors: Reference is inlet, turbines: Reference is outlet
Remark:
For rotors only.
Name:
CoefPressureStagLossDynAbs
Symbol:
ζt,abs
Definition:
pt,abs,in − pt,abs,out
pt,abs,ref − pref
Unit:
-
=
∆pt,abs
qabs,ref
Description: Total pressure loss coefficient in absolute frame of reference based on dynamic pressure.
Compressors: reference is inlet, turbines: reference is outlet
Remark:
For stators only
Name:
CoefSkinFriction
Symbol:
cf
Definition:
τwall /(pt,ref − pref )
Unit:
-
Description: Wall skin friction coefficient.
Remark:
Reference location is inflow at same position as the airfoil cut is located. For 2D surfaces
the relative mass flow is used for the reference location.
Name:
CoefVelocityMean
Symbol:
cv,mean
Definition:
|Ω|rmean
√
2|wht |
Unit:
-
10
Name:
CoefVelocityRatio
Symbol:
Definition:
cv,ratio
q
1
2 |Ω|rrotor /|vθ,abs,out,rotor − vθ,abs,in,rotor |
Unit:
-
Name:
CoefWork
Symbol:
ψ
Definition:
w/(Ωrmean )2
Unit:
-
Description: Work coefficient or enthalpy rise coefficient. rmean is the arithmetic mean of the mass
averaged radius at in- and outlet.
Name:
CoefWork2
Symbol:
ψ2
Definition:
2ψ
Unit:
-
Description: Alternative definition for work coefficient.
Name:
CoefWorkIsentropic
Symbol:
ψis
Definition:
ηis,noleak ψ
Unit:
-
Description: Isentropic work coefficient or isentropic enthalpy rise coefficient.
Name:
CoefWorkIsentropic2
Symbol:
ψis,2
Definition:
2ψis
Unit:
-
Description: Alternative definition for isentropic work coefficient.
Name:
CoefZweifel
Symbol:
CZw
Definition:
Zw nblades cx
Unit:
m2
Description: Blade loading criterion for turbines.
11
Name:
CoordinateMeanRTheta
Symbol:
rmean Θ
Definition:
(rLE + rTE )/2 · Θ
Unit:
m
Description: Turbomachinary coordinate which consists of the mean radius and local theta coordinate.
Remark:
Used in S1 blade slices.
Name:
CoordinateR
Symbol:
r
Unit:
m
Description: Radial, normally spanwise, coordinate.
Name:
CoordinateTheta
Symbol:
θ
Unit:
-
Description: Pitchwise coordinate.
Name:
CoordinateX
Symbol:
x
Unit:
m
Description: Cartesian coordinate x. Turbomachines: axial direction.
Name:
CoordinateY
Symbol:
y
Unit:
m
Description: Cartesian coordinate y.
Name:
CoordinateZ
Symbol:
z
Unit:
m
Description: Cartesian coordinate z.
Name:
CoordinateXi
Symbol:
ξ
Unit:
-
Description: Normalised coordinate in streamwise direction.
Name:
CoordinateEta
Symbol:
η
Unit:
-
Description: Normalised coordinate in spanwise direction.
12
Name:
DeHaller
Symbol:
DH
Definition:
vout /vin
Unit:
-
Description: Simple blade loading criterion for compressors.
Name:
Density
Symbol:
ρ
Unit:
kg/m3
Description: Density of fluid.
Name:
DiffusorExitPressureStagAbs
Symbol:
pt,abs,diffuser,out
Unit:
Pa
Description: If the diffusor is not part of the computational mesh the absolute total pressure at the exit of
the diffusor has to be given to determine the efficiency of the turbine-diffusor combination.
Name:
DisplacementX
Symbol:
δx
Unit:
m
Description: x component of grid point displacement (difference of coordinates)
Name:
DisplacementY
Symbol:
δy
Unit:
m
Description: y component of grid point displacement (difference of coordinates)
Name:
DisplacementZ
Symbol:
δz
Unit:
m
Description: z component of grid point displacement (difference of coordinates)
Name:
DisplacementMagnitude
Symbol:
Definition:
δxyz
p
Re{δx}2 + Re{δy}2 + Re{δz}2 + Im{δx}2 + Im{δy}2 + Im{δz}2
Unit:
m
Description: Magnitude of grid point displacement (difference of coordinates)
13
Name:
DistanceWallCoordinate
Symbol:
y+
Definition:
ρuτ
µ
Unit:
-
d
Description: Dimensionless wall distance computedp
from the distance d to the nearest viscous wall.
The friction velocity is given by uτ = τw /ρ with the wall shear stress τw .
Name:
EfficiencyIsentropicH
Symbol:
ηis
Definition:
Compressors:
∆ht,is
∆ht
=
ht,abs,out,is − ht,abs,in
ht,abs,out − ht,abs,in
Turbines:
∆ht
∆ht,is
=
ht,abs,out − ht,abs,in
ht,abs,out,is − ht,abs,in
Unit:
-
Description: Isentropic efficiency defined for rotors, stages, groups, turbomachines.
Remark:
Only used for spanwise relations, corresponding variable for 0D values is
EfficiencyIsentropicHWoLeak
Name:
EfficiencyIsentropicHLeak
Symbol:
ηis,leak
Definition:
Compressors:
Turbines:
Unit:
Pht ,is,leak
Pht ,leak
Pht ,leak
Pht ,is,leak
-
Description: Isentropic efficiency defined for rotors, stages, groups, turbomachines.
Remark:
All leakage flows included.
Name:
EfficiencyIsentropicHMLeak
Symbol:
ηis,mainleak
Definition:
Compressors:
Turbines:
Pht ,is,mainleak
Pht ,mainleak
Pht ,mainleak
Pht ,is,mainleak
Unit:
-
Remark:
All main leakage flows included.
Name:
EfficiencyIsentropicHWoLeak
Symbol:
ηis,noleak
Definition:
Compressors:
∆ht,is
∆ht
=
ht,abs,out,is − ht,abs,in
ht,abs,out − ht,abs,in
Turbines:
∆ht
∆ht,is
=
(ht,abs,out −ht,abs,in )
(ht,abs,out,is −ht,abs,in )
Unit:
-
Description: Isentropic efficiency defined for rotors, stages, groups, turbomachines.
Remark:
No leakage flows included.
14
Name:
EfficiencyIsentropicSwirl
Symbol:
ηis,swirl
Definition:
Compressors:
Pht ,is,leak
Pswirl
Turbines:
Pswirl
Pht ,is,leak
Unit:
-
Description: Isentropic efficiency based on swirl (rotors, stages, groups, turbomachines).
Remark:
Pht ,is,leak denotes the overall isentropic power output. Pswirl denotes the swirl power output.
For turbines both should be negative.
Name:
EfficiencyIsInclDiffusor
Symbol:
ηis,noleak,diffuser
Definition:
∆ht
∆ht,is
Unit:
-
=
ht,in − ht,out
ht,in − ht,out,is
with ht,out,is = f (pt,abs,diffuser,out )
Description: Isentropic efficiency including the total pressure loss in the diffusor.
Remark:
For turbines only. No leakage flows included.
Name:
EfficiencyIsLeaksDiffusor
Symbol:
ηis,leak,diffuser
Definition:
Pht ,leak
Pht ,is,leak
Unit:
-
with Pht ,out,is = f (pt,abs,diffuser,out )
Description: Isentropic efficiency including the total pressure loss in the diffusor.
Remark:
For turbines only. All leakage flows included.
Name:
EfficiencyPolytropic
Symbol:
ηpol
Definition:
Compressors:
κ−1
κ
log
pt,abs,out
pt,abs,in
/ log
ht,abs,out
ht,abs,in
Turbines:
κ
κ−1
log
ht,abs,out
ht,abs,in
/ log
pt,abs,out
pt,abs,in
Unit:
-
Description: Polytropic efficiency.
Name:
EfficiencyPolyInclDiffusor
Symbol:
ηpol,diffuser
Unit:
-
Description: Polytropic efficiency computed from inlet stagnation values, exit stagnation temperature
and a user defined exit stagnation pressure, to account for the additional losses of a diffusor
which is not part of the configuration.
15
Name:
EnergyDensityStagnation
Definition:
ρ e + 21 v 2 for translational,
ρ e + 12 (v 2 − (rΩ)2 for rotational configurations.
Unit:
Nm−2
Name:
EnergyDensityStagnationAbs
Definition:
2
ρ e + 21 vabs
Unit:
Nm−2
Name:
Enthalpy
Symbol:
h
Definition:
e+
Unit:
m2 /s2
Name:
EnthalpyStagnation
Symbol:
ht
Definition:
h + 12 v 2
Unit:
m2 /s2
p
ρ
Description: Relative stagnation enthalpy.
Name:
EnthalpyStagnationAbs
Symbol:
ht,abs
Definition:
2
h + 12 vabs
Unit:
m2 /s2
Description: Absolute stagnation enthalpy.
Name:
EnthalpyStagnationRot
Symbol:
ht
Unit:
m2 /s2
Description: Relative total enthalpy.
Remark:
The relative velocty is computed from the absolute average and the averaged radius.
Name:
Entropy
Symbol:
s
Definition:
cv (log(κp) − κ log(ρ))
Unit:
m2 /(Ks2 )
Description: Entropy of fluid.
Name:
FactorSwirl
Definition:
vθ,abs /(rΩ)
Unit:
-
16
Name:
FlowDirectionR
Unit:
-
Description: Radial component of velocity direction vector in cylindrical coordinates.
Name:
FlowDirectionTheta
Unit:
-
Description: Circumferential component of velocity direction vector in cylindrical coordinates.
Name:
FlowDirectionThetaAbs
Unit:
-
Description: Circumferential component of absolute velocity direction vector in cylindrical coordinates.
Name:
FlowDirectionX
Unit:
-
Description: Axial (x) component of velocity direction vector.
Name:
FlowDirectionY
Unit:
-
Description: y component of velocity direction vector in cartesian coordinates.
Name:
FlowDirectionYAbs
Unit:
-
Description: y component of absolute velocity direction vector in cartesian coordinates.
Name:
FlowDirectionZ
Unit:
-
Description: z component of velocity direction vector in cartesian coordinates.
Name:
FlowTurningAlpha
Definition:
∗
∗
|
− αΘ,in
|αΘ,out
Unit:
◦
Description: Relative pitchwise flow turning based on αΘ,MTU . Turbine: no absolute value !
Name:
FlowTurningAlphaAbs
Definition:
∗
∗
|αΘ,abs,out
− αΘ,abs,in
|
Unit:
◦
Description: Absolute pitchwise flow turning based on αΘ,MTU,abs . Turbine: no absolute value !
Name:
FlowTurningAlphaZ
Definition:
|αz,out − αz,in |
Unit:
◦
Description: Relative pitchwise flow turning based on αz . Turbine: no absolute value !
17
Name:
FlowTurningAlphaZAbsOK
Definition:
|αz,abs,out − αz,abs,in |
Unit:
◦
Description: Absolute pitchwise flow turning based on αz,abs . Turbine: no absolute value !
Name:
FlowTurningMeridional
Definition:
|αr,out − αr,in |
Unit:
◦
Description: Radial flow turning based on αr .
Name:
FlowTurningTheta
Definition:
|αΘ,out − αΘ,in |
Unit:
◦
Description: Pitchwise flow turning in relative frame of reference. Turbine: no absolute value !
Name:
FlowTurningThetaAbs
Definition:
|αΘ,abs,out − αΘ,abs,in |
Unit:
◦
Description: Pitchwise flow turning in absolute frame of reference. Turbine: no absolute value !
Name:
FractionSpecificWork
Definition:
wrotori /wcomp
Unit:
−
Description: Specific work of a rotor divided by the specific work of a contol volume (stage, component,
volume)
Name:
FrameVelocityTheta
Definition:
Ωr
Unit:
m/s
Description: Velocity of rotating frame of reference at the reference radius, e.g., the mass averaged radius.
Name:
FrequencyShift
Symbol:
δf
Definition:
κf
2π
Unit:
1/s
Description: Frequency shift of an eigenmode due to aerodynamic stiffness κ.
18
Name:
FrequencyUpdate
Symbol:
f
Definition:
-
Unit:
1/s
Description: Updated (guessed) frequency from a coupled FSI flutter simulation in the frequency domain.
Name:
Gamma
Symbol:
γ
Definition:
-
Unit:
-
Description: Transported variable in γ-Reθ transition model.
Name:
IdealGasConstant
Symbol:
R
Unit:
J · kg−1 · K−1
Description: Specific gas constant.
Name:
Mach
Symbol:
M
Definition:
v/a
Unit:
-
Description: Mach number in relative frame of reference.
Name:
MachAbs
Symbol:
Mabs
Definition:
vabs /a
Unit:
-
Description: Mach number in absolute frame of reference.
Name:
MachIsentropic
Symbol:
Mis
q
Definition:
Unit:
2
κ−1
[(
pt,ref
p
)
κ−1
κ
− 1]
-
Description: Isentropic Mach number.
Remark:
Tt,ref = Trot +
pt,ref = pref (
v2
2cp
Tt,ref
Tref
κ
) κ−1
Reference location for Trot , Tref and pref is the inflow at same position as the airfoil cut is
located. For 2D surfaces the relative mass flow is used for the reference location.
19
Name:
MachMeridional
Symbol:
Mmer
Definition:
vmer
a
Unit:
-
Description: Meridional Mach number.
Name:
MachRot
Symbol:
Mrot
Unit:
-
Description: Relative Mach number computed from the absolute velocities, the rotational speed, and the
reference radius.
Name:
MassFlow
Symbol:
ṁ
Definition:
−ρv · ~n dS
Unit:
kg/s
Remark:
w.r.t. full annulus for rotational configurations
Name:
MassFlowCorrected
Symbol:
Definition:
ṁISA
q
Tt,abs
ṁ
Tref
Unit:
kg/s
pref
pt,abs
Description: Mass flow corrected via ISA conditions:
pref = 101325 Pa, Tref = 288.15 K.
Remark:
w.r.t. full annulus for rotational configurations
Name:
MassFlowDefect
Symbol:
ζ
Definition:
|ṁin | − |ṁout |
|ṁin |
Unit:
-
Description: Mass flow defect.
Name:
MassFlowReduced
Symbol:
ṁred
Definition:
ṁ
Unit:
msK 2
√
Tt
pt
1
Description: Reduced mass flow rate.
Remark:
w.r.t. full annulus for rotational configurations
20
Name:
MassFlowUnsigned
Symbol:
|ṁ|
Unit:
kg/s
Remark:
w.r.t. full annulus for rotational configurations
Name:
MixingLossPressureStagnation
Symbol:
ωmix
Unit:
pW
− pt
t
pt − p
Description: −
Remark:
Mixing loss computed as total pressure loss. pt and p refer to the average reference value
under consideration (e.g. flux average), pW
t is the so-called work average, cf. [?].
Name:
MomentumDensityMeridRatio
Definition:
M V DR = (ρvmer )out / (ρvmer )in
Unit:
-
Description: Meridional velocity density ratio.
Name:
MomentumDensityR
Definition:
ρvr
Unit:
kg · m−2 · s−1
Description: Radial component of the momentum density.
Name:
MomentumDensityTheta
Definition:
ρvθ
Unit:
kg · m−2 · s−1
Description: Circumferential component of the relative momentum density.
Name:
MomentumDensityThetaAbs
Definition:
ρvθ,abs
Unit:
kg · m−2 · s−1
Description: Circumferential component of the absolute momentum density.
Name:
MomentumDensityX
Definition:
ρvx
Unit:
kg · m−2 · s−1
Description: Axial component of the momentum density.
Name:
NormalX
Symbol:
nx
Unit:
-
Description: x component of face normal vector
21
Name:
NormalY
Symbol:
ny
Unit:
-
Description: y component of face normal vector
Name:
NormalZ
Symbol:
nz
Unit:
-
Description: z component of face normal vector
Name:
NumberOfAirfoils
Symbol:
nblades
Unit:
-
Description: Number of blades across annulus.
Name:
NumberOfSegments
Symbol:
nsegments
Unit:
-
Description: Number of segments across annulus.
Remark:
Single passage: identical to nblades .
Name:
Pitch
Symbol:
s
Unit:
m
Description: Blade pitch for turbomachines and linear cascades.
Name:
PitchChordRatio
Definition:
s
c
Unit:
-
Description: Pitch to chord ratio for turbomachines and linear cascades.
Name:
PowerHLeak
Symbol:
Pht ,leak
Definition:
ṁout ht,abs,out − ṁin ht,abs,in + ṁbleed ht,abs,bleed − ṁcool ht,abs,cool
Unit:
W
Description: Overall power output accounting for all bleeds and leaks.
22
Name:
PowerHMLeak
Symbol:
Pht ,mainleak
Definition:
ṁout ht,abs,out − ṁin ht,abs,in + ṁmainbleed ht,abs,mainbleed − ṁmaincool ht,abs,maincool
Unit:
W
Description: Overall power output accounting for all main bleeds and leaks.
Name:
PowerHWoLeak
Symbol:
Pht ,noleak
Definition:
ṁout ht,abs,out − ṁin ht,abs,in
Unit:
W
Description: Overall power output based on main inlets and outlets only.
Name:
PowerIsentropicHLeak
Symbol:
Pht ,is,leak
Definition:
(ṁout − ṁcool ) ht,is,out − ṁin ht,in + ṁbleed ht,is,bleed + ṁcool ht,is,cool
Unit:
W
Description: Overall isentropic power output accounting for all bleeds and leaks.
Remark:
ht,is,out is computed from the stagnation pressure at the main outlet and the inlet stagnation
temperature and pressure at the main inlet. ht,is,cool is computed from the local (cooling)
stagnation values and the outlet stagnation pressure. ht,is,bleed is computed from the bleed
total pressure and the inlet stagnation temperature and pressure.
Name:
PowerIsentropicHMLeak
Symbol:
Pht ,is,mainleak
Definition:
(ṁout
− ṁcool ) ht,abs,is,out
ṁmaincool ht,abs,is,maincool
Unit:
W
−
ṁin ht,abs,in
+
ṁmainbleed ht,abs,is,mainbleed
Description: Overall isentropic power output accounting for all main bleeds and main leaks.
Name:
PowerIsentropicHWoLeak
Symbol:
Pht ,is,noleak
Definition:
ṁout ht,is,out − ṁin ht,in
Unit:
W
Description: Overall isentropic power output without accounting for bleeds and leaks.
Name:
PowerReducedHLeak
Symbol:
Pred,leak
Definition:
Unit:
Pht ,leak
pt,in,abs
√
Tt,in,abs
√
W/(Pa K)
23
+
Name:
PowerSwirl
Symbol:
Pswirl
Definition:
Ω(ṁout Kout,abs + ṁin Kin,abs )
Unit:
W
Remark:
Idealized shaft power based on angular momentum flux (ṁK) and wheel speed. Positive for
compressors , negative for turbines. General domains: Summation of Pswirl over all rotors.
Name:
Pressure
Symbol:
p
Unit:
Pa
Description: Static pressure.
Name:
PressureNormalized
Symbol:
p∗
Definition:
p
pt,in
Unit:
-
Description: Normalized pressure.
Name:
PressureRatio
Symbol:
π
Definition:
Compressors:
pout
pin
Turbines:
pin
pout
Unit:
-
Description: Static pressure ratio.
Name:
PressureStagnation
Symbol:
pt
Definition:
p 1+
Unit:
Pa
κ−1
2
2 M
κ
κ−1
Description: Stagnation or total pressure in relative frame of reference.
Name:
PressureStagnationAbs
Symbol:
pt,abs
Definition:
p 1+
Unit:
Pa
κ−1
2
2 Mabs
κ
κ−1
Description: Stagnation or total pressure in absolute frame of reference.
24
Name:
PressureStagnationAbsRatio
Symbol:
πt,abs
Definition:
Compressors:
pt,out,abs
pt,in,abs
Turbines:
pt,in,abs
pt,out,abs
Unit:
-
Description: Stagnation or total pressure ratio in absolute frame of reference.
Name:
PressureStagnationRatio
Symbol:
πt
Definition:
Compressors:
pt,out
pt,in
Turbines:
pt,in
pt,out
Unit:
-
Description: Stagnation or total pressure ratio in relative frame of reference.
Name:
PressureStagnationRot
Symbol:
Definition:
pt,rot
κ
κ−1
T
p t,rot
T
Unit:
Pa
Description: Relative total pressure.
Remark:
Computed from the absolute averages.
Name:
PressureStaticStagAbsRatio
Symbol:
πs,t,abs
Definition:
p
pt,abs
Unit:
-
Description: Static to total pressure ratio in absolute frame of reference.
Name:
ReactionEnthalpy
Symbol:
ρh
Definition:
hrotor,out − hrotor,in
hout − hin
Unit:
-
Remark:
Computed for stages.
Name:
ReactionPressure
Symbol:
ρp
Definition:
protor,out − protor,in
pout − pin
Unit:
-
Remark:
Computed for stages.
25
Name:
ReactionVelocity
Symbol:
ρv
Definition:
−
Unit:
-
Remark:
Computed for stages.
Name:
RelativeArcLength
Symbol:
srel
Definition:
s
S
Unit:
-
vθ,rotor,out + vθ,rotor,in
Ω(rout + rin )
where S denotes whole arc length
Description: Relative arc length of a point on a spatial curve.
Name:
RelativeChannelHeight
Symbol:
Hrel
Definition:
h
H
Unit:
-
Description: Relative channel height in S3 direction.
Name:
RelativeChord
Symbol:
c∗
Definition:
xc − xc,LE
c
Unit:
-
Description: Coordinate xc in chordwise direction.
Name:
RelativeChordAxial
Symbol:
c∗x
Definition:
x − xLE
cx
Unit:
-
Description: Coordinate x − xLE in axial direction normalised with axial chord length cx .
Name:
RelativeChordRadial
Symbol:
c∗r
Definition:
r − rLE
rTE − rLE
Unit:
-
Description: Relative chord in terms of radial coordinate r.
Name:
RelativeCoordinateR
Symbol:
rrel
Definition:
r − rmin
rmax − rmin
Unit:
-
Description: Relative radial coordinate.
26
Name:
RelativeMassFlow
Symbol:
ṁrel
Definition:
ṁ(h)
ṁ(H)
Unit:
-
Description: Relative mass flow rate across blade span - h=0: hub, h=H: shroud
Name:
RelativeMeridionalLength
Symbol:
s∗mer
Definition:
∆smer
Smer
Unit:
-
Name:
RelativePressure
Symbol:
prel
Definition:
p
psteady
Unit:
-
P p
P √
Description: With ∆smer = LE ∆x2 + ∆r2 = LE (x − xLE )2 + (r − rLE )2 and
PTE
Smer = LE ∆smer
Description: Relative pressure p/psteady in terms of the pressure field from the steady solution psteady .
Name:
RelativeTemperature
Symbol:
Trel
Definition:
T
Tsteady
Unit:
-
Description: Relative temperature T /Tsteady in terms of the temperature field from the steady solution
Tsteady .
Name:
RelativeVelocityMeridMag
Symbol:
vmer,rel
Definition:
vmer
vmer,steady
Unit:
-
Description: Relative meridional velocity vmer /vmer,steady in terms of the meridional velocity field from
the steady solution vmer,steady .
Name:
ReTheta
Symbol:
Reθ
Definition:
-
Unit:
-
Description: Transported variable in γ-Reθ transition model.
27
Name:
ReynoldsInflow
Symbol:
Re or Rein
Definition:
ρin c vin
µin
Unit:
-
Description: Reynolds number based on inflow conditions, used primarily in compressors.
Remark:
c is used from GMC settings.
Name:
ReynoldsOutflow
Symbol:
Re or Reout
Definition:
ρout c vout
µout
Unit:
-
Description: Reynolds number based on outflow conditions, used primarily in turbines.
Remark:
c is used from GMC settings.
Name:
RotatingFrameVelocityTheta
Definition:
rΩ
Unit:
m/s
Description: Frame velocity of rotating system.
Name:
RotationSpeed
Definition:
Ω
2π
Unit:
1/s
Description: Rotational speed of relative system in rounds per second.
Name:
Definition:
Unit:
RotationSpeedReduced
2π
√Ω
Tt,abs,in
√
1/(s K)
Description: Reduced wheel speed of rotor.
Name:
ShapeFactor12
Symbol:
H12
Definition:
δ1
δ2
Unit:
-
Description: Boundary layer shape factor: displacement thickness / momentum thickness.
Name:
ShapeFactor32
Symbol:
H32
Definition:
δ3
δ2
Unit:
-
Description: Boundary layer shape factor: energy thickness / momentum thickness.
28
Name:
SpatialAverageType
Description: Averaging type: 0 =flux, 1 =mass, 2 =area
Name:
SpecificHeatPressure
Symbol:
cp
Unit:
m2 /(s2 K)
Description: Specific heat capacity at constant pressure.
Name:
SpecificHeatRatio
Symbol:
κ
Definition:
cp
cv
Unit:
-
Description: Ratio of specific heats or heat capacity ratio.
Name:
SpecificWorkH
Symbol:
w
Definition:
Pht ,leak
|ṁin |
Unit:
J/kg
Description: Specific work of turbomachine or turbomachine component.
Remark:
In case of spanwise distribution Pht ,noleak is used instead of Pht ,leak .
Name:
SpecificWorkSwirl
Symbol:
wswirl
Definition:
Ω (Kabs,in − Kabs,out )
Unit:
J/kg
Description: Specific work based on swirl.
Name:
SwirlAbs
Symbol:
Kabs
Definition:
rvθ,abs
Unit:
m2 /s
Description: Flow swirl in absolute frame of reference.
Remark:
Equivalent to angular momentum per unit mass.
Name:
Temperature
Symbol:
T
Unit:
K
Description: Static temperature of flow.
29
Name:
TemperatureRothalpy
Symbol:
Trot
Definition:
Tt −
Unit:
K
Name:
TemperatureStagAbsRatio
Symbol:
τabs
Definition:
Compressors:
Tt,abs,out
Tt,abs,in
Turbines:
Tt,abs,in
Tt,abs,out
Unit:
v2
2cp
-
Description: Total temperature ratio in absolute frame of reference.
Name:
TemperatureStagnation
Symbol:
Tt
Definition:
T+
Unit:
K
v2
2cp
=T 1+
κ−1
2
2 M
Description: Total temperature in relative frame of reference.
Name:
TemperatureStagnationAbs
Symbol:
Tt,abs
Definition:
T+
Unit:
K
2
vabs
2cp
=T 1+
κ−1
2
2 Mabs
Description: Total temperature in absolute frame of reference.
Name:
TemperatureStagnationAbsDiff
Symbol:
∆Tt,abs
Definition:
Tt,out,abs − Tt,in,abs
Unit:
K
Description: Difference in total temperature in absolute frame of reference.
Name:
TemperatureStagnationRot
Symbol:
Tt,rot
Definition:
T+
Unit:
K
2
vrot
2cp
Description: Relative total temperature.
Remark:
Computed from the absolute averages.
30
Name:
TemperatureStagRatio
Symbol:
τ
Definition:
Compressors:
Tt,out
Tt,in
Turbines:
Tt,in
Tt,out
Unit:
-
Description: Total temperature ratio in relative frame of reference.
Name:
ThicknessBoundaryLayer
Symbol:
δ
Unit:
m
Description: Boundary layer thickness based on pt -criterion. Edge is defined as point where
pt = 0.995pt,max .
Name:
ThicknessDisplacement
Symbol:
Definition:
δ1
Rδ
Unit:
m
0
1−
ρ(y)u(y)
ρ0 u0
dy
Description: Boundary layer displacement thickness.
Name:
ThicknessDisplacementInc
Symbol:
Definition:
δ1,inc
Rδ
1−
0
Unit:
m
u(y)
u0
dy
Description: Boundary layer displacement thickness incompressible formulation.
Name:
ThicknessEnergy
Symbol:
Definition:
δ3
Rδ
Unit:
m
ρ(y)u(y)
ρ0 u0
0
2 ρ(y)u(y)
1−
dy
ρ0 u0
Description: Boundary layer energy thickness.
Name:
ThicknessMomentum
Symbol:
Definition:
δ2
Rδ
Unit:
m
ρ(y)u(y)
ρ0 u0
0
1−
u(y)
u0
dy
Description: Boundary layer momentum thickness.
31
Name:
ThicknessMomentumInc
Symbol:
Definition:
δ2
Rδ
Unit:
m
u(y)
u0
0
1−
u(y)
u0
dy
Description: Boundary layer momentum thickness incompressible formulation.
Name:
TurbulentDissipationRate
Symbol:
ω
Definition:
ε
k
Unit:
1/s
Description: Turbulent dissipation rate.
Name:
TurbulentDistance
Symbol:
d
Unit:
m
Description: Distance to nearest viscous wall.
Name:
TurbulentEnergyKinetic
Symbol:
k
Unit:
m2 /s2
Description: Turbulent kinetic energy.
Name:
TurbulenceIntensity
Symbol:
Tu
q2
3k
Definition:
Unit:
v
-
Description: Turbulence level or turbulence intensity of fluid with turbulent kinetic energy k and velocity
magnitude v.
Name:
TurbulenceIntensityAbs
Symbol:
T uabs
q 2
Definition:
Unit:
3k
vabs
-
Description: Turbulence level or turbulence intensity of fluid with turbulent kinetic energy k and velocity
magnitude vabs in absolute frame of reference.
32
Name:
TurbulentLengthScale
Symbol:
LT
√
k
ω
Definition:
Unit:
m
Description: Turbulent length scale corresponding to turbulent kinetic energy k and specific turbulent
dissipation rate ω (as used in Wilcox and Menter ω-equation).
Name:
VelocityAngleAlpha
Symbol:
∗
αΘ
Definition:
90 − arctan
Unit:
◦
◦
vθ
vm
◦
= 90 − αΘ
Description: Azimuthal or pitchwise flow angle.
◦
Remark:
Takes values between 0◦ and 180◦ , equals 90 when relative flow is purely axial. Undefined
when v = 0.
Name:
VelocityAngleAlphaAbs
Symbol:
∗
αΘ,abs
Definition:
90 − arctan
Unit:
◦
◦
vΘ,abs vm
◦
= 90 − αΘ,abs
Description: Absolute azimuthal or pitchwise flow angle.
◦
Remark:
Takes values between 0◦ and 180◦ , equals 90 when absolute flow is purely axial. Undefined
when vabs = 0.
Name:
VelocityAngleAlphaZ
Symbol:
αz
Definition:
90 − arctan
Unit:
◦
◦
vθ
vx
, if vx > 0.
Description: Azimuthal or pitchwise flow angle with respect to the axial velocity.
◦
Remark:
Takes values between −180◦ and 180◦ , equals 90 when relative flow is purely axial. Undefined when flow is purely radial.
Name:
VelocityAngleAlphaZAbs
Symbol:
αz,abs
Definition:
90 − arctan
Unit:
◦
◦
vθ,abs ,
vx
if vx > 0.
Description: Absolute azimuthal or pitchwise flow angle with respect to the axial velocity.
Remark:
◦
Takes values between −180◦ and 180◦ , equals 90 when absolute flow is purely axial. Undefined when absolute flow is purely radial.
33
Name:
VelocityAngleEpsilon
Symbol:
ε
Definition:
arctan
Unit:
◦
vr
vx
, if vx > 0.
Description: Radial flow angle.
◦
Remark:
Identical to αr . Takes values between −180 and 180◦ . Undefined when vmer = 0.
Name:
VelocityAngleEpsilonCylAbs
Symbol:
εcyl,abs
Definition:
arctan
Unit:
◦
vr
vabs
Description: Radial flow angle.
Remark:
Angle between the absolute velocity and direction obtained by projecting the absolute ve◦
locity into the x-θ plane. Takes values between −90 and 90◦ . Undefined when vabs = 0.
Name:
VelocityAngleR
Symbol:
αr
Definition:
arctan
Unit:
◦
vr
vx
, if vx > 0.
Description: Radial flow angle.
◦
Remark:
Identical to ε. Takes values between −180 and 180◦ . Undefined when vmer = 0.
Name:
VelocityAngleTheta
Symbol:
αθ
Definition:
arctan
Unit:
◦
vθ
vmer
◦
= 90 − αθ∗
Description: Azimuthal or pitchwise angle defined as angle between the velocity and the meridional
velocity.
◦
Remark:
Takes values between −90◦ and 90◦ , equals 0 when relative flow is purely axial. Undefined
when v = 0.
Name:
VelocityAngleThetaAbs
Symbol:
αθ,abs
Definition:
arctan
Unit:
◦
vθ,abs vmer
◦
∗
= 90 − αθ,abs
Description: Absolute azimuthal or pitchwise flow angle defined as angle between the absolute velocity
and the meridional velocity.
Remark:
◦
Takes values between −90◦ and 90◦ , equals 0 when absolute flow is purely axial. Undefined
when vabs = 0.
34
Name:
VelocityAngleY
Symbol:
αy
Definition:
arctan
Unit:
◦
vy vx
Description: Angle between y- and x-component of velocity vector in relative frame of reference.
Remark:
Default angle for translational cases.
Name:
VelocityAngleYAbs
Symbol:
αy,abs
Definition:
arctan
Unit:
◦
vy,abs vx
Description: Angle between y- and x-component of velocity vector in absolute frame of reference.
Remark:
Default angle for translational cases.
Name:
VelocityAngleZ
Symbol:
αz
Definition:
arctan
Unit:
◦
vz
vx
Description: Angle between z- and x-component of velocity vector.
Remark:
Default angle for translational cases.
Name:
VelocityMeridionalMagnitude
Symbol:
|vmer |
Definition:
|vmer | =
Unit:
-
p
2 + v2
vax
rad
Description: Norm of the vector of the meridional velocity vmer .
Name:
VelocityMagnitude
Symbol:
v
Definition:
|v|
Unit:
m/s
Description: Magnitude of relative velocity vector.
Name:
VelocityMagnitudeAbs
Symbol:
vabs
Definition:
|vabs |
Unit:
m/s
Description: Magnitude of absolute velocity vector.
35
Name:
VelocityMagnitudeRot
Symbol:
vrot
Definition:
|v|
Unit:
m/s
Description: Relative velocity magnitude
Remark:
Computed from the absolute averages.
Name:
VelocityMeridionalMagnitude
Symbol:
Definition:
vmer
p
vx2 + vr2
Unit:
m/s
Description: Meridional velocity.
Name:
VelocityR
Symbol:
vr
Unit:
m/s
Description: Velocity component in r-wise (radial) direction.
Name:
VelocityRatio
Symbol:
vout
vin
Unit:
-
Description: Deceleration ratio - loading criterion for compressors.
Name:
VelocitySound
Symbol:
a
√
Definition:
Unit:
κRT
m/s
Description: Speed of sound.
Name:
VelocityTheta
Symbol:
vθ
Unit:
m/s
Description: Velocity component in pitchwise direction
Remark:
For flux-averaged data, this is computed from the absolute velocity using the mass-averaged
radius.
Name:
VelocityThetaAbs
Symbol:
vθ,abs
Unit:
m/s
Description: Velocity component in pitchwise direction for absolute frame of reference.
36
Name:
VelocityThetaRot
Symbol:
vθ,rot
Definition:
vθ,abs − Ωr
Unit:
m/s
Description: Wheel velocity component in pitchwise direction
Name:
VelocityX
Symbol:
vx
Unit:
m/s
Description: Velocity component in x-wise direction.
Name:
VelocityXAbs
Symbol:
vx,abs
Unit:
m/s
Description: Velocity component in x-wise direction for absolute frame of reference.
Name:
VelocityY
Symbol:
vy
Unit:
m/s
Description: Velocity component in y-wise direction.
Name:
VelocityYAbs
Symbol:
vy,abs
Unit:
m/s
Description: Velocity component in y-wise direction for absolute frame of reference.
Name:
VelocityZ
Symbol:
vz
Unit:
m/s
Description: Velocity component in z-wise direction.
Name:
VelocityZAbs
Symbol:
vz,abs
Unit:
m/s
Description: Velocity component in z-wise direction for absolute frame of reference.
Name:
ViscosityEddy
Symbol:
µT
Unit:
kg/(ms)
Description: Eddy viscosity as returned by active turbulence model.
37
Name:
ViscosityEddyRatio
Definition:
µT
µ
Unit:
-
Description: Eddy viscosity normalised by local molecular viscosity.
Name:
ViscosityKinematic
Symbol:
ν
Unit:
m2 /s
Description: Kinematic fluid viscosity.
Name:
ViscosityMolecular
Symbol:
µ
Definition:
ρν
Unit:
kg/(ms)
Description: Molecular fluid viscosity.
Name:
WaveNumberTheta
Symbol:
m
Unit:
-
Description: Circumferential wave number.
Name:
WaveNumberY
Symbol:
ky
Unit:
1/m
Description: Wave number in y direction.
Name:
WorkReduced
Symbol:
Wred
Definition:
ht,abs,out − ht,abs,in
Tt,abs,in
- for component
w
Tt,abs,in
- for stage
Unit:
J/(kgK)
Description: Reduced work.
Name:
Zweifel
Symbol:
Zw
Definition:
|
Unit:
m
ṁin Kin,abs + ṁout Kout,abs
|
(pt,in − pout )Aairfoil
Description: Loading criterion for turbines.
Remark:
Aairfoil is the blade surface projected into the S2m plane x-r.
38
3
3.1
Flow Angles
Cylindrical Coordinates
For rotating engines the rotation axis is assumed to be the x-axis. Therefore the natural cylindrical coordinates are (x, r, θ) defined by
x=x
y = r cos θ
(1)
z = r sin θ
z
r
x
y
Figure 1: Cylindrical Coordinates - looking in x-wise direction
3.2
General Flow Angle Definition
v
v vm
z
y
er
r
v
,abs
vabs
x
,abs
e
Figure 2: Turbomachinery: Coordinate Systems, Velocity Triangle, Angle Definition. Axis of Rotation: x
39
vm
r
r
x
vr
vx
Figure 3: Turbomachinery: Definition of Meridional Velocity and Radial Flow Angle
3.3
Cartesian Coordinates: Flow Angle Definition
Figure 4: Cartesian-based for translational mode
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Figure 5: Cylindrical-based for translational mode
3.4
Cylindrical Coordinates: Flow Angle Definition
Figure 6: Cylindrical-based for rotational mode - Turbomachines
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Figure 7: Projection-based for rotational mode - Turbomachines
s
v z co
n
v y si
vz
vyz
v = vz cos - vy sin
v
vr = vz sin + vy cos
vr
n
v z si
s
v y co
vy
z
y
x
Figure 8: Turbomachinery: Transformation Cartesian to Cylindrical Velocity Components
Note that (x, r, θ) define right-handed coordinates. The corresponding velocity components are (vx , vr , vθ ),
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vz
vyz
vy = vr cos - v sin
vr
v
vz= vr sin + v cos
vr sin
v cos
vy
v sin
z
y
vr cos
x
Figure 9: Turbomachinery: Transformation Cylindrical to Cartesian Velocity Components
where
vr =
(cos θ)vy + (sin θ)vz
vθ =
−(sin θ)vy + (cos θ)vz .
(2)
The meridional velocity (magnitude) is defined as
vmer =
p
vx2 + vr2 .
In the following, let arg(z1 , z2 ) ∈ [−180◦ , 180◦ ) be the angular coordinate of the point z1 + iz2 in the complex
plane1 , i.e.,
q
z1 = r(z1 , z2 ) cos (arg(z1 , z2 )) , z2 = r(z1 , z2 ) sin (arg(z1 , z2 )) , r(z1 , z2 ) = z12 + z22 .
For instance, the angular coordinate is the argument in the y, z-plane, i.e.,
θ = arg(y, z).
We note that
sgn arg(z1 , z2 ) = sgn z2 ,
unless z2 = 0 and z1 < 0.
3.5
Cylindrical Flow Angles
Consider the cylindrical coordinate system defined above. At a point (x, y, z) with r > 0, the angles
αr = arg(vx , vr ),
αθ = arg(vmer , vθ )
are called radial and circumferential (azimuthal, tangential) flow angles2 , resp., see Fig. ?? and ??. The
radial flow angle is defined for non-vanishing meridional velocity, i.e., vmer > 0.
1 arg(z , z ) corresponds to the function atan2 in the standard library math.h of C, applied to (z , z ), i.e., with reversed
1 2
2 1
arguments.
2 In TRACE these flow angles are termed “meridional based”.
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Note that αθ has the same sign as the circumferential component vθ . Hence, αθ is positive if and only if the
swirl is positive, using the right-hand rule.
Other definitions of circumferential flow angles, often used in turbomachinery, are
αθ∗ = arg(vθ , vmer ),
αz = arg(vθ , vx ).
Note that when vx is positive, then αz takes values between 0◦ and 180◦ . Moreover, αθ∗ increases with
decreasing vθ . The same holds for αz when vx > 0. We have
αθ = 90◦ − αθ∗ .
Another common definition of the radial flow angle is
q
2
2
vx + vθ , vr .
εcyl = arg
ε is another notation3 for αr .
3 In
TRACE, the pair of flow angles (αz , ε) is called “projection based”.
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