Research ArticleSolution of Turbine Blade Cascade Flow Usingan Improved Panel Method
Zong-qi Lei and Guo-zhu Liang
Department of Aerospace Propulsion Beijing University of Aeronautics and Astronautics Beijing 100191 China
Correspondence should be addressed to Zong-qi Lei leizongqigmailcom
Received 27 September 2015 Revised 19 November 2015 Accepted 22 November 2015
Academic Editor Linda L Vahala
Copyright copy 2015 Z-q Lei and G-z Liang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An improved panel method has been developed to calculate compressible inviscid flow through a turbine blade row The methodis a combination of the panel method for infinite cascade a deviation angle model and a compressibility correction The resultingsolution provides a fast flexible mesh-free calculation for cascade flow A VKI turbine blade cascade is used to evaluate the methodand the comparison with experiment data is presented
1 Introduction
The design of modern aeroengine gas turbine adopts variousnumerical methods to increase design efficiency At the pre-liminary design stage the major work for numerical methodis repetitive calculations of flow fields over a wide rangeof blade geometries This task has been dominated by fieldmethods such as finite differentialmethods and finite elementmethods with the advent of computers However the use ofthese field methods requires an experienced user to generatea body-fitted mesh which is labor intensive On the otherhand panel method only requires boundary meshes that areone dimension lower than the flow field reducing the workand difficulty for mesh generation enormously This methodis based on boundary integral equation it formulated theflow about arbitrary configurations as integration of analyticsolutions of singularity distribution over boundary surface[1] It was initially developed for incompressible potentialflow [2] Soon the implement of linearised potential equationendowed the method with the capability of solving subsonicand supersonic external flow [3] Various panelmethodsweredeveloped using different kind of singularities and higherorder panel elements since then eventually evolved into seriesof computer codes commonly in industrial use [4ndash7]
The main drawback of the panel method is the limitationof its application to linear potential flow To be specific theflow should either be incompressible or possess a sole free
stream as linearization reference But modern aeroengine gasturbines generally work at high subsonictransonic conditionand adopt blades with large deflection implying that (1) theincompressible assumption is not applicable and (2) the freestreams upstream and downstream of the blades are quite dif-ferentThere are two schemes to overcome this restriction thefield panel method that uses a field mesh to account for non-linear effects [8] or the correction correlations that transformthe incompressible solution to compressible solution Sincethe aim of this paper is to develop a mesh-free method thecorrection correlations are chosen as the scheme to be used
There are several forms of corrections based on freestream Mach number [2] Their combination with the panelmethod is straightforward and reliable [9] But as mentionedbefore the free stream Mach numbers upstream and down-stream of aeroengine gas turbine blades are not the sameLieblein and Stockman developed a correction for this cir-cumstance [10] which is deduced from empirical observationon the compressible flow in a turbine nozzle passage How-ever the error of this method is very large at high subsonicMach number when compared with experiment data
A method to rapidly calculate turbine blade cascade flowis presented in this paper The flow field is solved with thepanel method at first to obtain an incompressible solutionThen the free stream velocities upstream and downstreamare modified with a deviation angle model The compressiblesolution is obtained by applying compressibility corrections
Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 312430 6 pageshttpdxdoiorg1011552015312430
2 International Journal of Aerospace Engineering
y
x
Vonset
Vonset
120572in
120572out
Vin
Vout
Figure 1 Flow through infinite cascade
at each cross section with local average Mach number onthe cross section as a reference value Examples are given todemonstrate the capabilities of the method
2 Modeling Method
21 Panel Method The flow through an infinite cascade isshown in Figure 1 The governing equations and boundaryconditions for inviscid incompressible flow through an infi-nite cascade are as follows
nabla sdot V = 0 (1)
V sdot nblade surface = 0 (2)
V 997888rarr Vin as 119909 997888rarr minusinfin (3)
The solution is developed using a velocity potential that is thesum of a constant onset velocity potential plus a disturbanceinduced by the cascade The quantities of both are unknown
Φ = 120601onset + 120601dist (4)
V = minusnablaΦ = Vonset + Vdist (5)
Vonset = constant (6)
The onset velocity is constant so (1) (5) and (6) yield
nabla sdot Vdist = nabla2120601dist = 0 (7)
The flow field is determined by solving (1) subject to bound-ary conditions (2) and (3)
Laplacersquos equation governs the disturbance potential (7)Since it is a linear equation simpler solutions of Laplacersquosequation may be added together to develop solutions withhigher complexity A general solution to flow over a body orcascade of bodies may be developed by using basic incom-pressible potential flow solutions for source and vortex flows
Nodes
EndpointControl point
m + 1 m
n
n
tt
Panels
Nminus 1
N
2
1
Figure 2 Panel representation of blade
distributed along the body surfaces and varying the strengthof the source and vortex singularities so that the problemrsquosboundary conditions are satisfied
In this paper the surface of the body is represented byinscribing a polygon as shown in Figure 2 Flat panel elementswith constant source and vortex singularity strengths are usedfor simplicity The source strength varies for each elementwhile the vortex strength is identical over the whole bladesurface A control point is selected on each element centroidwhere the normal velocity boundary condition is to beappliedThere will be 119873 element endpoints and119873minus1 controlpoints All the endpoints are arranged clockwise The trailingedge is left open to avoid a velocity peak in the inviscidcalculation
The variables n and t are the unit normal and tangent vec-tors of the local panel elements respectively The velocity inthe flow field could be expressed in complex form as follows
V = 119881119909 minus i119881119910 =
119873
sum
119895=1
120590119895A119895
+ 120574
119873
sum
119895=1
B119895
+ Vonset (8)
where 120590119895is the source strength on the 119895th panel element
and 120574 is the vortex strength over blade surface A119895and B
119895
are complex influence factors of the source and vortex at the119895th panel element According to Hess and Smith [11] theirexpressions are
A119895
= minus119890minusi120573
2120587ln(
sinh [(120587pitch) [119911119895+1
minus 120577]]
sinh [(120587pitch) [119911119895
minus 120577]]
)
B119895
= iA119895
(9)
where 119911119895 119911119895+1
are the endpoints of the 119895th element 120573 is theargument of 119889119911 = 119911
119895+1minus 119911119895 120577 is the evaluated point and
pitch stands for the value of pitchApplying (2) at those control points would yield
V119894sdot n119894= 0 119894 = 1 119873 (10)
International Journal of Aerospace Engineering 3
Another boundary condition is the upstream boundary con-dition (3) For a nominalized velocity field the inlet velocitycould be expressed as follows
Vin = cos120572in minus i sin120572in (11)
If the circulation over the blade is Γ (the sum of the vortexstrength over the blade) its equation is
Γ = 120574
119873
sum
119895=1
119897119895
Vin = 119881119909in minus i119881119910in = 119881119909onset minus i(119881119910onset +Γ
2pitch)
(12)
where 119897119895is the length of the 119895th panel element So the
upstream boundary condition could be expressed as
119881119909in = cos120572in
119881119910in = 119881119910onset +
120574 sum119873
119895=1119897119895
2pitch
(13)
For airfoil inviscid calculations a Kutta condition must beapplied at the trailing edge
(V1
sdot t1) + (V
119873sdot t119873
) = 0 (14)
Equations (10) (13) and (14) compose a linear equation groupthat would yield the values of the singularity strength andVonset fromwhich the velocity at any position can be obtainedby (8)
22 Compressibility Correction Liebleinrsquos correction forinternal flow is based on the flow status of each cross section
119881119888
= 119881119894(
120588119894
120588119888
)
119881119894119881119894
(15)
Liebleinrsquos formula was derived from empirical observationover a turbine nozzle [10] As shown later in the paper thisdoes not match with experimental data well However thisformula indicates the importance of considering the status oflocal flowpaths in the compressibility correction correlationsThus a new compressibility correction is developed in thispaper a reference Mach number at the evaluated cross sec-tion is calculated first and then is used to transform the localincompressible solution into a compressible solution usingthe formula for small disturbance flow such as Karman-Tsienformula
119862119901 =1198621199010
radic1 minus 1198722infin
+ (1198722infin
(1 + radic1 minus 1198722infin
)) (11986211990102)
(16)
Assume there is a virtual flow path where the blade thicknessis neglected and the mass flow rate and average flow angleare equal to those of real blades as shown in Figure 3 withdash-dotted line 119878119875 is the cross section in the flowpathwherethe compressibility correction to be applied 11987810158401198751015840 is the cross
120572in
120572out
120572ref
P998400
S998400
SMin
Mout
Mref
P
Figure 3 Cross section for compressibility correction
section of that virtual flow path at the same axial location120572ref and119872ref are the average flow angle and the averageMachnumbers at 119878
10158401198751015840 According to mass conservation there is
(1 + ((119896 minus 1) 2) 119872
2
out1 + ((119896 minus 1) 2) 119872
2
ref)
1(119896minus1)
119872ref119872out
=cos120572outcos120572ref
(17)
When119872ref is calculated using (17) (16) may be used to trans-form incompressible solutions into compressible solutions
23 Deviation Angle Model Equation (17) indicates that theexit flow angle 120572out must be obtained in advance to calcu-late 119872ref However in practice the downstream boundarycondition is usually back pressure 119901out or exit Mach number119872out rather than 120572out The panel method mentioned above isonly able to provide the incompressible exit flow angle thevalue of which is obviously different from compressible flowUnder this circumstance a deviation angle model based onmomentum balance is introduced to calculate 120572out
Consider the pressure distribution on the suction andpressure surface of a turbine blade row flow path shown inFigure 4The circumferential momentum equation of controlvolume 119860119861119862119863119864 is
where OP is the opening width the length of 119862119863
4 International Journal of Aerospace Engineering
120572out
120572op
Vout
Suction surface
p
D
B
E
A
C
pC = pE
Pressure surface
pD
Vop
x
Figure 4 Control volume
The expansion from 119862119863 to 119860119861 is assumed to be isen-tropic According to the compressible version of Bernoullirsquosequation
(
Vop
Vout)
2
=2
(119896 minus 1) 1198722119890
(1 minus (
119901op
119901119890
)
(119896minus1)119896
) + 1 (21)
Reorganizing (19) (20) and (21) yields
((sin120572outsin120572op
)
2
minus 1)119896 minus 1
21198722
119890
= 1 minus (
sin120572op
tan120572out
pitchOP
)
119896minus1
(22)
Since 120572op and pitchOP can be obtained from the bladegeometry (22) can be solved numerically to provide 120572out
3 Comparison of Results
VKI LS 59 turbine cascade data [12] is used to evaluatethe modeling method for that its geometry and workingcondition are similar to those of the aeroengine turbineblades Liebleinrsquos method is also used for referenceThe bladegeometry and general parameters are shown in Figure 5 andTable 1 A FORTRAN computer code of the new method wasdeveloped for the calculation The blade was approximatedwith 50 elements and the solution required less than 1 secondof computer time using a 26GHz Pentium CPU core
Table 1 Blade parameters
Parameter ValuePitchchord 071Install angle 120573
11990433∘
Inlet flow angle 120572in 30∘
120572in
120573s
Pitch
Chord
Figure 5 Blade geometry
31 Inlet Mach Number In Figure 6 the prediction of theinlet Mach number is compared between the new methodLiebleinrsquos method and experimental data As the experimentdata shows the mass flow will not increase with the exitMach number as the latter approaches unityThenewmethodshows better consistency with experimental results
32 Exit Flow Angle The comparison of the exit flow angle isshown in Figure 7 The exit angle of Liebleinrsquos method doesnot vary with exit Mach number since it conserves the massflow rate of the incompressible solution which is fixed for agiven inlet flow angle but disagrees with the true value whencompressibility effect is strong In this case the new methodalso provides better agreement
33 Surface Mach Number Figure 8 shows the comparisonof the blade surface Mach number distribution The Machnumber given by Liebleinrsquosmethod overpredicts the data overthe entire blade surface On the other hand the new methodcompares well with the experimental data for the majority ofthe blade surface
International Journal of Aerospace Engineering 5
Present methodLiebleinrsquos methodExperiment
015
020
025
030
035
Inle
t Mac
h nu
mbe
rMin
05 07 09 1103
Exit Mach number Mout
Figure 6 Inlet Mach number distribution
05 07 09 1103
Exit Mach number Mout
60
65
70
Exit
flow
angl
e120572ou
t
Present methodLiebleinrsquos methodExperiment
Figure 7 Exit flow angle distribution
4 Summary
The panel method has been adopted to calculate the flowthrough turbine blades The inherent computational speedand flexibility of the integral equation solution can make thismethod useful for design calculationsThemethod presentedcombines a panel method a deviation angle model and acompressibility correction to yield a compressible solutionComparison with experiment shows that this method is suf-ficiently accurate to provide a means of selecting aeroengineturbine blade designs for further analysis
Present methodLiebleinrsquos methodExperiment
02 04 06 08 1000
00
05
10
Surfa
ce M
ach
num
ber
Axial blade coordinate (xchord)
Figure 8 Blade surface Mach number distribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990
[2] J L Hess and A M O Smith ldquoCalculation of non-liftingpotential flow about arbitrary three-dimensional bodiesrdquo ES40622 Douglas Aircraft Division 1962
[3] F A Woodward ldquoAnalysis and design of Wing-Body combina-tions at subsonic and supersonic speedrdquo Journal of Aircraft vol5 no 6 pp 528ndash534 1968
[4] L Morino ldquoOscillatory and unsteady subsonic and supersonicaerodynamicsmdashproduction version (SOUSSA-P11) vol 1 the-oretical manualrdquo NASA CR-159130 1980
[5] R L Carmichael and L L Erickson ldquoPAN AIRmdasha higherorder panelmethod for predicting subsonic or supersonic linearpotential flows about arbitrary configurationsrdquo in Proceedingsof the 14th Fluid and Plasma Dynamics Conference AIAA Paper81-1255 Palo Alto Calif USA 1981
[6] L Fornasier ldquoHISSmdasha higer order subsonicsupersonic singu-larity method for calculating linearized potential flowrdquo AIAAPaper 84-1646 1984
[7] B Maskew ldquoProgram VSAERO theory documentrdquo NASA CR4023 1987
[8] L Gebhardt D Fokin T Lutz and S Wagner ldquoAn implicit-explicit dirichlet-based field panel method for transonic aircraftdesignrdquo in Proceedings of the 20th AIAA Applied AerodynamicsConference AIAA 2002-3145 St Louis Mo USA June 2002
[9] M Drela ldquoXFOIL an analysis and design system for lowreynolds number airfoilsrdquo in Low Reynolds Number Aerody-namics T J Mueller Ed vol 54 of Lecture Notes in Engineeringpp 1ndash12 Springer Berlin Germany 1989
6 International Journal of Aerospace Engineering
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
at each cross section with local average Mach number onthe cross section as a reference value Examples are given todemonstrate the capabilities of the method
2 Modeling Method
21 Panel Method The flow through an infinite cascade isshown in Figure 1 The governing equations and boundaryconditions for inviscid incompressible flow through an infi-nite cascade are as follows
nabla sdot V = 0 (1)
V sdot nblade surface = 0 (2)
V 997888rarr Vin as 119909 997888rarr minusinfin (3)
The solution is developed using a velocity potential that is thesum of a constant onset velocity potential plus a disturbanceinduced by the cascade The quantities of both are unknown
Φ = 120601onset + 120601dist (4)
V = minusnablaΦ = Vonset + Vdist (5)
Vonset = constant (6)
The onset velocity is constant so (1) (5) and (6) yield
nabla sdot Vdist = nabla2120601dist = 0 (7)
The flow field is determined by solving (1) subject to bound-ary conditions (2) and (3)
Laplacersquos equation governs the disturbance potential (7)Since it is a linear equation simpler solutions of Laplacersquosequation may be added together to develop solutions withhigher complexity A general solution to flow over a body orcascade of bodies may be developed by using basic incom-pressible potential flow solutions for source and vortex flows
Nodes
EndpointControl point
m + 1 m
n
n
tt
Panels
Nminus 1
N
2
1
Figure 2 Panel representation of blade
distributed along the body surfaces and varying the strengthof the source and vortex singularities so that the problemrsquosboundary conditions are satisfied
In this paper the surface of the body is represented byinscribing a polygon as shown in Figure 2 Flat panel elementswith constant source and vortex singularity strengths are usedfor simplicity The source strength varies for each elementwhile the vortex strength is identical over the whole bladesurface A control point is selected on each element centroidwhere the normal velocity boundary condition is to beappliedThere will be 119873 element endpoints and119873minus1 controlpoints All the endpoints are arranged clockwise The trailingedge is left open to avoid a velocity peak in the inviscidcalculation
The variables n and t are the unit normal and tangent vec-tors of the local panel elements respectively The velocity inthe flow field could be expressed in complex form as follows
V = 119881119909 minus i119881119910 =
119873
sum
119895=1
120590119895A119895
+ 120574
119873
sum
119895=1
B119895
+ Vonset (8)
where 120590119895is the source strength on the 119895th panel element
and 120574 is the vortex strength over blade surface A119895and B
119895
are complex influence factors of the source and vortex at the119895th panel element According to Hess and Smith [11] theirexpressions are
A119895
= minus119890minusi120573
2120587ln(
sinh [(120587pitch) [119911119895+1
minus 120577]]
sinh [(120587pitch) [119911119895
minus 120577]]
)
B119895
= iA119895
(9)
where 119911119895 119911119895+1
are the endpoints of the 119895th element 120573 is theargument of 119889119911 = 119911
119895+1minus 119911119895 120577 is the evaluated point and
pitch stands for the value of pitchApplying (2) at those control points would yield
V119894sdot n119894= 0 119894 = 1 119873 (10)
International Journal of Aerospace Engineering 3
Another boundary condition is the upstream boundary con-dition (3) For a nominalized velocity field the inlet velocitycould be expressed as follows
Vin = cos120572in minus i sin120572in (11)
If the circulation over the blade is Γ (the sum of the vortexstrength over the blade) its equation is
Γ = 120574
119873
sum
119895=1
119897119895
Vin = 119881119909in minus i119881119910in = 119881119909onset minus i(119881119910onset +Γ
2pitch)
(12)
where 119897119895is the length of the 119895th panel element So the
upstream boundary condition could be expressed as
119881119909in = cos120572in
119881119910in = 119881119910onset +
120574 sum119873
119895=1119897119895
2pitch
(13)
For airfoil inviscid calculations a Kutta condition must beapplied at the trailing edge
(V1
sdot t1) + (V
119873sdot t119873
) = 0 (14)
Equations (10) (13) and (14) compose a linear equation groupthat would yield the values of the singularity strength andVonset fromwhich the velocity at any position can be obtainedby (8)
22 Compressibility Correction Liebleinrsquos correction forinternal flow is based on the flow status of each cross section
119881119888
= 119881119894(
120588119894
120588119888
)
119881119894119881119894
(15)
Liebleinrsquos formula was derived from empirical observationover a turbine nozzle [10] As shown later in the paper thisdoes not match with experimental data well However thisformula indicates the importance of considering the status oflocal flowpaths in the compressibility correction correlationsThus a new compressibility correction is developed in thispaper a reference Mach number at the evaluated cross sec-tion is calculated first and then is used to transform the localincompressible solution into a compressible solution usingthe formula for small disturbance flow such as Karman-Tsienformula
119862119901 =1198621199010
radic1 minus 1198722infin
+ (1198722infin
(1 + radic1 minus 1198722infin
)) (11986211990102)
(16)
Assume there is a virtual flow path where the blade thicknessis neglected and the mass flow rate and average flow angleare equal to those of real blades as shown in Figure 3 withdash-dotted line 119878119875 is the cross section in the flowpathwherethe compressibility correction to be applied 11987810158401198751015840 is the cross
120572in
120572out
120572ref
P998400
S998400
SMin
Mout
Mref
P
Figure 3 Cross section for compressibility correction
section of that virtual flow path at the same axial location120572ref and119872ref are the average flow angle and the averageMachnumbers at 119878
10158401198751015840 According to mass conservation there is
(1 + ((119896 minus 1) 2) 119872
2
out1 + ((119896 minus 1) 2) 119872
2
ref)
1(119896minus1)
119872ref119872out
=cos120572outcos120572ref
(17)
When119872ref is calculated using (17) (16) may be used to trans-form incompressible solutions into compressible solutions
23 Deviation Angle Model Equation (17) indicates that theexit flow angle 120572out must be obtained in advance to calcu-late 119872ref However in practice the downstream boundarycondition is usually back pressure 119901out or exit Mach number119872out rather than 120572out The panel method mentioned above isonly able to provide the incompressible exit flow angle thevalue of which is obviously different from compressible flowUnder this circumstance a deviation angle model based onmomentum balance is introduced to calculate 120572out
Consider the pressure distribution on the suction andpressure surface of a turbine blade row flow path shown inFigure 4The circumferential momentum equation of controlvolume 119860119861119862119863119864 is
where OP is the opening width the length of 119862119863
4 International Journal of Aerospace Engineering
120572out
120572op
Vout
Suction surface
p
D
B
E
A
C
pC = pE
Pressure surface
pD
Vop
x
Figure 4 Control volume
The expansion from 119862119863 to 119860119861 is assumed to be isen-tropic According to the compressible version of Bernoullirsquosequation
(
Vop
Vout)
2
=2
(119896 minus 1) 1198722119890
(1 minus (
119901op
119901119890
)
(119896minus1)119896
) + 1 (21)
Reorganizing (19) (20) and (21) yields
((sin120572outsin120572op
)
2
minus 1)119896 minus 1
21198722
119890
= 1 minus (
sin120572op
tan120572out
pitchOP
)
119896minus1
(22)
Since 120572op and pitchOP can be obtained from the bladegeometry (22) can be solved numerically to provide 120572out
3 Comparison of Results
VKI LS 59 turbine cascade data [12] is used to evaluatethe modeling method for that its geometry and workingcondition are similar to those of the aeroengine turbineblades Liebleinrsquos method is also used for referenceThe bladegeometry and general parameters are shown in Figure 5 andTable 1 A FORTRAN computer code of the new method wasdeveloped for the calculation The blade was approximatedwith 50 elements and the solution required less than 1 secondof computer time using a 26GHz Pentium CPU core
Table 1 Blade parameters
Parameter ValuePitchchord 071Install angle 120573
11990433∘
Inlet flow angle 120572in 30∘
120572in
120573s
Pitch
Chord
Figure 5 Blade geometry
31 Inlet Mach Number In Figure 6 the prediction of theinlet Mach number is compared between the new methodLiebleinrsquos method and experimental data As the experimentdata shows the mass flow will not increase with the exitMach number as the latter approaches unityThenewmethodshows better consistency with experimental results
32 Exit Flow Angle The comparison of the exit flow angle isshown in Figure 7 The exit angle of Liebleinrsquos method doesnot vary with exit Mach number since it conserves the massflow rate of the incompressible solution which is fixed for agiven inlet flow angle but disagrees with the true value whencompressibility effect is strong In this case the new methodalso provides better agreement
33 Surface Mach Number Figure 8 shows the comparisonof the blade surface Mach number distribution The Machnumber given by Liebleinrsquosmethod overpredicts the data overthe entire blade surface On the other hand the new methodcompares well with the experimental data for the majority ofthe blade surface
International Journal of Aerospace Engineering 5
Present methodLiebleinrsquos methodExperiment
015
020
025
030
035
Inle
t Mac
h nu
mbe
rMin
05 07 09 1103
Exit Mach number Mout
Figure 6 Inlet Mach number distribution
05 07 09 1103
Exit Mach number Mout
60
65
70
Exit
flow
angl
e120572ou
t
Present methodLiebleinrsquos methodExperiment
Figure 7 Exit flow angle distribution
4 Summary
The panel method has been adopted to calculate the flowthrough turbine blades The inherent computational speedand flexibility of the integral equation solution can make thismethod useful for design calculationsThemethod presentedcombines a panel method a deviation angle model and acompressibility correction to yield a compressible solutionComparison with experiment shows that this method is suf-ficiently accurate to provide a means of selecting aeroengineturbine blade designs for further analysis
Present methodLiebleinrsquos methodExperiment
02 04 06 08 1000
00
05
10
Surfa
ce M
ach
num
ber
Axial blade coordinate (xchord)
Figure 8 Blade surface Mach number distribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990
[2] J L Hess and A M O Smith ldquoCalculation of non-liftingpotential flow about arbitrary three-dimensional bodiesrdquo ES40622 Douglas Aircraft Division 1962
[3] F A Woodward ldquoAnalysis and design of Wing-Body combina-tions at subsonic and supersonic speedrdquo Journal of Aircraft vol5 no 6 pp 528ndash534 1968
[4] L Morino ldquoOscillatory and unsteady subsonic and supersonicaerodynamicsmdashproduction version (SOUSSA-P11) vol 1 the-oretical manualrdquo NASA CR-159130 1980
[5] R L Carmichael and L L Erickson ldquoPAN AIRmdasha higherorder panelmethod for predicting subsonic or supersonic linearpotential flows about arbitrary configurationsrdquo in Proceedingsof the 14th Fluid and Plasma Dynamics Conference AIAA Paper81-1255 Palo Alto Calif USA 1981
[6] L Fornasier ldquoHISSmdasha higer order subsonicsupersonic singu-larity method for calculating linearized potential flowrdquo AIAAPaper 84-1646 1984
[7] B Maskew ldquoProgram VSAERO theory documentrdquo NASA CR4023 1987
[8] L Gebhardt D Fokin T Lutz and S Wagner ldquoAn implicit-explicit dirichlet-based field panel method for transonic aircraftdesignrdquo in Proceedings of the 20th AIAA Applied AerodynamicsConference AIAA 2002-3145 St Louis Mo USA June 2002
[9] M Drela ldquoXFOIL an analysis and design system for lowreynolds number airfoilsrdquo in Low Reynolds Number Aerody-namics T J Mueller Ed vol 54 of Lecture Notes in Engineeringpp 1ndash12 Springer Berlin Germany 1989
6 International Journal of Aerospace Engineering
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
Another boundary condition is the upstream boundary con-dition (3) For a nominalized velocity field the inlet velocitycould be expressed as follows
Vin = cos120572in minus i sin120572in (11)
If the circulation over the blade is Γ (the sum of the vortexstrength over the blade) its equation is
Γ = 120574
119873
sum
119895=1
119897119895
Vin = 119881119909in minus i119881119910in = 119881119909onset minus i(119881119910onset +Γ
2pitch)
(12)
where 119897119895is the length of the 119895th panel element So the
upstream boundary condition could be expressed as
119881119909in = cos120572in
119881119910in = 119881119910onset +
120574 sum119873
119895=1119897119895
2pitch
(13)
For airfoil inviscid calculations a Kutta condition must beapplied at the trailing edge
(V1
sdot t1) + (V
119873sdot t119873
) = 0 (14)
Equations (10) (13) and (14) compose a linear equation groupthat would yield the values of the singularity strength andVonset fromwhich the velocity at any position can be obtainedby (8)
22 Compressibility Correction Liebleinrsquos correction forinternal flow is based on the flow status of each cross section
119881119888
= 119881119894(
120588119894
120588119888
)
119881119894119881119894
(15)
Liebleinrsquos formula was derived from empirical observationover a turbine nozzle [10] As shown later in the paper thisdoes not match with experimental data well However thisformula indicates the importance of considering the status oflocal flowpaths in the compressibility correction correlationsThus a new compressibility correction is developed in thispaper a reference Mach number at the evaluated cross sec-tion is calculated first and then is used to transform the localincompressible solution into a compressible solution usingthe formula for small disturbance flow such as Karman-Tsienformula
119862119901 =1198621199010
radic1 minus 1198722infin
+ (1198722infin
(1 + radic1 minus 1198722infin
)) (11986211990102)
(16)
Assume there is a virtual flow path where the blade thicknessis neglected and the mass flow rate and average flow angleare equal to those of real blades as shown in Figure 3 withdash-dotted line 119878119875 is the cross section in the flowpathwherethe compressibility correction to be applied 11987810158401198751015840 is the cross
120572in
120572out
120572ref
P998400
S998400
SMin
Mout
Mref
P
Figure 3 Cross section for compressibility correction
section of that virtual flow path at the same axial location120572ref and119872ref are the average flow angle and the averageMachnumbers at 119878
10158401198751015840 According to mass conservation there is
(1 + ((119896 minus 1) 2) 119872
2
out1 + ((119896 minus 1) 2) 119872
2
ref)
1(119896minus1)
119872ref119872out
=cos120572outcos120572ref
(17)
When119872ref is calculated using (17) (16) may be used to trans-form incompressible solutions into compressible solutions
23 Deviation Angle Model Equation (17) indicates that theexit flow angle 120572out must be obtained in advance to calcu-late 119872ref However in practice the downstream boundarycondition is usually back pressure 119901out or exit Mach number119872out rather than 120572out The panel method mentioned above isonly able to provide the incompressible exit flow angle thevalue of which is obviously different from compressible flowUnder this circumstance a deviation angle model based onmomentum balance is introduced to calculate 120572out
Consider the pressure distribution on the suction andpressure surface of a turbine blade row flow path shown inFigure 4The circumferential momentum equation of controlvolume 119860119861119862119863119864 is
where OP is the opening width the length of 119862119863
4 International Journal of Aerospace Engineering
120572out
120572op
Vout
Suction surface
p
D
B
E
A
C
pC = pE
Pressure surface
pD
Vop
x
Figure 4 Control volume
The expansion from 119862119863 to 119860119861 is assumed to be isen-tropic According to the compressible version of Bernoullirsquosequation
(
Vop
Vout)
2
=2
(119896 minus 1) 1198722119890
(1 minus (
119901op
119901119890
)
(119896minus1)119896
) + 1 (21)
Reorganizing (19) (20) and (21) yields
((sin120572outsin120572op
)
2
minus 1)119896 minus 1
21198722
119890
= 1 minus (
sin120572op
tan120572out
pitchOP
)
119896minus1
(22)
Since 120572op and pitchOP can be obtained from the bladegeometry (22) can be solved numerically to provide 120572out
3 Comparison of Results
VKI LS 59 turbine cascade data [12] is used to evaluatethe modeling method for that its geometry and workingcondition are similar to those of the aeroengine turbineblades Liebleinrsquos method is also used for referenceThe bladegeometry and general parameters are shown in Figure 5 andTable 1 A FORTRAN computer code of the new method wasdeveloped for the calculation The blade was approximatedwith 50 elements and the solution required less than 1 secondof computer time using a 26GHz Pentium CPU core
Table 1 Blade parameters
Parameter ValuePitchchord 071Install angle 120573
11990433∘
Inlet flow angle 120572in 30∘
120572in
120573s
Pitch
Chord
Figure 5 Blade geometry
31 Inlet Mach Number In Figure 6 the prediction of theinlet Mach number is compared between the new methodLiebleinrsquos method and experimental data As the experimentdata shows the mass flow will not increase with the exitMach number as the latter approaches unityThenewmethodshows better consistency with experimental results
32 Exit Flow Angle The comparison of the exit flow angle isshown in Figure 7 The exit angle of Liebleinrsquos method doesnot vary with exit Mach number since it conserves the massflow rate of the incompressible solution which is fixed for agiven inlet flow angle but disagrees with the true value whencompressibility effect is strong In this case the new methodalso provides better agreement
33 Surface Mach Number Figure 8 shows the comparisonof the blade surface Mach number distribution The Machnumber given by Liebleinrsquosmethod overpredicts the data overthe entire blade surface On the other hand the new methodcompares well with the experimental data for the majority ofthe blade surface
International Journal of Aerospace Engineering 5
Present methodLiebleinrsquos methodExperiment
015
020
025
030
035
Inle
t Mac
h nu
mbe
rMin
05 07 09 1103
Exit Mach number Mout
Figure 6 Inlet Mach number distribution
05 07 09 1103
Exit Mach number Mout
60
65
70
Exit
flow
angl
e120572ou
t
Present methodLiebleinrsquos methodExperiment
Figure 7 Exit flow angle distribution
4 Summary
The panel method has been adopted to calculate the flowthrough turbine blades The inherent computational speedand flexibility of the integral equation solution can make thismethod useful for design calculationsThemethod presentedcombines a panel method a deviation angle model and acompressibility correction to yield a compressible solutionComparison with experiment shows that this method is suf-ficiently accurate to provide a means of selecting aeroengineturbine blade designs for further analysis
Present methodLiebleinrsquos methodExperiment
02 04 06 08 1000
00
05
10
Surfa
ce M
ach
num
ber
Axial blade coordinate (xchord)
Figure 8 Blade surface Mach number distribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990
[2] J L Hess and A M O Smith ldquoCalculation of non-liftingpotential flow about arbitrary three-dimensional bodiesrdquo ES40622 Douglas Aircraft Division 1962
[3] F A Woodward ldquoAnalysis and design of Wing-Body combina-tions at subsonic and supersonic speedrdquo Journal of Aircraft vol5 no 6 pp 528ndash534 1968
[4] L Morino ldquoOscillatory and unsteady subsonic and supersonicaerodynamicsmdashproduction version (SOUSSA-P11) vol 1 the-oretical manualrdquo NASA CR-159130 1980
[5] R L Carmichael and L L Erickson ldquoPAN AIRmdasha higherorder panelmethod for predicting subsonic or supersonic linearpotential flows about arbitrary configurationsrdquo in Proceedingsof the 14th Fluid and Plasma Dynamics Conference AIAA Paper81-1255 Palo Alto Calif USA 1981
[6] L Fornasier ldquoHISSmdasha higer order subsonicsupersonic singu-larity method for calculating linearized potential flowrdquo AIAAPaper 84-1646 1984
[7] B Maskew ldquoProgram VSAERO theory documentrdquo NASA CR4023 1987
[8] L Gebhardt D Fokin T Lutz and S Wagner ldquoAn implicit-explicit dirichlet-based field panel method for transonic aircraftdesignrdquo in Proceedings of the 20th AIAA Applied AerodynamicsConference AIAA 2002-3145 St Louis Mo USA June 2002
[9] M Drela ldquoXFOIL an analysis and design system for lowreynolds number airfoilsrdquo in Low Reynolds Number Aerody-namics T J Mueller Ed vol 54 of Lecture Notes in Engineeringpp 1ndash12 Springer Berlin Germany 1989
6 International Journal of Aerospace Engineering
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
The expansion from 119862119863 to 119860119861 is assumed to be isen-tropic According to the compressible version of Bernoullirsquosequation
(
Vop
Vout)
2
=2
(119896 minus 1) 1198722119890
(1 minus (
119901op
119901119890
)
(119896minus1)119896
) + 1 (21)
Reorganizing (19) (20) and (21) yields
((sin120572outsin120572op
)
2
minus 1)119896 minus 1
21198722
119890
= 1 minus (
sin120572op
tan120572out
pitchOP
)
119896minus1
(22)
Since 120572op and pitchOP can be obtained from the bladegeometry (22) can be solved numerically to provide 120572out
3 Comparison of Results
VKI LS 59 turbine cascade data [12] is used to evaluatethe modeling method for that its geometry and workingcondition are similar to those of the aeroengine turbineblades Liebleinrsquos method is also used for referenceThe bladegeometry and general parameters are shown in Figure 5 andTable 1 A FORTRAN computer code of the new method wasdeveloped for the calculation The blade was approximatedwith 50 elements and the solution required less than 1 secondof computer time using a 26GHz Pentium CPU core
Table 1 Blade parameters
Parameter ValuePitchchord 071Install angle 120573
11990433∘
Inlet flow angle 120572in 30∘
120572in
120573s
Pitch
Chord
Figure 5 Blade geometry
31 Inlet Mach Number In Figure 6 the prediction of theinlet Mach number is compared between the new methodLiebleinrsquos method and experimental data As the experimentdata shows the mass flow will not increase with the exitMach number as the latter approaches unityThenewmethodshows better consistency with experimental results
32 Exit Flow Angle The comparison of the exit flow angle isshown in Figure 7 The exit angle of Liebleinrsquos method doesnot vary with exit Mach number since it conserves the massflow rate of the incompressible solution which is fixed for agiven inlet flow angle but disagrees with the true value whencompressibility effect is strong In this case the new methodalso provides better agreement
33 Surface Mach Number Figure 8 shows the comparisonof the blade surface Mach number distribution The Machnumber given by Liebleinrsquosmethod overpredicts the data overthe entire blade surface On the other hand the new methodcompares well with the experimental data for the majority ofthe blade surface
International Journal of Aerospace Engineering 5
Present methodLiebleinrsquos methodExperiment
015
020
025
030
035
Inle
t Mac
h nu
mbe
rMin
05 07 09 1103
Exit Mach number Mout
Figure 6 Inlet Mach number distribution
05 07 09 1103
Exit Mach number Mout
60
65
70
Exit
flow
angl
e120572ou
t
Present methodLiebleinrsquos methodExperiment
Figure 7 Exit flow angle distribution
4 Summary
The panel method has been adopted to calculate the flowthrough turbine blades The inherent computational speedand flexibility of the integral equation solution can make thismethod useful for design calculationsThemethod presentedcombines a panel method a deviation angle model and acompressibility correction to yield a compressible solutionComparison with experiment shows that this method is suf-ficiently accurate to provide a means of selecting aeroengineturbine blade designs for further analysis
Present methodLiebleinrsquos methodExperiment
02 04 06 08 1000
00
05
10
Surfa
ce M
ach
num
ber
Axial blade coordinate (xchord)
Figure 8 Blade surface Mach number distribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990
[2] J L Hess and A M O Smith ldquoCalculation of non-liftingpotential flow about arbitrary three-dimensional bodiesrdquo ES40622 Douglas Aircraft Division 1962
[3] F A Woodward ldquoAnalysis and design of Wing-Body combina-tions at subsonic and supersonic speedrdquo Journal of Aircraft vol5 no 6 pp 528ndash534 1968
[4] L Morino ldquoOscillatory and unsteady subsonic and supersonicaerodynamicsmdashproduction version (SOUSSA-P11) vol 1 the-oretical manualrdquo NASA CR-159130 1980
[5] R L Carmichael and L L Erickson ldquoPAN AIRmdasha higherorder panelmethod for predicting subsonic or supersonic linearpotential flows about arbitrary configurationsrdquo in Proceedingsof the 14th Fluid and Plasma Dynamics Conference AIAA Paper81-1255 Palo Alto Calif USA 1981
[6] L Fornasier ldquoHISSmdasha higer order subsonicsupersonic singu-larity method for calculating linearized potential flowrdquo AIAAPaper 84-1646 1984
[7] B Maskew ldquoProgram VSAERO theory documentrdquo NASA CR4023 1987
[8] L Gebhardt D Fokin T Lutz and S Wagner ldquoAn implicit-explicit dirichlet-based field panel method for transonic aircraftdesignrdquo in Proceedings of the 20th AIAA Applied AerodynamicsConference AIAA 2002-3145 St Louis Mo USA June 2002
[9] M Drela ldquoXFOIL an analysis and design system for lowreynolds number airfoilsrdquo in Low Reynolds Number Aerody-namics T J Mueller Ed vol 54 of Lecture Notes in Engineeringpp 1ndash12 Springer Berlin Germany 1989
6 International Journal of Aerospace Engineering
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
The panel method has been adopted to calculate the flowthrough turbine blades The inherent computational speedand flexibility of the integral equation solution can make thismethod useful for design calculationsThemethod presentedcombines a panel method a deviation angle model and acompressibility correction to yield a compressible solutionComparison with experiment shows that this method is suf-ficiently accurate to provide a means of selecting aeroengineturbine blade designs for further analysis
Present methodLiebleinrsquos methodExperiment
02 04 06 08 1000
00
05
10
Surfa
ce M
ach
num
ber
Axial blade coordinate (xchord)
Figure 8 Blade surface Mach number distribution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990
[2] J L Hess and A M O Smith ldquoCalculation of non-liftingpotential flow about arbitrary three-dimensional bodiesrdquo ES40622 Douglas Aircraft Division 1962
[3] F A Woodward ldquoAnalysis and design of Wing-Body combina-tions at subsonic and supersonic speedrdquo Journal of Aircraft vol5 no 6 pp 528ndash534 1968
[4] L Morino ldquoOscillatory and unsteady subsonic and supersonicaerodynamicsmdashproduction version (SOUSSA-P11) vol 1 the-oretical manualrdquo NASA CR-159130 1980
[5] R L Carmichael and L L Erickson ldquoPAN AIRmdasha higherorder panelmethod for predicting subsonic or supersonic linearpotential flows about arbitrary configurationsrdquo in Proceedingsof the 14th Fluid and Plasma Dynamics Conference AIAA Paper81-1255 Palo Alto Calif USA 1981
[6] L Fornasier ldquoHISSmdasha higer order subsonicsupersonic singu-larity method for calculating linearized potential flowrdquo AIAAPaper 84-1646 1984
[7] B Maskew ldquoProgram VSAERO theory documentrdquo NASA CR4023 1987
[8] L Gebhardt D Fokin T Lutz and S Wagner ldquoAn implicit-explicit dirichlet-based field panel method for transonic aircraftdesignrdquo in Proceedings of the 20th AIAA Applied AerodynamicsConference AIAA 2002-3145 St Louis Mo USA June 2002
[9] M Drela ldquoXFOIL an analysis and design system for lowreynolds number airfoilsrdquo in Low Reynolds Number Aerody-namics T J Mueller Ed vol 54 of Lecture Notes in Engineeringpp 1ndash12 Springer Berlin Germany 1989
6 International Journal of Aerospace Engineering
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
[10] S Lieblein andN O Stockman ldquoCompressibility correction forinternal flow solutionsrdquo Journal of Aircraft vol 9 no 4 pp 312ndash313 1972
[11] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967
[12] R Kiock F Lehthaus N C Baines and C H Sieverding ldquoThetransonic flow through a plane turbine cascade as measuredin four european wind tunnelsrdquo Journal of Engineering for GasTurbines and Power vol 108 no 2 pp 277ndash284 1986
