research article turbulent kinetic energy production in the vane...

Research ArticleTurbulent Kinetic Energy Production in the Vane ofa Low-Pressure Linear Turbine Cascade with Incoming Wakes

V Michelassi and J G Wissink

Institut fur Hydromechanik Universitat Karlsruhe Kaiserstrasse 12 76128 Karlsruhe Germany

Correspondence should be addressed to V Michelassi vittoriomichelassigmailcom

Received 5 November 2014 Accepted 4 January 2015

Academic Editor Funazaki Ken-ichi

Copyright copy 2015 V Michelassi and J G Wissink This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Incompressible large eddy simulation anddirect numerical simulation of a low-pressure turbine at Re = 518times104 and 148times105 withdiscrete incoming wakes are analyzed to identify the turbulent kinetic energy generationmechanism outside of the blade boundarylayer The results highlight the growth of turbulent kinetic energy at the bow apex of the wake and correlate it to the stress-straintensors relative orientation The production rate is analytically split according to the principal axes and then terms are computedby using the simulation resultsThe analysis of the turbulent kinetic energy is followed both along the discrete incoming wakes andin the stationary frame of reference Both direct numerical and large eddy simulation concur in identifying the same productionmechanism that is driven by both a growth of strain rate in the wake first followed by the growth of turbulent shear stress afterThepeak of turbulent kinetic energy diffuses and can eventually reach the suction side boundary layer for the largest Reynolds numberinvestigated here with higher incidence angle As a consequence the local turbulence intensity outside the boundary layer can growsignificantly above the free-stream level with a potential impact on the suction side boundary layer transition mechanism

1 Introduction

Typically the time-averaged flow field of undisturbed planewakes consists of two equally strong slowly diverging vortexsheets of opposite orientation The rate at which the sheetsdiverge is proportional to 11988912 where 119889 is the distance to theorigin of the wake (see eg Schlichting [1]) indicating thatfor large 119889 the two vortex sheets will be roughly parallel Eachvortex sheet corresponds to a plane shear layer not unlikea flat plate boundary layer In a turbulent plane wake eachshear layer is associated with a peak in the turbulent kineticenergy 119896 indicating that the production of 119896 is concentratednear the regions of high shear

Castro and Bradshaw [2] and Gibson and Rodi [3]analysed the flow and turbulence structure of a highlycurved mixing layer The mixing layer under investigationbounds a normally impinging plane irrotational jet In theexperiment turbulence was discovered to be first attenuatedThen approximately after the first 50ndash60 of the curve thenormal and shear stresses were first amplified and exceeded

the plane-layer values reached in relaxation region furtherdownstream when the flow curvature vanishes The analysisindicated that the overshoot of turbulent quantities wasmostly due to shear-stress production

In the absence of strain Moser et al [4] observed a self-similar evolution of plane wakes The effect of the presenceof uniform mean strain was studied by Rogers [5] whoperformed direct numerical simulations of turbulent time-evolving strained wakes using a pseudo-spectral methodIn all his simulations the strain was applied to the sameself-similar wake flow field He found that though the mainflow reacts quickly on the applied strain the response ofturbulence to strain is slower changes in the turbulenceintensity could not keep pace with changes in the meanwake velocity Turbulence is produced by two competingmechanisms shear and strain Rogers found that when thedirection of compression is parallel to the centre-line of thewake (case C) the wake-width grows exponentially as thewake velocity deficit increasesThe normal Reynolds stresses119906

1015840119906

1015840 V1015840V1015840 11990810158401199081015840 and also 1199061015840V1015840 are all found to increase in

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2015 Article ID 650783 15 pageshttpdxdoiorg1011552015650783

2 International Journal of Rotating Machinery

Table 1 Details of the blade geometry and conditions

Test Re 120574-deg 1205731-deg 120573

2-deg 119905

119887119905 119880bar 119889

119887119862

L 518 times 104 3072 455 632 05 minus041 002H 148 times 105 3072 377 632 10 minus1204 002

time while the typical structure of the time-averaged wakethat is the two more or less parallel shear layers remainsintact When the direction of expansion is parallel to thecentre-line of the wake (case D) the wake-width the wakevelocity deficit and the Reynolds stresses all decrease in timeeventually degrading the structure of the wake

In summary the measurements by Castro and Bradshaw[2] allow studying the evolution of shear layers (ie halfportion of a wake) in presence of strong flow-core turningwhereas the DNS by Rogers [5] focus on the effect of strainon planar straight wakes Both are relevant to the flowin a turbomachine in which the wakes produced by thepreceding blade row are periodically ingested into a bladevane In particular while a plane wake travels through thepassage between two turbine blades it is severely strained anddistorted by the main flow In contrast to the study of Rogersthe actual direction of the mean strain relative to the centre-line of the wake varies with the actual location Moreover thedirection of shear individuated by the wakes differs from theflow direction because of the relative motion between bladesand wakes Hence differences arise with respect to the flowgeometry by Castro and Bradshaw too in which the directionof shear is aligned with the core flow

Wu and Durbin [6] performed the DNS of the flow ina low-pressure (LP) linear turbine rotor blade with periodicincomingwakesThewakes are subject to both flow curvature(as in Castro and Bradshaw [2]) and strain (as in Rogers[5]) The simulations revealed a peak in the turbulent kineticenergy located near the bow-apex of the wake where thedirection of compression is aligned with the centre-line ofthe wake corresponding to case C of Rogers Near thepressure side Wu and Durbin observed that the directionof expansion was almost aligned with the centre-line of thewake corresponding to caseD of Rogers In the present paperwe aim to further analyse the production of turbulence inthe plane turbulent wake as it travels through the free-streamregion of a turbine cascade

2 Description of the Test Cases

The geometry under investigation is that of the low-pressureaft-loaded turbine blade T106 The flow around this bladeassembled in a linear cascade was measured by Stadtmuller[7] and by Stadtmuller and Fottner [8] In the experimentsthe blade aspect ratio (hc) is 176 and the blade chord is100mm Therefore the flow at mid-span can be considerednearly two-dimensional and the three-dimensional computersimulations can be performed under the assumption of ahomogeneous flow in the span-wise direction The measure-ments were carried out at Re = 518 times 104 and 2 times 105 (basedon inlet conditions and axial chord) with the effect of anupstream row of blades simulated by a moving bar wake

minus05minus05

0 05 1 15 2

0

05

1

15

tb

U0

Ubar1205731

1205732Cax

t

x

y

120574

C

Figure 1 Flow configuration and grid for LES at Re = 518 times 104(every 6th grid line is shown in both stream-wise and pitch-wisedirections)

generator The bar diameter to pitch ratio is 119889119887119862 = 002

Unfortunately for the larger Re only the bar-to-blade pitchratio is not an integer number (040525) To yield an exactlyperiodic flow the simulations should hence be carried outby using eight blade vanes which would require an excessivecomputational effortThereforewewill focus on theDNSdataby Wissink [9] and the LES data by Michelassi et al [10]which refer both to the flow at the lower Reynolds numberwith a wake-to-blade pitch ratio of 05 (see test L in Table 1)The same geometry was also selected by Wu and Durbin [6]for a DNS at the significantly larger Re of 148 times 105 witha wake-to-blade pitch ratio of one This DNS was used as areference data set by Michelassi et al [10] who performedthe LES of the same flow The latter data set will also be usedfor further analysis under different operating conditions withrespect to [9 10] (see test H in Table 1)

3 Computational Details

31 Computational Grids The blade stagger angle 120574 theinlet flow angle 120573

1 and the outlet flow angle 120573

2 defined

with respect to the axial direction as displayed in Figure 1are summarised in Table 1 The grid employed in both theDNS and LES was carefully selected using the elliptic gridgeneration algorithm proposed by Hsu and Lee [11] Thiselliptic mesh generation ensured a nearly orthogonal gridclose to the blade walls and it was therefore adopted forall the simulations Table 2 summarises the adopted gridsizes The DNS grid [9] (see run D) consists of 1014 times 266times 64 nodes in the stream-wise pitch-wise and span-wise

International Journal of Rotating Machinery 3

Table 2 Summary of the test runs

Run Test Grid ℎ SGS modelD L 1014 times 260 times 64 02 times 119862ax NoneL1 L 454 times 144 times 32 015 times 119862ax Dyn SmagorinskyL2 H 646 times 256 times 64 015 times 119862ax Dyn Smagorinsky

directions respectively The grid extension in the span-wisedirection and downstream of the trailing edge is 020119862axand 10119862ax respectively The grid of the LES under the sameoperating conditions (see run L1) [10] consists of 448 times 144times 32 points in the stream-wise pitch-wise and span-wisedirections respectively and provides a fair representation ofthe DNS flow field [10] The LES grid extension in the span-wise direction and downstream of the trailing edge is 015119862axand 08119862ax respectively Further details on the grid resolutionare available in [9 10]The LES at Re = 148 times 105 adopts 646times256 times 64 nodes in the stream-wise pitch-wise and span-wisedirections respectively As in theDNS byWu andDurbin [6]the grid extends 05 times 119862ax upstream of the leading edge ofthe blades and 10 times119862ax downstream of the trailing edgeThespan-wise width is ℎ = 015 times 119862ax The grid in the span-wisedirection is uniform for all runs

32 Boundary Conditions and Phase Averaging A no-slipboundary condition is enforced at the surface of the bladeand a periodic flow condition is enforced in the pitch-wisedirection The latter condition is not critical since the sizeof the expected flow structures is a small percentage ofthe blade pitch Since the dynamically relevant motions areexpected to have a span-wise extent smaller than the size ofthe computational domain it is possible to enforce the peri-odicity of the instantaneous flow in the span-wise directiontoo The respective values of the span-wise extension of thecomputational domain H (see Table 2) had been carefullyselected to avoid undesired effects on the development of flowstructures The inflow boundary condition for the wakes isenforced by using the database made available by Wu andDurbin who generated the incoming turbulent wakes-likedata with preliminary LES [12] In all runs the wake dataclosely resembles those adopted in the DNS [6] in which thewake half-width is 004 times 119862ax and the maximum mean wakevelocity deficit is 18

Apart from the Reynolds number and the inlet flow anglethe two test cases L and H differ in the incoming wake-to-blade pitch ratio and wake tangential speed In particular intest case H the tangential speed of the wake is significantlylarger than in case L In order to properly resolve the wake inboth space and time one period119879 = 119905

119887119880119887was resolved using

9600 time steps in runD 4800 time steps in run L1 and 10240in run L2 In all runs this choice implies a maximum CFLnumber of approximately 030ndash050 In all the simulations theflow was allowed to develop for five periods 119879 that is fivewake passes After this start-up period the flow developed aperiodic behaviour and it was possible to initiate the phaseaveraging of the flow field Phase-averaging is performedover ten further periods Run D stored 240 phases whilethis number is reduced to 120 and 64 for runs L1 and L2

respectively In the followingΦ with 0 le Φ lt 1 correspondsto the phase of a phase-averaged flow field Angular bracketsdenote phase-averaged quantities

33 Calculation Method and Test Runs The DNS and LES ofthe flow around the T106 turbine blade have been performedusing the LESOCC code [13] The incompressible continuityand momentum equations are discretized by means of a cell-centred finite volume method Mass conservation is ensuredby the implicit solution of a Poisson equation for the pressurecorrection which is complemented with a Fourier solverin the span-wise direction that substantially reduces thecomputational effort The equations are solved by marchingin time with a three-stage Runge-Kutta algorithm The massconservation step was converged to a residual of 10minus8 TheSGSmodel used in the LES is the dynamicmodel byGermanoet al [14] with the modification of Lilly [15] It employs aneddy viscosity model together with a procedure of reducingthemodel constant whenever the flow iswell resolved For thepresent computation the SGS model employs filtering andaveraging in the homogeneous span-wise direction The testruns which are summarised in Table 2 refer to two differentReynolds numbers and incoming wake frequencies There-fore the relatively large data availability allows the flow insidethe blade vane to be analysed in different configurations andwith different modelling assumptions The next sections willdeal with the DNS at Re = 518 times 104 and the LES at both Re= 518 times 104 and Re = 148 times 105

4 Flow Visualisation

The dynamics of the wake while it is convected inside theblade vane has already been extensively discussed byWu andDurbin [6] for the Re = 148 times 105 case and by Wissink [9] aswell as byMichelassi et al [10] for theRe= 518times 104 case bothin terms of instantaneous and phase-averaged velocities anditwill not be repeated here In particular bothWuandDurbin[6] and Wissink [9] observed an accumulation of turbulentkinetic energy along the wake path inside the turbine vaneThis phenomenon is illustrated in the plots of the phase-averaged turbulent kinetic energy ⟨119896⟩ of Figure 2Thephase-averaged three-dimensional flow field around the T106 bladeis first averaged in the span-wise homogeneous directionThis allows extracting the two-dimensional phase-averagedsnapshots of the turbulence quantities The plots refer to twodifferent phases for both the LES and the DNS at Re = 518times 104 and the LES at Re = 148 times 105 In a DNS basically allvelocity fluctuations are resolved and thus contribute to theturbulent kinetic energy In the LES only the resolved velocityfluctuations are used to compute the turbulent kinetic energyThe absence of the SGS model contribution explains theslightly lower overall values of ⟨119896⟩ shown by LES with respectto DNS In all simulations at the inlet the wake is the onlysource of velocity fluctuations in an otherwise uniform flowfieldThis is the reason for the absence of any visible turbulentkinetic energy in between wakes in the first 40ndash50 of thecomputational domain Hence the wake is easily identifiedwith the flow regions in which the turbulent kinetic energydeparts from zero The figures illustrate how the turbulent


minus05 0 05 1 215x

y

minus05

0

05

1

15

00 02 0604 08 10

(a)

minus05 0 05 1 215x

y

minus05

0

05

1

15

P1

P2

00 02 0604 08 10

(b)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(c)

0 1 2x

y

minus05

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(d)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(e)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(f)

Figure 2 Phase-averaged turbulent kinetic energy (a) LES Re = 518 times 104 Φ = 0508 (b) LES Re = 518 times 104 Φ = 0008 (c) DNS Re =518 times 104 Φ = 0508 (d) DNS Re = 518 times 104 Φ = 0008 (e) LES Re = 148 times 105 Φ = 0375 and (f) LES Re = 148 times 105 Φ = 10


kinetic energy increases while the wake is gradually turnedand deformed inside the vane The gradual increase of 119896 isapproximately located around the wake apex

Apparently the turbulent kinetic energy keeps on grow-ing while the wake travels from the leading to the trailingedge of the blade This phenomenon does not seem to bevery sensitive to the Reynolds number since the same trendis obtained at Re = 518 times 104 as well as at Re = 148 times

105 The only noticeable qualitative difference between theresults obtained at different Reynolds numbers consists inthe increased proximity of the peak of 119896 to the suctionside of the blade at Re = 148 times 105 This however is mostlikely a consequence of the different inflow angle that isemployed in the simulations at Re = 518 times 104 compared tothose at Re = 148 times 105 At the lower Reynolds number theinflow angle is larger and the core of the flow region withlarge values of ⟨119896⟩ inside the vane never manages to hit thesuction side boundary layer Conversely thewake regionwithincreased ⟨119896⟩ clearly hits the suction side boundary layer inthe simulation with the larger Reynolds number and smallerinflow angle The plots also indicate that the portion of theflow field with the large values of turbulent kinetic energy inthe wake remains somewhat confined up to 119909119862ax cong 06ndash07Further downstream while approaching the exit region of theflow the spot with the large values of 119896 diffuses and mergeswith the wake downstream of the trailing edge Here anotherdifference arises between the simulations for different inflowangles and Reynolds numbers while for the higher-inflow-angle simulation the incoming wake merges with the trailingedge wake towards the pressure side for the smaller-inflow-angle simulation the incoming and trailing edge wakesmergenear the suction side

The observed large values of turbulent kinetic energyappear to stem from large values of local production asis illustrated in Figure 3 which shows the production ofturbulent kinetic energy computed by using the phase-averaged quantities ⟨119875

119896⟩ = minus⟨119906

119894119906119895⟩ sdot (120597⟨119880

119894⟩120597119909119895) All plots

reveal that the peak of turbulent kinetic energy coincides withlarge values of the production in the blade vane While thereis not much production in the portion of the wake betweenthe pressure side and the apex in the portion from the apexto the suction side ⟨119875

119896⟩ values are obtained up to 10 times as

large as those encountered in the wake near the inlet sectionThe circles on the lines ldquo119875rdquo of Figure 4 represent the

approximate location of the phase-averaged turbulent kineticenergy peaks observed in Figure 2 while the wake is swal-lowed into the blade vane for each of the three simulationsThe path of line ldquo119875rdquo corresponds to the path of the bowapex of the wake which also corresponds roughly to thelocation of the maximum wake curvature Moreover theanalysis of the unsteady flow field reveals that the location ofthe peak follows the path of a fluid element in thewake whichmoves towards the suction side on account of the cross-flowpressure gradient induced by the core flow turning Figure 4reveals that the peaks remain clearly outside the suction sideboundary layer although the turbulent kinetic energy clearlydiffuses close to wall in proximity to the trailing edge For119909119862ax gt 04ndash05 the distance from the wall becomes virtually

constant This can also be observed in the contour plots of 119896and 119875

119896in Figures 2 and 3

Now that the peaks of turbulent kinetic energy in the vaneare localised it is desirable to show the magnitude of 119896 and119875119896along 119875 at different phasesΦThis is done by interpolating

the desired quantities along the fixed line119875 at selected phasesand it is illustrated in Figure 5 the abscissa reports the axialcoordinate 119909 to avoid problems stemming from the slightlydifferent lengths of the 119875 curves for the three simulationsThe plots start at 119909119862ax = 0 which corresponds to the bladeleading edge and stop shortly downstreamof the trailing edgelocated at 119909119862ax = 1 For the Re = 518 times 104 case Figure 5shows five discrete curves which correspond to five phase-averaged snapshots of the flow field equally spaced to coverone full period They are selected from the database with 120stored phases for the LES and 240 for the DNS

For the Re = 148 times 105 simulation the same figure reportsfour equally spaced phases out of the 64 stored phases Both119896 and 119875

119896are computed by using the phase-averaged flow

fields The dashed curves draw the envelope of the peaksat different phases of both the selected quantities along the119875 line The dashed curves show that the peak of turbulentkinetic energy increases while the wake is swallowed intothe vane and it begins to decrease slightly upstream of thetrailing edge Remarkably this behaviour does not dependon the Reynolds number For the Re = 518 times 104 case thepeak of 119896 in the LES result is located somewhat upstreamof the position found in the DNS It must be pointed outthat some of the differences in the curves between DNS andLES at Re = 518 times 104 may stem from the difficulties intracing the peak of 119896 and selecting exactly the same phasefor the two simulations Observe that each phase-averagedcurve shows two peaks (see eg the peaks indicated by thelabels 1198751 and 1198752 in Figure 5(a)) the first one refers to thenew incoming wake whereas the second one is clearly theproduct of the previouswakewhich travels downstreamTheycorrespond qualitatively to thewake peaks labelled1198751 and1198752in Figure 2(b)The curves show that the peaks are quite sharpalong the first 50 to 60of the blade that is the large values of119896 are localised in a narrow region These peaks partly diffuseonce the trailing edge of the blade is approached An estimateof the growing strength of the localised peaks of 119896 along thewake path is given by the ratio of the turbulent kinetic energyencountered in the wake upstream of the blade 119896wake andthe maximum peak of 119896 encountered when the wake is insidethe blade vane 119896vane For the time phases considered here theratio 119896vane119896wake can exceed the value of 10 Conversely thegrowth of the turbulence intensity (defined as the ratio of thesquare root of the local turbulent kinetic energy and themeanlocal velocity Tu = radic(23)119896|119880|) is not as steep since Tureaches values approximately 50 to 60 larger than the levelsfound deep inside the wake upstream of the cascade

The production rates of turbulent kinetic energy 119875119896

shown in Figure 5 also increase continuously while the wakeis travelling in the first 50ndash60 of the blade vane The plotsreveal that the envelopes of the peaks of both 119896 and 119875

119896

substantially coincide in the first 50 to 60 of the blade vaneThis confirms the strong link between these two quantities


0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(a)

0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(b)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(c)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(d)

00 02 0604 08 10

0 1 2

x

y

minus05

05

15

0

1

(e)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(f)

Figure 3 Phase-averaged production of turbulent kinetic energy (a) LES Re = 518 times 104 Φ = 0508 (b) LES Re = 518 times 104 Φ = 0008(c) DNS Re = 518 times 104 Φ = 0508 (d) DNS Re = 518 times 104 Φ = 0008 (e) LES Re = 148 times 105 Φ = 0375 and (f) LES Re = 148 times 105Φ = 10


minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)

Figure 4 Approximate path of the peak of turbulent kinetic energy in the blade vane (a) LES Re = 518 times 104 (b) DNS Re = 518 times 104 and(c) LES Re = 148 times 105

regardless of the Reynolds number of the wake frequencyand of the nature of the simulation (DNS or LES) Furtherdownstream the rate of production of turbulent kineticenergy drops and consequently the dissipation of turbulentkinetic energy takes over Hence the gradient of 119896 revertsfrom positive to negative The drop of 119875

119896is the strongest in

the Re = 148 times 105 caseNow that the peaks of 119896 and119875

119896have been clearly identified

in both space and time by using the phase-averaged flowfields it is possible to show how they are linked to the strain

and stress tensors Prior to doing this it is convenient to recallthe expression for the phase-averaged production rate

⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878

119894119895⟩ are the turbulent stress and strain

tensors respectively The relative contribution of strain andstress to the production rate can be quantified by computingthe eigenvalues of the two tensors Analysing the 2D strain


00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)

Figure 5 Peaks of turbulent kinetic energy 119896 and production rate 119875119896 along line 119875 versus the axial coordinate in the blade vane (a) LES Re

= 518 times 104 (b) DNS Re = 518 times 104 and (c) LES Re = 148 times 105


00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax

Figure 6 Time averaged magnitude of the strain tensor eigenvaluealong line 119875 versus the axial coordinate in the blade vane (a) LESRe = 518 times 104 (b) DNS Re = 518 times 104 and (c) LES Re = 148 times105

tensor instead of the 3D tensor suffices since we assume span-wise flow periodicity and because of the prismatic nature ofthe blade (ie in the phase-averaged flow field 119878

33= 0) Since

the flow is incompressible implying that tr(119878119894119895) = 0 the 2D

strain tensor can be rewritten along its principal axes as

Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)

The same analysis can be carried out for the stress tensorIn this case the 3D tensor for the current flow configurationsreads

⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because

of span-wise periodicity Therefore for the present analysisonce again it suffices to refer to the 2D subtensor since thereis no direct phase-averaged production of turbulence fromspan-wise stresses The third eigenvalue is ⟨119906

31199063⟩ which

corresponds to the third eigenvector aligned with the span-wise direction Hence the 2D substress tensor reads

⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)

The two-dimensional stress tensor eigenvalues read

Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)

Now the magnitude of the eigenvalues along the 119875 lineallows investigating what is the contribution of the strain andturbulent stress tensors to the production rate of (1) in theplane 119909-119910 Most likely due to the limited number of phasesused for the phase-averaging the eigenvalues 120590

119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591

Figure 7 Time averaged magnitude of the stress tensor eigenvaluealong line 119875 versus the axial coordinate in the blade vane (a) LESRe = 518 times 104 (b) DNS Re = 518 times 104 and (c) LES Re = 148 times105

and 1205901120591and 1205902

120591of the phase-averaged 2D strain tensor and

stress tensor respectively extracted along the 119875 line show asomewhat chaotic behaviour Hence it was decided to plotthe average magnitude of the phase-averaged eigenvaluesalong the 119875 line for the three simulations as shown inFigures 6 and 7 The curves reveal that the peak of theproduction rate roughly corresponds to the maximum of thelargest strain tensor eigenvalue while the peak of the stresstensor eigenvalue although quite close is located 010ndash015times119862ax downstream Moreover the overall shape of the largeststrain tensor eigenvalue curve resembles the shape of theturbulence production curve plotted in Figure 5 Apparentlythe turbulent stress reacts to the turning and straining of thewake with some slight delay with respect to the strain tensorThis feature was already observed by Rogers [5] for linearwakes in which either the direction of compression or thatof stretching was aligned with the wake path

5 Analysis along the Wakes

The previous section proved that the peak of productionis located close to the strain tensor eigenvalue peak andsubstantially stems from the concerted action of the peaksof both strain and turbulent stresses In order to understandhow the wake triggers both production and turbulent kineticenergy it is convenient to concentrate the analysis on a linealigned with the phase-averaged wakes For this purpose weselected two different phasesΦ = 0008 and 0508 for the Re= 518 times 104 case and Φ = 0375 and 100 for the Re = 148 times105 case respectively The lines that identify the wakes followthe location of the maximum velocity magnitude defect andare illustrated in Figure 8 The 2D phase-averaged straintensor eigenvectors allow to identify the local directions ofcompression and stretching and to compare these directionswith those of the eigenvectors of the stress tensor along the


minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

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18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)

Figure 8 Eigenvectors of the strain tensor (left) and of the stress tensor (right) along two selected wakes (a) LES Re = 518 times 104 (b) DNSRe = 518 times 104 and (c) LES Re = 148 times 105 (for the strain tensor the red segment represents the direction of compression)


(a) Castro and Bradshaw [2]

120573

(b) Present

Figure 9 Sketch of the convection and shear layer directions Grey arrows indicate direction of convection thick black arrows indicateorientation of shear layer in proximity to the bow apex of the incoming wakes

selected wakesThis comparison is carried out in Figure 8 forboth the LES and the DNS at Re = 518 times 104 and the LES at Re= 148 times 105 Observe that the eigenvector length is chosen tobe proportional to the corresponding eigenvalue magnitudeThe plots show a remarkable feature of flow the wake isalmost perfectly aligned with the direction of compression inthe proximity of its apex This seems to be a common featurefor both wake positions and for both values of the Reynoldsnumbers regardless of the nature of the simulation (DNS orLES) Wu and Durbin [6] observed the same feature in theirDNS of the same flow at Re = 148 times 105

Apparently while the wake is smoothly turned anddeformed inside the blade vane the portion which isentrained in the fastest part of the flow field identified bythe bow apex of the wake gradually aligns with the directionof compression The portion of the wake shown in Figure 8close to the suction side is not aligned with either thedirection of compression or that of stretching whereas theportion from the apex to the pressure side is almost alignedwith the direction of stretching

The plots showing the eigenvectors of the stress tensoralong the wakes reveal a similar behaviour one of theeigenvectors of the stress tensor is clearly tangent to the wakein proximity to its apex This is again a common feature ofthe two Reynolds numbers and the two wake positions andat least for the lower Reynolds number case it is predicted byboth DNS and LES Observe that of the two the eigenvectorthat is almost aligned with the wake is the one associatedwith the largest eigenvalueThis interesting feature of the flowwas also observed for all other phase-averaged positions ofthe wake and it is not reported here for the sake of brevityFigures 4 and 7 reveal that the location of the turbulent kineticenergy peaks corresponds to the apex of the wakes On turnthe apex of the wakes is almost perfectly aligned with oneof the eigenvectors of both the strain and stress tensors Inother words the location of the turbulent kinetic energy peakcorresponds to the wake-eigenvectors alignment

The production rate of turbulent kinetic energy can besplit into normal and tangential contributions respectivelyUnfortunately in such a complex flow field the definitionof normal and tangential production is not straightforwardbecause of the relativemotion of the wakes with respect to thebackground flow Castro and Bradshaw [2] analysed a highlycurvedmixing layer and Gibson and Rodi [3] and Leschzinerand Rodi [16] formulated the rate of production of turbulentkinetic energy in terms of normal and tangential stressescontributions (ie production that stems from normal andtangential stresses resp) for such flow In the experiments byCastro and Bradshaw and in the annular and twin paralleljets computed by Leschziner and Rodi the direction ofconvection coincides with the direction of the shear layeras illustrated in Figure 9 In this case the decompositionbetween normal and tangential production is trivial and canbe carried out by following Gibson and Rodi However in thepresent flow configuration the direction of convection differsfrom the direction of the shear layer generated by the incom-ing wake (Figure 9) In particular the angle between the twodirections at the inlet section can be as large as 30 degreesand it reaches approximately 90 degrees inside the blade vaneThis is made evident in Figure 10 which shows on the sameplot at Φ = 000 both the streamlines and the velocity defectdefined as 119906

119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)

2In order to determine whether the large production rate

in proximity to the wake apex is mostly due to normal ortangential stresses the particular orientation of both thestrain and stress tensors eigenvectors with respect to the wakeneeds to be further investigated Prior to this it is convenientto analyse what happens in a turbulent boundary layer asa simplified reference condition This is done in Figure 11where the typical eigenvectors of a turbulent boundary layerrefer to the log-law region with moderate anisotropy Herethe shear layer is aligned with the wall and with the directionof the core flow Due to incompressibility the strain tensoreigenvectors in the turbulent boundary layer are such that


0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C

Figure 10 Phase-averaged streamlines and velocity defect isolines computed by the LESΦ = 000 Re = 148 times 105

Solid wall Wake path (A) Wake path (B C)

120573 120573120573

Figure 11 Sketch of the strain tensor eigenvectors (black) and stress tensor eigenvectors (grey) in a turbulent boundary layer (left) and intwo positions A and B as detailed in Figure 10 along a typical wake

120573 = 45 deg while for the stress tensor eigenvectors 120573 lt

45 deg and the production rate are due to tangential stresses(see Pope [17]) The strain and stress tensor eigenvectors forthe wake are extracted from the points labelled A and Bas documented in Figure 10 The wake in the turbine vanecan be imagined as the sum of two virtually parallel shear-layers as illustrated in Figures 11(A) and 11(B C)The rotationof the core flow only produces a very weak shear whoseeffect on the local production of turbulence can be neglectedwhen compared to the effect of the wake shear layers Inpoint A (see Figure 11(A)) which is located away from theapex and close to the suction side the orientation of theeigenvectors with respect to the wake is very similar to whatis found for the boundary layer both the strain and stresstensors eigenvectors are approximately at an angle 120573 of 40ndash45 deg with the wake and consequently with the main shearlayers Hence the production rate is mostly governed byshear stresses In point B the situation drastically changesone of the eigenvectors of both the strain and stress tensors is

almost alignedwith the shear layer produced by the wake (seeFigure 11(B)) This means that both tensors are diagonal in areference frame aligned with the wake shear layer Thereforethe absence of any significant contribution stemming fromoff-diagonal terms suggests that in point B the productionof turbulence is predominantly due to normal strain Inparticular it is the eigenvector that identifies the directionof compression that aligns with the bow apex of the wakeIn point C the eigenvectors are still almost aligned with thewake but here it is the direction of stretching that alignswith the shear layer Hence the strong straining of the shearlayer considerably reduces the production rate with respectto point B This is particularly evident for the first of the twowake positions reported in Figure 12 (Φ = 0008 at Re = 518times 104 andΦ = 0375 at Re = 148 times 105) In fact after the peaklocated at 119878119878max cong 05119875119896 drops downwhile approaching thepressure side

It is now possible to rearrange the production rate in theprincipal frame of reference identified by the eigenvectors


00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)

Figure 12 Turbulent kinetic energy 119896 production rate 119875119896 and strain tensor eigenvalue 120590 along two selected wakes (a) LES Re = 518 times 104

(b) DNS Re = 518 times 104 and (c) LES Re = 148 times 105

of the strain tensor and not in the direction of the flow Bydefining 120572 as the angle between the 119909-axis and the directionof compression the stress tensor components 120591

119894119895 are

⟨119906

2

119901⟩ = ⟨119906

2⟩ sdot cos2120572 + ⟨V2⟩ sdot sin2120572 + 2 sdot ⟨119906V⟩ sdot sin120572 sdot cos120572

⟨V2119901⟩ = ⟨119906

2⟩ sdot sin2120572 + ⟨V2⟩ sdot cos2120572 minus 2 sdot ⟨119906V⟩ sdot sin120572 sdot cos120572

⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906

2⟩ + ⟨V2⟩) sdot sin120572 sdot cos120572

+ ⟨119906V⟩ sdot (cos2120572 minus sin2120572)(6)

in which the subscript ldquo119901rdquo denotes values in the principalframe of reference In the same frame the strain tensoris obviously diagonal so that the production rate can becomputed by a term by term multiplication of (2) and (6)The following normal stress contribution results in

119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)

As it could be expected the contribution of the tangentialstresses to the production rate in the principal frame ofreference is analytically zero since

119875

119879

119896= minus [120590

119878sdot ⟨119906119901V119901⟩ minus 120590119878sdot ⟨119906119901V119901⟩] = 0 (8)


Therefore the production rate is mostly due to normalstresses where the wake is aligned with one of the principaldirections

It is now possible to plot 119896 119875119896 and 120590

119878along a phase-

averaged wake-wise coordinate 119878 defined in Figure 8(a) forboth phase-averaged positions of the wakes illustrated in thesame figure The origin of 119878 is located in the proximity ofthe suction side Figure 12 shows the values of 119896 along thetwo wakes The circles and triangles help in locating the peakof 119896 along the wakes of Figure 8 It is evident that the peakof 119896 and the largest production of turbulent kinetic energyare approximately positioned at the point where the wake isaligned with the direction of compression and with one ofthe eigenvectors of the stress tensor Both 119896 and 119875

119896gradually

increase along the axis of the wake starting from the suctionside portion They both reach a maximum and then decreasein the flow region where the wake is almost perfectly alignedwith the direction of stretching

For the Re = 518 times 104 case along the first wake (Φ =

0508) compared to DNS LES underpredicts 119896 and 119875119896

The reason for this may be partly the omission of SGScontributions in the LES As remarked before the maximumvalue of 119896 is generally located closer to the suction sideboundary layer than in the lower Re case When switching tothe secondwake (Φ = 0008) the discrepancies betweenDNSand LES are substantially reduced A possible explanation isthat at the previous phase there are not many significant flowstructures and inside the wakemostly small scale fluctuationsare present which are not resolved but modelled by LES(and neglected in the plots) When moving further in timeturbulence has developed and more flow structures appear(as evidenced by the flow visualisation) the contribution ofwhich is directly captured by the LES The plots related tothe flow at Re = 148 times 105 also reveal another interestingpoint At Φ = 0375 the location of the peaks of 119896 and 119875

119896

coincides but at Φ = 10 the location of the large value of 119896is different from that of 119875

119896 This apparently contradicts what

has been discovered so far However Figure 12(c) reveals thatthe maximum production rate at Φ = 10 is approximately14 of that at Φ = 0375 In other words the build-up ofturbulence is virtually complete when the apex of the wakereaches 119909119862ax cong 09 at Φ = 10 (Figure 8) Therefore thepeak of turbulent kinetic energy visible along this wake at119904119878max cong 02 (ie very close to the suction side) is the result ofconvection from upstream where the flow experiences largeproduction rates

6 Conclusions

The analysis conducted on both the DNS and LES data setsat two Reynolds numbers illustrates how the alignment of thewake with the strain tensor eigenvectors controls the positionof the peak of turbulent kinetic energy production whileits strength is mainly controlled by the magnitude of thestrain tensor eigenvalue In practice the largest productionof turbulent kinetic energy will take place at a particularposition along the wake where the maximum strain tensoreigenvalue approximately coincides with the location of thealignment of the wake with the direction of compression If

this condition is not fulfilled the turbulent kinetic energymay still grow locally (in fact the peak of turbulent kineticenergy is constantly increasing up to 119909119862ax = 085ndash09) butat a smaller rate

LES was also found in fair agreement with the DNS atRe = 518 times 104 suggesting that further analyses may beconducted by using much less computationally intensive LESas compared to DNS provided the discretization quality iscarefully controlled to appropriately resolve not only theboundary layers but the incoming wakes as well

The high-fidelity simulation data sets identified all themechanisms contributing the production rate of turbulentkinetic energy The analysis showed the relative importanceof the strain and stress tensors respectively together withnormal and tangential contribution to the production rateby comparing with Castro and Bradshaw results A furtheranalysis carried out in the strain tensor principal frame ofreference showed the main contributions to the productionrate of turbulent kinetic energy

The simulations also revealed differences in the locationand intensity of the turbulent kinetic energy peaks whenincreasing the Reynolds number from 518 times 10

4 to 148 times10

5 and incidence angle In particular at Re = 148 times 105the turbulent kinetic energy peak moves much closer to thesuction side boundary layer This phenomenon associatedwith the local growth of turbulent kinetic energy above thefree-stream level is expected to improve the suction sideboundary layer stability and it can provide very valuableinformation in the design of suction-side-separation-freelow-pressure turbine blades

Last but not least the anisotropic turbulence productionrate captured by the DNS and LES can be compared withthe prediction of lower order models to allow fixing eventualweaknesses by a strictly analytical manner

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor W Rodi forthe fruitful discussions on the analysis of the results Thiswork was supported by the German Research Foundation(DFG) within the joint Project ldquoPeriodic Unsteady Flow inTurbomachineryrdquo

References

[1] H Schlichting Boundary Layer Theory McGraw-Hill 7thedition 1979

[2] I P Castro and P Bradshaw ldquoThe turbulence structure of ahighly curved mixing layerrdquo Journal of Fluid Mechanics vol 73no 2 pp 265ndash304 1976

[3] M M Gibson andW Rodi ldquoA reynolds-stress closure model ofturbulence applied to the calculation of a highly curved mixinglayerrdquo Journal of Fluid Mechanics vol 103 pp 161ndash182 1981


[4] R D Moser M M Rogers and D W Ewing ldquoSelf-similarityof time-evolving plane wakesrdquo Journal of Fluid Mechanics vol367 pp 255ndash289 1998

[5] M M Rogers ldquoThe evolution of strained turbulent planewakesrdquo Journal of Fluid Mechanics vol 463 pp 53ndash120 2002

[6] X Wu and P A Durbin ldquoEvidence of longitudinal vorticesevolved from distorted wakes in a turbine passagerdquo Journal ofFluid Mechanics vol 446 pp 199ndash228 2001

[7] P Stadtmuller ldquoInvestigation of Wake-Induced Transition onthe LP turbine Cascade T106A-EIZrdquo DFG-VerbundprojectFo13611 Version 11 2001

[8] P Stadtmuller and L Fottner ldquoA test case for the numericalinvestigation of wake passing effects of a highly loaded LPturbine cascade bladerdquo ASME Paper 2001-GT-311 2001

[9] J G Wissink ldquoDNS of a separating low Reynolds numberflow in a turbine cascade with incoming wakesrdquo in Proceedingsof the 5th International Symposium on Engineering TurbulenceModelling andMeasurements Mallorca Spain September 2002

[10] V Michelassi J Wissink and W Rodi ldquoAnalysis of DNS andLES of flow in a low pressure turbine cascade with incomingwakes and comparisonwith experimentsrdquo Flow Turbulence andCombustion vol 69 no 3-4 pp 295ndash329 2002

[11] K Hsu and S L Lee ldquoA numerical technique for two-dimen-sional grid generationwith grid control at all of the boundariesrdquoJournal of Computational Physics vol 96 no 2 pp 451ndash4691991

[12] XWu R G Jacobs J C R Hunt and P A Durbin ldquoSimulationof boundary layer transition induced by periodically passingwakesrdquo Journal of Fluid Mechanics vol 398 pp 109ndash153 1999

[13] M Breuer andWRodi ldquoLarge eddy simulation for complex tur-bulent flows of practical interestrdquo in Flow Simulation with High-Performance Computers II vol 48 of Notes on Numerical FluidMechanics (NNFM) pp 258ndash274Vieweg+TeubnerWiesbaden1996

[14] MGermanoU Piomelli PMoin andWHCabot ldquoA dynamicsubgrid-scale eddy viscosity modelrdquo Physics of Fluids A vol 3no 7 pp 1760ndash1765 1991

[15] D K Lilly ldquoA proposed modification of the Germano subgrid-scale closure methodrdquo Physics of Fluids A vol 4 no 3 pp 633ndash635 1992

[16] M A Leschziner and W Rodi ldquoCalculation of annular andtwin parallel jets using various discretization schemes and tur-bulence-model variationsrdquo Transaction of the ASME Journal ofFluids Engineering vol 103 no 2 pp 352ndash360 1981

[17] S B Pope Turbulent Flows Cambridge University Press Cam-bridge UK 2001

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Table 1 Details of the blade geometry and conditions

Test Re 120574-deg 1205731-deg 120573

2-deg 119905

119887119905 119880bar 119889

119887119862

L 518 times 104 3072 455 632 05 minus041 002H 148 times 105 3072 377 632 10 minus1204 002

time while the typical structure of the time-averaged wakethat is the two more or less parallel shear layers remainsintact When the direction of expansion is parallel to thecentre-line of the wake (case D) the wake-width the wakevelocity deficit and the Reynolds stresses all decrease in timeeventually degrading the structure of the wake

In summary the measurements by Castro and Bradshaw[2] allow studying the evolution of shear layers (ie halfportion of a wake) in presence of strong flow-core turningwhereas the DNS by Rogers [5] focus on the effect of strainon planar straight wakes Both are relevant to the flowin a turbomachine in which the wakes produced by thepreceding blade row are periodically ingested into a bladevane In particular while a plane wake travels through thepassage between two turbine blades it is severely strained anddistorted by the main flow In contrast to the study of Rogersthe actual direction of the mean strain relative to the centre-line of the wake varies with the actual location Moreover thedirection of shear individuated by the wakes differs from theflow direction because of the relative motion between bladesand wakes Hence differences arise with respect to the flowgeometry by Castro and Bradshaw too in which the directionof shear is aligned with the core flow

Wu and Durbin [6] performed the DNS of the flow ina low-pressure (LP) linear turbine rotor blade with periodicincomingwakesThewakes are subject to both flow curvature(as in Castro and Bradshaw [2]) and strain (as in Rogers[5]) The simulations revealed a peak in the turbulent kineticenergy located near the bow-apex of the wake where thedirection of compression is aligned with the centre-line ofthe wake corresponding to case C of Rogers Near thepressure side Wu and Durbin observed that the directionof expansion was almost aligned with the centre-line of thewake corresponding to caseD of Rogers In the present paperwe aim to further analyse the production of turbulence inthe plane turbulent wake as it travels through the free-streamregion of a turbine cascade

2 Description of the Test Cases

The geometry under investigation is that of the low-pressureaft-loaded turbine blade T106 The flow around this bladeassembled in a linear cascade was measured by Stadtmuller[7] and by Stadtmuller and Fottner [8] In the experimentsthe blade aspect ratio (hc) is 176 and the blade chord is100mm Therefore the flow at mid-span can be considerednearly two-dimensional and the three-dimensional computersimulations can be performed under the assumption of ahomogeneous flow in the span-wise direction The measure-ments were carried out at Re = 518 times 104 and 2 times 105 (basedon inlet conditions and axial chord) with the effect of anupstream row of blades simulated by a moving bar wake

minus05minus05

0 05 1 15 2

0

05

1

15

tb

U0

Ubar1205731

1205732Cax

t

x

y

120574

C

Figure 1 Flow configuration and grid for LES at Re = 518 times 104(every 6th grid line is shown in both stream-wise and pitch-wisedirections)

generator The bar diameter to pitch ratio is 119889119887119862 = 002

Unfortunately for the larger Re only the bar-to-blade pitchratio is not an integer number (040525) To yield an exactlyperiodic flow the simulations should hence be carried outby using eight blade vanes which would require an excessivecomputational effortThereforewewill focus on theDNSdataby Wissink [9] and the LES data by Michelassi et al [10]which refer both to the flow at the lower Reynolds numberwith a wake-to-blade pitch ratio of 05 (see test L in Table 1)The same geometry was also selected by Wu and Durbin [6]for a DNS at the significantly larger Re of 148 times 105 witha wake-to-blade pitch ratio of one This DNS was used as areference data set by Michelassi et al [10] who performedthe LES of the same flow The latter data set will also be usedfor further analysis under different operating conditions withrespect to [9 10] (see test H in Table 1)

3 Computational Details

31 Computational Grids The blade stagger angle 120574 theinlet flow angle 120573

1 and the outlet flow angle 120573

2 defined

with respect to the axial direction as displayed in Figure 1are summarised in Table 1 The grid employed in both theDNS and LES was carefully selected using the elliptic gridgeneration algorithm proposed by Hsu and Lee [11] Thiselliptic mesh generation ensured a nearly orthogonal gridclose to the blade walls and it was therefore adopted forall the simulations Table 2 summarises the adopted gridsizes The DNS grid [9] (see run D) consists of 1014 times 266times 64 nodes in the stream-wise pitch-wise and span-wise














minus05 0 05 1 215x

y

minus05

0

05

1

15

00 02 0604 08 10

(a)

minus05 0 05 1 215x

y

minus05

0

05

1

15

P1

P2

00 02 0604 08 10

(b)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(c)

0 1 2x

y

minus05

0

05

1

15

00 02 0604 08 10

(d)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(e)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(f)







119896⟩ = minus⟨119906

119894119906119895⟩ sdot (120597⟨119880

119894⟩120597119909119895) All plots














119896



0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(a)

0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(b)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(c)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(d)

00 02 0604 08 10

0 1 2

x

y

minus05

05

15

0

1

(e)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(f)



minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)








⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878




00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















minus05 0 05 1 215x

y

minus05

0

05

1

15

00 02 0604 08 10

(a)

minus05 0 05 1 215x

y

minus05

0

05

1

15

P1

P2

00 02 0604 08 10

(b)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(c)

0 1 2x

y

minus05

0

05

1

15

00 02 0604 08 10

(d)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(e)

0 1 2x

y

minus05

0

05

1

15

00 01 0302 04 05

(f)







119896⟩ = minus⟨119906

119894119906119895⟩ sdot (120597⟨119880

119894⟩120597119909119895) All plots














119896



0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(a)

0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(b)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(c)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(d)

00 02 0604 08 10

0 1 2

x

y

minus05

05

15

0

1

(e)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(f)



minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)








⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878




00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119896⟩ = minus⟨119906

119894119906119895⟩ sdot (120597⟨119880

119894⟩120597119909119895) All plots














119896



0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(a)

0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(b)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(c)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(d)

00 02 0604 08 10

0 1 2

x

y

minus05

05

15

0

1

(e)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(f)



minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)








⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878




00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(a)

0

0

1

1

2

x

y

minus05minus05

05

05

15

15

00 01 0302 04 05

(b)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(c)

0

0

1

1

2

x

y

minus05

05

15

00 04 1208 16 20

(d)

00 02 0604 08 10

0 1 2

x

y

minus05

05

15

0

1

(e)

0 1 2

x

y

minus05

0

05

1

15

00 02 0604 08 10

(f)



minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)








⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878




00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus050 1 2

x

y

0

05

1

15

P

(a)

minus050 1 2

x

y

0

05

1

15

P

(b)

minus050 1 2

x

y

P

0

05

1

68 78 1808 28

38

48 58

68 78

18

2838

4858

68

78

15

(c)








⟨119875119896⟩ = minus ⟨120591

119894119895⟩ sdot ⟨119878

119894119895⟩ (1)

in which ⟨120591119894119895⟩ and ⟨119878




00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

P1

P2

k

xCax

Pk

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(a)

00 02 04 06 08 10 120000

0004

0008

0012

0016

0020

Peaks

00 02 04 06 08 10 12000

002

004

006

008

010

Peaks

k

xCax

Pk

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

xCax

Φ = 010Φ = 210Φ = 410

Φ = 610Φ = 810

(b)

00 02 04 06 08 10 12 140000

0004

0008

0012

0016

0020

Peaks

0828

4868

k

00 02 04 06 08 10 12 14000

002

004

006

008

010

Peaks

xCax xCax

0828

4868

Pk

(c)




00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 02 04 06 08 10 120

2

4

6

(a)(b)

(c)

120590s

xCax



33= 0) Since



Λ

2D119878= (

120590119904

0

0 minus120590119904

) (2)


⟨120591119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩ 0

⟨11990621199061⟩ ⟨11990621199062⟩ 0

0 0 ⟨11990631199063⟩

) (3)

in which ⟨11990631199061⟩ = ⟨119906

11199063⟩ = ⟨119906

31199062⟩ = ⟨119906

21199063⟩ = 0 because


31199063⟩ which


⟨120591

2D119894119895⟩ = (

⟨11990611199061⟩ ⟨11990611199062⟩

⟨11990621199061⟩ ⟨11990621199062⟩

) (4)


Λ

2D120591= (

120590

1

1205910

0 120590

2

120591

) (5)


119878and minus120590

119878

00 02 04 06 08 10 120

1

2

3

4

5

xCax

120590120591lowast

Relowast10

2

(a) 1205901120591(a) 1205902120591(b) 1205901120591

(b) 1205902120591(c) 1205901120591(c) 1205902120591


and 1205901120591and 1205902






minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

Suction side

Pressure side

Strain Stress

S

S

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 0508

Φ = 0008 Φ = 0008

Φ = 0508

(a)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

axΦ = 0508

Φ = 0008

Φ = 0508

Φ = 0008

(b)

Strain Stress

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

minus02 00 02 04 06 08 10

04

06

08

10

12

14

16

18

20

xCax

yC

ax

Φ = 38

Φ = 88

Φ = 38

Φ = 88

(c)




120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120573

(b) Present






119889= radic(119906

1minus ⟨1199061⟩0)

2+ (V2minus ⟨V2⟩0)




0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

1

1

2x

y

minus05

05

15

Streamlines

B

A

C



120573 120573120573







00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 02 04 06 08 1000

02

04

06

08

10

12

14

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

Φ = 0008

Φ = 0508

120590

Pklowast100

klowast100

SSmax

SSmax

SSmax

(a)

00 01 02 03 04 05 06 07 08 09 10

000510152025303540

Pklowast100

SSmax

00 02 04 06 08 1000

02

04

06

08

10

12

14

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 0008

Φ = 0508

(b)

00 01 02 03 04 05 06 07 08 09 1000

01

02

03

04

05

klowast100

SSmax

00 01 02 03 04 05 06 07 08 09 10

00

05

10

15

20

Pklowast100

SSmax

00 01 02 03 04 05 06 07 08 09 100

1

2

3

4

5

120590

SSmax

Φ = 38

Φ = 88

(c)




119894119895 are

⟨119906

2

119901⟩ = ⟨119906


⟨V2119901⟩ = ⟨119906


⟨119906119901V119901⟩ = ⟨V

119901119906119901⟩ = (minus ⟨119906




119875

119873

119896= 119875119896= minus [120590

119878sdot ⟨119906

2

119901⟩ minus 120590119878sdot ⟨V2119901⟩] (7)


119875

119879

119896= minus [120590







119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119896gradually





119896




6 Conclusions






4 to 148 times10





Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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