vortex structure and kinematics of encased axial turbomachines...

Vortex Structure and Kinematics of Encased AxialTurbomachinesPeter F. Pelz1*, Paul Taubert1, Ferdinand-J. Cloos1

SYM

POSI

A

ON ROTATING MACHIN

ERY

ISROMAC 2017

InternationalSymposium on

Transport Phenomenaand

Dynamics of RotatingMachinery

Maui, Hawaii

December 16-21, 2017

Abstractis investigation models the kinematics of the vortex system of an encased axial turbomachine atpart load and overload applying analytical methods. So far, the inuence of the casing and the tipclearance on the kinematics was solved separately. e vortex system is composed of a hub, boundand tip vortices. For the nominal operating point ϕ ≈ ϕopt and negligible induction, the tip vorticestransform into a screw. For part load operation ϕ→ 0, the tip vortices wind up to a vortex ring, i.e.the pitch of the screw vanishes. e vortex ring itself is generated by bound vortices rotating atthe angular frequency Ω. e hub vortex induces a velocity on the vortex ring causing a rotationat the sub-synchronous frequency Ωind = 0.5Ω. Besides, the vortex ring itself induces an axialvelocity. Superimposed with the axial main ow, this results in a stagnation point at the tube wall.is stagnation point may wrongly be interpreted as dynamic induced wall stall. For overloadoperation ϕ→∞, the vortex system of the turbomachine forms a horseshoe, i.e. the pitch of thescrew becomes innite. Both, hub and tip vortices, are semi-innite, straight vortex laments. etip vortices rotate against the rotating direction of the turbomachine, due to the induction of thehub vortex, yielding the induced frequency Ωind = −0.5Ω/s, with the tip clearance s.Keywordsvortex dynamics — potential ow — part load — overload — sub-synchronous frequency —kinemtatic induced noise1Chair of Fluid Systems, Technische Universitat Darmstadt, Darmstadt, Germany*Corresponding author: [email protected]

INTRODUCTION

By now, the common understanding is, that rotating stalland the resulting noise and vibration within a turbomachineis a dynamic eect. is means, that frictional forces lead toboundary layer separation and eventually stall in rotatingmachines [1, 2]. is understanding is recently conrmedby Cloos et al. [3] both, experimentally and analytically, forthe most generic machine, a ow through a coaxial rotatingcircular tube. According to Cloos et al. [3], ”wall stall” –a term coined by Greitzer [1] in contrast to ”blade stall” –is caused at part load by the interaction of axial boundarylayer and swirl boundary layer ow, i.e. the inuence ofcentrifugal force on axial momentum. For ”wall stall”, theaxial velocity component uz vanishes at the line z = −z0,r = a (axial coordinate in mean ow direction z, distancez0 from the reference point on the line of symmetry r = 0,radial coordinate r , tube radius a; cf. gure 1).

is paper analyzes the ow situation in encased axialturbomachines for small viscous friction. is work shows,that ”wall stall”, i.e. uz(−z0, a) = 0, can also be a resultof kinematics only, due to induced velocities of the vortexsystem superimposed with the axial main ow. e structureof the vortex system, especially the tip vortices, dependson the operating point of the turbomachine. Furthermore,the tip vortices rotate with a sub-synchronous frequencyΩind (Karstadt et al. [4], Zhu [5]). e aim of the presentinvestigation is to analyze the inuence of the vortex systemin encased axial turbomachines and its circulation strengthon the observed phenomena, yielding the research questions:

1. Is it possible to explain by means of analytical methodsthe sub-synchronous frequencies observed for turbo-machines?

2. Can ”wall stall” be a result of kinematics only?

To answer these questions, this work rst employs vortextheory for an encased axial turbomachine, followed by theapplication of fundamental solutions. For machines withoutcasing like wind turbines and screw propellers, vortex theoryis well described by Betz [6], Goldstein [7], Glauert [8] andvan Kuik [9]. e method is not yet developed in such a de-gree for encased turbomachines requiring the ow potentialof a vortex ring inside a tube. Pelz et al. [10] recently derivedthis solution.

In addition, to enlarge the investigation on the wholeoperating range of a turbomachine, we investigate the struc-ture and kinematics of the vortex system at heavy overload,applying theory of functions (complex analysis). For turbo-machines, the ow number ϕ := U/Ωa denes the operatingpoint (e.g. part load or overload). e ow number is theratio of axial free-stream velocityU to the circumferential ve-locity Ωa, where Ω = 2πn is the rotational speed (the scalingto Ωa and not to Ωb, with the blade tip radius b, is commonin the context of turbomachines and therefore used here aswell [11]).

For the nominal operating point ϕ ≈ ϕopt and negligibleinduction, the vortex system of an encased axial turboma-chine consists of a hub, Z bound and Z tip vortices, with Zthe number of blades. e tip vortices transform into heliceswith a pitch of 2πϕa.

Vortex Structure and Kinematics of Encased Axial Turbomachines — 2/7

For part load operation ϕ→ 0, see gure 1, the Z helices”roll up” and form a vortex ring, i.e. the pitch of the helicesvanishes. e vortex ring is continuously generated by thebound vortex system. Hence, the coaxial vortex ring strengthis transient. A nice picture for this vortex ring is that ofa thread spool rolling up and gaining strength over time.is picture will explain some transient phenomena usingkinematic arguments only.

e case of heavy overload occurs for innitely high ownumbers ϕ → ∞; see gure 2. e hub, the bound and thetip vortices form a horseshoe, i.e. the pitch of the helicesbecomes innite. Both, hub and tip vortices are semi-innite,straight vortex laments. In real turbomachines, the ownumber cannot be adjusted to innity, but is limited to amaximum value ϕ due to ow rate limitations and geometricrestrictions. Nevertheless, the analysis of this limiting case isimportant for the basic understanding of the vortex systemin axial turbomachines.

To develop physical understanding of the whole picturein detail, the paper is organized as follows. Section 1 givesa short literature overview. Section 2 uses vortex theory todetermine the strength of the vortices. Subsequently, sec-tion 3 derives the velocity potential of a coaxial vortex ringwithin a circular tube at part load and the induced rotatingfrequency. e ow potential and the induction at overloadis introduced in section 4. e paper closes with a short out-look to potential applications in section 5 and a discussionin section 6.

1. LITERATURE REVIEWInvestigations of vortex systems in uid dynamics traceback to the work of Helmholtz [12], who formulated theHelmholtz’s theorems as a basis for the research concerningrotational uid motion.

Didden [13] performed measurements of the rolling-upprocess of vortex rings and compared the results with simi-larity laws for the rolling-up of vortex sheets.

Besides the investigation of vortex kinematics, a broadresearch eld on vortex structures in turbomachines is theexperimental and numerical analysis of acoustic and noiseemission of tip vortices [14, 15, 16, 17]. e noise of a fan isnoticeable by a CPU, car or a rail vehicle cooler. All three ex-amples are met in the everyday life. One of the main reasonsfor the noise is the gap s := (a − b)/a between the housingand the impeller tip. With increasing gap, the noise emis-sion and the energy dissipation increase [18, 19]. Karstadt etal. [20] and Zhu [5] investigated noise and dissipation due totip vortices.

e kinematics for an encased axial turbomachine oper-ating at part load or overload, are not fully understood yet.Especially the basic kinematics of these phenomena are notsuciently analyzed.

2. VORTEX THEORYAn encased axial turbomachine with Z impellers is consid-ered. e sketched vortex system (see gures 1 and 2) results

𝑈𝑈

𝑍𝑍Γ

Γ𝑡𝑡

𝛺𝛺

𝛺𝛺ind

Figure 1. A coaxial vortex ring of transient strength Γt andradius b in a circular tube of radius a, according to the caseϕ→ 0. e sketch is for Z = 1, i.e. one bound vortex onlyto improve clarity.

𝑈𝑈

𝑍𝑍Γ

Γ

𝛺𝛺

𝛺𝛺ind

Γ

Figure 2. A horseshoe vortex of strength Γ and radius b ina circular tube of radius a, according to the case ϕ→∞.e sketch is for Z = 1, i.e. one bound vortex only toimprove clarity.


from the Z bound vortices. e generation of a bound vortexwas explained by Prandtl [21] using arguments of boundarylayer theory and Kelvin’s circulation theorem. e presenceof viscosity is essential for the creation of the bound vortex,but the generation phase is not in the scope of this paper. Forvortex generation, we would like to refer the reader to thework of Prandtl [21].

By vortex theory, each blade 1...Z of length b is repre-sented by its bound vortex of strength Γ. For simplicity, thisinvestigation assumes Γ to be constant in radial directionalong the blade from r = 0 to r = b. As a vortex lament can-not end in a uid due to Helmholtz’s vortex theorem, a free,trailing vortex springs at each blade end r = 0 and r = b; seegures 1 and 2. ese vortices are of the same strength as thebound vortex. At the inner end r = 0, a straight semi-innitevortex line 0 ≤ z < ∞ of strength ZΓ – the so called hub vor-tex – aaches to the blade. e tip vortices at the outer endr = b are helices. e axial distance of the each helix wind-ing, i.e. the helix pitch, is given by U/n = 2πaϕ. Dependingon the load, these helices either ”wind up” (ϕ → 0), form-ing a vortex ring, or stretch to innity (ϕ→∞), yielding astraight, semi-innite vortex line.

Regardless of the ownumber ϕ, the semi-innite straightvortex line at r = 0 induces the circumferential velocityZΓ/(4πb) at z = 0, r = b due to the Biot-Savart law. Hence,the induced rotational speed is Ωind = ZΓ/(4πb2).

In a next step, this analysis calculates the vortex strengthZΓ, employing the angular momentum equation and theenergy equation. On the one hand, the axial componentof the angular momentum equation is ZΓ/2π = dM/d Ûm.Here, M is the axial torque component and Ûm the mass ux.Multiplying the momentum equation by Ω yields ZΓn =dP/d Ûm. P = MΩ is the power applied to the uid by meansof the rotating bound vortices. On the other hand, the energyequation for an adiabatic ow reads dP/d Ûm = 4ht, with 4htbeing the dierence in total enthalpy experienced by a uidparticle passing the cross-section z = 0. Both argumentsresult in the relation ZΓn = 4ht.

From turbomachine theory, the expression4ht = (Ωb)2(1 − ϕ/ϕ) can be derived from the equationmentioned above. e dimensionless design parameter ϕequals the tangent of the blade’s trailing edge angle β2, i.e.ϕ = tan β2. Hence, the relation between ZΓ and Ω yields

ZΓn

(Ωb)2= 1 − ϕ

ϕ. (1)

As equation 1 shows, the total change in circulation ZΓ alongthe plane of the machine is linked to the ow number ϕ byEuler’s turbine equation.

3. THE VORTEX SYSTEM AT HEAVY PARTLOADFor the limiting case of interest ϕ→ 0, the relation betweenZΓ and Ω, equation 1, yields 4ht = ZΓn = (Ωb)2. is

results in an induced sub-synchronous frequency

Ωind

Ω=

12, for ϕ→ 0. (2)

is induced frequency is in surprisingly good agreementwith measured sub-synchronous frequencies 0.5Ω...0.7Ω ofrotating stall of compressors, fans and pumps at part loadoperation [11] and may result in a rethinking of rotating stallfrom a kinematic perspective.

is investigation is now set to analyze the kinematicsof coaxial vortex rings of radius b and maximal strengthΓt = ZΓnt < (Ωb)2t as sketched in gure 1. By doing so,Laplace’s equation

1r∂

∂r

(r∂Φ

∂r

)+∂2Φ

∂z2= 0 (3)

for the velocity potential Φ, with ®u = ∇Φ, is solved for anincompressible, axisymmetric ow. Pelz et al. [10] recentlyderived the velocity potential for a coaxial vortex ring la-ment in a circular tube, yielding

φ(r, z) := ΦUa=

=za− 2τβ2

∞∑n=1

J1(knβ)knJ20 (kn)

J0(kn

ra

)exp

(−kn|z |a

), (4)

with J0, J1 the Bessel function of orders 0 and 1, respectively,and kn the zeros n = 1...∞ of the functionJ ′0(kn) = −J1(kn) = 0. e dimensionless velocity potential φdepends on the dimensionless ring radius β := b/a and the di-mensionless vortex strength τ := Γt/ 2bU. Since Γt increaseslinearly in time, τ can also be interpreted as a parametrictime of the process.

Stokes stream function for this ow is (using the integra-bility conditions ∂ψ/∂z = −r ∂φ/∂r and ∂ψ/∂r = r ∂φ/∂z[10])

ψ(r, z) := ΨUa= −1

2

(1 − r2

a2

)+

+ 2τβ2∞∑n=1

J1(knβ)knJ20 (kn)

J1(kn

ra

) raexp

(−kn|z |a

). (5)

With the stream function, the radial velocity component

ur (r, z)U

= 2τβ2∞∑n=1

J1(knβ)J20 (kn)

J1(kn

ra

)exp

(−kn|z |a

)(6)

and the axial velocity component

uz(r, z)U

= 1+2τβ2∞∑n=1

J1(knβ)J20 (kn)

J0(kn

ra

)exp

(−kn|z |a

)(7)

are given. e velocity eld (equation 6 and 7) takes theinduction of the vortex ring into account. At the stagnationpoint z = ±z0, the axial velocity uz vanishes for

1 = −2τβ2∞∑n=1

J1(knβ)J20 (kn)

J0(kn) exp(−kn|z0 |a

). (8)


1−1 0 0.5−0.5

AXIAL COORDINATE ⁄𝑧𝑧 𝑎𝑎

1

1

0

0.5

0.5

RAD

IAL

CO

ORD

INATE

⁄ 𝑟𝑟𝑎𝑎

1

1

0

0.5

0.5

1

1

0

0.5

0.5

1−1 0 0.5−0.5 1−1 0 0.5−0.5

𝜏𝜏 = 0.8 𝜏𝜏 = 2.5 𝜏𝜏 = 5.0

⁄−𝑧𝑧0 𝑎𝑎 ⁄−𝑧𝑧0 𝑎𝑎 ⁄−𝑧𝑧0 𝑎𝑎

Figure 3. Contour plots of the stream function (equation 5) for a vortex ring with the strength τ = 0.8 (le), τ = 2.5(middle), τ = 5.0 (right) for β = 0.8.

Equation 8 is implicit for the stagnation point z0 = z0(τ, β).Figure 3 shows Stokes stream function and the stagnationpoints at z = ±z0 for dierent circulation strengths τ of thevortex for β = 0.8.

4. THE VORTEX SYSTEM AT HEAVY OVER-LOAD

MIRROREDTIP VORTEX

TIP VORTEX

MIRROREDTIP VORTEX

Γ

Γ

Γ𝜉𝜉

i𝜂𝜂 𝑣𝑣ind

Ω

A-AEXCLUDING HUB VORTEX

A-AINCLUDING HUB VORTEX

𝑈𝑈

𝑍𝑍Γ

Γ

𝛺𝛺

𝛺𝛺ind

Γ

Figure 4. e vortex system of an axial turbomachine atoverload, Z = 1.

Figure 4 visualizes the vortex system of an axial turboma-chine composed of Z = 1 impeller blade at heavy overloadϕ → ∞ (Z = 1 is chosen to improve clarity only). For thislimiting case, the ow at cross section A-A far downstream ofthe machine is a plane potential ow. It can be described us-ing theory of functions (complex analysis) [22]. Mirrored tipvortices are necessary, to fulll the kinematic boundary con-dition on the tube wall. ese mirrored vortices are locatedin the housing and on the rotational axis of the turbomachine.Considering the tip vortex and its mirrored conjugates only,i.e. neglecting the hub vortex as a rst step, one obtains thesystem visualized in gure 4 boom le. e vortex on theaxis and the hub vortex feature identical magnitude but op-posed rotating direction. Adding the hub vortex, yielding thecomplete system, hence, results in the annulation of thesetwo vortices; see gure 4 boom right.

In the following notation, the complex coordinatesζ = ξ+ iη = r exp (iθ) are composed of a real part ξ = r cos θand an imaginary part η = r sin θ, with i =

√−1. e com-

plex potential F(ξ, η) is divided into the real part, which isthe velocity potential <[F (ξ, η)] = Φ(ξ, η) and the imagi-nary part, which is the stream function= [F (ξ, η)] = Ψ(ξ, η).F is the complex conjugate of F.

For the considered potential ow, the tip vortex at radialposition b = (1 − s)a yields the complex potential

F1(ζ) = −iΓ2π

ln (ζ − b) . (9)

Here, s is the dimensionless gap. e Milne-omson circletheorem [23] is applied, to derive the complex potential sat-isfying the kinematic boundary condition at the wall. istheorem postulates a resulting complex potential

F2(ζ) = F1(ζ) + F1

(a2

ζ

)(10)


for a potential F1 and the mirrored potential at the surround-ing wall. Adding the potential of the mirrored tip vortex onthe axis of the turbomachine ζ = 0, see gure 4 boom right,yields for the complex ow potential

F ′(ζ) = − iΓ2π

[ln (ζ − b) − ln

(ζ − a2

b

)+ ln ζ

]+const. (11)

e tip vortex at ζ = b with the circulation Γ necessitates amirrored vortex at ζ = 0 with the same magnitude of circula-tion and a mirrored vortex in the housing atζ = a2/b = a/(1 − s) with the same magnitude and inverteddirection. Up to now, the hub vortex is excluded from theconsiderations. Considering the hub vortex, as visualized ingure 4 boom right, yields for the complex potential

F(ζ) = − iΓ2π

[ln (ζ − b) − ln

(ζ − a2

b

)]+ const. (12)

In the following, this analysis shows, that an inducedmovement of the gap vortex occurs against the rotating di-rection of the turbomachine at heavy overload. A potentialvortex induces a velocity on the surrounding ow. e ve-locity components of a given potential F(ζ) are calculatedby

dF(ζ)dζ

= uind ®eξ − ivind ®eη . (13)

A straight vortex lament does not induce a velocity on itsown, due to the Biot-Savart law, so the induced velocity atζ = b is only due to the mirrored tip vortex at ζ = a/(1 − s).e resulting induced velocity at the position of the tip vortexyields

uind ®eξ − ivind ®eη =ddζ

[iΓ2π

ln(ζ − a2

b

)] ζ=b

= − iΓ2πb

b2

a2 − b2®eη . (14)

Assuming a turbomachine with Z impeller blades, the rotat-ing velocity of the tip vortex is

vind =ZΓ2πb

b2

a2 − b2=

ZΓ2πas

1 − s2 − s

= Ωa1 − ϕ/ϕ

s(1 − s)32 − s

. (15)

is is the rotating direction against the rotating directionof the turbomachine. For symmetry reasons, the rotatingtrajectory denes a circular path at radius b = a(1 − s).Hence, the induced frequency at ζ = b is

Ωind =vind

b=Ω

s(1 − s)22 − s

(1 − ϕ

ϕ

). (16)

For high ow number ϕ → ϕ and small gap s 1, theinduced frequency yields

Ωind

Ω=

12s, for ϕ→∞ (17)

against the rotating direction of the turbomachine.

5. APPLICATION FOR ACOUSTICAL INVES-TIGATIONS

SOUND FREQUENCY 𝑓 in HzFLO

W N

UM

BER 𝜑 𝑠 = 0.38%, Z = 9, n = 42 s−1, 𝑅𝑒 = 3.46 ⋅ 106

Figure 5. Frequency spectra of the tip clearance noisedepending on the ow number ϕ [4].

Previous investigations by Karstadt et al. [4] analysedthe tip clearance noise in axial turbomachines. Figure 5shows the frequency spectra over the complete operatingrange. Remarkable are the peaks at 42 Hz and 375 Hz, whichcorrespond to the rotational speed n and the blade passingfrequency Zn. Fukano and Yang [17] show, that the circum-ferential frequency of the tip clearance noise shis to lowervalues with increasing tip clearance s and decreasing ownumber ϕ due to the larger extent of the gap vortex. epresent paper investigates the frequency of the tip clearancenoise depending on the operating point applying analyticalmethods. At heavy part load, equation 2 indicates that aninduced frequency at half the rotation speed of the turbo-machine should appear. is frequency was also observedby Karstadt et al. [4] as gure 5 shows a high intensity inthe region of 20 Hz. Equation 17 indicates, that for heavyoverload, the induced frequency will increase with decreas-ing tip clearance. Furthermore, we expect a noise of highfrequency due to the small value of s < 1%, which is commonfor turbomachines. e broadband drop in the sound powerfor all ow numbers is clearly visible.

Muller [24] applied the continuity and the momentumequation and deduced, that sound inside a uid volume isonly emied, if the rotation of the velocity eld changes intime. e present study applies a similar approach to analysethe tip clearance noise of a turbomachine. Time-consumingsimulations, as performed by Carolus et al. [25] surely allowa more profond and accurate insight into the acoustics ofturbomachines. e development of an analytical model,which predicts main frequencys, is yet interesting to generatea deeper understanding of the acoustics in turbomachines.

ese ndings and the presented analytical model inthis paper could be an ecient tool for acoustic design ofturbomachines.

6. SUMMARY AND CONCLUSIONAn interplay between dynamic and kinematic eects explainsow structures and phenomena. Using computational uiddynamics, a clear distinction of both eects is oen impos-sible. In contrast, analytical methods allow a more focusedpicture of uid mechanics, i.e. they allow a clear distinctionof eects. Of course, only generic ows are accessible toanalytical methods.


is paper focused on an analytic model for wall stall,so far being explained by dynamics only: boundary layerseparation is indeed a dynamic eect. Nevertheless boundarylayer separation is not necessarily the only reason for wallstall. It is shown, that kinematics may also explain at leastsome eects of wall and rotating stall. e used picture fora ow at small ow numbers is a thread spool, rolling upthe tip vortices resulting from rotating bound vortices. Fromthe uid mechanics perspective, the thread spool is a coaxialvortex ring of increasing strength connected to a semi-innitehub vortex (gure 1).

So far, the velocity potential of a coaxial vortex ring insidea tube was unknown. e solution of Laplace’s equationresults in the velocity potential for the vortex lament withina tube [10]; see equation 4.

is paper gained three main results, which are due tokinematics only. First, at part load operation, the hub vor-tex induces a sub-synchronous rotation of the vortex ring.e derived rotational speed Ωind = 0.5Ω of the vortex ringis surprisingly consistent with observed sub-synchronousspeeds of rotating stall; cf. [11]. Second, the vortex ring in-duces an upstream axial velocity at the wall. Together withthe undisturbed ow velocity, this results in a stagnationpoint upstream and downstream at the wall, which may beinterpreted as ”wall stall” (gure 3). ird, at overload opera-tion, the induced rotational direction is inverted to the caseat part load. e semi-innite straight vortex lament at theouter blade end rotates against the rotating direction of theturbomachine, due to the induction of the hub vortex. einduced frequency yields Ωind = −0.5Ω/s.

e presented analytical model may give new argumentsand improves the understanding of the vortex system in tur-bomachines, but is also intended to motivate generic experi-ments. Hence, a test rig will validate the models presentedin this paper in the near future.

As a next step, the velocity potential for a coaxial vortexring lament in a circular tube (equation 4) will be extendedto a coaxial vortex layer, yielding a transient behavior of thevortex system. is behavior leads to a change in the circula-tion over time being responsible for noise emission [24].

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