Vortex Structure and Kinematics of Encased Axial Turbomachines

This paper models the kinematics of the vortex system of an encased axial turbomachine at part load and overload applying analytical methods. Thus far, the influence of the casing and the tip clearance on the kinematics have been solved separately. The vortex system is composed of a hub, bound and tip vortices. For the nominal operating point φ ≈ φ opt and negligible induction, the tip vortices transform 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. The vortex ring itself is generated by bound vortices rotating at the angular frequency Ω . The hub vortex induces a velocity on the vortex ring causing a rotation at the sub-synchronous frequency Ω ind = 0.5 Ω . Besides, the vortex ring itself induces an axial velocity. Superimposed with the axial main flow this results in a stagnation point at the tube wall. This stagnation point may wrongly be interpreted as dynamic induced wall stall. For overload operation φ → ∞ the vortex system of the turbomachine forms a horseshoe, i.e., the pitch of the screw becomes infinite. Both hub and tip vortices are semi-infinite, straight vortex filaments. The tip vortices rotate against the rotating direction of the turbomachine due to the induction of the hub vortex yielding the induced frequency Ω ind = − 0.5 Ω / s with the tip clearance s.


INTRODUCTION
By now, the common understanding is, that rotating stall and the resulting noise and vibration within a turbomachine is a dynamic effect. is means, that frictional forces lead to boundary layer separation and eventually stall in rotating machines [ , ]. is understanding is recently confirmed by Cloos et al. [ ] both, experimentally and analytically, for the most generic machine, a flow through a coaxial rotating circular tube. According to Cloos et al. [ ], "wall stall"a term coined by Greitzer [ ] in contrast to "blade stall"is caused at part load by the interaction of axial boundary layer and swirl boundary layer flow, i.e. the influence of centrifugal force on axial momentum. For "wall stall", the axial velocity component u z vanishes at the line z = −z 0 , r = a (axial coordinate in mean flow direction z, distance z 0 from the reference point on the line of symmetry r = 0, radial coordinate r, tube radius a; cf. figure ).
is paper analyzes the flow situation in encased axial turbomachines for small viscous friction. is work shows, that "wall stall", i.e. u z (−z 0 , a) = 0, can also be a result of kinematics only, due to induced velocities of the vortex system superimposed with the axial main flow. e structure of the vortex system, especially the tip vortices, depends on the operating point of the turbomachine. Furthermore, the tip vortices rotate with a sub-synchronous frequency Ω ind (Karstadt et al. [ ], Zhu [ ]). e aim of the present investigation is to analyze the influence of the vortex system in encased axial turbomachines and its circulation strength on the observed phenomena, yielding the research questions: . Is it possible to explain by means of analytical methods the sub-synchronous frequencies observed for turbomachines? . Can "wall stall" be a result of kinematics only?
To answer these questions, this work first employs vortex theory for an encased axial turbomachine, followed by the application of fundamental solutions. For machines without casing like wind turbines and screw propellers, vortex theory is well described by In addition, to enlarge the investigation on the whole operating range of a turbomachine, we investigate the structure and kinematics of the vortex system at heavy overload, applying theory of functions (complex analysis). For turbomachines, the flow number ϕ := U/ Ωa defines the operating point (e.g. part load or overload). e flow number is the ratio of axial free-stream velocity U to the circumferential velocity Ωa, where Ω = 2πn is the rotational speed (the scaling to Ωa and not to Ωb, with the blade tip radius b, is common in the context of turbomachines and therefore used here as well [ ]).
For the nominal operating point ϕ ≈ ϕ opt and negligible induction, the vortex system of an encased axial turbomachine consists of a hub, Z bound and Z tip vortices, with Z the number of blades. e tip vortices transform into helices with a pitch of 2πϕa.
For part load operation ϕ → 0, see figure , the Z helices "roll up" and form a vortex ring, i.e. the pitch of the helices vanishes. e vortex ring is continuously generated by the bound vortex system. Hence, the coaxial vortex ring strength is transient. A nice picture for this vortex ring is that of a thread spool rolling up and gaining strength over time.
is picture will explain some transient phenomena using kinematic arguments only. e case of heavy overload occurs for infinitely high flow numbers ϕ → ∞; see figure . e hub, the bound and the tip vortices form a horseshoe, i.e. the pitch of the helices becomes infinite. Both, hub and tip vortices are semi-infinite, straight vortex filaments. In real turbomachines, the flow number cannot be adjusted to infinity, but is limited to a maximum valueφ due to flow rate limitations and geometric restrictions. Nevertheless, the analysis of this limiting case is important for the basic understanding of the vortex system in axial turbomachines.
To develop physical understanding of the whole picture in detail, the paper is organized as follows. Section gives a short literature overview. Section uses vortex theory to determine the strength of the vortices. Subsequently, section derives the velocity potential of a coaxial vortex ring within a circular tube at part load and the induced rotating frequency. e flow potential and the induction at overload is introduced in section . e paper closes with a short outlook to potential applications in section and a discussion in section .

. LITERATURE REVIEW
Investigations of vortex systems in fluid dynamics trace back to the work of Helmholtz [ ], who formulated the Helmholtz's theorems as a basis for the research concerning rotational fluid motion.
Didden [ ] performed measurements of the rolling-up process of vortex rings and compared the results with similarity laws for the rolling-up of vortex sheets.
Besides the investigation of vortex kinematics, a broad research field on vortex structures in turbomachines is the experimental and numerical analysis of acoustic and noise emission of tip vortices [ , , , ]. e noise of a fan is noticeable by a CPU, car or a rail vehicle cooler. All three examples are met in the everyday life. One of the main reasons for the noise is the gap s := (a − b)/a between the housing and the impeller tip. With increasing gap, the noise emission and the energy dissipation increase [ , ]. e sketch is for Z = 1, i.e. one bound vortex only to improve clarity. from the Z bound vortices. e generation of a bound vortex was explained by Prandtl [ ] using arguments of boundary layer theory and Kelvin's circulation theorem. e presence of viscosity is essential for the creation of the bound vortex, but the generation phase is not in the scope of this paper. For vortex generation, we would like to refer the reader to the work of Prandtl [ ].
By vortex theory, each blade 1...Z of length b is represented by its bound vortex of strength Γ. For simplicity, this investigation assumes Γ to be constant in radial direction along the blade from r = 0 to r = b. As a vortex filament cannot end in a fluid due to Helmholtz's vortex theorem, a free, trailing vortex springs at each blade end r = 0 and r = b; see figures and . ese vortices are of the same strength as the bound vortex. At the inner end r = 0, a straight semi-infinite vortex line 0 ≤ z < ∞ of strength Z Γ -the so called hub vortex -a aches to the blade. e tip vortices at the outer end r = b are helices. e axial distance of the each helix winding, i.e. the helix pitch, is given by U/n = 2πaϕ. Depending on the load, these helices either "wind up" (ϕ → 0), forming a vortex ring, or stretch to infinity (ϕ → ∞), yielding a straight, semi-infinite vortex line.
Regardless of the flow number ϕ, the semi-infinite straight vortex line at r = 0 induces the circumferential velocity Z Γ/(4πb) at z = 0, r = b due to the Biot-Savart law. Hence, the induced rotational speed is In a next step, this analysis calculates the vortex strength Z Γ, employing the angular momentum equation and the energy equation. On the one hand, the axial component of the angular momentum equation is Z Γ/2π = dM/d m.
Here, M is the axial torque component and m the mass flux. Multiplying the momentum equation by Ω yields Z Γn = dP/d m. P = MΩ is the power applied to the fluid by means of the rotating bound vortices. On the other hand, the energy equation for an adiabatic flow reads dP/d m = △h t , with △h t being the difference in total enthalpy experienced by a fluid particle passing the cross-section z = 0. Both arguments result in the relation Z Γn = △h t .
From turbomachine theory, the expression △h t = (Ωb) 2 (1 − ϕ/φ) can be derived from the equation mentioned above. e dimensionless design parameterφ equals the tangent of the blade's trailing edge angle β 2 , i.e. ϕ = tan β 2 . Hence, the relation between Z Γ and Ω yields As equation shows, the total change in circulation Z Γ along the plane of the machine is linked to the flow number ϕ by Euler's turbine equation.

. THE VORTEX SYSTEM AT HEAVY PART LOAD
For the limiting case of interest ϕ → 0, the relation between Z Γ and Ω, equation , yields △h t = Z Γn = (Ωb) 2 . is results in an induced sub-synchronous frequency is induced frequency is in surprisingly good agreement with measured sub-synchronous frequencies 0.5 Ω...0.7 Ω of rotating stall of compressors, fans and pumps at part load operation [ ] and may result in a rethinking of rotating stall from a kinematic perspective.
is investigation is now set to analyze the kinematics of coaxial vortex rings of radius b and maximal strength Γ t = Z Γnt < (Ωb) 2 t as sketched in figure . By doing so, Laplace's equation with J 0 , J 1 the Bessel function of orders 0 and 1, respectively, and k n the zeros n = 1...∞ of the function J ′ 0 (k n ) = −J 1 (k n ) = 0. e dimensionless velocity potential φ depends on the dimensionless ring radius β := b/a and the dimensionless vortex strength τ := Γ t / 2bU. Since Γ t increases linearly in time, τ can also be interpreted as a parametric time of the process. Stokes stream function for this flow is (using the integrability conditions ∂ψ/∂z = −r ∂φ/∂r and ∂ψ/∂r = r ∂φ/∂z [ ]) With the stream function, the radial velocity component and the axial velocity component are given. e velocity field (equation and ) takes the induction of the vortex ring into account. At the stagnation point z = ±z 0 , the axial velocity u z vanishes for   Mirrored tip vortices are necessary, to fulfill the kinematic boundary condition on the tube wall. ese mirrored vortices are located in 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 first step, one obtains the system visualized in figure bo om le . e vortex on the axis and the hub vortex feature identical magnitude but opposed rotating direction. Adding the hub vortex, yielding the complete system, hence, results in the annulation of these two vortices; see figure bo om right.
For the considered potential flow, the tip vortex at radial position b = (1 − s)a yields the complex potential Here, s is the dimensionless gap. e Milne-omson circle theorem [ ] is applied, to derive the complex potential satisfying the kinematic boundary condition at the wall. is theorem postulates a resulting complex potential for a potential F 1 and the mirrored potential at the surrounding wall. Adding the potential of the mirrored tip vortex on the axis of the turbomachine ζ = 0, see figure bo om right, yields for the complex flow potential e tip vortex at ζ = b with the circulation Γ necessitates a mirrored vortex at ζ = 0 with the same magnitude of circulation and a mirrored vortex in the housing at ζ = a 2 /b = a/(1 − s) with the same magnitude and inverted direction. Up to now, the hub vortex is excluded from the considerations. Considering the hub vortex, as visualized in figure bo om right, yields for the complex potential In the following, this analysis shows, that an induced movement of the gap vortex occurs against the rotating direction of the turbomachine at heavy overload. A potential vortex induces a velocity on the surrounding flow. e velocity components of a given potential F(ζ) are calculated by A straight vortex filament does not induce a velocity on its own, due to the Biot-Savart law, so the induced velocity at ζ = b is only due to the mirrored tip vortex at ζ = a/(1 − s). e resulting induced velocity at the position of the tip vortex yields Assuming a turbomachine with Z impeller blades, the rotating velocity of the tip vortex is is is the rotating direction against the rotating direction of the turbomachine. For symmetry reasons, the rotating trajectory defines a circular path at radius b = a(1 − s). Hence, the induced frequency at ζ = b is For high flow number ϕ →φ and small gap s ≪ 1, the induced frequency yields , that for heavy overload, the induced frequency will increase with decreasing tip clearance. Furthermore, we expect a noise of high frequency due to the small value of s < 1%, which is common for turbomachines. e broadband drop in the sound power for all flow numbers is clearly visible.
Müller [ ] applied the continuity and the momentum equation and deduced, that sound inside a fluid volume is only emi ed, if the rotation of the velocity field changes in time. e present study applies a similar approach to analyse the tip clearance noise of a turbomachine. Time-consuming simulations, as performed by Carolus et al. [ ] surely allow a more profond and accurate insight into the acoustics of turbomachines.
e development of an analytical model, which predicts main frequencys, is yet interesting to generate a deeper understanding of the acoustics in turbomachines. ese findings and the presented analytical model in this paper could be an efficient tool for acoustic design of turbomachines.

. SUMMARY AND CONCLUSION
An interplay between dynamic and kinematic effects explains flow structures and phenomena. Using computational fluid dynamics, a clear distinction of both effects is o en impossible. In contrast, analytical methods allow a more focused picture of fluid mechanics, i.e. they allow a clear distinction of effects. Of course, only generic flows are accessible to analytical methods. is paper focused on an analytic model for wall stall, so far being explained by dynamics only: boundary layer separation is indeed a dynamic effect. Nevertheless boundary layer separation is not necessarily the only reason for wall stall. It is shown, that kinematics may also explain at least some effects of wall and rotating stall. e used picture for a flow at small flow numbers is a thread spool, rolling up the tip vortices resulting from rotating bound vortices. From the fluid mechanics perspective, the thread spool is a coaxial vortex ring of increasing strength connected to a semi-infinite hub vortex (figure ).
So far, the velocity potential of a coaxial vortex ring inside a tube was unknown.
e solution of Laplace's equation results in the velocity potential for the vortex filament within a tube [ ]; see equation .
is paper gained three main results, which are due to kinematics only. First, at part load operation, the hub vortex induces a sub-synchronous rotation of the vortex ring. e derived rotational speed Ω ind = 0.5 Ω of the vortex ring is surprisingly consistent with observed sub-synchronous speeds of rotating stall; cf. [ ]. Second, the vortex ring induces an upstream axial velocity at the wall. Together with the undisturbed flow velocity, this results in a stagnation point upstream and downstream at the wall, which may be interpreted as "wall stall" (figure ). ird, at overload operation, the induced rotational direction is inverted to the case at part load. e semi-infinite straight vortex filament at the outer blade end rotates against the rotating direction of the turbomachine, due to the induction of the hub vortex. e induced frequency yields Ω ind = −0.5 Ω/s. e presented analytical model may give new arguments and improves the understanding of the vortex system in turbomachines, but is also intended to motivate generic experiments. Hence, a test rig will validate the models presented in this paper in the near future.
As a next step, the velocity potential for a coaxial vortex ring filament in a circular tube (equation ) will be extended to a coaxial vortex layer, yielding a transient behavior of the vortex system. is behavior leads to a change in the circulation over time being responsible for noise emission [ ].