DEVELOPMENT OF A MATHEMATICAL MODEL FOR THE SWIRLING FLOW EXITING THE RUNNER OF A HYDRAULIC TURBINE

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1 6 th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, September 9-11, 2015, Ljubljana, Slovenia DEVELOPMENT OF A MATHEMATICAL MODEL FOR THE SWIRLING FLOW EXITING THE RUNNER OF A HYDRAULIC TURBINE Romeo SUSAN-RESIGA * Politehnica University Timișoara, Research Centre for Complex Fluid Systems Engineering, Romania ABSTRACT The paper introduces a mathematical model for computing the swirling flow exiting the Francis turbine runner without actually computing the three-dimensional flow in the inter-blade channels. Such models are useful for a-priori assessment of the draft tube hydrodynamics, as well as for runner outlet optimization to achieve a best match with a given draft tube. Moreover, the model can be used within a range of operating regimes, provided that no severe flow detachment from the runner blade occurs. Since we assume an inviscid, steady and axisymmetric swirling flow, and the model is restricted to an arbitrary hub-to-shroud line, we assess the impact of these assumptions on the model accuracy and point out the direction for further refinements. KEYWORDS Hydraulic turbine, inviscid, steady, axisymmetric swirling flow, mathematical model 1. INTRODUCTION When designing a new runner or refurbishing an old one for modern hydraulic turbines one should account for a range of operating regimes instead of focusing on a single operating point. Moreover, particularly for refurbishment problems, the runner should be a best match for the existing draft tube. As a result, a critical issue is to robustly compute the swirling flow exiting the runner, and further ingested by the draft tube, prior actually designing the runner. If properly parameterized, such flow configurations can be optimized for best draft tube performance within a given operating range, then the runner blades can be designed accordingly through an inverse design method. A first model for computing the swirling flow at the draft tube inlet, i.e. on a plane cross section, normal to the machine axis, located in the upstream region of the discharge cone was developed and validated against experimental data in [1]. This model also introduced the concept of the swirl-free velocity profile, which is a fictitious quantity with a clear and intuitive physical interpretation and conveniently replaces the relative flow angle. The model was used in [2] for assessing the draft tube losses within a range of operating regimes without actually computing the runner. Moreover, it was shown that small adjustments of the swirl-free velocity profile could diminish the weighted average draft tube losses within a given range of discharge values. It was also recognized that instead of computing the swirling flow on a plane cross-section normal to the machine axis, it would be preferable to choose a conical section closer to the runner blades trailing edge. This prompted a further development in [3], where the swirling flow exiting a Francis runner was computed * Corresponding author: Bvd. Mihai Viteazu, No. 1, Timișoara, Romania, phone: , romeo.resiga@upt.ro

2 along a straight line connecting the hub and the shroud in the meridian half plane, and in [4] where an arbitrary line from hub-to-shroud was considered, thus getting as close as possible to the actual runner blades trailing edge. An important addition to such models is the capability of modelling the vortex breakdown, i.e. the development of a stagnant region near the axis particularly for operating points far from the design one. Based on a variational formulation that actually minimizes the swirl number, the theory of swirling flows with stagnant region is detailed in [5]. A full example of optimizing the runner outlet using such models for swirling flows downstream the runner is presented in [6]. This paper attempts to review the current status of developing robust and accurate swirling flow models to be used for runner outlet optimization with respect to the minimization of weighted averaged draft tube losses within a range of operating points. In Section 1 we revisit the two-dimensional through-flow model, together with the variational formulation of the corresponding boundary value problem. Then, in Section 2, we restrict this model to an arbitrary hub-to-shroud line, emphasizing the information that one should provide for a correct swirling flow computation. Section 3 includes a model validation and assessment against numerical results for three-dimensional flow computation in a Francis runner, an discusses the cause of the discrepancies near the hub. Finally, Section 4 summarizes the conclusions and outline the directions for further refine such models. 2. THE THROUGHFLOW THEORY FOR TURBOMACHINES REVISITED An early steady through-flow theory for inviscid fluid flow in turbomachines, with arbitrary hub and shroud wall shapes, was developed by Wu [7]. This theory was intended for both direct and inverse problems, within the assumption of an infinite number of blades, infinitely thin. The continuity equation was automatically satisfied by introducing the streamfunction, and the equation of motion was casted as the so-called principal equation. A synopsis of the computational methods for turbomachinery flows until early 80s is presented by Hirsch [8]. Later, Denton and Dawes [9] confirm that the axisymmetric hub-to-shroud calculation, often called throughflow calculation has become the backbone of turbomachinery design, and coupled with blade-to-blade calculations on streamsurfaces of revolution form the quasithree-dimensional approach. a) streamsurfaces in a radial-axial turbomachine b) meridian half-plane cross-section Fig. 1 Streamsurfaces for axisymmetric swirling flows in turbomachineries and a typical meridian cross-section for a Francis turbine.

3 We briefly revisit the inviscid, steady, axisymmetric swirling flow theory for turbomachinery runners, as being the most general model from which one can derive the equation for non-rotating blades or for blade-less regions. Figure 1a) shows a typical flow passage for a radial-axial turbomachine, in particular a Francis turbine, with two revolution surfaces for hub (crown) and shroud (band). The shroud is continued downstream with a discharge cone. The flow is considered axial-symmetric, with the velocity vector in cylindrical coordinates v v a,v r,v u ( ). The velocity field can be conveniently expressed using the streamfunction ψ( z,r) for the meridian vector projection v m = v a ẑ + v r ˆr and the circulation function κ r v u for the azimuthal (tangential) velocity, as: v = v m + v u ˆθ = ψ θ + κ θ. (1) Note that in cylindrical coordinates we have the unit vectors ( ẑ, ˆr, ˆθ ) and θ = ˆθ r. The axisymmetric streamsurfaces (S1) from Fig. 1 are simply surfaces with ψ = constant. The vorticity vector can also be expressed using the two scalar functions ψ and κ, v = r ψ r 2 ˆθ + κ θ. (2) A hub-to-shroud streamsurface within the bladed region, (S 2 ) in Fig. 1a) can be geometrically described by the azimuthal coordinate for points on the surface, ϕ, as function of the meridian coordinates z,r ( ). As a result, the equation for a streamsurface S 2 is: α( z,r,θ) = θ ϕ( z,r) = constant. (3) The velocity relative to a rotating frame of reference, w = w m + w u ˆθ, has the same meridian projection as the absolute velocity, i.e. w m = v m, while the circumferential component is w u = v u r. One should keep in mind that the last relationship uses dimensionless velocities, with the reference velocity ΩR ref, with Ω the runner angular speed and R ref the runner outlet radius. As a result, the dimensionless transport velocity becomes ΩR ( ) ( ΩR ref ) = r, i.e. the dimensionless radius. The flow tangency condition on the streamsurface (3) is simply the orthogonality condition between the unit normal to S 2 and the relative velocity, w α = 0 ( ψ θ) ϕ = w u r. (4) The Euler equation for steady axisymmetric flow of a perfect fluid (incompressible and inviscid) in the rotating frame of reference is conveniently projected [10] along two orthogonal directions in the tangent plane to S 2. The projection along the relative velocity w leads to the conclusion that the relative total pressure χ p + v 2 2 r v u is a function of the streamfunction only, i.e. we have χ(ψ). This is a Bernoulli-like theorem, leading in its integral form to the turbomachinery fundamental equation. On the other hand, the projection along the direction w α, Fig. 1a), leads to the principal equation of the turbomachinery throughflow theory,

4 ψ r 2 + ( κ θ ) ϕ = dχ dψ. (5) Equations (4) and (5) involve three scalar functions ψ(z,r), κ(z,r) and ϕ(z,r). For design problems (the so-called inverse problem) the circulation schedule κ(z,r) is given, and the streamfunction is computed, together with the blade shape ϕ(z,r). For analysis purpose (the so-called direct problem) ϕ(z,r) is given, and both ψ(z,r) and κ(z,r) are computed. Then the velocity field follows from Eq. (1), and the pressure follows from the known function χ(ψ). The partial differential equation (5) should be completed with boundary conditions on the boundary of the meridian domain D occupied by the runner blades, Fig. 1b), essential condition: ψ = f e on ( D), e (6a) natural condition: v τ + κ ϕ τ = f n on D ( ) n, (6b) where τ is the tangential coordinate along the boundary in counter-clockwise direction. The boundary with essential conditions D ( ) e corresponds to the crown and band, Fig. 1b), where the streamfunction is given. The boundary with natural conditions ( D) corresponds to the n leading edge and trailing edge lines in the meridian half plane. The rather unusual natural condition (6b) was derived by Lurie et al. [11] while examining the flow at the leading edge. More precisely, when using the through-flow model for analysing the flow in a turbine operated at various regimes, it is obvious that the circulation κ generated by the adjustable guide vanes does not match the circulation resulting from the flow tangency condition Eq. (4) within the runner blades. As a result, there will be a jump in circulation at the leading edge κ vτ +. Correspondingly, there will be a jump in the meridian velocity along the leading edge, and Eq. (6b) provides the relationship vτ + κ ϕ τ ( ) = 0. On the other hand, at the trailing edge the circulation jump + κ vanishes thanks to the Kutta-Jukowski condition and the streamlines do not have an angular point. The variational formulation of the boundary value problem (5) (6) can be stated as: given the blade shape ϕ(z,r), the circulation schedule κ(z,r) and the relative total pressure χ(ψ) function, find the streamfunction ψ(z,r) that satisfies the essential conditions (6a) and minimizes the functional 1 ( ψ) 2 F(ψ) = + κ( ψ θ) ϕ + χ(ψ) r dd + 2 r ψ f 2 n dτ. (7) D ( D) n This variational formulation provides the mathematical foundation for the finite element method which is quite robust and accurate for such elliptic boundary value problems. 3. SWIRLING FLOW MODEL ON AN ARBITRARY HUB-TO-SHROUD LINE The functional (7) can be adapted for a hub-to-shroud line as follows.

5 First, let us consider a computational domain in the meridian half-plane with a small stream-wise width, enclosing a hub-to shroud line located in the neighbourhood of the blade trailing edge. Figure 1b) shows two such lines (red, dashed lines) downstream the trailing edge of a Francis runner. On the crown and band we have essential boundary conditions, ψ = 0 and ψ = q 2, respectively, where q Q ( πr 2 ref ΩR ref ) is the dimensionless discharge (discharge coefficient). Since the circulation function variation along the streamlines vanishes at the trailing edge, i.e. further downstream in the bladeless region we have a function κ(ψ), the natural conditions (6b) on the upstream/downstream hub-to-shroud boundaries exactly cancel as the domain stream-wise width vanishes. As a result, the boundary term in (7) vanishes as the domain collapses to a hub-to-shroud line. Second, the geometry of the hub-to-shroud line can be described with two functions z(ξ) and r(ξ), where ξ is the arc-length coordinate running from 0 at the crown up to l at the band. The intrinsic coordinates and the corresponding unit vectors are defined as normal: ˆσ = ẑ cosδ + ˆr sin δ, tangent: ˆξ = ẑ sin δ + ˆr cosδ, where cosδ = dr dξ and sin δ = dz dξ. Third, on a streamline in the meridian half-plane we define the unit vectors normal: ˆn = ẑ sin γ + ˆr cos γ, tangent: ˆm = ẑ cos γ + ˆr sin γ, where cos γ = tan 2 γ and sin γ = tan γ 1 + tan 2 γ, tan γ = v r v a. (8) (9) Using the above geometrical considerations, we can express the meridian velocity magnitude, 1 v m = cos γ δ ( ) 1 r ψ ξ. (10) One can see that solving for the swirling flow on a hub-to-shroud line requires the meridian flow direction, v r v a as input information. The flow tangency condition, Eq. (4), can be written as w m r ϕ m since the relative flow angle with respect to a meridian plane is, by definition tan β w u w m ( ) = w u, and = v r u ( ψ θ) ( r ϕ) = v v m tan β. (11) m In the end, the functional from Eq. (7) becomes on a hub-to-shroud line, l F TE (ψ) = r v 2 m 2 + κv tan β + rχ(ψ) dξ, with v m m from Eq. (10). (12) 0 In the above functional the function χ(ψ) is given, and r, tan β, cos(γ δ), as well as κ, are considered known functions of ξ. The unknown function ψ(ξ) is approximated with cubic Hermite polynomials on subintervals ξ i,ξ i+1, then a standard finite element method is employed to minimize F. TE The circulation κ(ξ) is iteratively updated using the flow tangency condition v u = r + v m tan β. In order to achieve convergence, this iterative correction is under-relaxed.

6 4. NUMERICAL RESULTS A numerical example is considered for the GAMM Francis turbine model. The threedimensional turbulent flow computations are performed using one interblade channel for the distributor (stay vanes and guide vanes) and runner, using a mixing interface technique [12]. Guide vane opening Discharge coefficient [-] Energy coefficient [-] Tab.1 Operating points for the GAMM Francis turbine model The operating points considered for the present example are at constant head and variable discharge, as shown in Tab. 1. The numerical results from the 3D runner computations are circumferentially averaged on surfaces of revolutions generated by the hub-to-shroud lines shown in Fig. 1b). The resulting axial, radial and circumferential velocity profiles, shown in Figs. 2 5 with circles labelled NUM, as well as the pressure, are used to provide the information required for minimizing the functional from Eq. (12). There are no additional approximations involved, as we intended to check the axisymmetric swirling flow solver against full 3D computations. a) hub-to-shroud line 1 a) hub-to-shroud line 2 Fig. 2 Velocity profiles downstream de runner, for guide vane opening 24.

7 a) hub-to-shroud line 1 a) hub-to-shroud line 2 Fig. 3 Velocity profiles downstream de runner, for guide vane opening 25. a) hub-to-shroud line 1 a) hub-to-shroud line 2 Fig. 4 Velocity profiles downstream de runner, for guide vane opening 26. a) hub-to-shroud line 1 a) hub-to-shroud line 2 Fig. 5 Velocity profiles downstream de runner, for guide vane opening 28.

8 The numerical results obtained with the present methodology derived from the turbomachinery throughflow theory (TTT) are shown in Figs. 2 5 with solid lines. One can see a very good agreement between NUM data and TTT data, particularly towards the hub (band). On the other hand, there are some discrepancies in the hub (crown) neighbourhood. A closer look show that these discrepancies are larger on the hub-to-shroud line no.1, close to the trailing edge, and smaller on the hub-to-shroud line no.2, see Fig. 1b). 5. CONCLUSION We present a mathematical model for computing the inviscid, steady, axisymmetric swirling flow exiting the Francis turbine runner on an arbitrary hub-to-shroud line in the meridian half plane. The model uses a variational formulation that incorporates information regarding both meridian and tangential flow direction. Three-dimensional numerical flow simulation results, circumferentially averaged, are used for validating the model and assessing its accuracy on two hub-to-shroud lines. We have found a general good agreement, except in the neighbourhood of the crown. We conjecture that these discrepancies are related to the meridian streamlines curvature, which is currently neglected by the model. Further refinements should account for this streamline curvature, which is larger near the crown than near the band. 6. ACKNOWLEDGEMENTS The author acknowledges the support from a grant of the Romanian Ministry of National Education and Research, CNCS-UEFISCDI, project number PN-II-ID-PCE The numerical data from 3D flow computations were provided by Dr. Sebastian Muntean. 7. REFERENCES [1] Susan-Resiga, R., Muntean, S., Avellan, F., and Anton, I.: Modelling of swirling flow in hydraulic turbines for the full operating range, Applied Mathematical Modelling, 2011, 35, pp [2] Susan-Resiga, R., Muntean, S., Ciocan, T., Joubarne, E., Leroy, P. and Bornard, L.: Influence of the velocity field at the inlet of a Francis turbine draf tube on performance over an operating range, 26th IAHR Symposium on Hydraulic Machinery and Systems, 2012, IOP Conf. Series.: Earth and Environmental Science 15, paper [3] Susan-Resiga, R., Muntean, S., Ciocan, T., de Colombel, T., and Leroy, P.: Surrogate runner model for draft tube losses computation within a range of operating points, 27th IAHR Symposium on Hydraulic Machinery and Systems, 2014, IOP Conf. Series.: Earth and Environmental Science 22, paper [4] Susan-Resiga, R., Ighișan, C., and Muntean, S.: A Mathematical Model for the Swirling Flow Ingested by the Draft Tube of Francis Turbines, WASSERWIRTSCHAFT, 2015, 105(1), pp [5] Susan-Resiga, R., Muntean, S., Stuparu, A., Bosioc, A. I., Tănasă, C., and Ighișan C.: A variational model for swirling flow states with stagnant region, European Journal of Mechanics B/Fluids, 2015 (accepted). [6] Ciocan, T., Susan-Resiga, R., and Muntean, S.: Modelling and Optimization of the Velocity Profiles at the Draft Tube Inlet of a Francis Turbine within an Operating Range, Journal of Hydraulic Research, 2015 (accepted).

9 [7] Wu, C.-H.: A General Through-flow Theory for Fluid Flow with Subsonic or Supersonic Velocity in Turbomachines of Arbitrary Hub and Casing Shapes, Technical Note 2302, National Advisory Committee for Aeronautics, [8] Hirsch, Ch.: Computational Models for Turbomachinery Flows, Technical Report NPS , Naval Posgraduate School, Monterey, California, [9] Denton, J. D. and Dawes, W. N.: Computational fluid dynamics for turbomachinery design, Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science, 213(2), 1998, pp [10] Bosman, C.: The Occurrence and Removal of the Indeterminacy from Flow Calculations in Turbomachines, RM 3746, Aeronautical Research Council, London, [11] Lurie, K. A., Fedorov, A. V., and Klimovich, V. I.: Conditions along the boundaries of bladed zones within the flow tracts of turbines, International Journal for Numerical Methods in Fluids, 2(3), 1982, pp [12] Muntean, S., Susan-Resiga, R., and Anton, I.: Mixing Interface Algorithm for 3D turbulent flow analysis of the GAMM Francis turbine, Modelling Fluid Flow, J. Vad, T. Lajos and R. Schilling, eds., Springer-Verlag, 2004, pp