Conference on Modelling Fluid Flow (CMFF’09) The 14th International Conference on Fluid Flow Technologies

Budapest, Hungary, September 9-12, 2009

THREE-DIMENSIONAL VERSUS TWO-DIMENSIONAL AXISYMMETRIC ANALYSIS FOR DECELERATED SWIRLING FLOWS

Romeo F. SUSAN-RESIGA1, Sebastian MUNTEAN2, Constantin TĂNASĂ3, Alin BOSIOC4

1 Corresponding Author. Department of Hydraulic Machinery, “Politehnica” University of Timişoara, Bvd. Mihai Viteazu, No. 1, Timişoara, 300222, Romania. Tel.: +40 256 403692, Fax: +40 256 403692, E-mail: resiga@mh.mec.upt.ro 2 Center of Advanced Research in Engineering Sciences, Romanian Academy-Timisoara Branch . E-mail: seby@acad-tim.tm.edu.ro 3 Department of Hydraulic Machinery, “Politehnica” University of Timişoara . E-mail: costel@mh.mec.upt.ro 4 Department of Hydraulic Machinery, “Politehnica” University of Timişoara . E-mail: alin@mh.mec.upt.ro

ABSTRACT We assess the validity of averaging the Navier-Stokes equations in a cylindrical geometry, resulting in axisymmetric swirling flow simulation, when the actual flow is unsteady three-dimensional with precessing helical vortex. This particular flow configuration occurs in the draft tube cone of Francis hydraulic turbines operated at partial discharge, as a result of the self-induced instability of the decelerated swirling flow downstream the runner. We compare the axisymmetric swirling flow simulation with the time averaged unsteady three-dimensional numerical solution. It is concluded that when a strong unsteady three-dimensional precessing vortex rope is developed, the 2D axisymmetric results are in reasonable agreement with the 3D time averaged ones. When stabilizing the swirling flow using an axial jet injection, both 3D and 2D axisymmetric results are practically identical. As a result, the 2D axisymmetric simulation can be reliably used to obtain the base swirling flow for stability analysis. This approach provides valuable information on the self-induced flow instability and allows a rapid assessment of various flow control techniques.

Keywords: Swirling flow, helical vortex breakdown, axisymmetric flow simulation, time averaging, unsteady three-dimensional precessing vortex

NOMENCLATURE , ,z rV V V [m/s] axial, radial, and circumferential

velocity components p [Pa] static pressure [kg/m3] fluid density [Pa·s] dynamic viscosity

1. INTRODUCTION The helical vortex breakdown (also known as

precessing vortex rope) is a self-induced instability of a swirling flow, encountered in the draft tube cone of hydraulic turbines operated far from the best efficiency, Figure 1. Extensive experimental [1] and numerical [2] investigations have offered a comprehensive understanding of this complex flow phenomenon, with accurate evaluation of the main parameters (precession frequency, amplitude of the wall pressure fluctuations, vortex rope shape), as well as various details of the hydrodynamic field.

Figure 1. Helical vortex breakdown in the draft tube cone of Francis turbine at partial discharge.

Nishi et al. [3] put forward a qualitative model for the precessing vortex rope, based on their experimental investigations. They suggest that the circumferentially averaged velocity profiles in the cone could be represented satisfactorily by a model comprising a dead (quasi-stagnant) water region surrounded by the swirling main flow, Figure 2. This model is also supported by the measured averaged pressure, which remains practically constant within the quasi-stagnation region.

Figure 2. Stagnant region model for the precessing vortex rope, Nishi et al. [3].

All these considerations led to the conclusion that the spiral vortex core observed in the draft tube of a Francis turbine at part load is a rolled-up vortex sheet which originates between the central stalled region and the swirling main flow.

The development of a central stagnant region in decelerated swirling flows has been proved theoretically by Keller et al., [4, Fig. 8a], who employed the inverse Euler equations to compute the incompressible, inviscid, steady axisymmetric flow with swirl in a diffuser. Similar results have been obtained by Goldshtik and Hussain, [5]. As a result, we have implemented a stagnant region model within the framework of an incompressible, turbulent, axisymmetric flow solver, [6], and the numerical results for the flow with precessing vortex rope in a Francis turbine discharge cone were in very good agreement with the LDV measured axial and circumferential velocity profiles. Moreover, we have shown that the vortex rope is exactly wrapped around the central stragnant region as computed with our model.

Operating Francis turbines at partial discharge is hindered by the occurrence of the precessing vortex rope, with associated severe pressure fluctuations. Susan-Resiga et al. [7] have introduced a novel control technique for swirling flows, where a water jet is injected along the symmetry axis in order to remove the stagnant region depicted in Fig. 2. This idea was further investigated numerically by Zhang et al., [8], with additional considerations on the stability of the circumferentially averaged swirling flow in the cone are presented.

In order to assess the efectiveness of this flow control technique, one has to perform a parametric study on the jet, for a given swirling flow configuration, in order to find the optimum jet velocity and discharge for mitigating the unsteady pressure by stabilizing the swirling flow. This can be done numerically [9] using a full three-dimensional unsteady turbulent flow simulation, with the major drawback of large computing resources required. This particular problem rises well know difficulties for current three-dimensional unsteady turbulent flow solvers that require high performance parallel computing to obtain accurate

results, as shown by Ruprecht et al. [10] and Sick et al. [11]. The second approach is to assess the stability properties of the axisymmetric swirling flow. However, one may question the use of an axisymmetric flow model for a fully three-dimensional flow as shown in Figs. 1 and 2. A first answer to this fundamental question is given in [6], where the numerical results for axisymmetric swirling flow computed in the draft tube cone of a Francis turbine are validated against Laser Doppler Velocimetry measurements for axial and circumferential velocity profiles.

In this paper, we use a time averaging procedure for the unsteady three-dimensional flow simulation in a swirling flow apparatus [9], then we compare these results against two-dimensional axisymmetric swirling flow simulation. We show that averaging the flow governing equations provides a reasonable agreement with the three-dimensional averaged results, even when the flow has a strong unsteady 3D character.

2. NUMERICAL SIMULATIONS The axisymmetric computational domain used

for three-dimensional flow simulations [9], with inlet swirl given in [12], is shown in Figure 3. A structured grid of approximately 2E6 cells is used, together with an unsteady realizable k turbulence model.

Figure 3. Axisymmetric computational domain for the swirling flow apparatus.

Figure 4. Axial velocity map in a meridian cross-section of the swirling flow apparatus, for 3D unsteady swirling flow with precessing vortex.

Although the computational domain is axisymmetric, Fig. 3, and the inlet boundary conditions are steady and axisymmetric (radial profiles for axial and circumferential velocity components), the decelerated swirling flow in the conical diffuser develops an instability, with a precessing helical vortex as shown in Figure 4. Such flow simulations require very large computing resources and computing time. As a result, from practical point of view it is preferable to use a more tractable approach to assess the stability properties of the swirling flow, and to estimate to occurrence of unsteady velocity and pressure field. This is very important for large hydraulic turbines, where the pressure fluctuations generated by the vortex rope may lead to structural damage.

Figure 5. Computational domain in the meridian half-plane for axisymmetric swirling flow computations.

Since the geometry considered in this study has rotational symmetry, it would be convenient to compute the flow using axisymmetric governing equations. In doing so, the three-dimensional problem becomes a two-dimensional one, to be solved in the domain shown in Figure 5. This 2D computational domain is obtained by intersecting

the 3D domain from Fig. 3 with a meridian half-plane. From the annular inlet section, the hydraulic passage has a convergent part upstream the throat, followed by a conical diffuser discharging in a downstream pipe. The hub ends with a nozzle where the control jet is injected. Two survey sections are considered, one located at the throat, and the second at the middle of the conical diffuser. A grid of 3.2E4 quadrilateral cells, with boundary layer refinement near the walls is considered. It is obvious that the grid is two orders of magnitude smaller than the one used for the full 3D simulation.

The governing equations for axisymmetric, turbulent swirling flows of an incompressible fluid are obtained by writing both the continuity and the momentum equations in cylindrical coordinates, then discarding the derivatives with respect to the circumferential coordinate: i) the continuity equation,

V 0z r rV V Vz r r

(1)

ii) axial momentum equation,

1 1

1 1 2

1

zz z r z

T z

T z r

V rV V rV Vt r z r r

Vp rz r z z

V Vrr r r z

(2)

iii) radial momentum equation,

2

2

1 1

1

1

1 2 2

rz r r r

T r z

T r T r

V rV V rV Vt r z r r

Vpr r

V Vrr z z r

V Vrr r r r

(3)

iv) circumferential momentum equation,

32

1 1

1

1

z r

r T

T

V rV V rV Vt r z r rV V Vr

r r z z

Vrr r r r

(4)

The effective dynamic viscosity is written as the sum of the molecular, , and the so-called “turbulent” viscosity, T , the second being computed using various turbulence models. The above axisymmetric swirling flow model, available

in the FLUENT 6.3 code, is used for the present computations together with the realizable k (RKE) turbulence model. The term “realizable” means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of the flow. When compared with the standard k and RNG k models, the RKE model is predicting more accurately the spreading rate of both planar and round jets, [13]. As a result, the RKE turbulence model is chosen for the present investigations, in particular since we are using axial water jet injection to stabilize the swirling flow.

The radial equilibrium condition used on the outlet section of the computational domain follows from the radial momentum equation (3) for vanishing radial velocity. When setting 0rV in Eq. (3), all terms containing the radial velocity dissapear. Moreover, the first viscous term becomes negligible if the flow does not evolve anymore in the axial direction, i.e. the axial derivative is vanishing as well. As a result, we obtain the well known condition for the outlet section,

2Vpr r

. (5)

Note that Eq. (5) does not explicitly specify a pressure profile. Instead, it correlates the radial pressure gradient with the circumferential velocity. When the flow does not have swirl, Eq. (5) reduces to a constant pressure condition on the outlet.

On the inlet section we prescribe the same axial and circumferential velocity profiles, together with the turbulence quantities, as the ones used for the 3D numerical simulation.

Figure 6. Axial and circumferential velocity profiles on the annular inlet section for both 3D (Fig. 3) and 2D (Fig. 5) computational domains, [12].

Note that both 3D and 2D axisymmetric computations have similar boundary conditions and turbulence models. However, it is obvious that the simplified axisymmetric flow model cannot capture the swirling flow three-dimensional and unsteady

character when the precessing helical vortex is developed as a result of the self-induced instability, Fig. 4.

3. NUMERICAL RESULTS In order to compare the results of the 3D

unsteady flow simulation with the simplified 2D axisymmetric flow model, a time averaging procedure is used for the 3D flow field.

a)

b)

Figure 7. Axial velocity map for 2D axisymmetric simulation (upper half plane) and 3D time averaged results (lower half plane): a) without control jet, b) with control jet discharge 10.6%.

a)

b)

Figure 8. Circumferential velocity map for 2D axisymmetric simulation (upper half plane) and 3D time averaged results (lower half plane): a) without control jet, b) with control jet discharge 10.6%.

First, a meridian cross-section is considered through the computational domain from Fig. 3. Then, the data files from successive time steps over

two periods (two complete rotations of the helical vortex) are loaded, and velocity components and pressure are saved in separate files. These files are loaded in TECPLOT, and an averaged flow field is computed as arithmetic average of each variable.

Figures 7, 8 and 9 show the axial and circumferential velocity, and pressure, respectively, computed with the 2D axisymmetric flow model (upper half plane) and the above averaging procedure (lower half plane). The iso-value lines in both half-plane for each figure have the same value to facilitate the comparison.

Each figure includes the case without control jet, when the 3D flow has a well developed precessing helical vortex, and the case with a control jet injection with 10.6% discharge with respect to the discharge through the inlet section. A qualitative assessment of the 3D versus 2D axisymmetric comparison shown in Figs. 7…9 indicates a strong similarity in both cases without and with control jet. However, when the swirling flow is stabilized with the control jet and becomes practically axisymmetric after mitigating the helical vortex, the 3D and 2D results become quasi identical, as expected.

a)

b)

Figure 9. Circumferential velocity map for 2D axisymmetric simulation (upper half plane) and 3D time averaged results (lower half plane): a) without control jet, b) with control jet discharge 10.6%.

Figures 10…21 show a quantitative comparison between the radial profiles of axial and circumferential velocity, and pressure, respectively, obtained in the two survey sections from Fig. 5, by time averaging the 3D flow field, on one hand, and by computing the 2D axisymmetric flow field, on the other hand.

Figure 10. Axial velocity radial distribution at the throat, without control jet.

Figure 11. Circumferential velocity radial distribution at the throat, without control jet.

Figure 12. Static pressure radial distribution at the throat, without jet.

Figure 13. Axial velocity radial distribution at the middle of the cone, without control jet.

Figure 14. Circumferential velocity radial distribution at the middle of the cone, without control jet.

Figure 15. Static pressure radial distribution at the middle of the cone, without control jet.

Figure 16. Axial velocity radial distribution at the throat, with 10.6% control jet discharge.

Figure 17. Circumferential velocity radial distribution at the throat, with 10.6% control jet discharge.

Figure 18. Static pressure radial distribution at the throat, with 10.6% control jet discharge.

Figure 19. Axial velocity radial distribution at the middle of the cone, with 10.6% control jet discharge.

Figure 20. Circumferential velocity radial distribution at the middle of the cone, with 10.6% control jet discharge.

Figure 21. Static pressure radial distribution at the middle of the cone, with 10.6% control jet discharge.

Figs. 10…15 correspond to the case without control jet, when the 3D flow has a strong three-dimensional and unsteady character, while the axisymmetric simulation leads to a steady circumferentially averaged picture. One can see that in general we have a good agreement at the throat section, while further downstream the 2D results depart more significantly from the 3D averaged ones. This result shows that averaging the three-dimensional results is not fully equivalent with circumferentially averaging the governing the governing equations if the flow departs significantly from axial symmetry.

On the other hand, Figs. 16…21 clearly show that when the helical vortex is mitigated by injecting a control jet, both 3D averaged results and the 2D axisymmetric ones are virtually identical.

Going back to the original motivation of the present study, it is clear that we can replace the very expensive full three-dimensional unsteady flow simulation by a far more tractable two-dimensional flow computation, associated with an evaluation of the flow stability.

In addition, the axisymmetrical flow simulation allows a very convenient and inexpensive evaluation of new flow control techniques. For example, after observing that for practical applications a 10% jet discharge is too large and inacceptable from the energy point of view, we have developed a flow feedback method, [12], as shown in Fig. 22.

Figure 22. Streamlines for the swirling flow in the swirl apparatus, without flow control (upper half plane) and with flow feedback control (lower half plane), [12].

The preliminary assessment of the jet discharge, when supplied with a fraction of the main flow discharge collected near the wall at the end of the conical diffuser, as well as the mitigation of the central stagnant region, can be rapidly done with an axisymmetrical turbulent swirling flow solver.

5. CONCLUSIONS The paper examines two numerical approaches

for computing swirling flows in axisymmetric cylindrical domains.

The first approach employs a full three-dimensional unsteady turbulent flow model, which can correctly capture the self induced flow instability resulting in unsteady flow field generated by a distinct precessing helical vortex. This type of flows is specific to the draft tube cone of hydraulic turbines operated far from the best efficiency point. Both velocity and pressure fluctuations can be mitigated by an original flow control technique that injects a water jet along the symmetry axis.

The second approach considers an axisymmetric flow model, resulting in a two-dimensional problem in the meridian half-plane, with the main benefit of reducing the computational effort by two orders of magnitude. However, this simplified model cannot capture the flow unsteadiness since the mathematical model assumes circumferential averaging in the first place. As a result, the axisymmetric flow field must be examined in conjunction with a stability analysis, such that the occurrence or lack of unsteadiness is predicted within a parametric study of the control jet.

We conclude that time-averaging the three-dimensional flow field provides a flow picture quite close to the axisymmetric steady one provided by the two-dimensional problem. As a result, one can reliably use the 2D axisymmetric turbulent swirling flow model to obtain the base flow for a stability analysis, as an efficient approach for the parametric studies of techniques for stabilizing swirling flows. As a first example, we present a novel flow-feedback method, where the control jet is supplied by a fraction of the mean flow discharge collected near the wall at the cone outlet. In this case, no additional power is required to supply the jet.

ACKNOWLEDGEMENTS This work has been supported by the Swiss

National Science Foundation under the SCOPES Joint Research Project IB7320-110942, and by the Romanian National University Research Council under the Exploratory Research Project 799/2009.

REFERENCES [1] Ciocan, G. D., Iliescu, M. S., Vu, T. C.,

Nennemann, B., and Avellan, F., 2007, “Experimental Study and Numerical Simulation of the FLINDT Draft Tube Rotating Vortex”, J Fluids Engineering, Vol. 129, pp. 146-158.

[2] Stein, P., 2007, “Numerical Simulation and Investigation of Draft Tube Vortex Flow”, PhD Thesis, Coventry University, U.K.

[3] Nishi, M., Matsunaga, S., Okamoto, M., Uno, M., and Nishitani, K., 1988, “Measurement of three-dimensional periodic flow on a conical draft tube at surging condition”, in Rohatgi, U. S., et al., (eds.) Flows in Non-Rotating

Turbomachinery Components, FED, Vol. 69, pp. 81-88.

[4] Keller, J.J., Egli, W., and Althaus, R., 1988, “Vortex breakdown as a fundamental element of vortex dynamics”, J. Applied Mathematics and Physics (ZAMP), Vol. 39, pp. 404-440.

[5] Goldshtik, M., and Hussain, F., 1998, “Analysis of inviscid vortex breakdown in a semi-infinite pipe”, Fluid Dynamics Research, Vol. 23, pp. 189-234.

[6] Susan-Resiga, R., Muntean S., Stein, P., and Avellan, F., 2008, “Axisymmetric Swirling Flow Simulation of the Draft Tube Vortex in Francis Turbine at Partial Discharge”, Proc. 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brasil, paper 13.

[7] Susan-Resiga, R., Vu, T. C., Muntean, S., Ciocan, G. D., Nennemann, B., 2006, “Jet Control of the Draft Tube Vortex Rope in Francis Turbines at Partial Discharge”, Proc. 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan, paper 192.

[8] Zhang, R.-K., Mao, F., Wu, J.-Z., Chen, S.-Y., Wu, Y.-L., Liu, S.-H., 2009, “Characteristics and Control of the Draft Tube Flow in Part-Load Francis Turbine”, J Fluids Engineering, Vol. 131(2), pp. 021101-1...13.

[9] Muntean S., Susan-Resiga, R., and Bosioc, A. I., 2009, “3D Numerical Analysis of Unsteady Pressure Fluctuations in a Swirling Flow Without and With Axial Jet Control”, Proc. Conf. Modelling Fluid Flow CMFF’09.

[10] Ruprecht, A., Helmrich, T., Aschenbrenner, T., and Scherer, T., 2002, “Simulation of Vortex Rope in a Turbine Draft Tube”, Proc. 22nd IAHR Symposium on Hydraulic Machinery and Systems, Lausanne, Switzerland, Vol. 1, pp. 259-276.

[11] Sick, M., Stein, P., Doerfler, P., Sallaberger, M., and Braune, A., 2005, “CFD prediction of the part-load vortex in Francis turbines and pump-turbines”, Int. J. Hydropower and Dams, Vol. 22, No. 85.

[12] Susan-Resiga, R., and Muntean, S., 2008, “Decelerated Swirling Flow Control in the Discharge Cone of Francis Turbines”, Proc. 4th Int. Symposium on Fluid Machinery and Fluid Engineering, Beijing, China, pp. 89-96.

[13] Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z., and Zhu, J., 1995, “ A New k-ε Eddy Viscosity Model for High Reynolds Number Turbulent Flows – Model Development and Validation”, Computers and Fluids,Vol. 24, No. 3, pp. 227-238.