Energy transformation and flow topology in an elbow draft tube

Applied and Computational Mechanics 6 (2012) 93–106 Energy transformation and flow topology in an elbow draft tube D. ˇ Stefana,∗, P. Rudolfa,A.Skot ´ akb, L. Motyˇ c´ akb aFaculty of Mechanical Engineering, Brno University of Technology, Technick´a 2896/2, 616 69 Brno, Czech Republic bCKD Blansko Engineering, a.s., ˇ Capkova 2357/5, 678 01 Blansko, Czech Republic Received 20 January 2012; received in revised form 30 April 2012 Abstract Paperpresents a computationalstudy of energy transformationin two geometricalconfigurationsof Kaplan turbine elbow draft tube. Pressure recovery, hydraulic efficiency and loss coefficient are evaluated for a series of flow rates and swirl numbers corresponding to operating regimes of the turbine. These integral characteristics are then correlated with local flow field properties identified by extraction of topological features. Main focus is to find the reasons for hydraulic efficiency drop of the elbow draft tube. c 2012 University of West Bohemia. All rights reserved. Keywords: draft tube, efficiency, pressure recovery, flow topology 1. Introduction Draft tubes are diffusers placed at the outlet of the hydraulic turbine runner, see Fig. 1. Their main purpose is transformation of the residual kinetic energy into pressure energy with maximum efficiency. Although geometrically relatively simple the internal flow can be quite complex due to adverse pressure gradient, possible boundary layer separation, swirling and streamline curvature. The elbow draft tubes are usually used for a vertical arrangement of Kaplan turbines. The most problematic part of the draft tube with negative influence on flow properties is the elbow. Paper summarizes results of a computational study of energy transformation in two geometrically different configurations of Kaplan turbine elbow draft tube. Fig. 1. Longitudinal cross-section view of hydraulic power plant ∗Corresponding author. Tel.: +420 541 143 477, e-mail: [email protected].cz. 93 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 2. Longitudinal cross-section view of DFT-A Fig. 3. Longitudinal cross-section view of DFT-B Draft tube geometrical configuration A (DFT-A) – The elbow draft tube for vertical Kaplan turbineincludes one horizontalrib (inelbowpart) and one vertical rib(in horizontal part), Fig.2. This draft tube is installed in hydro-power plant Stˇrekov, Czech Republic. Draft tube geometrical configuration B (DFT-B) – The elbow draft tube for vertical Kaplan turbine without any ribs, Fig. 3. This draft tube is result of shape modification of DFT-A for one specific velocity profile (different than velocity profile used in this work) [12]. Part of turbine runner hub (at the inlet of the draft tube), see Fig. 2 and Fig. 3, has been included and modelled as a stationary wall for both configurations of the draft tube. The numerical computation has been carried out in commercial CFD software ANSYS FLUENT r.12. employing Reynolds Averaged Navier-Stokes equations to solve computational domain by finite volume method. We used Realizable k−εmodel of turbulence (RKE) with non-equilibrium wall function to perform steady state solution. The RKE model of turbulence and steady state of computation are chosen because of suitable capturing of flow properties and computational demands for evaluation, where dynamics properties of flow are not extracted. Each draft tube has been computed for a series of flow rates corresponding to operating regimes of the turbine. The best efficiency point of the turbine (100 % QBEP ) corresponds to the mass flow Qm= 440 kg/s. Inlet velocity profiles in Fig. 5 are the result of numerical computation of the new design of Kaplan turbine runner for hydro-power plant Stˇrekov and data were obtained from ˇ CKD Blansko Engineering. Investigation of the three-dimensional separation, carried out by software ANSYS CFDPOST, and its influence on energy transformation is discussed. Skin friction lines, surface streamlines and critical points are used to identify the topological features of the flow. Fig. 4. Efficiency and hydraulic loss curves Fig. 5. Inlet velocity profile for CFD computation, flow rate 100 % QBEP 94 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 2. Energy transformation The hydraulic efficiency (1), pressure recovery factor (2) and hydraulic loss coefficient (3) have been evaluated in several cross-sections for series of flow rates and swirl numbers η=ps(2) −ps(1) pd(1) −pd(2) ,(1) cp=ps(2) −ps(1) 1 2ρ¯v2 (1) ,(2) ξ=2 ¯v2 (2) α(1)¯v2 (1) −α(2)¯v2 (2) 2+ps(1) −ps(2) ρ,(3) where psis static pressure, pdis dynamic pressure, ¯vis mean velocity and αis Coriolis number. The number inparentheses represents 1=inletand 2=outlet. The global developmentof these integral characteristics (1–3) is plotted in Fig. 6 for DFT-A and in Fig. 7 for DFT-B. 2.1. Energy transformation in DFT-A Changes of DFT-A cross-sectional area are nearly linear from inlet to outlet of the draft tube. For flow rates 95.5 % QBEP and 100 % QBEP decrease of pressure recovery factor at the elbow part is observed and as will be shown in section 2.1, it is caused by flow separation under horizontal rib. For 109.1 % QBEP , the flow separation is significantly reduced, therefore, any considerable decrease of pressure recovery factor is not apparent. The other decrease in pressure recovery factor is evident between cross-sections 9 and 11 for all three flow rates and is caused by the beginning of the vertical rib. The volume of vertical rib reduces cross-sectional area and thereby increases velocity and frictional losses. This effect is a consequence of the design so therefore it occurs for every flow rate. The smallest back-flow region at horizontal part of draft tube for whole range of flow rates is observed for 109.1 % QBEP . This is the reason for steeper slope of the pressure recovery curve than for other operating points. The highest value of DFT-A efficiency (1) is derived from efficiency curve plotted in Fig. 4 and corresponds to flow rate 103 % QBEP . 2.2. Energy transformation in DFT-B The shape modification of DFT-B has had appreciable influence on the development of crosssectional area at the elbow part of draft tube, see area between cross-section 3 and 6 in Fig. 7. The flow cross-section at the end of elbow part has been reduced because of separation risk on the inner curved wall. This modification was done for some past design of the draft tube with slightly different inlet velocity profile. It was anticipated that it might be beneficial also in combination with new runner design, which has to some degree altered outlet blade angles and hence also the outlet velocity components. Unfortunately, this assumption was not confirmed, flow inside the draft tube is rather sensitive to inlet boundary conditions and no positive impact for the DFT-B modification was observed. DFT-B draft tube features rather steep increase of pressure recovery factor at the inlet part, which is connected with large wall divergence leading into reduction of velocity and abrupt transformation of kinetic energy into pressure energy. This is an undesirable effect, because too low kinetic energy of flow stream brings risk of flow separation at the downstream parts of the draft tube. In Fig. 13, it is shown that a large part of kinetic energy is dissipated just at the inlet part of draft tube and as will be shown in section 6.1, the flow separation occurs at the elbow 95 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 part of the draft tube. The highest value of DFT-B efficiency (1) is derived from efficiency curve plotted in Fig. 4 and corresponds to flow rate 92 % QBEP . 3. Back-flow regions The back-flow regions block flow area, increase velocity and cause higher hydraulic losses. DFT-A: The first significant back-flow region is situated under the horizontal rib and occupies right (for flow rate lower than that of the best efficiency point of turbine, Fig. 8) or left (for flow rate higher than that of the best efficiency point of turbine, Fig. 9) side of the draft tube. The highest suppression of this region is reached for flow rate 109.1 % QBEP , Fig. 10. The second back-flow region is situated on the top of horizontal part of the draft tube and is the most suppressed between operating points 104.5 % QBEP and 109.1 % QBEP . The highest hydraulic efficiency of DFT-A is reached for 103 % QBEP when the back flow regions are not the smallest. It shows influence of higher hydraulic losses caused by surface friction on the ribs when the flow rate increases. DFT-B: Two back-flow regions occur. The first one is situated at the end of elbow part and second one at the horizontal part of draft tube. For flow rate 104.5 % QBEP , they are the largest and connected together, as shown in Fig. 10. The largest suppression of these regions is observed close to flow rate 95.5 % QBEP , Fig. 11. 4. Contours of dissipation The dissipation function Dfor computation of turbulent flow by RANS equation is defined as follows D=2  V(μ+μt)∂vx ∂x 2 +1 2∂vx ∂y +∂vy ∂x 2 +1 2∂vx ∂z +∂vz ∂x 2 + ∂vy ∂y 2 +1 2∂vy ∂z +∂vz ∂y 2 +∂vz ∂z 2dV. (4) For both draft tubes, the contours of dissipation (4) were computed at several cross-sections. Areas with the highest value of dissipation are coloured from black to white/vanish (over-range of colormap). The range of the colormap has been set and reduced so that only the dissipation inside of the volumeis visible, because the highest value of dissipationoccurs in boundary layer regions. DFT-A: Several important areas with high value of dissipation are observed. The first one is at the inlet part of the draft tube. Considerable dissipation is caused by high velocity gradient of stream coming out from turbine runner. The second area is under the horizontal rib and corresponds with the back-flow region, compare Fig. 12 and Fig. 8. This result confirms that the back-flow regions are highly dissipative. The third area is located behind the trailing edge of the horizontal rib and is caused by sweeping of boundary layer into the main stream. DFT-B: The main area of high dissipation is located close to the inner bend radius, Fig. 13. It is caused by very high velocity gradients induced by flow inside the elbow part. As in the case of DFT-A, significant dissipation caused by stream coming out from the turbine runner is observed at the inlet part of the draft tube. 96 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 6. Hydraulic efficiency, pressure recovery factor and loss coefficient evaluated in several crosssections for three flow rates 95.5 % QBEP , 100 % QBEP and 109.1 % QBEP in case of DFT-A 97 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 7. Hydraulic efficiency, pressure recovery factor and loss coefficient evaluated in several crosssections for three flow rates 95.5 % QBEP , 100 % QBEP and 109.1 % QBEP in case of DFT-B 98 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 8. Back-flow regions (in grey colour) in case of DFT-A, flow rate 104.5 % QBEP Fig. 9. Back-flow regions (in grey colour) in case of DFT-A, flow rate 109.1 % QBEP Fig. 10. Back-flow regions in case of DFT-B flow rate 104.5 % QBEP Fig. 11. Back-flow regions in case of DFT-B flow rate 95.5 % QBEP Fig. 12. Contours of dissipation in case of DFT-A Fig. 13. Contours of dissipation in case of DFT-B 5. Topology of the three-dimensional separation Asmentioned inTobak and Peak[13] and alsoin Depardonet al. [2], thehypothesisproposedby Legendre in 1956 brought a mathematical framework for description of the three-dimensional flow separation. The hypothesis is based on shear-stress patterns and critical points situated in the flow field and corresponding with the three-dimensional separated flow. Fig. 14. Skin friction lines and surface streamlines in case of DFT-A Fig. 15. Skin friction lines and surface streamlines in case of DFT-B 99 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 16. Types of critical points The three types of critical points in the flow field are identified, see Fig. 16. Focus F is a point where the vortex filament of the three-dimensional separation core is concentrated. Surface friction lines go into (stable) or out (unstable) of the centre of focus. Saddle point Sis a place where surface friction lines converge from one side and diverge to the other side. Node N: Surface friction lines go into (stable) or out (unstable) of the centre of node. Node usually lies near or directly onto solid surfaces. For the exact determination of critical points in particular flow cross-section computation of velocity gradient tensor eigenvalue (5) is used [3,4] vij|xi,s =∂vx ∂x ∂vx ∂y ∂vy ∂x ∂vy ∂y .(5) Each critical point (saddle, stable and unstable node, stable and unstable focus) is defined by sign of the eigenvalue of the tensor (5). In case of elbow draft tube, several regions with risk of the three-dimensional separation occur. The beginning of the three-dimensional flow separation lies onto draft tube body surface where boundary layer separates and is carried out by main stream. The vortex region originates from this separation and causes blockage effect leading to the flow acceleration. Exact visualization and evolution of separation core (so-called dividingsurface) in flow field is rather difficult, especially when dealing with 3D data. One of the possibilities is to use the Sujudi-Haimes algorithm [9]. The computational algorithm looks for the points in the velocity field where a single real eigenvector exists and this is parallel to the velocity vector [5, 6]. There are also other approaches to visualize separation surfaces emanating from critical points, see [7,10]. Some of them are specifically focused on swirling flows, see [1]. In this work, prediction of the core evolution is solved only by visualization method based on searching of foci in particular draft tube cross-sections (surface streamlines) and on the solid surfaces (skin friction lines), Fig. 14 and Fig. 15. This method is simplification of finding possible occurrence of core but not very suitable for tracking the spatial evolution of the core. Setting the cross-section orientation represents a very difficult task, because foci are properly visible only on cross-section which is almost exactly perpendicular to the separation core. It is also advised to observe back-flow regions, which are related to global flow separation and lead to the hydraulic efficiency drop. 5.1. Global flow separation In case of elbow diffusers, the main part with risk of the three-dimensional separation is the bend. For rectangular curved diffuser, the authors of [8] investigated three kinds of global flow separations: massive,typical and simple. The typical idea of global flow separation: “The global flow separation begins, where the flow is separated from the wall and formed the back100 D. ˇ Stefan et al. / Applied and Computational Mechanics 6 (2012) 93–106 Fig. 17. Right view on elbow part of DFT-A for flow rate 95.5 % QBEP Fig. 18. Right view on elbow part of DFT-A for flow rate 100 % QBEP Fig. 19. Right view on elbow part of DFT-A for flow rate 104.5 % QBEP Fig. 20. Left view on elbow part of DFT-A for flow rate 109.1 % QBEP flow regions” is stated in [8]. The global flow separation is highly dissipative phenomenon, increases hydraulic losses and reduces hydraulic efficiency. In case of turbulent flow in the draft tube (unsteady, three-dimensional with rotational character), the flow topology of the separation is very complex and directly corresponds with shape of the draft tube. This statement is especially characteristic of the draft tube containing ribs. Types of the three-dimensional separation corresponding to each of the investigated draft tubes will be shown in sections 5 and 6. 5.2. Local flow separation In contrary to the case of global flow separation, the local flow separation is not related to backflow regions. Hence no significant negative effect, as lower efficiency due to higher dissipation, has been documented. The example of local type of three-dimensional separation is observed in section 5.2 for the DFT-A draft tube. 6. Three-dimensional separation in DFT-A draft tube 6.1. Separation under horizontal rib This separation is mainly caused by leading edge of horizontal rib which suppresses stream rotation at the inlet of the draft tube. This separation is in combination with back-flow region that means it is the global flow separation which decreases energy transformation and deteriorates efficiency. Flow rate: (Figs. 17–20) •95.5 % QBEP :The back-flow region is very large and starts from leading edge of the horizontal rib, see Fig. 17, and develops downstream to the draft tube. The global flow separation is represented by saddle point Sin combination with focus F. 101