Improving channel flow with deflectors optimized using a genetic algorithm.

Fucak, S. ; Carija, Z. ; Mrsa, Z. 等

1. Introduction

The HPP Vinodol started operating in 1952 and the turbines were revitalized in 1998. It now houses six Pelton turbines with refitted runners and a combined water flow of 18.7 [m/s]. The outflow system was originally designed for a lower throughput and this has caused the water level to rise in turbine chambers. Air is blown into turbine housings to ensure there is no contact between outflow water and the runners. It was suggested that the next point of improvement should be the drainage channel (Fig. 1). It is uniquely S-shaped with increasing width to collect turbine outflow and two 90[degrees] bends which connect it to the final oval section leading into the outer open channel. The high water level during peak HPP operation causes spillage into the engineering room above the S-channel as the channel is completely flooded. The ceiling vents and the open section in the second bend allow the engineering room floor to fill with over 10 cm of water making it a difficulty for operators and equipment.

[FIGURE 1 OMITTED]

The layout of the S-channel cannot be changed due to its armature and the structural integrity considerations for the facility above. The perpendicular positioning of the T-junctions, which connect turbine chambers and the main channel section, along with the two bends contribute to flow losses and turbulences inside the channel causing the water level to rise due to slower outflow.

The idea of improving channel flow was to be performed using sheet metal deflectors which would be mounted inside the S-channel, as its shape could not be widened or remodeled. Two pairs of deflectors were designed positioned in the channel bends and one at each T-junction in order to steady the flow and allow for a smoother mixing of the streams in hopes of reducing pressure losses and swirl, making a total of ten deflectors.

Numerical modeling and optimization were chosen because the six water inlets, the widening channel profile and the S-bends make the flow difficult to estimate analytically. Optimization is easier to perform by varying parameters in numerical simulations rather than actual models as they allow a greater degree of flexibility and need less time, leaving actual prototypes for the task of validating the final results.

2. Physical model design

The 3D CAD geometry of the channel was created according to available original designs. Additional corrections were included into the geometry, obtained from photographs and control measurements taken in visits to the drained S-channel during regular HPP maintenance. The resulting geometry was imported into the Gridgen mesher as a base for creating the 2D and 3D computational grids intended for fluid flow simulations performed in Fluent.

A 3D water flow simulation was performed to get the basic data needed to determine problem zones in the channel flow. The resulting pressure losses, velocities and streamlines were used to design the deflector shapes taking into account the results obtained in the study (Carija & Mrsa, 2005) where just the S-bend deflectors and the sixth turbine T-junction were modeled in a 3D Fluent VoF (Volume of Fluid) simulation.

[FIGURE 2 OMITTED]

While the Sk11 through Sk22 deflectors (Fig. 2) remained cylindrical and positioned according to the guidelines set in (Idel'chik, 1966), the T1 through T6 deflectors had to have a more creative shape due to their demanding role in directing inflow streams while not constricting the main channel flow. The initial T-deflector design concept was set to follow the streamlines taken from the base simulation as well as correspond to smoothed channel widenings from 2 to 2.5 m at T3 and 2.5 to 3 m at T5 (which would also be tapered by adding material to channel walls).

The main channel cross section is basically a rectangle with the bottom edges rounded more than the top edges, and the trapezoidal T-inlets widen upwards. Each Pelton turbine is run by a single needle-valve nozzle oriented opposite to the outflow. The T-junctions have a shallow sweep oriented downstream along the main channel section and the upstream side of each join has a concrete lip designed to redirect flow. This feature needed to be improved by adding deflectors which would shape the T-junctions in a more streamline way to better direct contributing flows from each turbine into the main stream.

3. Numerical model and optimization parameters

The rectangular shape of the channel inspired a transition from a 3D to a 2D domain whose defining plane (Fig. 2) lies at mid-height of the trapezoidal velocity inlets T1 through T6 (visible in the meshes on Fig. 4 and Fig. 5). Since this longitudinal section is the furthest from the inlet floor and ceiling it was judged to be less influenced by flow features in the vertical Z direction and chosen as the 2D computational domain for optimization. This simplification allowed more efficiency in the time needed to run a large number of simulations with varying parameters by reducing the complexity of the numerical model and utilizing some of Gridgen's capabilities for 2D mesh optimization. The channel section upstream from the T1 turbine has a negligible inflow of cooling water and can be considered a dead zone for the purposes of flow simulation so it was excluded and replaced with a wall to reduce mesh size.

A 2D unstructured mesh of triangular elements was selected (Fig. 5 detail) as it showed greater flexibility in meshing the changing deflector geometry automatically. Simulations were run to determine optimal wall element sizes and 10 mm was selected with a 60% size growth bias towards maximum interior element size of 140 mm. The k-epsilon turbulent viscosity model in Fluent was selected for its robustness (Ferziger & Peric, 2002) and, after trial runs to monitor continuity of characteristic values, the residual convergence criterion was fixed at 10E-4. Mesh adaption by gradients was not used in the automated optimization as it negatively influenced simulation stability and sometimes lead to crashes and was instead performed just on select cases for comparison.

Total pressure was computed at control cross sections in the channel to monitor energy losses between the inlets and the outlet. Since the mass flow is constant for all simulations, calculating energy losses [P.sub.loss] from total pressure reports in Fluent can be performed using the following equations:

[P.sub.loss] = [DELTA][p.sub.tot] * [??] [W] (1)

where [??] is the total volume flow rate through the channel and Aptot is the difference between mass-averaged total pressure on the outlet and all inlet surfaces.

Since each turbine was set to supply the same volume flow rate, a unit mass flow nit can be defined where the index i = 1 ... n stands for n inlets in the model. As mass and density are conserved, the total mass flow equals the inflow and outflow:

[[??].sub.outflow] = [[??].sub.inflow] = n * [??]. [[kgs.sup.-1]] (2)

The integral for mass averaged total pressure can therefore be represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Simulations were run where the initial T-deflector shape was varied in the 2D domain and results showed that energy losses were more influenced by the aperture, measured from the deflector tip to the side wall, than variations in its general shape. The outline was therefore left to follow a streamlined shape, and control points for the splines forming the deflector were positioned so that the shape did not distort in its most open position (furthest from the downstream right-hand side wall, where the turbine inlets are situated). The general shape was adapted for T1 which has almost zero upstream flow in the main channel, and the T6 deflector which is a special case since it is under the heaviest main flow and is practically in the Z1 bend. The vertical shape of T-deflectors was thought as a straight extrusion following the trapezoidal TJunction wall covering the full height of the main channel, and was further addressed in subsequent 3D simulations.

As the mid section of the S-part of the channel rises in height and serves to somewhat steady the flow, the Sk21 and Sk22 deflectors were left in their original positions from (Carija & Mrsa, 2005). Although in a critical spot, varying their radii did not seem to reduce energy losses and more often had detrimental influence. Their exclusion from the optimization also served to reduce the number of variables. Deflectors Sk11 and Sk12, being semicircular, were defined by their distance from the convex side of the Z1 bend (downstream left-hand side) along the radius from the bend curvature origin. As the position and size of the T6 deflector directly influence the flow in the Z1 bend, the Sk12 deflector could not cover the full 90[degrees] quarter-circle as the others. Its length was halved leaving the possibility of adjustment for later optimizations. The distance and shape of the Sk-deflectors were left uniform and semicircular to prevent them from acting as nozzles as their main role would be to reduce swirl and roll and steady the flow by directing it towards straight sections that follow downstream.

4. Genetic Algorithm and Optimization

The genetic algorithm is an alternative to objective search functions and is well suited for cases where the evaluation is complex and not easy to represent algorithmically. It takes the design parameters and varies them within predefined limits running a fixed objective function to evaluate their fitness to find the optimal set. The objective function in this case is the energy loss in the channel which needs to be minimized and the evaluation is performed through cfD simulation.

The genetic algorithm (GA) operates on principles of evolution through selection (Goldberg, 1989). The one used here was a multi-variable compiled executable of the GALOPPS GA libraries (Goodman , 1996) using binary encoding of parameters and pseudorandom functions that allow repeatability of the GA optimization run.

It takes a preset number of design parameter sets forming the initial population of individuals (i.e. parameter sets) and randomly assigns values for each parameter. For all subsequent generations, the individuals have their chromosomes (i.e. parameters) combined to form the next generation using crossover and mutation. The elitist algorithm preserves the best unit and the rest are subject to selection according to their fitness using a roulette wheel selection in which fitter individuals have a larger chance of surviving until the next generation as new ones are introduced. The whole process is repeated for a predetermined number of generations and the parameters that are encoded in the best individual are the resulting output.

Each deflector setup was represented by encoding deflector design parameters into binary genes that form chromosomes representing individuals involved in the GA selection whose fitness function was set as the difference of total pressure in the whole channel, between all inlets and the outlet

A custom C++ application was written to handle evaluation calls that ran the scripted meshing and CFD simulation, gathering statistics from output files before returning the fitness value to its parent process, the GA executable. The design parameters were selected according to previous simulations and simplified to reduce the number of variables. Those were deflector tip x-coordinates, representing the distance from the wall, for deflectors T1 through T5 and both x and y-coordinates for deflector T6. The x-coordinate was later left constant for T6 as less influential. Deflector Sk11 had its arc length varied by shortening it from its full 90[degrees] at the upstream edge, but the full length proved optimal in the end.

[FIGURE 3 OMITTED]

The fitness function application (outlined in Fig. 3) used a pre-generated mesh for each optimization phase considerably speeding up the meshing part of each chromosome evaluation. The application generated the meshing script recalculating the new deflector shape from coordinates selected by the GA, called the Gridgen mesher to alter the pre-generated case mesh accordingly and finally exported it for Fluent. The application then generated the script to run the Fluent simulation, which also saved graphical representations for post-processing, and subsequently extracted data from Fluent result transcripts.

Each GA run consisted of 32 individuals in a population over 32 generations. A file with all the model names, parameters and results was updated in the run directory and GA fitness evaluation calls were checked versus existing previously evaluated results to prevent duplicate simulations. This also allowed the GA run to continue after crashes and merge results from several machines in one master data file for subsequent GA runs. The variation of parameters was performed in 10 mm increments as the total pressure loss sometimes differed little with varying deflector results. All this helped reduce the time spent on computationally costly evaluations. Since the mass flow was constant in all case variations, the surface of the domain area showing velocity magnitudes greater than 4.5 [ms-1] was taken as a measure for flow uniformity and used during post-processing as a secondary fitness function to evaluate cases that had similar total pressure losses.

5. Optimization phases and results for the 2D model

The optimization was performed in several phases constantly improving the model and parameterization. Deflector representations in the mesh were reduced to lines of zero thickness (until the final 2D phase) as initial steps were intended to determine optimization parameter (i.e. deflector position) ranges from the selection of most successful results within equivalent simulation conditions.

5.1 Upstream incremental optimization

[FIGURE 4 OMITTED]

In the first phase, zero-thickness deflectors were optimized individually for each T-inlet mesh segment and accompanying wake segment (Fig. 4), always taking the upstream portion of the channel into account. Due to the upstream influence of each deflector, their results differed from results obtained after attaching downstream channel segments. This attempt at reducing the number of parameter combinations and simulation times was thus abandoned and the final result was used as a starting point for the next phase.

5.2 Optimizing segments T1 through T5

The second phase consisted of optimizing the part of the channel upstream from the bend Z1, including segments T1 through T5. The initial mesh was joined together from best case segments and the outlet was placed before the T6 segment (end of T5 wake). Zero-thickness deflectors, mesh walls and interface lines are outlined white in the T1-T5 mesh (Fig. 4).

5.3 Whole channel optimization

The whole channel was modeled in the third phase using best-case positions from the previous phase and varying the positions of T6, Sk11 and Sk12. The domain was extended 10m from the end of Z2 bend, slowly resembling the final 2D mesh (shown Fig. 5). The Sk-12 deflector was shortened to fill a 45[degrees] arc, unlike in the previous study (Carija & Mrsa, 2005), as it was noticed that it negatively affected the T6 stream. The length of the Sk-11 deflector was set as an optimization parameter by shortening the upstream end.

5.4 Whole channel optimization improvement

The fourth phase optimized the whole channel again, but varied all the parameters within smaller value ranges as the beneficial ranges were obtained from phases 2 and 3. An improved pre-generated mesh was used and segment boundaries were altered for smoother transitions. Sk-11 was left at full length as the results from the previous phase seemed erratic in regard to its length, showing large variations in the top ten results. The aim was to reduce the parameter ranges and prepare for the final phase.

5.5 Final 2D optimization and model

The final phase added thickness to the deflectors and extended them further over the edge of T-sections so the parameter value ranges had to be expanded again for the optimization. The deflector position ranges were subsequently narrowed for the final run. The top results were then checked by varying all the parameters within a small range to verify if the GA runs had indeed found the actual function minimum given the specified conditions.

[FIGURE 5 OMITTED]

The 2D simulation of the original state shows a total pressure loss of 2615.2 [Pa]. The energy loss is 30250 [W], obtained by multiplying the total pressure loss with volume flow rate (which is lower than 18.7 [ms-1] due to unit height of the 2D model). The best case optimization result shows a 16.7% reduction in losses with a d([P.sub.TOT]) of 2178.5 [Pa], and Eloss of 25199 [W] in the whole channel computational domain.

[FIGURE 6 OMITTED]

Taking each segment individually, the largest energy losses occur in the Z1 bend and around the T6 deflector. The main flow there is the strongest and the sudden change of direction, along with the deflectors, leads to increased velocities and uneven flow. Figure 6 shows the velocity magnitude contours in the Z1 bend and the resulting improvement with deflectors. Due to the grayscale representation it may not be apparent that the left-hand side of the bend shows velocities up to 6 [ms-1] while the optimized state is in the range of 4-4.5 [ms-1], except around the deflectors. The energy losses before and after the Z1 bend were reduced by over 50%, contributing to the overall channel improvement despite constrictions introduced by the deflectors.

6. Simulation of the 3D channel model with deflectors

[FIGURE 7 OMITTED]

The inlet velocity field of each T-junction actually represents a cross-section through the turbine chamber. Its boundary conditions were set as a simplified uniform inflow with an average velocity that would produce the appropriate volume flow of 3.116 [ms-1]. The complexity of the high velocity flow and splashing in the chamber under a Pelton-type turbine would require a whole separate CFD analysis to define the inlet velocity field properly (Wilcox, 1998), which was not among the initial aims of this paper.

The best case parameters from the 2D optimization were used for the deflectors in a steady state 3D simulation of the whole channel filled with water. The outflow segment was extended another 10 [m] during meshing and the section at the previous outflow position was used to calculate the drop in total pressure and, consequently, energy losses so they could be compared to previous models. The 3D model was updated after visits to the HPP Vinodol (Fig. 7).

6.1 Full height deflectors

The initial design of the T-deflectors was intended for minimum complexity in order to be optimized in 3D in a future paper. The simplest solution was to make them full height as this would reduce the adjustments in mesh geometry to a sweep of deflector tips intersecting with the main channel section floor and ceiling. The profile from 2D optimization results was translated parallel to inlet section walls to create smooth deflector surfaces.

Meshing was again performed in sections using hexahedral elements for straight sections and tetrahedral elements for segments around the deflectors and transitions from thin wall elements to the main flow (especially in the steep fast flow part of the oval outlet segment shown in Fig. 7).

[FIGURE 8 OMITTED]

Particle trajectories shown in Fig. 8 show how the T2 inlet stream literally bounces off the deflector and the main flow, as the T-sections are almost a meter above the channel floor, creating swirl and additional losses. Simulation results nevertheless showed a 18.5% reduction in total pressure drop when compared to the 3D simulation of the original channel.

6.2 Box-style deflectors

One of the main new insights from the 3D simulation was the problem of additional swirl caused by the deflectors themselves and amplified in the Z1 and Z2 bends. Although the deflectors steadied the flow by reducing the constriction performed by the main stream mainly on downstream inlets and directing the inflow towards the main water flow, the height difference between the turbine chambers and the deflectors positioned directly in front of the inlets caused the inlet water to roll and swirl as it joined the main stream. The main swirl occurs in the Z1 and Z2 bends due to a 90[degrees] change of direction, but a similar effect is present around the T-deflectors, although to a lesser degree.

Therefore a change in the deflector design was introduced to allow a smoother transition of inflowing water to alleviate the drop from the turbine chambers (new design shown in Fig. 9). The area along the bottom of the channel was left unobstructed, as the flow velocity magnitudes were larger there. Unfortunately, although the resulting streamlines showed a reduction in inlet swirl, the energy losses using this type of deflectors were somewhat higher than the full height ones resulting in a 17.4% reduction of total pressure drop. Additionally, the forces in the flow showed larger stresses acting on these deflectors which would require reinforcements added to their design and other changes to the model.

[FIGURE 9 OMITTED]

Additional design tests would be needed to find an optimal deflector shape, not excluding a computationally costly full 3D GA optimization.

7. Deflector 3D VoF (Volume of Fluid) model

The final test of the 3D deflector design was to verify water level decrease, which was the actual objective sought through energy loss reduction and steadying of the flow.

The VoF model is notoriously sensitive to mesh imperfections and special care was taken to pave model walls with thin shell elements to increase accuracy of boundary layer simulation (Chung, 2002). The chosen mesher was Gambit and full height deflectors positioned according to the 2D optimization best case set were used to create a new hybrid mesh of hexahedral and prismatic elements. As an addition to the full channel model, the domain was expanded by adding a simplified representation of the engineering room above in form of a prismatic volume connected to the channel through vents positioned along the axes of the three generators.

The solver used was Star-CD and the free surface (Fig. 10) was obtained using the k-epsilon turbulent viscosity model in a steady-state simulation with separate water and air phases. The actual channel model (without deflectors) was remeshed and likewise simulated using the VoF model for comparison.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The water level was lowered from over 180 mm to below 35 mm showing that the reduced energy losses indeed had a significant effect. Fig. 11 shows the water levels in the engineering room with and without deflectors graphed on minimum (-0.5 m) to maximum x-coordinate (+1.5 m) of the computational volume.

8. Conclusion

The 2D optimization results show better velocity field uniformity, indicative of the 16.7% reduction in energy losses, and provide a solid basis for 3D deflector design. Initial 3D fluid flow simulations of the channel, with deflectors extruded to their full height, have shown a comparable 18.5% improvement with the 3D model. The gravitational influences and the swirl in the flow need to be studied further to enhance the design of deflectors under the dynamic body forces during load cycling and maximum turbine output. Numerical results of the VoF simulation show promise and give a clear indication that deflectors would be effective in lowering the water level inside the channel and reducing overflowing water level in the engineering room by 75% even though the main channel section is still filled with water. A 3D optimization of the channel would help adjust deflector design for optimum performance as the initial deflector shape was idealized and would need to be augmented for structural strength and better flow control. The examined box-style design has its drawbacks and an optimal design would be determined by future 3D simulations and prototype models. This would include possible additions to the original concept, such as guiding fins, aimed at lowering the swirl and increasing flow uniformity to further decrease the water level.

Validation of these 3D CFD results on a scale model of the channel and deflector prototypes would lead to the final design shapes and their implementation in HPP Vinodol.

DOI: 10.2507/daaam.scibook.2009.70

9. References

Chung, T.J. (2002). Computational Fluid Dynamics, Cambridge University Press, ISBN 0-521-59416-2, Cambridge

Ferziger F. H.; Peric M. (2002). Computational Methods for Fluid Dynamics, Springer Verlag, ISBN 3540420746, Berlin

Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Publishing Company Inc, ISBN 0-201-15767-5, Boston

Goodman, D. E. (1996). An introduction to GALOPPS, Michigan State University, East Lansing

Idel'chik, I. E. (1966). Handbook of Hydraulic Resistance, Israel Program for Scientific Translation, Jerusalem

Mrsa, Z.; Carija, Z. (2005). Hydraulic loss reduction capability analysis for the HPP Vinodol drainage channel, Tehnicki fakultet u Rijeci, Rijeka

Wilcox, D.C. (1998) Turbulence Modeling for CFD, DCW Industries Inc., ISBN 0-9636051-5-1, La Canada

This Publication has to be referred as: Fucak, S[anjin]; Carija, Z[oran] & Mrsa, Z[oran] (2009). Improving Channel Flow with Deflectors Optimized Using a Genetic Algorithm, Chapter 70 in DAAAM International Scientific Book 2009, pp. 721-734, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

Authors' data: Dipl. Ing. Fucak, S[anjin]; Doc. Dr. Sc. Carija, Z[oran]; Univ. Prof. Dr. Sc. Mrsa, Z[oran], Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia, [email protected], [email protected], [email protected]