Runner cone boundary layer control

Transactions on Mechanics Tom 52(66), Fascicola 6, 2007

in Hydraulic Machinery and Systems Timisoara, Romania

October 24 - 26, 2007

RUNNER CONE BOUNDARY LAYER CONTROL

Michel J. CERVANTES * Division of Fluid Mechanics Luleå University of Technology

*Corresponding author: Luleå University of Technology, 971 87 Luleå, Sweden E-mail: michel.cervantes@ltu.se

ABSTRACT

The runner cone plays an essential role in the performance of elbow draft tube and de facto of low head machines. An earlier separation on the runner cone deteriorates the pressure recovery and thus the overall efficiency of the machine. Control of the separation point on the runner cone is therefore of interest to improve efficiency at any regime. One alternative to control the separation on the runner cone may be to rotate the runner cone with a differ- ent angular velocity than the runner blades.

In the present work, the effect of runner cone angular rotation on elbow draft tube, typical in Kaplan turbine, is investigated using numerical simulations.

The Turbine-99 test case (T-99) is used as benchmark at the top of the propeller curve. Simulations are performed for 4 different angular rotations: -595 (stipulated in T-99), 0, +600 and +1200 rpm. The

results indicate a delay of the separation on the cone at 0, +600 and +1200 rpm. The mean pressure recovery increases in all cases. The improvement reaches 6.6 % for the mean pressure recovery for an angular velocity of +600 rpm where separation disappear, while the loss factor decreases with 23.6 %.

INTRODUCTION

Draft tube flow simulations are challenging for the numerical community due to the different flow phenomena appearing simultaneously such e.g.

turbulence, separation, swirl and unsteadiness. The Turbine-99 workshops aim to determine state of the art within the area by proposing a test case, which consists of a model draft tube of the Hölleforsen hydropower station with detailed pressure and ve- locity measurements at different cross-sections, see Fig. 1.

Figure 1. Draft tube model used for the Turbine-99 workshops, Engström [1].

For the third IAHR/ERCOFTAC Turbine-99 work- shop [2], many participants performed unsteady simulations. A vortex rope issued from the runner cone appeared in most of the cases. This is surprising since the machine is of Kaplan type and thus doubly regulated. The vortex rope is the results of an early

separation on the runner cone. Figure 2 shows simu- lation of the Turbine-99 test case. As the axial wall shear stress vanishes, the flow releases the runner cone and rotates since it has a tangential component.

The tangential wall shear stress does not vanish since

the runner cone rotates faster than the tangential

velocity of the flow. In fact, the runner cone entrains the flow. Similar mechanism is expected for the creation of vortex rope in turbines of type Francis.

The importance of the runner cone region to simulate the flow in the draft tube was also pointed out by Cervantes [3]. Different turbulence models (zero equation model, k-ε and shear stress transport models) were tested. The results pointed out different values for the pressure recovery. The main reason of the differences was attributed to the separation point on the runner cone which was different function of the turbulence model used.

These facts point out the importance of the runner cone on the pressure recovery of elbow draft tube.

Early separation on the runner cone may give rise to a vortex rope and a recirculation zone below the runner cone, see Fig. 2. The recirculation region is an obstruction which decreases the pressure recovery since the available area decreases. Furthermore, the losses increase in the rest of the draft tube since the recirculation zone accelerates the neighborhood flow and thus the square of the velocity increases which is proportional to the losses.

Figure 2. Flow in the cone and elbow in the draft tube of the Turbine-99 test case at the top of the

propeller curve, see Cervantes [2].

Several solutions to modify the flow issue from the runner cone, essentially in Francis turbines, have been proposed. The continuous or pulsated injection of air or water in the center of the runner cone has been proposed. The most recent contribution concerns the injection of water, see Susan-Resiga [3].

A control of the boundary layer on the runner cone may allow a delayed separation and thus an increase of the draft tube performance, i.e. an increase of the machine overall performances. Such control will solve the problem at its source and thus avoid the need of air or water injections afterwards. Control involves the addition of a new degree of freedom in the machine.

The present work presents firstly a qualitative analysis of the boundary layer on the runner cone to identify the appropriate parameter which may control the separation point: the angular velocity of the runner cone. The method used to performed simulation of the Turbine-99 test case for various angular velocities (-595 to +1200 rpm) as well as the results obtained follow.

RUNNER CONE

An elbow draft tube is composed of a cone, an elbow and a diffuser, see Fig. 1. Most of the pressure recovery occurs in the cone where about 80% of the pressure recovery occurs. The losses are equally distributed between the cone, the elbow and the diffuser at best efficiency for the Turbine-99 test case. The flow issued from the cone conditions the flow in the rest of the draft tube. A flow leaving the cone with a high velocity will pass the elbow with a large velocity and thus increase the overall losses. It is therefore imperative to avoid or minimize any recirculation zone.

The cone is a diffuser with a large angle. The presence of the runner cone allows a more rapid deceleration of the flow. This region is in fact a double diffuser since the area increase both inward and outward in the radial direction from the inlet draft tube down- stream. For example, the area in the T-99 test case increases from 0.145 to 0.228 m

in 0.19 m. The distance 0.19 m represents about 20% of the draft tube height. The flow has to be attached to the shroud and runner cone otherwise the function of the cone is deteriorated, i.e. pressure has to increase toward the shroud and the runner cone since the streamlines have to bend. Separation may create a recirculation zone which will accelerate the average mean velocity and thus increase the losses. The boundary layer on the rotating runner cone is de facto of great interest to understand and improve the performance of an elbow draft tube. Boundary layers on rotating bodies have been extensively study between the 60s and 80s, principally for projectiles. However, no work was found concerning boundary layer on rotating bodies in swirling flows.

A curvilinear system of coordinates is now assumed, where x represents the distance along the meridian from the beginning of the domain and y the coordinate at right angle to the surface, cf. Fig. 3. The quantity r(x) is the radius of the body from the axis of revolution.

At point M, U and V are the x and y components of the velocity respectively and W is the transverse velocity component due to the runner cone spin.

Assuming an incompressible, axis-symmetric and

steady flow and a radius of curvature of the body

large compared to the boundary layer thickness, the

boundary layer equations [4] are given:

( )

2 2

2 2

2

2 2

0

1

2

rU rV

x y

U U W dr d U

U V U W

x y r dx dx y

W W UW dr W

U V

r dx y

⎧∂ ∂

+ =

⎪ ∂ ∂

⎪ ⎪ ∂ ∂ ∂

⎪ + − = + + ν

⎨ ∂ ∂ ∂

⎪ ⎪ ∂ ∂ ∂

+ + = ν

⎪ ∂ ∂ ∂

⎪⎩

(1)

Figure 3. Boundary layer on a rotating runner cone.

The boundary conditions are:

0

outerflow outerflow

0 : U=0, V=0, W=r W : U=U , V 0, W=W y

y

= ⋅ω =

⎧ ⎨ +∞ =

⎩ 6 (2)

Assuming that the runner blades cannot be changed;

there are only two parameters which may be varied to delay the separation on the runner cone: r(x) and ω.

Adaptive runner cone seems difficult. A similar solution was proposed at the end of the draft tube by Cervantes and Videhult [5], where a variable floor was tested on a Kaplan model. A gain of 0.5 % on the efficiency was gained.

A variable angular velocity of the runner cone is the most appropriate parameter to control. Its influence is tested in the following using numerical simulations.

NUMERIC

The commercial code CFX 11.0 was used to per- form the simulations. The code uses the finite volume method and has a coupled unstructured solver.

Turbulence models

The shear stress transport (k-ω) was used for the simulations. The shear stress transport (k-ω) based model is a two equations model similar to the k-ε model. The transport equation for the turbulent dis- sipation is replaced by an equation for the turbulent

frequency. The turbulent kinetic energy and frequency are related through the turbulent viscosity such as

ρ ω

μ

= ^k . (3)

The k-ω formulation is advantageous for near wall treatment compared to the standard k-ε. The SST model is a development of the Wilcox model and baseline (BSL). It is known to give accurate predic- tions of the onset and the amount of flow separation under adverse pressure gradients [6].

Boundary conditions

The boundary conditions used for the calculations are these proposed for case 1 at the 3

T-99 work- shop, see www.turbine-99.org for more details.

The angular velocity is varied in the simulations.

The followings values are used:

Table 1. Angular velocity used

Case 1 Case 2 Case 3 Case 4 ω (rpm) - 595 0 + 600 +1200

The geometry proposed by the organizers of the

T-99 workshop ends immediately after the straight

diffuser. Thus, recirculation is expected. The outlet

boundary condition opening was used to allow flow

in both directions.

Grid

The grid y

=1 furnished by the organizers for case 1 was used, visit www.turbine-99.org for more information. The characteristics of the grids are:

1002360 nodes, 981424 hexahedral elements and a minimum face Angle of 20.8°. The grid has a maxi- mum edge length ratio of 132 and a maximum element volume ratio of 8.9.

Discretization

The schemes used for the discretization of the different equations are presented in Table 2. The use of the upwind scheme is motivated by the simulations performed by Marjavaraa et al. [7]. Marjavaraa et al.

found no differences between upwind and high reso- lution for the turbulent equations when using high resolution for the momentum and the continuity equations with CFX for the T-99 test case. No con- vergence difficulties were encountered.

Convergence was not achieved with second order accurate scheme for the momentum and the continuity equation with the present model. The reason is unclear but may be attributed to the unsteady behavior of the flow in the draft tube. The objective of the paper is to give a trend on the influence of the runner cone angular velocity on the pressure recovery. Therefore, the present schemes are acceptable.

Table 2. Scheme order used

Continuity Momentum K-e Tef

Scheme HR HR UPW UPW

HR: high resolution UPW: upwind RESULTS Convergence

The residuals obtained for the simulations are pre- sented in Table 3. The maximum residual is of about 10

, while rms values are around 10

or below.

Table 3. Residual reached

Case 1 Case 2 Case 3 Case 4 Umom.rms 1.9E-7 2.2E-7 5.0E-7 2.2E-7 Umom.max 1.1E-4 1.7E-4 3.5E-4 1.1E-4 Vmom.rms 7.2E-7 8.24E-7 9.7E-7 7.2E-7 Vmom.max 3.5E-4 2.3E-4 6.9E-4 3.7E-4 Wmom.rms 6.7E-7 8.4E-7 4.3E-7 7.2E-7 Wmom.max 5.4E-4 6.4E-4 2.5E-4 6.5E-4 P-Mass. rms 7.9E-9 1.8E-9 2.4E-9 1.9E-9 P-Mass max 6.0E-7 8.3E-7 1.2E-6 8.7E-7 Ke rms 1.0E-7 3.7E-8 3.2E-8 2.5E-8 Ke max 2.3E-5 3.7E-5 2.2E-5 1.9E-5 Tef rms 1.0E-7 4.0E-8 5.7E-8 5.3E-8 Tef max 5.8E-5 4.0E-5 3.6E-5 4.0E-5

The minimum average and maximum values of y

on the wall and the hub are presented is Table 4.

Higher values are obtained on the hub. The grid does not seem to be enough fine to allow y

around 1.

Table 4. y

values

Case 1 Case 2 Case 3 Case 4 y

hub 38.2 19.6 199 42.5 y

hub 111.1 64.9 119 191.6 y

hub 220.1 174.4 313 452.6

y

wall 0 0 0 0

y

wall 1.3 1.2 1.2 1.2 y

wall 5.7 5.9 5.2 5.9 Engineering quantities

The engineering quantities are presented in Table 5.

They are defined by:

= ⎛ ⎞

⎜ ⎟

⎝ ⎠

∫∫ ∫∫

wall 2

1 1

dA- dA

1 2

out in

out A in A

p

in

P P

A A

C

Q ρ A

, (6)

⎛ ⎞ ⎛ ⎞

⎜ + ⎟ ⋅ − ⎜ + ⎟ ⋅

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

=

⋅

∫∫ ∫∫

∫∫

U U

U U

U U

2 2

2

ˆ ˆ

2 2

2 ˆ

in out

in

A A

A

P ndA P ndA

ndA

ρ ρ

ζ

ρ

. (7)

Both the mean pressure recovery and the loss factor change significantly. The maximum variation occurs for ω = +600 rpm:

• + 6.6 % in pressure recovery,

• 23.6 % in loss factor.

Table 5 . Engineering quantities

ω (rpm) - 595 0 + 600 + 1200

Pressure recovery (Pa) 5651 5867 6023 5929

C

0.88 0.914 0.938 0.923

Variation C

(%) 0 3.9 6.6 4.9

ζ 0.178 0.153 0.136 0.146

Variation ζ (%) 0 -14 -23.6 -18 Flow visualization

Figure 4 represents Streamlines issue from the runner cone, contour of the axial velocity and vector plot of the tangential velocity.

Function of the angular velocity the streamlines

follows the runner cone wall differently. For ω = -595,

0 and +1200 rpm, the streamlines leave the runner

cone early while they are attached to the wall for

ω = +600 rpm. For a stationary runner cone, the stream-

lines indicate the presence of a swirl issue from the

runner. As the runner cone angular velocity increases,

the swirl decreases. At ω = + 600 rpm, the stream- lines are straight and attached to the runner cone.

The runner cone does not entrain the flow anymore, it decrease the intensity of the swirl issued from the runner.

a) ω = -595 rpm

b) ω = 0 rpm

c) ω = 600 rpm

d) ω = 1200 rpm

Figure 4. Streamlines issue from the runner cone, contour of the axial velocity and vector plot of the

tangential velocity.

The size of the recirculation zone below the runner is different function of the angular velocity.

Depending on the angular velocity, the position of the separation point on the runner cone will occur later. As the recirculation zone decreases, the cone operates better: the pressure recovery increases.

Figure 5 presents the contours of the axial wall shear stress on the runner cone. For ω = -595, 0 and +1200 rpm, the wall shear stress has positive values near the end of the cone indicating separation, the z axis is directed upward. Separation is obtained earlier for ω = -595 rpm. This corroborates the fact that the streamlines leaves the runner cone earlier for this angular velocity. The separation occurs later for a stationary runner cone. This means a better utilization of the cone, i.e. a better pressure recovery as found in Table 5. For ω = +600 rpm, the separation has vanished. The flow is attached to runner cone until its end. An optimal use of the runner cone is obtained.

The highest pressure recovery is obtained.

a) ω = -595 rpm

b) ω = 0 rpm

c) ω = 600 rpm

d) ω = 1200 rpm

Figure 5. Contour of the axial wall shear stress on the cone.

CONCLUSION

Simulations of the Turbine-99 test case have been performed with different angular velocity of the runner cone. The results indicate that the angular velocity has a strong influence on the separation point on the runner cone. An analytical analysis is necessary to understand perfectly the mechanism.

A contra rotating runner cone is found to be optimum in the case study: the pressure recovery increases with 6.6 %. This increase is equivalent to 0.8 % efficiency increase on the T-99 test case on the top of the propeller curve.

REFERENCES

[1] Engström T. F., Gustavsson L. H., and Karlsson R. I., 2001, “Proceedings of Turbine-99 - Workshop 2”, June 2001. Luleå University of Technology. Proceedings online available from http://www.sirius.ltu.se/strl/

Turbine-99/index.htm.

[2] Cervantes M.J., Engström T.F., and Gustavsson L.H., 2005, “Proceedings of Turbine-99 III”. Luleå University of Technology, number 2005:20.

[3] Susan-Resiga R. et al., 2006, “Jet Control of the Draft Tube Vortex Rope in Francis Turbines at Partial dis- charge”. Proceedings of the 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan, October, 2006.

[4] Laminar Boundary Layers, Oxford University Press 1963.

[5] Cervantes M.J. and Videhult S. ,2006, “Adaptive draft tube for increased efficiency”, Water Power and Dam Construction, Vol. 58, Number 3, pp. 43-45

[6] ANSYS CFX 10.0, help manual.

[7] Marajavaraa et al., 2005, “Steady and Unsteady Simu-

lations of the Turbine-99 Draft Tube using CFX-5 and

Stream”. Proceedings of Turbine-99 III, Luleå University

of Technology, number 2005:20.