Turbine-99 III Proceedings of the third IAHR/ERCOFTAC workshop on draft tube flow

8-9 December 2005, Porjus, Sweden Paper No. xxx

NUMERICAL SIMULATION OF THE FLOW IN TURBINE-99

DRAFT TUBE

Buntić Ogor I., Dietze S., Ruprecht A. Institute of Fluid Mechanics and Hydraulic Machinery

University of Stuttgart Pfaffenwaldring 10, D-70550 Stuttgart, Germany

buntic@ihs.uni-stuttgart.de

ABSTRACT In this paper numerical simulation of the flow in the turbine-99 draft tube using the in-house finite element code FENFLOSS (Finite Element based Numerical FLOw Simulation System) is presented.

Besides modeling with standard k-ε model, main focus is on the simulations with the extended k-ε model of Chen-Kim and Very Large Eddy Simulation (VLES) known as promising tool for prediction of unsteady phenomena. Both turbulence modeling approaches are explained in details as well as the applied adaptive filtering technique which can distinguish between numerically resolved and unresolved parts of the flow. Numerical results are analyzed and presented.

INTRODUCTION The increasing consumption and demand on energy again takes into account hydropower as important energy source. Therefore the increase of old hydropower plants efficiency through refurbishment and upgrading is a challenging task. The quality and the design of the draft tube are very important. The power output of a hydraulic turbine is specially affected by the performance of its draft tube whose role is to convert the kinetic energy behind the runner into static pressure. There is a potential of improving the pressure recovery in the draft tube by modifying and optimizing its geometry. Flow in a draft tube is characterized as very intricate turbulent flow followed with appearance of different flow phenomena, e.g. unsteadiness, flow separation, swirling flow etc. Thus its simulation is complicated and time consuming requiring high computational power. Additionally, adequate turbulence modeling is needed which is able to predict such flows satisfactorily. An example of complexity of numerical simulations which are connected to the real industrial applications is Turbine-99 (Engström [1]). Nowadays Computational Fluid Dynamics (CFD) is a powerful tool for simulation and analyzing complex flows such as flows in the elbow draft tubes. It reduces the time and the costs which are necessary for design and optimization of the turbines i.e. draft tubes. The most important part of the modeling is turbulence modeling. It is still one of the fundamental CFD

problems. Widely spread in industry are Reynolds-averaged Navier-Stokes (RANS) based turbulence models which are usually not capable to reproduce complex 3D turbulent flows, especially unsteady flows. The highest accuracy for resolving complete turbulence is offered by a Direct Numerical Simulation (DNS). Unfortunately its industrial application is not possible in the foreseeable future. Lately Large Eddy Simulation (LES) starts to be a mature technique despite its high computational cost. Very Large Eddy Simulation (VLES), or also known as Detached Eddy Simulation (DES), starts to expand as a promising compromise for simulation of industrial flow problems with reasonable computational time and costs.

SIMULATION METHOD Governing equations and numerical method The governing equations describing incompressible, viscous and time dependant flow are the Navier-Stokes equations. They express the conservation of mass and momentum. In the RANS approach, these equations are time or ensemble averaged leading to the well known RANS equations:

j

iji

ij

ij

i

xU

x

P

x

UU

t

U

∂

∂−∇+

∂

∂−=

∂

∂+

∂

∂ τν 2 (1)

0=∂

∂

i

i

x

U (2)

In RANS τij expresses the Reynolds stress tensor which is unknown and has to be modelled. The task of turbulence modelling is the formulation and determination of suitable relations for Reynolds stresses. Details of the new VLES approach are described in next section. The calculations are performed using the program FENFLOSS which is developed at the Institute of Fluid Mechanics and Hydraulic Machinery, University of Stuttgart. It is based on the Finite Element Method. For spatial domain discretisation 8-node hexahedral elements are used. Time discretisation involves a three-level fully implicit finite difference approximation of 2

nd

order. For the velocity components and the turbulence quantities a trilinear approximation is

applied. The pressure is assumed to be constant within element. For advection dominated flow a Petrov-Galerkin formulation of 2

nd order with skewed

upwind orientated weighting function is used. For the solution of the momentum and continuity equations a segregated algorithm is used. It means that each momentum equation is handled independently. They are linearised and the linear equation system is solved with the conjugated gradient method BICGSTAB2 with an incomplete LU decomposition (ILU) for preconditioning. The pressure is treated with the modified Uzawa pressure correction scheme (Ruprecht [2]). The pressure correction is performed in a local iteration loop without reassembling the system matrices until the continuity error is reduced to a given order. After solving the momentum and continuity equations, the turbulence quantities are calculated and a new turbulence viscosity is gained. The k and ε equations are also linearised and solved with BICGSTAB2 algorithm with ILU preconditioning. The whole procedure is carried out in a global iteration until convergence is obtained. For unsteady simulation the global iteration has to be performed for each time step. The code is parallelised and computational domain is decomposed using overlapping grids. In that case the linear solver BICGSTAB2 has a parallel performance and the data exchange between the domains is organised on the level of the matrix-vector multiplication. The preconditioning is then local on each domain. The data exchange uses MPI (Message Passing Interface) on computers with distributed memory. On the shared memory computers the code applies OpenMP. For more details on the numerical procedure and parallelisation the reader is referred to Maihöfer [3] and Maihöfer et al.[4]. Turbulence modeling RANS equations are established as a standard tool for industrial simulations, although it means that the complete turbulence behaviour has to be enclosed within appropriate turbulence model which takes into account all turbulence scales (from the largest eddies to the Kolmogorov scale). DNS is capable to resolve all turbulence scales but it requires a very fine grid resolution. Hence carrying out 3D simulations of the flow with high Reynolds number is very time consuming even for high performance computers (Fig. 1).

Figure 1 - Degree of turbulence modelling and computational effort for the different approaches.

LES starts to get more practical importance. With LES all anisotropic turbulent structures are resolved in the computation and only the smallest isotropic scales are modelled (schematically shown in Fig. 2). The models used for LES are simple compared to those used for RANS because they only have to describe the influence of the isotropic scales on the resolved anisotropic scales. With increasing Reynolds number the small anisotropic scales strongly decrease becoming isotropic and therefore not resolvable. If there is a gap in the turbulence spectrum between the unsteady mean flow and the turbulent flow, ”classical” RANS i.e. URANS models can be applied, as they are developed for modelling the whole range of turbulent scales (Fig. 2). They show excessive viscous behaviour and very often damp down unsteady motion quite early. It also means that they are not suitable for prediction and analysis of many unsteady vortex phenomena.

Figure 2 - Modelling approaches for RANS and LES.

Figure 3 - Modelling approach in VLES.

Contrary, if there is no spectral gap and even one part of the turbulence can be numerically resolved, VLES can be used. It is very similar to the LES, with

the difference that a smaller part of the turbulence spectrum is resolved and the influence of a larger part of the spectrum has to be expressed with the model (Fig. 3). It requires the use of an appropriate filtering technique which distinguishes between resolved and modelled part of the turbulence spectrum. Because of its adaptive characteristic it can be applied for the whole range of turbulence modelling approaches from the RANS to the DNS (Fig. 4).

Figure 4 - Adjustment for adaptive model.

Lately several hybrid methods have been proposed in the literature. All of them are based on the same idea to represent a link between RANS and LES. They try to keep computational efficiency of RANS and the potential of LES to resolve large turbulent structures, even on coarser grids and with high Reynolds number. Various VLES methods slightly differ in filtering techniques, applied model and interpretation of the resolved motion, but broadly speaking they all have a tendency to solve relevant part of the flow and model the rest (Fig. 5).

Figure 5 - Distinguishing of turbulence spectrum by VLES.

The basis of the adaptive model is the extended k-ε model of Chen and Kim (Chen et al. [5]). It is chosen due to its simplicity and capacity to better handle unsteady flows. Its transport equations for k and ε are given as

εσ

νν −+

∂

∂

+

∂

∂=

∂

∂+

∂

∂k

jk

t

jjj P

x

k

xx

kU

t

k (3)

44 344 21termadditional

3

2

21 kk

k

j

t

jjj

Pk

Pc

kcP

kc

xxxU

t

⋅

+−

+

∂

∂

+

∂

∂=

∂

∂+

∂

∂

εεε

ε

εε

ε

σ

νν

εε

(4)

with following coefficients: σk = 0.75, σε = 1.15, c1ε = 1.15, c2ε = 1.15 and c3ε = 0.25. Additionally, these extended k-ε equations need to be filtered. The applied filtering technique is similar to

Willems [6]. The smallest resolved length scale ∆ used in filter is according to Magnato et al. [7] dependant on the local grid size or the computational time step and local velocity. According to the Kolmogorov theory it can be assumed that the dissipation rate is equal for all scaled. This leads to

εε ˆ= (5)

This is not acceptable for turbulent kinetic energy. It is filtered according to

∆−⋅=

Lfkk 1ˆ (6)

As a suitable filter

<∆

∆−

≥∆

=L

L

L

ffor1

for03/2

(7)

is applied where

∆

∆=

∆⋅⋅=∆

3D for

2D for h witmax

3maxmax V

Vh

h

tuα (8)

contains model constant α in a range from 1 to 5.

Then the Kolmogorov scale L for the whole spectrum is given as

ε

2/3kL = . (9)

Modelled length scales and turbulent viscosity are

ε̂

ˆˆ

2/3kL = (10)

εν µ

ˆ

ˆˆ

2kct ⋅= (11)

with cµ = 0.09. The filtering procedure leads to the final equations

εσ

νν −+

∂

∂

+

∂

∂=

∂

∂+

∂

∂k

jk

t

jjj P

x

k

xx

kU

t

k ˆˆ (12)

kk

k

j

t

jjj

Pk

Pc

kcP

kc

xxxU

t

ˆˆ

ˆ

ˆ

3

2

21 ⋅

+−

+

∂

∂

+

∂

∂=

∂

∂+

∂

∂

εεε

ε

εε

ε

σ

νν

εε

(13)

with the production term

j

i

i

j

j

itk

x

U

x

U

x

UP

∂

∂

∂

∂+

∂

∂= ν))

. (14)

For more details of the model and its characteristics the reader is referred to Ruprecht [8]. Furthermore, VLES, of course depending on the grid size, is not valid all the way to the wall. Modelling procedure close to the wall is based on standard wall functions, which are most commonly used in industrial practice. Standard wall functions are as well used in case of extended k-ε model of Chen and Kim.

COMPUTATIONAL DETAILS

Geometry, boundary conditions and simulation

setup Turbine-99 draft tube is an elbow sharp-heel draft tube developed for use in Kaplan turbines. Draft tube geometry given by workshop organisers is fixed, as well as defined cross sections (for more details see [9]). Original computational domain starts at cross section CS-Ia and ends at cross section CS-IVb. It also includes a part of the runner hub at the inlet. Computational domain used for simulations presented in this paper is slightly modified. The 2.1m straight extension was added due to the outlet boundary condition. Used computational mesh was also provided by workshop organisers. Computations were performed at the coarsest available mesh (1 million elements, y+ ≈ 50, see Döbbener [10]), due to the interest to investigate behaviour and characteristics of VLES at coarser grids which are computationally “cheaper”. Due to geometry extension at the outlet, real mesh consisted of ca. 1.2 million elements. Inlet boundary conditions are as well predefined by workshop organisers (for details see [9]). They are represented with velocity profiles which are available by means of extensive LDV laboratory measurements. They provide the mean axial (U) and tangential (W) velocity components (Fig. 6 and Fig.

7) and three turbulent quantities ( 'u , 'w and ''wu )

(Fig. 9) in radial direction at CS-Ia. Radial velocity component (V) is not recorded and therefore no measured data are available. The same is valid for

turbulent quantities 'v , ''vu and ''wv . Therefore

assumption for the radial velocity is made using following (Fig. 8):

)tan()()( Θ⋅= rUrV

( )coneconewall

conewallcone Rr

RR−

−

Θ−Θ+Θ=Θ

with Rcone≤ r ≤ Rwall, Θcone = -12.8° and Θwall = +2.8°. The unknown turbulent quantities are assumed as follows:

'' wv =

'''''' wuwvvu ==

As turbulence kinetic energy is needed for the calculations, it is estimated using (Fig. 10)

++= 222 '''

2

1wvuk .

Estimation of dissipation rate (Fig. 11) is performed using

l

kc

5.1

µε =

with constant cµ = 0.09 and l = 0.1.

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Mean a

xia

l velo

city [m

/s]

Figure 6 – Profile of measured mean axial velocity.

0

1

2

3

4

5

6

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Mean tangential velo

city [m

/s]

Figure 7 - Profile of measured mean tangential velocity.

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Radia

l velo

city [m

/s]

Figure 8 - Profile of assumed mean radial velocity.

-0,20

0,00

0,20

0,40

0,60

0,80

1,00

1,20

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Turb

ule

nt quantities

u'

w'

u'w'

u' scale

w' scale

u'w' scale

Figure 9 - Profile of measure 'u , 'w and 'w'u .

0

0,2

0,4

0,6

0,8

1

1,2

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Turb

ule

nce k

inetic e

nerg

y [m

2/s

2]

Figure 10 - Turbulence kinetic energy profile.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,09 0,11 0,13 0,15 0,17 0,19 0,21 0,23 0,25

Radius r [m]

Dis

sip

ation rate

[m

2/s

3]

Figure 11 - Dissipation rate profile.

Correction of values was not performed due to the fact of significant discrepancy of ca. 6.5% between stated flow rate and the one integrated from measured velocity. Consequently, stated Reynolds number is 1.7 million and Reynolds number in simulation is 1.6 million. Difference can be seen in Fig 6. to Fig. 11. (lines with filled square are original not scaled data and lines with not filled square are scaled data). Slight modification of the profile at the runner point in axial direction was necessary due to implementation of rotating wall boundary condition and stability of convergence. Boundary condition set at the runner hub is rotating wall with runner speed of N = 595 rpm. For all other cone and side walls no-slip wall boundary condition in conjunction with standard wall function was set. At the outlet section constant pressure was imposed. Simulated operational point was operational mode T (defined by workshop organisers) corresponding to 60% load which is close to the best efficiency of the given system. Defined flow rate in mode T is Q = 0.522 m

3/s.

RESULTS AND DISCUSSION One of the main aims of the investigation is assessment of FENFLOSS code and implemented turbulence models to predict the flow phenomena and pressure recovery in Turbine-99 draft tube. Valuable knowledge is also general understanding of the flow which appears in draft tubes.

Calculations of Turbine-99 were performed as steady and unsteady calculations with already mentioned

boundary conditions. Steady cases were performed for calculations with k-ε model and extended k-ε model of Chen and Kim. In both cases previously mentioned wall function was used. Unsteady calculations were performed for further investigation of VLES. Used time steps were 0.1s and 0.01s. Important was to investigate the influence of time step on the filtering approach used in VLES.

Steady calculation of k-ε model showed good and quick convergence. Velocity distributions in symmetry and cut planes are shown in Fig. 12 and Fig. 13, respectively. Streamlines of the flow behind the runner and at the outlet of the draft tube are presented in Fig.14 and Fig.15.

Figure 12 - Velocity distribution in symmetry plane,

steady k-ε

Figure 13 - Velocity distribution in planes Ib, II, III and IVb,

steady k-ε

Figure 14 – Streamlines behind the runner,

steady k-ε

Figure 15 – Streamlines at cross section IVb,

steady k-ε

Due to the additional terms, extended k-ε model of Chen and Kim is expected to show in study case slightly different result i.e less damping of swirl behind the runner. As it can be seen in Fig. 16 to Fig. 18 swirling flow behind the runner is a bit brighter and the region of higher velocity on the right side (in the flow direction) is a bit longer. It can be also seen that the structure of the vortex developing behind runner is different (Fig. 14 and Fig. 18). Velocity distributions in cross sections II, III and IVb do not show significant difference for investigated models. In both cases vortices are suppressed after the elbow. At the place where flow enters the horizontal part of the draft tube, Chen and Kim model shows still existing small vortex on the right side. On the other hand, both cases show the appearance of the recirculation zone on the right side (in the flow direction) at the cross section IVb (Fig. 15 and Fig. 19).

Figure 16 - Velocity distribution in symmetry plane,

steady Chen and Kim model

Figure 17 - Velocity distribution in planes Ib, II, III and IVb,

steady Chen and Kim model

Figure 18 - Streamlines after the runner, steady Chen and Kim model

Figure 19 - Streamlines at cross section IVb, steady Chen and Kim model

Further investigations were concentrated on unsteady calculations using VLES. In presented VLES approach unsteady calculations were necessary due to the filtering technique. Two time steps were used 0.1s and 0.01s to investigate the influence of time step on implemented filtering. According to Helmrich [11] model constant α was

set to 2. Due to the previously presented definition of the filtering approach it is expected that for coarse mesh filter function becomes 0 and thus whole turbulence is expected to be modelled with Chen and Kim turbulence model. On the other hand for fine mesh filtering function approaches 1 resulting in complete resolving of the turbulence. Used mesh is relatively fine, times steps are coarse and relatively fine. Therefore the question is which part influences

the filtering - mesh size or tu ∆⋅ .

It was expected that time step 0.1s would lead to the more modeling of the whole turbulence spectrum and that result would develop toward the results calculated with Chen and Kim model. Unfortunately, for this calculation convergence problems occurred. As initialization for VLES with time step 0.01s a short unsteady calculation with Chen and Kim with the same time step was used. It showed the convergence to the same steady state which is previously presented. VLES calculation using time step 0.01s was time averaged and presented in Fig. 20 to Fig. 23.

Figure 20 - Velocity distribution in symmetry plane, VLES

Figure 21 - Velocity distribution in planes Ib, II, III and IVb,

VLES

Figure 22 - Streamlines after the runner, VLES

Figure 23 - Streamlines at cross section IVb, VLES

As it can be seen VLES shows different flow picture compared to the two previous cases. The flow shows a much more unsteady and turbulent characteristic. Appeared vortex behind the runner does not show in cross section Ib circular shape. Vortex stretches throughout whole draft tube after the elbow, especially on the right side (in the flow direction). As

well, strong recirculation zone in cross section IVb is observed. Fig. 24. presents the zone of low pressure (light colored left side in flow direction) where unsteadiness seems to appear and continues to spread through draft tube always on the right side.

Figure 24 - Vortex streamlines Calculated pressure recovery factor (based on the wall pressure) and mean pressure recovery factor (based on the entire inlet cross section) for investigate cases are presented in Table 1. Table 1. Pressure recovery factors

Case Cpr Cprm

k-ε steady 1.047 0.790

Chen & Kim steady 1.036 0.731

VLES 0.1s - -

VLES 0.01s 1.056 0.714

As the simulation of the flow in draft tube is time consuming, most of the calculations are performed on CRAY Opteron and NEC Xeon clusters at High Performance Computing Center HLRS, University of Stuttgart.

CONCLUSIONS In this paper simulations with standard k-ε model, extended k-ε model of Chen and Kim and VLES are presented. Predefined mesh and boundary conditions were used with the exception of the outlet region. All steady state calculations converged well. Slight difference of results between k-ε and Chen and Kim model can be noticed, especially in vortex behind the runner. Contrary to both k-ε models, VLES, which is based on Chen and Kim k-ε model, shows much more unsteady phenomena throughout the draft tube. It means that one part of the flow is really resolved and other part modelled.

ACKNOWLEDGEMENT

The authors would like to acknowledge the help of Mr. Volker Brost in data transformation and also thank other institute’s colleagues for their valuable comments and suggestions.

NOMENCLATURE f filter function hmax local grid size, m k turbulent kinetic energy, m

2/s

2

L Kolmogorov length scale, m Pk production term u local velocity, m/s Ui filtered velocity, m/s

U averaged velocity, m/s

P averaged pressure, Pa τij Reynolds stresses, Pa α model constant ∆ resolved length scale, m ∆t time step, s ε dissipation rate, m

2/s

3

ν kinematic viscosity, m2/s

ν t turbulent viscosity, m2/s

∆V size of the local element, m2 or m

3

N runner speed, rpm Q flow rate, m

3/s

U axial velocity, m/s V radial velocity, m/s W tangential velocity, m/s Cpr pressure recovery factor Cprm mean pressure recovery factor ^ modelled ~ resolved i covariant indices, i = 1,2,3

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