1

Abstract Hydro power plants generate 50% of the electric power in Sweden and most of the power

plants are 50 years old. Refurbishing and modernization of the power plants are

increasing in the present days to increase the efficiency of the older power plants. In

Vattenfall there is a plan to refurbish three hydro power plants every year for the next 10

years and EON recently launched a similar plan.

Sharp heel draft tubes with are common in hydro power plants constructed 50 year ago

and studies have shown that efficiency improvement can be realized by minor

modification of these older designs. However, to find these improvements in a cost

effective procedure, two things are needed:

1. Accurate computer simulations of the draft tube flow (CFD)

2. An effective optimization algorithm for choosing and evaluating different designs.

This Master Thesis studies how different turbulence models and boundary conditions

affect the predicted flow in different geometries.

As for a previous study performed within a PhD-project at Luleå Technical University

(LTU) only very small improvements in the draft tube efficiency could be seen, while the

modification of the draft tube to get the optimal design is in the same order as for the

experiments. This can be seen as a quality assurance that the two different CFD-

programmes used at LTU and Vattenfall Utveckling (i.e. CFX and FLUENT) gives the

same result.

The sensitivity analysis of the inlet conditions for draft tube shows that the optimization

result is sensitive to the inlet profile. Therefore either more information on how/if the

boundary conditions are changing at the inlet of the draft tube is needed or the runner has

to be included in the simulations.

2

3

Acknowledgements This thesis work was carried out at Vattenfall Utveckling AB, Älvkarleby, Sweden, during July 2005 to December 2005. I would like to thank my supervisor Urban Andersson for his guidance, support and motivation during the project and Daniel Marjavaara, Luleå University of Technology for providing the CAD models. I would like to thank Ass Prof. Håkan Nilsson for the encouragement and guidance in writing the report. I would like to thank the CFD group at Vattenfall Utveckling AB for the assistance in GAMBIT and FLUENT software during the project work.

4

5

Nomenclature D = Runner diameter [m] N = Runner speed [rpm] Q = Flow rate [m3/s] ρ = Density [kg/ m3] H = Test head [m] Qint = Flow rate (integrated from velocity profiles) Umean = Q/A [m/s] Pdyn = (ρQ2/(2A2) [Pa]

pC = Pressure recovery coefficient

BulkPC , = Integrated Pressure recovery coefficient

wallPC , = Wall Pressure recovery coefficient Ia, Ib, III, IVb = Cross-sections C.S. = Cross-sections T(r) = operational mode

6

7

Contents 1 Introduction...................................................................................................................... 9

1.1 Hydroelectric power in Sweden................................................................................ 9 1.2 Hydroelectric Power ............................................................................................... 10 1.3 The Draft Tube........................................................................................................ 12 1.4 Computational fluid Dynamics ............................................................................... 14 1.5 Optimization ........................................................................................................... 15

1.5.1 Adapted Design................................................................................................ 16 1.5.2 Profile design ................................................................................................... 16 1.5.3 Evaluated designs............................................................................................. 17

2. Governing Equations .................................................................................................... 19 2.1 Equation of Motion ................................................................................................. 19 2.2 The Averaged Equation – RANS............................................................................ 19 2.3 Boussinesq Approach.............................................................................................. 20 2.4 The Standard K-ε model ......................................................................................... 21 2.5 The SST- K-ω Turbulence model ........................................................................... 22 2.6 Wall approach ......................................................................................................... 23 2.7 Standard Wall Function .......................................................................................... 24

3. Computational Approach .............................................................................................. 25 3.1 Computational Domain........................................................................................... 26 3.2 Cross sections.......................................................................................................... 27 3.3 Boundary conditions ............................................................................................... 28 3.4 Numerical procedure............................................................................................... 29 3.5 Grid ......................................................................................................................... 30 3.6 Grid convergence .................................................................................................... 33

4. Results and Discussion ................................................................................................. 35 4.1. Comparison of Flow Field for Sharp heel and Modified radius draft tube............ 35

4.1.1 Flow field of Sharp heel draft tube .................................................................. 35 4.1.2 Flow field of Modified radius draft tube.......................................................... 42

4.2 Optimization of Draft tube...................................................................................... 45 4.2.1 The pressure recovery factor............................................................................ 45 4.2.2 Comparison between Cp from the present work and Marjavaara and Lundström [3]. .......................................................................................................... 46 4.2.3 Pressure comparison with Dahlbäck [1] .......................................................... 48 4.2.4 Cp along lower Centre line for all the draft tube ............................................. 50

4.3 Sensitivity Analysis - Change in inlet boundary condition .................................... 52 5 Conclusion ..................................................................................................................... 55 References......................................................................................................................... 57

8

9

1 Introduction

Hydro power plants generate about 50% of the electrical power produced in Sweden.

Even a small improvement of the hydrodynamic design and efficiency can contribute a

great deal to the supply of the electric power in Sweden. The efficiency of a hydropower

plant depends on a number of parameters of which few are listed here: Turbine

efficiency, Draft tube efficiency and Generator efficiency etc. Most of the past studies

have focused on the runner for increasing the efficiency of the plant. But a good runner

design is not enough. Recent studies have shown that efficiency improvement can also be

realized by minor modification on the older design in the rest of the waterway i.e., in the

draft tube and spiral casing [1]. Redesigning the sharp heel draft tube to a rounded elbow

improved the efficiency of 50 MW hydro units in the order of 0.5%. Improvement like

this in the draft tube will contribute to a considerable economical value [1]. Draft tubes

with a sharp heel are quite common in hydro power stations constructed 50 years ago.

This design reduced both construction time and investment. Previous studies have shown

that there is potential for increasing unit performance by a moderate modification of such

draft tubes. Sharp heel draft tube have been found to have an efficiency loss of 0.03 –

2.3% due to the sharp heel. There are more that 50 sharp heel draft tubes in Sweden in

hydro power stations representing 6700GWh/year in electrical generation. A small

increase in performance in these power stations represents a considerable economic

value. This project will examine the modified sharp heel draft tube with different radius

and to determine the most efficient draft tube (best sharp heel radius draft tube). The

analysis is based on CFD simulations. The purpose of this master’s thesis is to study the

flow field and global engineering quantities, especially pressure recovery factor, in the

existing hydropower draft tube.

1.1 Hydroelectric power in Sweden

Hydropower has always been an important resource in Sweden and will form the

backbone of the country’s electricity supply for many years to come. It was first put into

use for electricity production in the 1880s. At first, small hydropower generating plants

10

supplied local electricity networks. Starting in the 1930s it became technically possible to

transmit electricity over long distances, and a major expansion of hydropower facilities

began. It began to make economic sense to transmit hydropower from the rivers of

northern Sweden to more heavily populated areas further south. Expansion of the

Swedish network ended only in the 1980s with Klippen, an example of an

“environmentally sound” hydropower plant. Today there are more than 200 plants with

outputs of more than 10 MW, as well as nearly 2,000 smaller hydropower stations.

Until the 1960s, the only obstacle to the expansion of hydropower facilities was the

availability of financial and labor resources. Starting in the late 1950s, hydropower

became increasingly controversial for environmental and aesthetic reasons. In 1969, the

Swedish Parliament thus decided that the four major rivers in northern Sweden with no

hydropower stations - the Torne, Kalix, Vindeln and Pite Rivers - would be left that way.

Most other waterways are also protected against expansion of hydropower facilities.

Today Swedish hydropower generates more than 65 TWh of electricity during a year

with normal precipitation. The potential is much greater, however, and is estimated at

more than 100 TWh.

1.2 Hydroelectric Power Water turbines are designed to extract energy from the water. The potential energy of

water is proportional to the static head and by letting gravity work on the water; the

potential energy is converted to kinetic energy and pressure energy. This energy is in turn

converted to electrical energy by leading the water through a runner, connected to a

generator. Figure 1.1 shows a schematic description of hydro power plant and defines

some important terms. Depending on the head, different types of turbines are used. As

example of two different types, the Francis turbine and the Kaplan turbine can be

mentioned. At head ranging from 40 to700 m, Francis turbines are usually preferred.

Kaplan turbines are used up to 60 m, a range that includes many of Swedish hydropower

plants. For a schematic description of the two types, The Kaplan turbine is characterized

11

by the fact that not only the guide vanes, but also the turbine blades are adjustable and

can therefore be matched to the current flow. However, the draft tube cannot be adjusted.

The efficiency of the draft tube is very important for a water turbine working at low head,

and it is determined by how well the flow responds to the geometry. The designs of many

draft tubes in use today are far from satisfactory, but when refurbishing old hydro power

plants there are possibilities to modify the draft tubes. A hydrodynamically improved

design can increase the overall efficiency by 1.5% and yield a more reliable power plant.

Usually the runner and wicket gate are refurbished. However the importance of adjusting

the draft tube to the new flow condition should not be underestimated. If this is

disregarded, the stability of the flow and the efficiency may be less than expected.

Frequently the power plant has to run at non-optimal operating condition (off-design). At

off-design the water exits the runner with a strong vertical flow. The vertical flow gives a

strong unsteady vortex core. In Francis turbines, the oscillation of the vortex core can

give rise to pressure fluctuation and vibration of a magnitude that may dramatically

decrease the efficiency, but may also cause structural damage to the turbine. The same

kind of oscillation is present in Kaplan turbine as well, but has lower amplitude and will

not cause structural damage to the turbine, but may have a serious impact on the

efficiency. The Kaplan turbine draft tubes are more sensitive to flow separation, which

can be triggered by the pressure fluctuations. The customer demand warranties with

respect to both efficiency and vibration/noise. It is very important to be able to give

warranties accurate enough for making reliable economical estimate of the investments.

����������������������������������������

����������������������������������������

�������������������������

�������������������������

Axial diffusor

Static head

Pressure conduitTurbine

GeneratorShaft

Draft tube

Power distribution

Headwater

Tailwater

Figure 1.1 Overview of the hydropower plant referred from [10]

12

1.3 The Draft Tube The purpose of the draft tube of a water turbine is to reduce the exit velocity with a

minimum loss of energy. The draft tube ‘converts’ the dynamic pressure (kinetic energy)

into static pressure. Not all energy will be recovered; the total pressure is decreases

through the diffuser due to losses. Geometrically the draft tube is a fairly simple device, a

bending pipe diverging in the streamwise direction, figure 2. However the dynamical

processes of the flow in a draft tube is very complex and many unsteady effects have

been observed.

All the design of a hydropower system, the draft tube is an important component that

significantly affects both the efficiency and cost, especially in low-head systems. Because

of the effects on overall efficiency, even a slight increase in performance could result in a

substantial energy savings. Draft tubes can be large and expensive, therefore more

compact designs offer the potential of lower cost. The optimum trade-off between

efficiency and cost requires a thorough knowledge of diffuser performance. For

conventional systems, designers have a large amount of experience, but the possibility for

improvement is still there.

Efficient diffusion of flow has been a recognized problem for some 200 years, and it has

received a large amount of attention. Much of the research in this area is basic and does

not apply quantitatively for draft tubes. This work does, provide baseline data and

contribute to physical understanding, which is important in the design of anything that

involves fluid flow. The basic research includes the effects of non-uniform inlet

condition, swirling flow, and curved flow. A diffuser is a duct of which the cross

sectional area increase in the streamwise direction, i.e. a diverging channel. Diffusers are

used in many applications where a transfer of kinematic energy into static pressure

energy is desired, including most kinds of turbo- and hydraulic machinery.

13

Although far from complete, information that applies more directly to draft tube design is

available. The dimensions of many large systems as well as standardized units have been

published. These final designs are generally the results of experiments, but complete

results and performance of separate components are limited. The results of some

systematic experiments with the turbines have been published, and are extremely

valuable.

Figure 1.2 Sharp heel draft tube (Original Geometry) Draft tubes can be designed in slightly different ways, but some design variables are of

less importance than others. The shape of the outlet, circular or rectangular, is often of

less importance than the outlet area. However, the shaping of the elbow is one of the most

intricate problems with draft tubes. The challenge is to change the shape with minor

losses of energy and without risks for damaging mechanisms such as severe cavitations.

Earlier, the design of the draft tube was governed by a few hydro-mechanical principles

with great consideration of structural and constructional application. The sharp heel draft

tube in the ERCOFTAC test case is example of this.

In the design of a diffuser there are two major phenomena to take into account. Too rapid

expansion can make wall boundary layer separate, which leads to large losses. If the

expansion is too slow the diffuser must be made longer and consequently the fluid will be

exposed to an excessive area of walls. This will lead to large wall friction losses,

separation and a more expensive construction. The optimal rate of expansion is obviously

where these losses are minimized. Many times there are spatial restrictions of the size of

the diffuser that also increase the importance of an optimal design.

14

A streamlined shape with smooth curvature in the elbow was too expensive and time-

consuming to build. It has been used in newer hydropower system but still the design is

mainly based on other considerations than flow optimization. Also cavitations-free

operation is preferred, and the runner has often a deep setting in relation to tail-water.

This is why the draft tube entrance is often below the tail-water, which is especially true

for old design.

1.4 Computational fluid Dynamics Computational Fluid dynamics (CFD) is the analysis of the system involving the fluid

flow, heat transfer and associated phenomena such as chemical reaction by means of

computer based simulation. This technique is very powerful and spans a wide range of

industrial and non-industrial application area. The availability of affordable high

performance computing hardware and the introduction of user friendly interface have led

to recent upsurge of interest. The investment costs of CFD capability are not small, but

the total expense is not normally as great as that of a high quality experimental facility.

CFD codes can produce extremely large volumes of results at virtually no added expense

and it is very cheap to perform parameter studies, for instance to optimize equipment

performance.

CFD codes are structured around the numerical algorithm that can tackle fluid flow

problem. In order to provide easy access to their solving power all commercial CFD

packages include sophisticated user interface to input problem parameter and to examine

the results. Hence all codes contain three main elements: (1) pre processor (2) solver and

(3) post processor.

The Vattenfall Utveckling AB organized the Turbine 99 Workshop jointly with Luleå

University of Technology. The workshops were sponsored by ERCOFTAC and IAHR.

The workshops studied how to simulate the flow in the draft tube. The goal with the

workshop is to determine state-of-the-art of CFD simulations in hydraulic turbine draft

15

tubes by comparison with accurate pressure and laser Doppler velocity data. The

complexity of this flow offers serious challenges to CFD calculations as it contains

turbulence, separation, swirl, strong streamline curvature etc. In the process it is intended

to identify shortcomings of the current models and suggest directions for future research.

1.5 Optimization Draft tube design has to a large extent been based on the intuition and on the experience

of the design engineer. Recent studies have shown that efficiency improvement can also

been realized by minor modification to the geometry of the waterway (draft tube) [1]. Its

purpose is to convert the kinetic energy of the flow leaving the runner into the pressure

energy by an increase of the area perpendicular to the main flow direction. Such kind of

modified draft tube was proposed in [1] and new design has been installed in a 50 MW

hydro unit and an efficiency improvement in the order of 0.5% has been verified through

accurate measurement.

Figure 1.3 Modified draft tube with 499- Radius

In a close future, CFD simulations coupled with optimization algorithms will assist in the

search for an optimal technical solution. Such a shape optimization technique to redesign

an existing draft tube is presented by Majavaara and Lundström, Luleå University of

Technology [3]. In that method, the design was evaluated in terms of predefined

objective functions – the pressure recovery factor and the energy loss factor. The

optimization was performed with the Response Surface Method (RSM) and implemented

16

in the commercial code iSIGHT7.0, and CFD simulations were made with CFX 4.4. The

design of the draft tube can either be created from the Adapted design or profile design.

1.5.1 Adapted Design

The original shape of the ERCOFTAC Turbine 99 draft tube is described by a number of

traditional design parameters. A parametric study of this original shape can then be done

by changing the traditional design parameters and using some powerful CAD tools

existing in market today. The alteration of the elbow geometry can easily be done by

typical CAD action such as cutting the geometry. Draft tube geometries of Sharp heel,

290, 440, 499 and 620 mm radii were provided by Marjavaara, Luleå University of

Technology. Draft tubes are named based on the radius of cut at the sharp heel, for

example 290 mm radius cut as 290-R draft tube. Figure 1.4 (a) and (b) shows the draft

tubes at the sharp heel corner.

(a) (b)

Figure 1.4Draft tube at the sharp heel corner (a) Sharp heel draft tube (b) 440-radius draft tube

1.5.2 Profile design

In this case the shape of the ERCOFTAC turbine-99 draft tube is described by a number

of profile or cross sections with different shape, location and orientation, instead of the

traditional design parameters. Each profile is in next turn described by a number of

17

design parameters. The total number of profiles and the number of the design parameters

corresponding to each profile can vary from case to case depending on the design of the

draft tube. The intricate task is to parameterize these profiles with as few design

parameters as possible. The outer surface of the draft tube is at last obtained by either

straight lines between the different profiles or by smooth curvature based on spline

approximation.

The results from Marjavaara and Lundström [3] show that it is possible to carry out shape

optimization to design or redesign the waterway of a hydro power plant. Both

parameterization models, Adapted and Profile design, predicts an optimal geometry based

on the objective function Cp. Nevertheless it had been concluded that Adapted design is

better to use when designing the old hydropower plants, while profile design is the main

choice when constructing the new ones. Another remarkable result regarding the Adapted

design Cp varies only with 0.1% for small radius. It implies that the optimum for the

Adapted design can be anywhere in the range between R=35 and R=400 mm (true

optimum is at 200 to 300). This is if Cp, pressure recovery factor, is efficiency

determination factor.

1.5.3 Evaluated designs

The ERCOFTAC draft tube is the prototype and modified sharp heel draft tube with radii

440, 499 and 620mm were tested in Vattenfall Utveckling AB laboratory. Experimental

results show that the modified draft tube with radius 499mm is (i.e. has the highest Cp).

The main focus of the present work is to find what the best optimum radius is and also to

find the efficiency improvement based on the pressure recovery factor. Variation in

Workshop profile for axial and tangential velocity was studied. A case study of simplified

profile for axial profile along with three types of tangential profile, namely simplified

tangential profile, and ±50% variation of simplified tangential profile.

18

19

2. Governing Equations

2.1 Equation of Motion A flow can be considered incompressible if the density is constant in time and

space. From the principle of mass conservation, the continuity equation for

incompressible flow can be derived.

0=∂

∂

j

j

xu

(2.1)

The incompressible Navier-Stokes can be expressed as

ijji

ij

j uxx

px

uxu

t ∂∂

∂∂

+∂∂

−=∂

∂+

∂∂ υ

ρ1)(

(2.2)

2.2 The Averaged Equation – RANS

In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-

Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged)

and fluctuating components. For the velocity components:

iii uuu ′+= (2.3) Where iu and iu′ are the mean and fluctuating velocity components (i = 1, 2, 3). Likewise, for pressure and other scalar quantities:

φφφ ′+= (2.4) Where φ denotes a scalar such as pressure, energy, or species concentration. Substituting

expressions of this form for the flow variables into the instantaneous continuity and

momentum equations and taking a time (or ensemble) average (and dropping the overbar

on the mean velocity,u ) yields the ensemble-averaged momentum equations. They can

be written in Cartesian tensor form as:

20

0)( =∂∂

+∂∂

ii

uxt

ρρ

(2.5)

)((32)()( ′′−

∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂

∂+

∂∂

∂∂

+∂∂

−=∂∂

+∂∂

jiji

iij

i

j

j

i

jiji

ji uu

xxu

xu

xu

xxpuu

xu

tρδμρρ

(2.6)

The above equations are called Reynolds-averaged Navier-Stokes (RANS) equations.

They have the same general form as the instantaneous Navier-Stokes equations, with the

velocities and other solution variables now representing ensemble-averaged (or time-

averaged) values. Additional terms now appear that represent the effects of turbulence.

These Reynolds stresses )(( ′′− ji uuρ must be modeled in order to close Equation 2.6.

2.3 Boussinesq Approach The Reynolds-averaged approach to turbulence modeling requires that the Reynolds

stresses in Equation 2.6 are appropriately modeled. A common method employs the

Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients:

′′−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

jiiji

it

i

j

j

it uu

xu

kxu

xu

ρδμρμ32

(2.7) The Boussinesq hypothesis is used in the Spalart-Allmaras model, the k-ε models, and

the k-ω models. The advantage of this approach is the relatively low computational cost

associated with the computation of the, tμ turbulent viscosity. In the case of the Spalart-

Allmaras model, only one additional transport equation (representing turbulent viscosity)

is solved. In the case of the k-ε and k-ω models, two additional transport equations (for

the turbulence kinetic energy, k, ε and either the turbulence dissipation rate,ε or the

specific dissipation rate,ω ) are solved, and tμ is computed as a function of k andε . The

disadvantage of the Boussinesq hypothesis as presented is that it assumes tμ is an

isotropic scalar quantity, which is not strictly true.

21

2.4 The Standard K-ε model The simplest “complete models” of turbulence are two-equation models in which the

solution of two separate transport equations allows the turbulent velocity and length

scales to be independently determined. The standard k-ε model in FLUENT falls within

this class of turbulence model and has become the workhorse of practical engineering

flow calculations in the time since it was proposed by Launder and Spalding [11].

Robustness, economy, and reasonable accuracy for a wide range of turbulent flows

explain its popularity in industrial flow and heat transfer simulations. It is a semi-

empirical model, and the derivation of the model equations relies on phenomenological

considerations and empiricism.

In these equations, kG represents the generation of turbulence kinetic energy due to the

mean velocity gradients. bG is the generation of turbulence kinetic energy due to

buoyancy. MY represents the contribution of the fluctuating dilatation in compressible

turbulence to the overall dissipation rate. kS and εS are user-defined source terms. Below

the complete k-ε model is shown. The k equation is written as

kMbkjk

t

jii SYGG

xk

xxku

tk

+−−++⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=∂∂

+∂∂ ρε

σμ

μρ

(2.8) and

εερεεε

εεεσμ

μεερ Sk

CGCGk

Cxxx

ut bk

j

t

jii +−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=∂∂

+∂∂ 2

231 )( (2.9)

′′∂

∂−=

=

jii

jk

t

uuxu

G

kC

ρ

ερμ μ

2

22

2

22

Pr

akM

MY

xTgG

t

tM

it

tib

=

=

∂∂

=

ρε

μβ

The Coefficients are usually given the values

44.11 =εC 92.12 =εC

09.0=μC 0.1=kσ 3.1=εσ

2.5 The SST- K-ω Turbulence model The shear-stress transport (SST) k-ω model was developed by Menter [11] to effectively

blend the robust and accurate formulation of the k-ε model in the near-wall region with

the free-stream independence of the k-ε model in the far field. The SST k-ω model is

similar to the standard k-ω model, but includes the following refinements:

kkkj

kj

ii

SYGxx

kux

kt

+−+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

Γ∂∂

=∂∂

+∂∂ )()( ρρ (2.11)

and

ωωωωωωρωρω SDYGxx

uxt jj

ii

++−+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

Γ∂∂

=∂∂

+∂∂ )()( (2.12)

In these equations, kG represents the generation of turbulence kinetic energy due to mean

velocity gradients. ωG represents the generation ofω , kΓ and ωΓ represent the effective

diffusivity of k and ω respectively, which are calculated as described below. kY and

23

ωY Represent the dissipation of k and ω due to turbulence. ωD represents the cross-discussion

term, calculated as described below. kS and ωS are user-defined source terms.

k

tk σ

μμ +=Γ

ωω σ

μμ t+=Γ

kt

GGυα

ω =

′′∂

∂−= ji

i

jk uu

xu

G ρ

2

#*

βωρβ

ωβρβ

ω fY

kfYk

=

=

( )jj xx

kFD∂∂

∂∂

−=ω

ωρσωω

112 2,1

0828.0075.0

31.0168.10.10.2176.1

2,

1,

1

2,

2,

1,

1,

=

==

=

=

=

=

i

i

k

k

a

ββ

σσσσ

ω

ω

2.6 Wall approach Traditionally, there are two approaches to modeling the near-wall region. In one

approach, the viscosity-affected inner region (viscous sub-layer and buffer layer) is not

resolved. Instead, semi-empirical formulas called “wall functions” are used to bridge the

viscosity-affected region between the wall and the fully turbulent region. The use of wall

functions obviates the need to modify the turbulence models to account for the presence

of the wall. In another approach, the turbulence models are modified to enable the

viscosity-affected region to be resolved with a mesh all the way to the wall, including the

viscous sub-layer. For purposes of discussion, this will be termed the “near-wall

modeling” approach.

24

2.7 Standard Wall Function The standard wall functions in FLUENT are based on the proposal of Launder and

Spalding [11] and have been most widely used for industrial flows. The momentum is set

as law-of-the-wall for mean velocity yields,

*)ln(1* EyUκ

=

μκρ μ PP yC

y21

41

* ≡

κ = Von Kármán constant (=0.4187) E = Empirical constant (= 9.793)

PU = Mean velocity of the fluid at that point

Pk = Turbulence kinetic energy at that point

Py = Distance from that point to the wall μ = Dynamic viscosity of the fluid

The logarithmic law for mean velocity is known to be valid for 30< *y <300. In FLUENT,

the log-law is employed when *y >11.225. When the mesh is such that *y >11.25 at the

wall-adjacent cells, FLUENT applies the laminar stress-strain relationship that can be

written as ** yU =

The major advantage of the Wall function approach is that high gradient shear layer near

walls can be, modeled with relatively coarse grid, yielding substantial saving in CPU

time and storage.

25

3. Computational Approach

This chapter describes the numerical considerations. The set-up of boundary condition

and grid design are also discussed. The commercial CFD code FLUENT 6.2.16 is used

to solve the flow in the draft tube, with assumption of steady, incompressible and

turbulent water flow. A total of 16 simulations were carried out with same coarseness of

the grid in order to find the optimum radius of the draft tube. Grids were made for draft

tubes with sharp heel, 290, 440, 490 and 620-radii. The turbulence in the draft tube is

modeled with the K-ε and K-ω turbulence model using the standard wall function. The

K-ε Turbulence model was mainly examined in the present work since it is compared

with the previous work “Parameterization and Flow Design Optimization of Hydraulic

Turbine Draft tubes” by Daniel Marjavaara and Lundström [3] and Turbine 99 Workshop

[11]. Complete list of the cases can be seen in the table 3.1.

Case Grid (draft tube) Condition

1:1, 1:2, 1:3, 1:4, 1:5 Sharp heel, 290, 440, 490, 620-

radii draft tube

K-ε Turbulence model and

Workshop boundary condition.

2:1, 2:2, 2:3, 2:4, 2:5 Sharp heel, 290, 440, 490, 620-

radii draft tube

SST K-ω Turbulence model

and Workshop boundary

condition.

3:1, 3:2, 3:3 Sharp heel draft tube K-ε Turbulence model,

Simplified Axial profiles,

Simplified Tangential profile

and 50± Tangential profile.

4:1, 4:2, 4:3 440-radius draft tube K-ε Turbulence model,

Simplified Axial profiles,

Simplified Tangential profile

and 50± Tangential profile.

Table 3.1 Different cases

26

3.1 Computational Domain The computational domain is build and modified with the commercial CAD software I-

DEAS as mentioned in [1]. Often it starts as a cylindrical diffuser (connected to the

runner casing) followed by an elbow. Throughout the elbow the flow is generally

contracting. After the elbow the draft tube ends with a diffuser (often rectangular in older

plants). Figure3.1 shows the change in area (of a cross section) for a convectional sharp

heel draft tube.

Figure3.1 The area (normalized with area of outlet) of the sharp heel draft tube from [5]

Considerable attention has been given to modify the conventional sharp heel draft tube

due to the characteristic discontinuity peak at the elbow as shown in figure3.1. In this

work, the modification proposed by Dahlbäck [1] is considered. The modification of the

sharp heel draft tube is obtained by using the CAD action “cut” to remove a part of the

original geometry according to the value of radius. This results in a reduction of area only

at the elbow. The decrease in the area is based on the radius of cut made to the sharp heel

draft tube. Computations are carried out to optimize the draft tube for different radii 290,

440,499 and 620MM. The least area at the bend is for the 620 radius draft tube.

The flow domain of the draft tube is extended 1.5m further downstream to avoid

recirculation at the outlet boundary, as was done in the workshop grid. Moreover, it

27

ensures that constant average static pressure is an acceptable assumption at the outlet of

the draft tube. Figure3.2 and 3.3 shows the draft tube with extension and without

extension.

Figure3.2 Sharp heel draft tube without extension

Figure3.3 Sharp heel draft tube with extension

3.2 Cross sections

This section describes the cross sections that have been considered whole throughout the

work to analyze the flow and to calculate the pressure recovery factor to optimize the

draft tube. Cross sections, C.S. Ia and C.S. Ib are circular while C.S. II, C.S. III, C.S. IVa

and C.S. IVb are rectangular. Figure 3.4 shows location of all the cross sections used for

analyzing the flow in the Turbine 99 workshops for the Sharp heel draft tube. The present

work also focuses on the same cross sections for analyzing and calculation purpose of the

modified radius draft tubes.

28

Figure 3.4 Cross section locations in the Sharp heel draft tube

3.3 Boundary conditions

The boundary conditions set in the CFD simulation are the same as those from the

operational mode T reported in the second ERCOFTAC Turbine-99 workshop. This

mode represents the point with highest efficiency on the propeller curve, i.e. optimum

operation condition.

At the inlet the most of the boundary condition is given by LDA measurement, but the

radial velocity for instance is assumed to vary linearly with the flow angle due to

measurement problems. The “pressure outlet” boundary condition is used for the

extended outlet of the draft tube.

The runner cone was set as a wall boundary condition, which is rotating with 595 rpm.

The remaining surfaces of the draft tube are considered as stationary walls with surface

roughness of 10μ m and roughness factor of 0.5. The boundaries between the end of draft

tube and the outlet are assigned as the symmetry boundary condition. Figure 3.5 refer to

the different boundary conditions of the draft tube.

29

Figure3.5 Boundary condition of 620-radius draft tube

3.4 Numerical procedure

The CFD code, FLUENT 6.2.16, considers all meshes as hybrid and discretization is

done with a cell centred, finite volume method.

Since the flow in the draft tube is assumed to be turbulent, stationary and incompressible,

the Reynolds –Averaged Navier-stokes equations are used. In order to solve these

governing equations in Fluent 6.2.16, the segregated solver has been utilized in this

study. The SIMPLE algorithm is used for the Pressure-velocity coupling procedure.

The standard k-e model and SST k-w model was used to model the Reynolds stress terms

and to close the governing equations. This isotropic model relates the Reynolds stresses

to the mean rate of strain through the eddy viscosity as suggested by Bossinesq. The

momentum equations are discretizied with the QUICK scheme. Discretization for

pressure is done with a second order.

The standard logarithmic rough wall function has been imposed on all walls. The

roughness is set to 10μ m at all draft tube walls. For the extension downstream, which is

present in order to increase the numerical robustness, the boundaries are set to symmetry

condition. A rotating wall with the speed of 595 rpm is applied at the runner cone

30

(clockwise seen from the top). The fluid density is set to 32.998mKg and the dynamic

viscosity is set to 2310*006.1

mNs− . The under relaxation factors for pressure and

momentum is set to 0.3 and 0.5 respectively to prevent oscillations in the convergence.

The experimental values of the inlet boundary conditions are read into Fluent as profiles.

The value of the vertical velocity is linearly corrected to get the correct mass flow rate of

522 Kg/s. A pressure outlet boundary condition is set at the end of the extension.

3.5 Grid

The geometries were constructed using I-DEAS and exported as IGES file. Grids were

constructed using the Gambit 6.2 software. The draft tube volume is then sub-divided

into a number of volumes. This is called volume decomposition and is shown in figure

3.6. Hybrid grid was made for all the draft tubes. The grid consists of hexahedron,

tetrahedron, pyramid and wedge elements. All the computational draft tubes are divided

into a number of sub-volumes as shown in the figure 3.6.

Figur3.6 Different sub volumes of 290R draft tube

To minimize the numerical errors and the effects of differences in the grid topologies, the

same grid topology was used for each radius as of the sharp heel draft tube corner. This

implies that the generated grids are altered only in the sharp heel corner. This is important

since it has been shown that the result of a CFD calculation is closely connected to the

topology and the quality of the grid, especially at the inlet and cone of the draft tube [3].

31

To also ensure that grid topology has as good quality as possible, the grid resolutions

vary as function of spatial coordinates. All the draft tubes have the same number of

hexahedron in all the sub-volumes except the elbow volume. The elbow of the draft tubes

has tetrahedron, wedge and pyramid cells. The boundary layers mesh for all the sub-

volumes except the elbow has hexahedron cells. The elbow has pyramid cells in the

boundary layer. The face which connects the elbow volume and the volumes next to it

has wedge elements. The remaining portion of the elbow volume has tetrahedrons

elements.

The quality of the grid was checked in gambit. The hexahedrons has a skewness less than

0.6 which is below the maximum allowable 0.85 and the tetrahedrons are much more

skewed (0.89) than the hexahedrons and is in the safe limit of 0.9. Aspect ratio is 0 to 30,

is in the allowable range of 1 to 100. Grids with 1 Million cells for Sharp heel, 290, 440,

499 and 620 radii were generated. All the grids were generated with same first cell

location. Figure 3.7 shows the surface mesh of the 440 radius draft tube at the elbow. It

can be seen that the tetrahedrons at the elbow and hexahedrons at the cone and at the

outlet diffuser.

32

Figure 3.7 Mesh at bend for 440-radius draft tube (shows Tetrahedron cells in the

bend)

The grids were designed according to the wall functions for every case. This means that

the first interior node should be placed in the log-law region +y (30 to 300). Post-

processing verification of this criteria was done to analyze the distribution of the first

interior wall node. The average value of +y at the near wall node was found to be

95(ranges form 0 to 275) for the draft tube wall and about 150(ranges from 30 to 275) for

the runner cone wall. The region with low +y value corresponds to the area with

separated flow, where velocity is very low (close to zero). Additional grid with 1.2

million cells was made to check the grid independence. Figure 3.8 shows the +y

distribution of the sharp heel draft tube in different views.

33

Figure3.8 +y distribution on the sharp heel draft tube, dark color shows low +y

value and light color for high +y value.

The +y distribution of the other draft tube was found to be same as that of the sharp heel,

since the flow fields are the same. There was very minor variation in the +y values in

case of the other draft tubes.

3.6 Grid convergence

All the CFD simulations are assumed converged when all the residuals are less than 610− ,

which is sufficient for most engineering problems. The velocity at points at the inlet, the

centre and at the outlet is monitored and when there is no change in the results are

considered converged. The distinct rise in the residual plot is due to the change in the

differencing schemes and the under-relaxation factor (see the figure3.9). The

34

convergence of the standard k-ε turbulence model is very good, all the residuals drop

below 710− and the monitor points flatten out as in figure 3.10(a). In the case of the SST

k-ω turbulence model, the residuals are instable, fluctuating about a mean value shown

in figure 3.10(b). A lot of different numerical alterations were made, such as different

parameter and different differencing schemes but the convergence problem remains the

same.

Figure3.9 Residuals plot

(a) (b) Figure 3.10 Residual plots of Sharp heel draft tube (a) k-ε and (b) SST k-ω turbulence model

35

4. Results and Discussion

The results are discussed in three sections:

• Comparison of flow for Sharp heel draft tube and modified draft tube of 440-

radius.

• Discussion of Optimum radius based on Pressure recovery factor ( pC ), pC at

lower centre line, comparison of pressure at mid plane and comparison between

experimental and computational flow fields.

• Sensitivity analysis: Change in the inlet boundary condition: Variation in

Workshop profile for axial and tangential velocity was studied. A case study of

simplified profile for axial profile along with three types of tangential profile,

namely simplified tangential profile, and ±50% variation of simplified tangential

profile.

4.1. Comparison of Flow Field for Sharp heel and Modified radius draft tube 4.1.1 Flow field of Sharp heel draft tube In this section analysis of the flow feature is presented through the use of Iso-surface,

path lines and streamlines. Internal faces, path lines, vector and vortex rope of the draft

tube is displayed to understand the flow. Cross sections mentioned in the Turbine99

workshop II are discussed. The dynamic processes of draft tubes are very complex and

many unsteady effects have been observed. Asymmetric flow with swirl, recirculation

and separation is present in the draft tube. Turbulence k-ε model was mainly examined

in the present work since it has been used for the previous work “Parameterization and

Flow Design Optimization of Hydraulic Turbine Draft tubes” by Marjavaara and

Lundström [3] and the Turbine 99 Workshop.

Overview of the flow field:

36

Velocity (magnitude) contours of all the cross section along with the path lines below the

runner cone are shown in figure 4.1. The flow is fairly symmetrical in the cone of the

draft tube; it is in the elbow that the main asymmetry in the flow is formed. These

asymmetric and secondary flows then continue through out the rest of the outlet diffuser.

The path line shows the trace of the particle below the centre of the runner cone

throughout the draft tube. They indicate that the flow is slightly bent towards the left side

of the draft tube (seen from upstream). For the sharp heel draft tube there are three

regions with separated flow, one beneath the runner, second at the sharp heel and the

third at the upper left wall just before the elongation at the outlet. The result of the

calculated flow for the original draft tube is similar to what has been derived in the

previous studies.

Figure 4.1 Path lines and Velocity magnitude contours of sharp heel Draft tube

Detailed description of flow at inlet:

In order to view the secondary flow pattern, velocity vectors in the cross section are

plotted. Figure 4.2 and 4.3 shows the velocity magnitude contours at the inlet cone at

C.S.Ia and C.S.Ib respectively. Velocity contour at C.S.Ia shows symmetric flow and

velocity vectors rotate in clockwise direction due to rotation of the runner cone. From the

cross section C.S.Ib it can be observed that the radial velocity is least at the wall,

37

increases and gradually decreases as it moves towards the center of the draft tube cone.

Moving further down from the cone to the elbow, asymmetry is found to develop along

with the secondary flow.

Figure 4.2 Velocity magnitude contours at cross section C.S.Ia

38

Figure 4.3 Velocity magnitude contours at cross section C.S.Ib A small recirculation zone is located below the rotating runner cone, which is as shown in

figure 4.4. This recirculation originates due to the separation of the fluid at the blunt

bottom of the runner cone. Due to the rotation of the runner cone at a high speed and the

swirling fluid flow, a strong vortex is generated at this location.

Figure 4.4 Path Lines below runner cone

Detailed description of flow at elbow:

A second recirculation zone is located in the corner of the sharp heel. Figure 4.5 shows

the velocity vectors in the mid-plane of the draft tube in the elbow region. Flow at the

sharp heel gets separated into two and starts flowing towards the sides of the draft tube,

as shown in figure 4.6.

Figure 4.5 Vector in mid-plane section of draft tube

39

Figure 4.6 Path lines at sharp heel

Detailed description of flow at outlet diffuser:

Contour plot for the velocity fields at the cross section C.S.II can be seen in figure 4.7.

Figure 4.7 shows a maximum calculated axial velocity of 1.9 m/s at the right side of the

draft tube (seen from downstream position). Variation of velocity along the plane is

continuous, with lowest velocity at the upper left portion of the section. Velocity vectors

show that the large vortex is centered at the left side of the draft tube.

Figure 4.7 Velocity magnitude contours and vector at C.S. II

Figure 4.8 shows the velocity counters and velocity vectors at cross section C.S.III.

Velocity magnitude at the right side decreases as it moves form C.S.II to C.S.III.

Maximum calculated axial velocity of 1.4 m/s is at the right side of the section. The

lowest velocity in C.S.II, at the upper left portion, moves to the upper left corner in

C.S.III. Velocity vectors show a small vortex at the upper left corner in C.S.III and a

40

large vortex at the left side as in C.S.II. The streamlines appear to be moving in a

counter-clockwise manner around the entire draft tube (seen from downstream).

Figure 4.8 Velocity magnitude contours and vector at C.S. III

Figure 4.9 shows the velocity magnitude contours and vectors at the C.S.IVb. It is shown

that the velocity is larger at the right side and lowers at the left side, viewed from the

downstream. Velocity vectors show the existence of one dominant vortex-structure with a

counter-clockwise rotation. It is observed that there exist two vortices, one at the upper

left corner and the other at upper right corner. The flow features are also shown by the

experimental results by Andersson [7] and Turbine-99 workshops.

41

Figure 4.9 Velocity magnitude contours and vector at C.S. IVb

Figure 4.10 shows the large separation region located close to the outlet. This separation

extends almost to the outlet of the draft.

Figure 4.10 Recirculation close to the outlet

42

4.1.2 Flow field of Modified radius draft tube The flow field of the k-ε turbulence model for the modified radius draft tube is discussed

in this section. In general, flows of the 290, 440, 499, 620-radii draft tubes are similar and

hence only the 440-radius draft tube is discussed. The flow field characteristics of the

modified draft tube are found to be similar in all the cross sections as compared to the

sharp heel draft tube. But at the elbow due to the modification of the geometry, particular

attention to the flow characteristics in this region has been made. Figure 4.11 shows the

velocity magnitude contours of all the cross sections along with the path lines below the

runner cone.

Figure 4.11 Path lines and Velocity magnitude contours of440 radius Draft tube For the Sharp heel draft tube there are three regions with separated flows (as discussed in

the previous section). In the case of modified radius draft tube the recirculation region at

the sharp heel disappeared and the other two separated regions remain. Path lines indicate

that the flow is slightly bent towards the left side of the draft tube as that of the sharp heel

draft tube (seen from upstream). There are no notable variations in the velocity field

between the Sharp heel draft tube and the modified draft tube. But there is considerable

43

change in the pressure at the sharp heel of the draft tubes. The path lines at the sharp heel

of the modified radius show some difference compared to the sharp heel draft tube.

(a) (b)

Figure 4.12 Velocity vector at the elbow (a) Sharp heel draft tube and (b) 440-R draft tube Figure 4.12(a) and (b) show the velocity vectors at the sharp heel and near the modified

geometry of both the draft tubes. The sharp heel gives a recirculation zone at the edge

while it disappears in the modified draft tube, as shown in the above figures. In the

modified draft tube, flow goes along the curvature of the bend. Size of the velocity vector

shows the magnitude of the vector. It can be seen that the velocity is low at the bend of

the sharp heel draft tube.

(a) (b)

Figure 4.13 Static pressure contour lines at the elbow (a) Sharp heel draft tube and (b) 440-R draft tube (in Pascal)

44

Figure 4.13 (a) and (b) show the static pressure contour line at the elbow of the sharp heel

draft tube and the 440-radius draft tube. Static high pressure region of the draft tubes are

shown in the figures. The recirculation at the elbow of the sharp heel draft tube decreases

the velocity which in turn increases the static pressure. When comparing the static

pressure at elbows there is a considerable decrease in the static pressure at the elbow of

the 440-radius draft tube. The static high pressure of the 440-radius draft tube at the

elbow is decreased by 15% when compared to the sharp heel draft tube. And the static

low pressure at the inlet of the 440-radius draft tube increases by 0.35% as that of the

sharp heel draft tube. This contributes to the increase in the pressure recovery factor ( PC )

of the 440-radius draft tube.

Figure 4.14 Path lines at the 440-radius draft tube

Figure 4.14 shows the path lines at the elbow of 440-radius draft tube. The path lines at

the centre of the bend are traced. It can be seen that the particles flow towards the right

side of the draft tube. The particles out of the mid-plane are also observed to move

towards right side. (Seen from downstream)

45

4.2 Optimization of Draft tube This section deals with optimization of the draft tube based on the pressure recovery

factor. The validation is based on the overall pressure recovery of the draft tube, pressure

contours, pressure recovery along the centre line of the draft tube and comparison with

experimental flow field.

4.2.1 The pressure recovery factor

To evaluate the performance of the draft tube, the pressure recovery factor is calculated.

Often the optimization efficiency function takes care of not only the physical parameters

but also the economical aspects. For draft tube flow the target values of the efficiency

function can be pressure recovery factor and the energy loss factor. Another issue

concerning the efficiency function is to relate the target values to each other in a suitable

manner. One way to solve this is to optimize for each target value by itself, and then try

to find the optimal geometry i.e. to evaluate one of the parameters in the equation 4.1, at

a time.

ζ21 cCcf p += (4.1)

The efficiency function is set to pressure recovery factor PC , i.e. 1c is set to one, 2c to

zero in (4.1). This was done since the result from [8] show that energy loss factor, ζ , is

unstable, difficult to predict and use of wall functions for near wall calculation increases

the uncertainty. The pressure recovery factor PC in this case is a better measure of draft

tube efficiency and is defined as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

Ia

Ia

IaIVbBulkP

AQ

PPC

2

_ρ

(4.2)

where IVbP and IaP are the integrated pressure at the cross section C.S.IVb and C.S.Ia, IaQ

is flow rate at inlet (C.S.Ia), IaA is cross sectional area of the inlet (C.S.Ia), ρ is density

of water. Experimental determination of BulkPC , is difficult, since internal pressure

measurement is not possible. Instead wall pressures are used to determine the pressure

46

recovery factor on the wall. Computationally it is easy to produce both wallPC , and BulkPC , .

BulkPC , is mainly considered throughout the present work. Detailed study of pressure

recovery factor of the Sharp heel draft tube made in workshop II shows that the wallPC , is

in the range of 1.14-1.66 and BulkPC , is in the range of 0.89-0.99.

4.2.2 Comparison between Cp from the present work and Marjavaara and Lundström [3]. As previously mentioned the pressure recovery factor is used as the target efficiency

function. Equation 4.1 thus becomes

pCf = Results are drawn from the well-converged cases and the actual flow fields are similar to

what has been derived in the previous section. Important result from Marjavaara and

Lundström [3] is that the shape of BulkPC , seems to be independent of the grid size for

both adapted and profile design and indicate that the optimum can be found on the

coarser grids.

Figure 4.15 BulkPC , as function of Radius R from Marjavaara and Lundström [3].

It can be observed from figure 4.15 that BulkPC , value changes are based on the grid size

and similar for both adapted and profile design of same grid size. And the grid size does

not vary the best optimum radius for adapted and profile design. Optimum radius for the

47

adapted design is 190 mm and for profile design is at 290 mm. Results drawn are based

on the radius, where the BulkPC , is maximum. Another remarkable result regarding the

adapted design is that BulkPC , varies only with 0.1% for small radius. Small improvements

in the efficiency function BulkPC , , not more than 0.02% with in the interval 10≤R≤410

mm in [3].

−100 0 100 200 300 400 500 600 7000.905

0.91

0.915

0.92

Radius

Cp

Cp Vs Radius

290,440,490,620 radius draft tubeSharp heel draft tube

Figure 4.16 BulkPC , as function of Radius R from the present work

The K-ε turbulence model converges well and optimum can be found and SST K-ω

turbulence model has convergence problem and could not predict the optimum radius.

Table 4.1 below shows the pressure recovery factor for two turbulence models for all the

draft tubes. In the present work the best optimum radius is at 440mm with BulkPC , value

0.9123 and sharp heel draft tube with 0.9120 for k-ε turbulence model. Efficiency

increase by 0.025% compared to sharp heel draft tube. Considerable attention was given

while making the grid at the inlet of the draft tube. Figure 4.16 BulkPC , for k-ε turbulence

model.

Draft tubes

Sharp heel

290R

440R

499R

620R

)(, ε−kC BulkP

0.9121

0.9121

0.9123

0.9121

0.9100

)(, ω−kC BulkP

0.8848

0.8848

0.8849

0.8830

0.8871

Table 4.1 BulkPC , for the all the draft tubes

48

The wall +y values range from 1 to 275 for present work, which is in the allowable range

of 30 to 300 for wall function whereas in [3] the maximum +y at near wall cell are about

900 for different grid, which is higher than the recommended ones. Much detailed

explanation about the influence of +y at the runner cone was discussed in [9]. Pressure

recovery factor for the smallest radius (10 mm), which is assumed as Sharp heel draft

tube in [3], is higher then the 410-radius draft tube for all the grid sizes. Whereas in the

present work, the 440-radius draft tube has BulkPC , 0.025% higher than Sharp heel draft

tube. This when compared with the experimental results show that the efficiency

improvement in the turbine is around 0.5%, and indicate that the improvement of the

pressure recovery factor should be higher than that (about 1-2%). The modification of the

draft tube geometry may also reflect in the alteration of the inlet velocity profile, so that

the inlet velocity profile of the corresponding draft tube predict the accurate pressure

recovery factor.

4.2.3 Pressure comparison with Dahlbäck [1] In Dahlbäck the pressure measurement at the draft tube, were used to calculate pressure

recovery factor. A redesign of the sharp heel draft tube was installed and showed an

efficiency improvement of 0.5% for the plant and increase of 1-2% for the draft tube.

Using wall pressure measurements and an interpolation program Dahlbäck constructed

the pressure field plots shown in figure 4.17. The reference level is set to zero at the draft

tube outlet. The improvement is clearly shown as a low pressure after the runner. This

yields a higher net head at the same flow situation and thus a higher output and

efficiency. The pressure plots shown are in cm of water. The pressure fields shown below

are qualitative measure due to the lack of knowledge of rotating velocity component.

49

(a) (b)

Figure 4.17 Pressure field for (a) Sharp heel (b) modified draft tube from Dahlbäck [1]

The normalized pressure plot of Sharp heel and 440-radius draft tube of present

simulation are shown in figure 4.18 (a) and (b) respectively. The pressure is normalized

as Inletdynamic

LocalstaticoutletStatic

PPP

,

,, − to find the difference in BulkPC , for both Sharp heel and modified

440-radius draft tube. The reference level at the outlet is set as one. Difference in

pressure distribution occurs at the sharp heel corner at the inlet on the draft tube. The

normalized pressure distribution at sharp heel corner of Sharp heel draft and 440-radius

draft tube are shown in figure 4.12 (a) and (b). The mean Pressure at the inlet of the sharp

heel draft tube is 8232 Pascal and the 440-R draft tube yields a mean inlet pressure of

8240 Pascal. Change in the geometry at the sharp heel corner for 440-radius draft tube

increases the static low pressure at the inlet (after the runner). The pressure thus increases

by 0.4% at inlet which increases the pressure recovery factor 0.025%.

(a) (b)

Figure 4.18 Normalized pressure (a) Sharp heel draft tube and (b) 440-radius draft tube

50

From the above figures it is difficult to see the difference in BulkPC , accurately at the

sharp heel corner; hence the sharp heel corner is zoomed. The normalized pressure at the

sharp heel corner of the Sharp heel and 440-radius draft tube are shown in figure 4.19 (a)

and (b) respectively. It can be seen that the BulkPC , increases at the sharp heel corner of

the 440-radius draft tube compared to the Sharp heel draft tube.

Figure 4.19 Normalized pressure at the sharp heel corner of (a) Sharp heel draft tube and

(b) 440-radius draft tube

4.2.4 Cp along lower Centre line for all the draft tube The variations in wall pressure at the two cross sections, Ia and IVb for all the draft tube

is high. This indicates that there is a large relative distribution of velocities at the outlet,

since the dynamic pressure is much lower at the outlet section. Figure 4.20 shows the

wallPC , along lower centre line of all the draft tubes.

The different parts of the draft tube are separated by dashed vertical lines.

0.00< L<0.094 the draft tube cone

0.094<L<0.33 the elbow

0.33 <L<0.68 the outlet diffuser, part I

0.68<L<1.00 the outlet diffuser, part II

51

The figure 4.20 shows the variation of wallPC , along the length of the draft tubes.

Characteristic length of the Sharp heel draft tube can be seen in figure 4.20 (see the

difference in the peak at L=4.2 for Sharp heel draft tube and L=3.8 for modified radius

draft tubes). Most of the pressure recovery takes place in the first part (L < 0.06) of the

draft tube (see figure 4.20). The calculated pressure recovery at the cone of the draft tube

is bigger than the BulkPC , pressure recovery of the draft tube. The same phenomenon is

observed by Andersson [5]. The possible explanation is the additional energy that enters

the draft tube (e.g. swirl).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

L [ ]

Pre

ssur

e R

ecov

ery

[ ]

Cp Along Lower Centre Line

sharp heel290440499620experimental sharp heel

Figure 4.20 PC along the lower centre line of the all the draft tubes

The lower centre line at the elbow indicate that the flow is decelerated and maximum of

the pressure is reached at the before the corner of the elbow (shown in figure 4.13) this as

well was observed in experiment [5]. Computational wallPC , along the lower centre line at

the elbow of the sharp heel draft tube does not match well with the experimental by

Andersson [5]. This is due to inability of the k-ε turbulence model to predict exact flow

at the elbow (due to stagnation and complex flow). Both experiment and computation are

in good agreement at the cone and at the diffuser.

52

4.3 Sensitivity Analysis - Change in inlet boundary condition

This section deals with the sensitivity analysis of the inlet boundary condition and to

check how it varies the BulkPC , , Pressure recovery factor. The inlet profile used for the

cases 1.1 to 1.5 and 2.1 to 2.5 are same as that of the Turbine-99 Workshop II profile.

The mass flow rate for the workshop profile issec

522.03m . A case study of simplified

profile for axial profile along with three types of tangential profile, namely simplified

tangential profile, and ±50% variation of simplified tangential profile for the Sharp heel

draft tube and 440-radius draft tube (optimum draft tube). Change in the boundary

condition for cases 3.1, 3.2, 3.3, 4.1, 4.2 and 4.3 increases the mass flow rate

tosec

5311.03m . This change in the inlet profile may be due to increase in the runner speed,

change in the blade angle or increase in the net head. Figure 4.21 shows the inlet profile

used for the cases 1.1 to 1.5 and 2.1 to 2.5 and the Turbine-99 Workshop II profile.

Figure 4.22 shows the change in the inlet profile for the axial and tangential velocity

profile.

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24−1

0

1

2

3

4

5

6

Radius

Wor

ksho

p pr

ofile

Workshop Profile Vs Radius

Tangential velAxail velTurbulent KineticepsilonRadial vel

Figure 4.21Workshop II profile used for cases 1.1 to 1.5 and 2.1 to 2.5

53

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24−1

0

1

2

3

4

5

6

Radius

Tan

gent

ial v

eloc

ity

simplified Axial profileSimplified Tgt profile+50% simplified tgt simplified−50% simplified tgt profile

Figure 4.22 Altered profile used for the sensitivity analysis

See the list of the cases in table 4.1 for the reference. It is found that the 440-radius draft

tube is more efficient for the simplified tangential velocity and for decrease in 50% of

simplified tangential velocity. Efficiency increases by 0.046 % and 0.155% for 440-

radius draft tube (case 4.1, case 4.3) compared to the Sharp heel draft tube (case 3.1 and

case 3.3). But unexpectedly case 3.2 is 0.014% higher then the case 4.2 (for 50% increase

in simplified tangential velocity profile).

Change in inlet boundary Vs Draft tubes

Simplified tangential velocity profile

+50% simplified tangential velocity profile

-50% simplified tangential velocity profile

Sharp heel draft tube

Case 3.1 Case 3.2 Case 3.3

440-radius draft tube

Case 4.1 Case 4.2 Case 4.3

Table 4.2 List of cases for the sensitivity analysis

54

Change in inlet boundary Vs BulkPC ,

Simplified Tangential profile

+50% Tangential Velocity

-50% Tangential Velocity

BulkPC , of the Sharp heel draft tube

0.8954

0.9320

0.8858

BulkPC , of the 440-radius draft tube

0.8958

0.9319

0.8872

Percentage increase over Sharp heel draft tube

0.046%

---

0.155% Percentage increase over 440-radius draft tube

---

0.014%

---

Table 4.3 BulkPC , values for the cases in table 4.2

Case 3.2 and case 4.2 have 4% higher BulkPC , compared to the case 3.1 and 4.1

respectively. Case 3.1 and 4.1 have 1% higher BulkPC , compared to the cases 3.3 and 4.3

respectively. Hence from the case study it can be concluded that the draft tube is very

sensitive to the inlet boundary condition. Detailed experimental study of the inlet

boundary condition of the modified draft tubes would be of interest.

55

5 Conclusions Efficiency improvement of the present computation of the modified draft tube is in the

same range as that of the previous computations done by Daniel Marjavaara and

Lundström. But this improvement is much smaller when comparing with experiments

conducted by Dahlbäck.

The optimum radius of the present computation (440 mm radius) and experiments (499

mm radius) is closer. The results from the standard K-ε turbulence model could predict

the optimum radius, while the SST K-ω turbulence model had convergence problem due

to the unsteadiness in the flow.

On possible reason for very small improvement in the efficiency is due to the use of the

sharp heel draft tube’s inlet profile for the all modified draft tubes. Modified inlet profiles

of the corresponding draft tubes will result in better predictions of the efficiency

improvement comparable to the improvements found in the experiments.

Sensitivity analysis of the inlet conditions for draft tube shows that the draft tube flow is

very sensitive to the inlet profile and that the optimization result also is sensitive to the

inlet profile. Therefore either more information on how/if the boundary conditions are

changing at the inlet of the draft tube is needed or the runner has to be included in the

simulations.

56

57

References [1] Dahlbäck.N. Redesign of Sharp Heel Draft Tube – Results from Tests in the Model and Prototype, Proceeding of XVII IHAR Symposium on Hydraulic Machinery and cavitations. [2] Hal L. Moses. Diffuser Performance for Draft Tube Application. [3] Marjavaara BD, Lundström TS. Automatic shape optimization of a hydropower plant draft tube. Proceedings of the 4th ASME _JSME Joint Fluid Engineering Conference 2003. [4] Marjavaara BD, Lundström TS. Flow Design Optimization of a Sharp Heel Draft tube. Luleå University of Technology. [5] Andersson U. An Experimental study of the Flow in a sharp Heel Draft Tube. Licentiate Thesis, Luleå University of Technology. [6] Redesign of an Existing Hydropower Draft tube, I. Gunnar J. Hellstrom. [7] Andersson U. Test case T – Some new results and updates since Workshop – I. Proceeding of Turbine 99 –Workshop II. [8] T.F. Engström, L.M. Gustavsson and R.I Karlsson. Proceeding of Turbine 99 – Workshop 2. [9] Jonzén S, Hemström B and Andersson U. Turbine 99 – Accuracy in CFD Simulation of the Draft tube flow. [10] Håkan Nilsson. Numerical Investigation of Turbulent Flow in Water turbine, Licentiate Thesis, Chalmers University of Technology. [11] Fluent manual [12] Turbine Workshop III, http://www.turbine-99.org/ [13] http://www.sweden.se/