MASTER'S THESIS. CFD Validation of Pressure Fluctuations in a Pump Turbine. Alf Gunnar Backman. Luleå University of Technology

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1 MASTER'S THESIS 2008:017 CIV CFD Validation of Pressure Fluctuations in a Pump Turbine Alf Gunnar Backman Luleå University of Technology MSc Programmes in Engineering Engineering Physics Department of Applied Physics and Mechanical Engineering Division of Fluid Mechanics 2008:017 CIV - ISSN: ISRN: LTU-EX--08/017--SE

2 CFD validation of pressure fluctuations in a Pump Turbine Gunnar Backman Department of Applied Physics and Mechanical Engineering Luleå University of Technology SE Luleå Sweden January 11, 2008

3 Preface The work that lays the ground for this thesis has been conducted at Rainpower Norway AS, previously GE Hydro, at Kjeller under supervision of Jan Tore Billdal. The work is a part of the Hydrodyna research project managed by Ecole Polytechnique Federale de Lausanne, Switzerland. Supervisor at Luleå University of Technology has been Michel Cervantes. I would like to thank Michel, Jan Tore and the staff at the Hydraulics division for their support during the work and Sebastian Videhult and Jan Tore for giving me the opportunity to come to GE Hydro in the first place. I would also like to thank Bernd Nennemann and Thi Vu at GE in Lachine, Canada, for their support. i

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5 Abstract With growing environmental awareness, power producers turn their attention to renewable energy sources. Favorable government support has in many countries lead to a dramatic increase of wind power development. This has in turn raised demand for hydro pumped storage plants to guarantee the stability of the power grids. To allow for increased reliability and operating range, manufacturers focus on learning more about the dynamic behavior of pump turbines. The Hydrodyna research project is a collaboration project between turbine manufacturers and Ecole Polytechnique Federale de Lausanne, Switzerland, which aims to improve the understanding of high head pump turbine dynamics. It is not unusual that modern pump turbines operate at heads around m. For such high heads, rotor stator interaction is known to be of greater significance due to higher water velocities and thicker geometries in stay and guide vanes. The Hydrodyna research project aims to break through a technology gap in the understanding of pressure fluctuations due to rotor stator interaction. This work describes the results from simulations of the turbine operating at full load and its comparison to laboratory measurements. The commercial software Ansys CFX 5 has been used to perform transient iii

6 iv simulations with second order backward Euler transient scheme and the k ɛ turbulence model on structured, high density meshes. The simulations have been run on a twelve CPU cluster with 0.2 degree runner rotation in each time step. The dynamic behavior of rotor stator interaction is visualized and examined at key frequencies on the pressure and suction side of a runner blade. In order to find the best setup, different simulations have been run on guide vanes and runner with combinations including the stay vanes and draft tube. The results are validated against laboratory pressure measurements on a runner blade. The transient pressure readings, from transducers with measurement error below bar, make it possible to determine the accuracy of the CFD code. A comparison is also made with Laser Doppler Velocimetry (LDV) measurements of flow velocities in a guide vane channel.

7 Contents Preface Abstract i iii 1 Introduction Background Previous work Hydro power Pump turbines Limitations Goal Lab results Pressure measurements LDV measurements Data processing Experimental results v

8 3 Theory Introduction to CFD Turbulence modeling The k ɛ model Wall function Rotor Stator Interaction CFD Method Frame change Mesh and time step Modeling Basic model Extended inlet model Stay vane model Turbo Grid mesh model Results Basic model Extended inlet model Stay vanes model Turbo grid mesh model Discussion 49 7 Conclusions 51 Bibliography 54

9 CHAPTER 1. INTRODUCTION Chapter 1 Introduction 1.1 Background In the climate meeting in Kyoto, 1997, a greater majority of the countries of the world agreed to reduce the emissions of green house gasses, primarily CO 2, in order to stop the global warming. According to the Kyoto protocol, the industrial nations must reduce their CO 2 emissions at an average of 5% by At the same time the EU countries have agreed on an eight percent average reduction of the CO 2 emission level compared to the Kyoto protocol base year of 1990 [14]. These, already high goals compared to the current state in 2007, must be followed by even more ambitious levels in order to meet the common goal of the EU countries to limit the global temperature rise to two degrees. In order to meet these goals, large scale investments in renewable energy sources are crucial. In Europe and North America the large scale development of hydro power is almost completed, leaving the demand for other renewables, for instance wind power, high. Targeted political efforts and increased prices in energy have contributed to making renewable energy investments more and more appealing. During the past ten years the wind power industry has expe- 1

10 1.1. BACKGROUND CHAPTER 1. INTRODUCTION rienced ever increasing growth with the top notation in 2006 when the wind energy sector grew by 32%, corresponding to an 18 billion Euro investment [5]. In 2006 the installed capacity world wide amounted to 74 GW out of which 48 GW were operated in Europe [5]. As a consequence of the large investments in renewable energy sources, were the power supplied to the grid is known to fluctuate, parallel investments are needed in storage capacity to secure the power grid stability. The only large scale technology available for this application is storage pumps or reversible pump turbines [5]. These units can use surplus energy, when the load on the grid is low, to fill up an upper reservoir that can later be used to run the pump backwards as a turbine to generate energy. In the last couple of years an increase in refurbishment work as well as construction of new pumped storage plants has been initiated around the world, for instance the 2.8 GW Kannagawa plant taken into commission in 2005 in Japan or the 1 GW Linthal - Limmern project in Switzerland, due Not only does the future point towards increased demand in pumped storage plants, the trend is also that plant owners need more flexible units that can operate with an increased frequency of starts and stops, preferably over a longer lifetime [5]. As the focus of demand has turned to pumped storage plants, manufacturers aim to provide increased reliability and operating range through intensified R&D. The Hydrodyna project is a collaboration between turbine manufacturers and Ecole Polytechnique Federale de Lausanne, Switzerland, with the aim to break through a technology gap in the understanding of dynamic behavior of pump turbines. Potential benefits of the project range from methodologies to assess life spans of pump turbine units to knowledge that could allow for development of technologies that can increase operating range and the expected life span of machines [5]. One part of the Hydrodyna project is to validate simulations on rotor stator interaction with measurements on the Hydrodyna model, this project work is a part of that work. When measuring the signals from pressure sensors, two types of signal categories can be seen, deterministic signals and random signals. Deterministic signals can be described by mathematical expressions and can be either periodic and non periodic. The random signals can only be described by probability and statistical averages [6]. In the case of complex flow in a water turbine were both these types of signals are present in different shapes and places, signal analysis techniques are necessary to sort the valuable information from the noise. 2

11 CHAPTER 1. INTRODUCTION 1.2. HYDRO POWER When the runner rotates in relation to the stator the flow in the distributor and runner blade channels is periodically perturbed by each passing of the runner blades. Experimental results has shown that such pressure disturbances are created after each time that the leading edge of a runner blade passes the trailing edge of a guide vane [6]. The pressure disturbances propagates both upstream as well as downstream of the point of origin between the rotor and the stator [11]. The periodic forces generated by such pressure fluctuations are considerable, especially in high head machines were the guide and stay vanes have to be made thicker. The thicker geometries in the water passage, the larger wakes will be produced behind the object trailing edge. Due to the periodic forces acting on the runner, areas of high stress could in worst case lead to damage from fatigue Previous work Nennemann and Vu has shown in previous work at GE in Lachine that it is possible to reproduce pressure fluctuations on the Hydrodyna rotor blade for a part load operation point. In their work X-mesh generated grids were used for guide vanes and runner. Nennemann and Vu also initiated an examination of the operation point that is handled in this work. A more extensive investigation on the same operation point that is dealt with in this work has been done at EPFL, Lausanne, with paper presented at IAHR in Yokohama 2006 [15]. In this work the geometry from spiral casing to draft tube was included. 1.2 Hydro power Hydro power is one of the oldest forms of extracting energy from nature for industrial applications. The general principle is to convert the potential energy of water to electrical energy. In order to do so the water needs to exert its power on a turbine to create rotational, or mechanical, energy. The mechanical energy is then transformed into electricity in a generator mounted on the same shaft as the turbine, see figure 1.1 for a schematic representation of a hydro power plant. The magnitude of the electricity production is determined by the distance between the head water level and the tail water level and the 3

12 1.2. HYDRO POWER CHAPTER 1. INTRODUCTION Figure 1.1: Typical hydro power plant, figure from [8]. flow rate through the power plant, see equation 1.1. P = ηρgqh (1.1) Where P is the power in Watts, η is the efficiency of the power plant, ρ is the density of water, g is the gravitation constant, Q is the flow rate and H is the head of the power plant Pump turbines Already in the first half of the 20th century pumped storage schemes were installed with separate pumps, mounted on the same shaft as the turbine [1]. Along with the development of reversible pump turbines, where turbine and pump mode were both run on the same machine, and increased electricity use the number of pumped storages rose during the second half of the century. In modern time practically all installed units have been of reversible pump turbine type [1]. Comparing the reversible pump turbine unit to a separate pump and turbine, running on the same shaft, the main advantage of the former is reduced construction cost. Not only in constructing turbine and pump but also in reduced needs for excavation. The main disadvantage is a slightly lower turbine efficiency (by one to two percent) and longer pump starting times [1]. An important reason why reversible pump turbines have been attractive for power producers is that the machine is a simple construction, not much different from a normal Francis turbine. This ensures mechanical robustness with high reliability, a quality which is highly valued. 4

13 CHAPTER 1. INTRODUCTION 1.2. HYDRO POWER In difference to a Francis turbine, which only operates in one mode, a compromise is needed to account also for pumping aspects. In fact the pumping mode generally dominates the design criterions due to its limits for cavitation and stability when pumping under high heads [1]. The most striking differences between a pump turbine compared to a normal Francis unit are: Stay and guide vanes being shaped for two way flow, with rounded edges for the pumping mode leading edges. Rounded runner blade pump mode inlet edges. Fewer number of runner blades. A Francis turbine would normally be equipped with blades, while a reversible pump turbine would have 6-9 blades [1]. Increased outer runner diameter in relation to the band diameter. Comparing the reversible pump turbine to a conventional centrifugal pump, the largest difference is the pump turbine s diffuser with adjustable guide vanes. The guide vanes can be made good use of not only in turbine mode but also in pump mode and during change over modes [1]. Specific speed Specific speed is a measure of a machine s hydraulic characteristics and is valid for pumping and turbining. There are several definitions of the specific speed, one being described in 1.2 n q = n Q H 3/4 (1.2) were n q is the specific speed, n is the rotational speed in rpm, Q is the discharge in m 3 /s and H is the head. Two machines with the same specific speed will have similar appearance regardless of their size. Given that the construction of a machine is strong enough, the only hydraulic limitation for choosing a high specific speed is cavitation. High specific speed machines are characterized by small dimensions, high speed and thus lower prize. As the cavitation factor will decrease together with decreasing specific speed, a high head turbine generally demands lower specific speed. High head machines also require deeper 5

14 submergence (for more information on this topic read [9]), this is usually not an economic issue since most pumped storage units are under ground and the extra cost of low setting is negligible [1]. More advantages with a high specific speed machine involve higher efficiency in pumping mode and better access to the runner for maintenance and construction. All this being said, different projects have their own requirements resulting in different choice of specific speed. Typical specific speeds for reversible pump turbines are within the range for heads between meters [1]. 1.3 Limitations In this work the main objective has been to compare simulated results with pressure measurements on the Hydrodyna model runner. Due to limitations in time, no full grid sensitivity analysis has been undertaken. 1.4 Goal The goal for this project is to make transient simulations and reach computational results on the time dependent turbulent flow through the Hydrodyna model turbine. The accuracy of the method will then be valued by comparisons to experimental measurements. The simulations are based on meshes done in grid generators available at Rainpower and are meant to help gaining experience and to develop the simulation methodology.

15 CHAPTER 2. LAB RESULTS Chapter 2 Lab results The experimental measurements have previously been made on the Hydrodyna test rig at EPFL in Lausanne, Switzerland. The independent test facility has a number of universal test rigs that can be set up to test different types of turbine models, such as pump turbines or turbines with axial or radial inflow. The Hydrodyna model has been thoroughly checked, before the experimental measurements were started, for anomalies with regard to uniformity. The testing was done by creating a digital map over selected surfaces that later were used to create the CAD model. The conclusion from the geometry tests was that the inconsistencies were well under the levels that were needed for the purpose of the project. New technology has made it possible to perform more complex model tests by means of advanced instrumentation and data processing. Onboard measurements in hydraulic runners is an example of such tests. The measurements that have been used in this work are pressure and 2D Laser Doppler Velocimetry measurements. 7

16 2.1. PRESSURE MEASUREMENTS CHAPTER 2. LAB RESULTS 2.1 Pressure measurements The method that was used for the measurements in the Hydrodyna project has been developed at EPFL and is generally used for a variety of machines such as pump turbines. The method rely on miniature piezo resistive pressure transducers to collect pressure data. The transducers are mounted inside of model turbine blades as thin as two millimeters, see figure 2.1, without altering the shape of the surface. The transducer chamber is connected to the flow channel through a one millimeter thin pressure tap that is filled with a plastic compound. The compound has similar density to water and provides good transmission of pressure at the same time as it prevents air from being captured in the transducer chambers. Figure 2.1: The pressure transducer chambers as seen from the inside of the blade, before mounting. Pressure transducer 1-6 registering the pressure side and transducers 7-11 registering the suction side of the blade. Figure from [12]. The signals are gathered by the onboard electronic in the runner crown that consists of 32 preamplifiers, 32 anti aliasing programmable filters, eight acquisition boards, each with four channel inputs. The equipment enables monitoring of 32 channels in the rotating frame of reference. The maximum sampling frequency is 20 khz and the memory can handle up to 64 k-samples per channel [13]. The data is transferred 8

17 CHAPTER 2. LAB RESULTS 2.2. LDV MEASUREMENTS from the runner to a host computer through a slip ring. The transducers are calibrated statically by mounting the runner in a rotating cylinder filled with water. Transient calibration has been done with help of a spark generated bubble and by comparing the outputs to results generated from a different type of transducer. Pressure measurements generated by the method described above yields results with errors below bar absolute [5]. The measurements that have been used in this work have been sampled over 750 runner rotations and later decimated to 720 data samples over one rotation, or one pressure reading for every 0.5 degree phase. That way a lot of noise and non harmonic signals have been filtered out. 2.2 LDV measurements Laser Doppler Velocimetry measurements are obtained by detecting the reflected light from a test particle each time that it passes through a laser interference fringe. For this method to be successful, the laser beams need to be monochromatic and coherent. This is often done by splitting a laser beam, from a single laser source, in two. The laser light is then aligned so that the two beams intersect at their laser focal point (their waist). Through interference a number of straight fringes will appear, with known spacing. By letting a test particle move through the fringes it will reflect the laser light each time it moves through a fringe, thereby making it possible to measure the velocity perpendicular to the fringes. In the measurements done for the Hydrodyna project, hollow glass particles of approximately 10 micrometer diameter were used as tracers. For the Hydrodyna measurements, testing has been done for two velocities; radial and tangential. Optical access was given to the laser by mounting a transparent window at the bottom surface of the stator. Because of some limitations of experimental facilities, non dimensional numbers Φ and Ψ were respected instead of exact values of rotational velocity and flow rate. The measured velocities were therefor scaled using normalized velocity values according to 2.1 and 2.2. ( ) ( ) Cu Cu = C u1 r ω 1 r ω 2 C u 2 = 980 = (2.1) 720 9

18 2.3. DATA PROCESSING CHAPTER 2. LAB RESULTS ( ) ( ) Cr Cu = C r1 = = (2.2) Q/S 1 Q/S 2 C r Were C r and C u are the radial and tangential velocities, respectively. The measurements have been made for a cylindrical matrix, with constant r- and θ-values for eight rows and 27 columns in the guide vane region, see figure 2.2. Figure 2.2: The LDV measurement points after being imported as monitor points to CFX. The LDV data was phase averaged so that the result was given in 720 data readings for one rotation, with 0.5 rotation for each new reading. The individual phase readings were then presented in text files, each containing the velocity readings for all the measurement points and their location. 2.3 Data processing The post processing of measurement data has in my work been done in MATLAB 6.1 release 12.1 and covers numerical operations, automation of writing output control to CCL, rotation of point sets, Fourier transforms, mean values and graphing. The quality of MATLAB s discrete Fourier transform was examined with a simple sinus signal hidden in numerical noise. The quality test gave a good idea of the importance of different input variables, such as number of included samples and interval length compared to expected 10

19 CHAPTER 2. LAB RESULTS 2.3. DATA PROCESSING frequencies. The number of samples in relation to the sampling frequency affects the resolution of the over all results. As can be seen in figure 2.3, the resolution in this case was rather low with 720 samples and a sampling frequency of If these aspects are regarded, the MATLAB Fast Fourier Transform is successful with margins of error small enough to neglect. While the the Fourier transform is able to reproduce all frequencies from zero to half the sampling frequency, the Fourier transforms that have been presented in this work are all one sided and limited to only include the physically relevant interval of frequencies, as is the case in the presented example below. &!! $!! 2, /!1!!$!!!&!!!!"!#!"!$!"!%!"!&!"!'!"!(!"!) *+,-./01 #$! #!! 2, /!1 8! (! &! $!!! '! #!! #'! $!! $'! %!! %'! &!! 9:-;6-<=>./?@1 Figure 2.3: Time signal and Fourier transform. Amplitude should be 100 and the frequency random(:,1)=100*randn(720,1); %To create random vector 3 that doesn t regenerate. 4 5 vector = 1:720; 6 sampfreq = 1/(60/980/720) %same sampling frequency 7 as in real case 11

20 2.3. DATA PROCESSING CHAPTER 2. LAB RESULTS 8 9 hidden_frequency = signal = 100*sin(vector *2*pi*hidden_frequency/sampfreq) 12 +random(1:720,1); %The signal with amplitude 100 and frequency hidden in random noise Y=fft(signal); 18 N=length(Y(:,1)); Y(1)=[]; Yabs(:,1)=abs(Y(1:N/2,1))/(N/2); nyquist = 1/2; freq3=sampfreq*nyquist*(1:n/2)/(n/2); figure(1) 30 subplot(2,1,1) 31 plot(vector *(60/980/720),signal),grid on 32 xlabel( Time [s] ) 33 ylabel( Amplitude [-] ) 34 subplot(2,1,2) 35 plot(freq3,yabs),axis([ ]),grid on 36 xlabel( Frequency [Hz] ) 37 ylabel( Amplitude [-] ) Similar principles have been regarded in the work with the LDV measurements, but in this case only one dominant frequency was sought and the amplitude for that was interpolated and plotted on a surface spanned by the LDV measurement points. 12

21 CHAPTER 2. LAB RESULTS 2.4. EXPERIMENTAL RESULTS 2.4 Experimental results The rotor stator interaction was measured as pressure fluctuations on the pressure and suction side of one rotor blade as shown in figure 2.1. The result has been presented in figure 2.4. The pressure values have been normalized with respect to the runner peripheral velocity, see equation 2.3. C p = p p 1/2ρUper 2 (2.3) The amplitude was greatest at the leading edge of the blade and faded further into the runner blade channel. The guide vane passing frequency of f r n gv = = Hz can be seen with two harmonies. In figure 2.5, the mean values of the phase averaged LDV measurements have been plotted together with a fourier transform. The velocity was perturbed at the runner blade frequency, 147 Hz, in the guide vane channel, especially after the guide vane trailing edge, where the wake was forced to shift position at each passing of the runner blade. While the velocity fluctuations travel upstream of their place of origin, they fade out rather fast and have almost disappeared in level with the guide vane leading edges. 13

22 2.4. EXPERIMENTAL RESULTS CHAPTER 2. LAB RESULTS (a) Figure a. (b) Figure b. Figure 2.4: Figure a: (Upper) pressure fluctuations on the pressure side, leading edge. (Lower) waterfall plot of amplitudes on the pressure side. Figure b: (Upper) pressure fluctuations on the suction side, leading edge. (Lower) waterfall plot of amplitudes on the suction side. 14

23 CHAPTER 2. LAB RESULTS 2.4. EXPERIMENTAL RESULTS (a) Figure a. (b) Figure b. Figure 2.5: Figure a: Experimental mean velocity from Laser Doppler Velocimetry measurements. Figure b: Fourier transform of velocity fluctuations, the amplitude of the runner blade passing frequency of 147 Hz is ploted for each spatial measurement point. 15

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25 CHAPTER 3. THEORY Chapter 3 Theory 3.1 Introduction to CFD Computational Fluid Dynamics is a tool used to numerically analyze the flow of fluids. The equations that are solved are Navier Stokes equations for x,y and z momentum and the continuity equation. These equations are partial differential equations and in order to solve them numerically a discretization is of essence. An example of such discretization method is the finite volume method (FVM), which allows for the partial differential equations to be solved in discrete space and time. As of today modern CFD manages to solve laminar flows with ease and good reliability provided that the discretization of space, or the grid, is well resolved and the boundary conditions are properly specified. Turbulent flow on the other hand is more difficult to simulate since the finer features of the flow always are unsteady and three-dimensionally random. Turbulent eddies in all possible orientations will appear in turbulent flow. Depending on the choice of method used to model the flow, more or less computational power and time will be needed. The technique called Direct Numerical Simulation is the most difficult way to solve a turbulent CFD case. This method attempts to resolve the turbulent motion in all scales of the flow. The difference in length and time scale between the large and the small scale eddies can vary several orders of magnitude [4], furthermore this difference increases with 17

26 3.1. INTRODUCTION TO CFD CHAPTER 3. THEORY increasing Reynolds number. The method requires a lot of computational power and time and is therefor not feasible for practical engineering problems with high Reynolds number as is the case with a hydro power turbine. The method will likely not be applicable for yet many decades despite the fast improvements of computer power. The intermediate method is Large Eddy Simulation. This technique only resolves the large scale eddies while the small scale eddies are modeled to behave in a deterministic manner. The small eddies are assumed to be isotropic and behave in the same way, independent of the turbulent flow field and coordinate system orientation. While LES requires less computer resources than DNS it is still impractical to apply on engineering cases [4] Turbulence modeling Today the method of choice for engineering CFD is to model all the unsteady turbulent eddies with a turbulence model. Instead of resolving the unsteady features of the eddies, mathematical models are adopted to reproduce the enhanced mixing and diffusion originally caused by the eddies. By expressing the variables in Navier Stokes equation as sums of a statistically averaged component and a fluctuation component, the Reynolds Average Navier Stokes equation can be formed. The averaging of any variable can be written according to equation 3.1, φ(x i, t) = φ(x i ) + φ (x i, t) (3.1) where over bar denotes a statistical average and prime denotes a fluctuation component. Using Reynolds Averaging, the continuity and momentum equations can be written for incompressible flows without body forces according to equations 3.2 and 3.3, [7]. δ(ρu i ) δt δ(ρu i ) δx i = 0 (3.2) + δ ) (ρu i u j + ρu i δx u j j where τ ij are the mean viscous stress tensor components. ( δui τ ij = µ + δu ) j δx j δx i 18 = δp δx i + δτ ij δx j (3.3) (3.4)

27 CHAPTER 3. THEORY 3.1. INTRODUCTION TO CFD Deriving models for the Reynolds stresses, ρu i u j, is problematic since equation 3.3 contains more unknowns than equations. In order to have closure the unknowns are approximated by introducing a turbulence model The k ɛ model The k ɛ equation is a two equation turbulence model in the sense that it adds two additional transport equations that are solved at the same time as the equation of mass and linear momentum. The ɛ represents the turbulent dissipation rate while the k represents the turbulent kinetic energy. The turbulence eddy dissipation is calculated according to equation 3.5 ɛ = C µk 2 ν (µ T /µ) (3.5) where: C µ is a non dimensional constant k is the turbulent kinetic energy ν is the kinematic viscosity µ T /µ is the eddy viscosity The turbulence velocity scale is found from the solution of the transport equation after first calculating the turbulent kinetic energy. The turbulent length scale in a flow field is estimated from the turbulent kinetic energy and its dissipation rate. The eddy viscosity is modeled according to equation 3.6, this is what has given the name to the turbulence model, [2]. ν T = C µ k 2 ɛ (3.6) To capture the profiles of the turbulent kinetic energy and its dissipation it is suggested in [7] that a finer grid could be used for the turbulence quantities compared to the mean flow. The risk with using the same sized grid for both quantities is that the resolution could be insufficient to produce a stable solution for the turbulent quantities, possibly leading to negative values for these quantities in the region. Most of the time this issue is solved by blending the second order difference scheme with a more simple, upwind, difference scheme (first order) for the convective terms in local problematic areas. 19

28 3.1. INTRODUCTION TO CFD CHAPTER 3. THEORY Usually the advection scheme is a series expansion, in order to get higher accuracy more terms have to be included. In practical applications as CFD the number of terms that can be included is limited. As mentioned earlier the number of included terms is in the interval of one to two terms from the expansion. The additional accuracy that could be won by using many terms from the expansion would be lost in computer round off errors. In Ansys CFX 5, the importance of the second order terms can be regulated by setting a coefficient value, the specified blend factor, ranging between zero to one. The value of the blend factor can either be set manually for the entire simulation domain or be set to vary automatically so that only problematic areas, were the variable gradients are high, are solved under lower influence of the second order terms. The k ɛ model is numerically stable and gives accurate results for many cases without being too demanding on computational resources, this is why the model has become the most popular within industrial applications [2] Wall function In difference to laminar flow the instantaneous velocity profile of a turbulent flow, close to a stationary wall, varies with time. However, if the boundary layer profile is averaged over time it will reveal a much steeper increase of the average velocity close to the wall, see figure 3.1, this is due the the mixing effect. It has been shown in theory as well as experimentally that the boundary layer consists of two different layers, the viscous sub layer closest to the wall, were the viscosity plays a dominant role, and the logarithmic layer further from the wall, were the mixing turbulence has the greatest influence. In a thin region between the two layers, viscosity and turbulent mixing have equal influence, this region is called the buffer layer. When solving the governing model equations, boundary conditions are needed. At walls this poses a problem due to the rapid change of the dependent variables, such as velocity and wall shear stress. If the equations are solved directly right up the wall, the no slip condition would be used. However, the kinetic energy, k, is never zero in a turbulent flow, instead the conditions 3.7 or 3.8 can be used [7]. ( 2 ) k ɛ = ν y 2 wall 20 (3.7)

29 CHAPTER 3. THEORY 3.1. INTRODUCTION TO CFD Figure 3.1: Boundary layer for a flow before and after transition from laminar to turbulent flow, figure from [3]. ( ) 2 k 1/2 ɛ = 2ν (3.8) y wall At flows with a high Reynolds number, the viscous sub layer is very thin and it can be difficult to resolve the grid close to the wall with a sufficient number of nodes, this can be avoided through use of a wall function based on the logarithmic velocity profile. The near wall tangential velocity depends on the wall shear stress, τ ω, through a logarithmic relation. The method with wall functions uses empirical formulas to describe the velocity field in the boundary layer. In this way, near wall boundary conditions, such as the wall shear stress, are provided for the mean flow and for the transport equations at the near wall node of the mesh. The near wall velocity is given by relation 3.9 [2] where y + is given by 3.10, u τ by 3.11 u + = U t u τ = 1 κ ln(y+ ) + C (3.9) y + = ρ yu τ µ (3.10) u τ = ( ) 1/2 τω, (3.11) ρ u + is the near wall dimensionless velocity, U t is the known velocity tangent to the wall at distance y (in the case of scalable wall function 21

30 3.2. ROTOR STATOR INTERACTION CHAPTER 3. THEORY this value is set to y = n/4), y + is the dimensionless distance from the wall, τ ω is the wall shear stress, κ is the von Karman constant and C is an empirical constant related to the thickness of the viscous sub layer. The latter constants typically range from and 5-5-5, respectively [4]. Unfortunately, equation 3.9 does not work close to the wall, were ln(0) is undefined, another weakness is that it does not correspond well to experimental measurements close to the boundary layer edge [4]. In CFX this is solved by assuming that the flow is in local equilibrium with the same level of production and dissipation of turbulence. If that is true the alternative velocity scale, u can be used instead of u +, according to equation 3.12 [2]. u = C 1/4 µ k 1/2 (3.12) with k and C µ as described in section With this expression along with 3.9, an expression can be derived that connects the velocity at the first grid point with the wall shear stress, see equation 3.13 τ ω = ρu τ u = ρc 1/4 µ κ k U t ln y E (3.13) where E = e κc. This approach is valid under the condition that the first node lies within the logarithmic layer at 30 < y + < 100. A drawback in using this method is that the solution is depending on the location of the first node from the wall and thereby is sensitive to the near wall meshing. By using scalable wall functions, method developed in 1998 by Grotjans & Menter, this problem can be overcome and the model can be applied on arbitrary fine meshes, i.e. the lower limit in the recommended y + interval may be ignored. [10] 3.2 Rotor Stator Interaction Modern demands on hydro power turbine design pushes manufacturers to produce turbines with wider ranges of operation at lower costs. To achieve these goals it becomes necessary to understand the dynamic fluctuations around the turbine and make design rules based on this understanding. Important phenomena include the draft tube rope at part load and rotor stator interaction between wicket gates and runner blades. In the case of pump turbines that operate under high heads, the rotor stator interaction becomes more significant due to higher water velocities 22

31 CHAPTER 3. THEORY 3.2. ROTOR STATOR INTERACTION and less radial gap in the guide vane passages [11]. Steady state CFD has been available as a tool for turbine designers for a long time, but only recently, improved computer technology and CFD software has made it possible to simulate unsteady conditions. This has lead to increased understanding of the significance of the unsteady flow field. Three different effects contribute to the unsteady load on a runner blade [11]: Runner channels passing through varying static inflow pressures; if there is a phase difference between two adjacent channels the blade will be subjected to an unsteady load, see equation ( ) φ = 360 ngv 1 (3.14) n rb where n gv is the number of guide vanes and n rb is the number of runner blades. The incoming water hitting the blade with different angles of incidence over the guide vane passages. The varying magnitude of the water velocity passing over the blade. Pressure fluctuations in the water passages occur at different dominant frequencies depending on where a sensor is located. The regions upstream of the runner inside the stator feel the pressure fluctuations generated by the runner blade passing frequency, see equation 3.15, while the regions downstream of the stator, in the rotating domain inside the runner, primarily are subjected to pressure fluctuations at the guide vane passing frequency, equation As previously mentioned in section 1.1 pressure fluctuations generated in the area between the rotor and the stator will propagate in all directions, upstream as well as downstream. f rb = n rb f r (3.15) f gv = n gv f r (3.16) Where: f rb is the runner blade passing frequency n rb is the number of runner blades f r is the rotation frequency of the runner f gv is the guide vane passing frequency 23

32 3.3. CFD METHOD CHAPTER 3. THEORY n gv is the number of guide vanes. Another effect of pressure fluctuations on the pressure and suction side of a given runner blade is that the blade can experience different magnitudes of torque, even if the average torque acting on the shaft is constant. 3.3 CFD Method When simulating a turbine with, or close to, geometrical symmetry it could be interesting to set up a run with a fraction of the full domain. A fractional setup would reduce simulation time considerably but would still reproduce dynamic torque of individual runner blades as compared to a full geometry simulation [11]. The computationally simplest setup would be a single guide vane and runner blade channel, but since any hydro turbine consist of an unequal number of runner blades to guide vanes, the pitch ratio, or the fractional change in area between two domain interface sides, would be at unacceptable levels. It has been shown that as long as the pitch ratio is in the span , a simulation will give acceptable values on runner blade torque [11]. For the Hydrodyna pump turbine, with a 20:9 guide vane to runner blade ratio, the only possible setup that satisfies the pitch ratio criterion is a full geometry setup. A full geometry set up also eliminates possible sources of errors associated with a fractional set up. In Francis turbine rotor stator interaction, viscous effects are secondary, why the k ɛ turbulence model is sufficient for prediction of the pressure fluctuations, [11] Frame change In a hydro power turbine the flow passes through different frames of reference, stationary to rotating and back to stationary in the draft tube. CFX has different ways to transfer the flow from one interface side into a different frame of reference on the other side. For a transient simulation the domain interface setting is Transient Rotor Stator. This setting allows the runner domain to rotate relative to the stator and accounts for interaction effects between components that are in relative motion to each other. In order to start a transient simulation, initial values needs to be specified through an initial guess or through a prediction from a Frozen Rotor simulation. The Frozen Rotor frame change model 24

33 CHAPTER 3. THEORY 3.3. CFD METHOD will produce results that contain most of the flow features and is a good initial guess since it accounts for the relative position of the stator and runner meshes at the starting point. This method allows the transient simulation to converge in the fewest number of time steps, [2]. A Frozen Rotor simulation setting changes the frame of reference between rotor and stator, but allows the solution to converge without changing the relative position of the components. In the same way as the transient rotor stator frame change model, this setting distributes the flow from the up stream nodes to the down stream nodes on equal span position. Because of this it is important to have similar meshes on both sides of a domain interface with change in frame of reference Mesh and time step In [11] it has been shown that the influence of time step is just as important as the mesh resolution. The result show that a plateau for the dynamic torque was reached at 0.2 phase difference between two consecutive time steps (see figure 3.2 from [11]). While the recommended y + -values for a k ɛ simulation is below 100, it has been shown that the influence of the y + -values is of less importance for rotor stator interaction simulations where viscosity effects are proportionally small [11]. Figure 3.2: Figure describing the method of determining influence of mesh size and time steps. The simulation domain above was different from the Hydrodyna geometry, but the mesh has been generated with similar method in X-mesh. 25

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35 CHAPTER 4. MODELING Chapter 4 Modeling The simulated operation point corresponds to full load turbine mode, 43% over the flow at best efficiency point with 33 guide vane opening. At this operation point, the guide vanes were at maximum opening and the distance between the guide vane trailing edge and the runner blade leading edge was at a minimum. This setup should produce strong rotor stator interaction. The inlet boundary condition was set to mass flow, kg/s and the outlet condition to average static pressure. The runner rotational speed was 980 rpm. The simulations were run with the k ɛ turbulence model, scalable wall functions, high resolution advection scheme and with second order Backward Euler as discretisation algorithm for the transient term. Initial convergence has been reached with a frozen rotor simulation prior to starting the transient simulations. The solver was Ansys CFX 5.5, running on the 12 CPU cluster at Rainpower, Norway. For setup and post processing, a linux computer has been available with extra memory capacity to handle large models. A time step corresponding to a 0.2 rotation has been used in all simulations with a maximum of three coefficient loop iterations for each time step. The target residual was set well below the actual levels, at approximately , so that the maximum number of iterations per time step always were filled. For the last two simulations including draft tube and stay vanes this resulted in about five to six minutes simulation time per time step, running on six dual core CPUs. 27

36 4.1. BASIC MODEL CHAPTER 4. MODELING 4.1 Basic model The first model that was simulated, see figure 4.1, included guide vanes and runner, where the runner domain extended only a fraction of the outlet diameter into the suction cone. Inlet flow angle was set to 20.7, the same as simulations previously run at GE in Lachine. The meshes that were used for the basic model were supplied by B. Nennemann and T. Vu at GE Hydro, Lachine, Canada. The meshes for the guide vane and runner domain were hexa mesh of H-type and have been created in GE s in house software for structured meshing, X-mesh, see figure 4.2. The mesh domain consist of 2.35 million nodes with mean y + -values between The mesh has been created in accordance to GE Hydro s validated methods for meshing. Figure 4.1: Simulation model with X-mesh generated grids. Figure 4.2: X-mesh grid for the guide vane and runner domain. 28

37 CHAPTER 4. MODELING 4.2. EXTENDED INLET MODEL 4.2 Extended inlet model In an attempt to enable a calmer inflow to the guide vanes an extension mesh was created in ICEM CFD, see figure 4.3. The extension was copied in CFX to fill a 360 ring around the guide vane domain. The domain interface between the extension mesh and the guide vanes was set to fluid fluid coupling without frame change model. Unlike the stay vane geometry, the extension mesh has plane parallel crown and band. The simulation settings were the same as in the case with the basic model. Figure 4.3: Simple extension to add outside of guide vane mesh to move the inlet boundary condition further from the guide vane leading edges. 29

38 4.3. STAY VANE MODEL CHAPTER 4. MODELING 4.3 Stay vane model After concluding that the inlet condition played an important role for the rotor stator interaction, it was motivated to add stay vane grid to the original computational domain. This way a more realistic inflow would be provided to the guide vane domain. Since the X-mesh generated guide vane grid had had its inlet extended into the theoretical stay vane domain, it would not be possible to add a stay vane domain around it without having overlapping nodes. This called for a completely new stator mesh including a new guide vane mesh. The new meshes were created in Ansys Turbo Grid In order to account for the non plane parallel stay vanes, new turbo grid curves were exported from Unigraphics NX 4. The stay vane mesh consisted of 50 k nodes and the guide vane mesh of 120 k nodes per blade channel. The reason why the guide vane grid was constructed with a higher mesh density was to provide for a finer transition from stationary to rotational domain and to catch fine features of the velocity field, see figure 4.5. Since the draft tube was meshed relatively coarse, the number of nodes in the entire computational domain was held down to 4.7 million nodes. The simulation was running on six CPUs and needed approximately minutes per time step. The stator meshes had minimum angles in the range and produced mean y + values around By adding draft tube to the computational domain it was avoided that CFX put walls on the outlet to prevent back flow, this however gave rise to a small numeric instability that can be seen in the time domain plot of pressure fluctuations. Any non harmonic signals were then filtered out by MATLAB discrete fourier transform. The inlet flow angle to the stay vanes was set to 17. This angle was approximately measured in the spiral casing from a CAD model. 30

39 CHAPTER 4. MODELING 4.3. STAY VANE MODEL Figure 4.4: Simulation domain: Stay vane model. Figure 4.5: New stay vane and guide vane meshes used in the stay vane model. 31

40 4.4. TURBO GRID MESH MODEL CHAPTER 4. MODELING 4.4 Turbo Grid mesh model The last simulation model was a test to see whether the interface between stator and rotor domain would be better matched if the meshes on both sides of the domain interface were created with the same mesh generator. It was also interesting as a quality comparison between different mesh generators. A runner blade mesh was created along with two new stator meshes with emphasis on creating matching node to node conditions between fluid fluid interfaces. The minimum angle of the runner blade grid was approximately 20 and the mean y + values ranged from on the blades through out the entire computational domain. The runner blade grid consisted of 160k nodes, the guide vane grid of 70k nodes and the stay vane grid of 45k nodes. The mesh can be sen up close in figure 4.6. The main difference to the X-mesh generated runner grid was larger minimum angles and the absence of a region with elements normal to the domain interface, were the node density is higher, compare with figure 4.2. The draft tube mesh was the same as the one used in the previous stay vane model. The inlet boundary condition was similarly to previous stay vane models set to 17 inflow angle as measured from the CAD model. Figure 4.6: Complete Turbo Grid generated computational domain, figure describing stay vanes, guide vanes and domain interface to the runner. 32

41 CHAPTER 5. RESULTS Chapter 5 Results 5.1 Basic model The results presented in figure 5.1 show that the pressure fluctuations over predicted the results from laboratory measurements by several times. The solution also had walls placed at a fraction of the outlet area under the runner cone. Tests with adding draft tube did however not show significant changes in pressure fluctuations. Instead the results from a comparison to LDV measurements, figure 5.2, showed a large variation of the velocity distribution in the guide vane channel. Therefore it was concluded that in this simulation domain the inlet boundary was set too close to the guide vanes. This lead to non physical, high velocities around the guide vane leading edges that in turn caused the high pressure fluctuations in the runner domain. 33

42 5.1. BASIC MODEL CHAPTER 5. RESULTS (a) Figure a. (b) Figure b. Figure 5.1: Figure a: (Upper) simulated pressure fluctuations on the pressure side, leading edge. (Lower) waterfall plot of amplitudes on the pressure side. Figure b: (Upper) simulated pressure fluctuations on the suction side, leading edge. (Lower) waterfall plot of amplitudes on the suction side. 34

43 CHAPTER 5. RESULTS 5.1. BASIC MODEL (a) Figure a. (b) Figure b. Figure 5.2: Figure a: Simulated mean velocity in one guide vane channel. Figure b: Difference between simulated mean velocity and experimental mean velocity. 35

44 5.2. EXTENDED INLET MODEL CHAPTER 5. RESULTS 5.2 Extended inlet model The result from the pressure measurement can be seen in figure 5.3. The result show substantially lessened values of pressure fluctuations due to rotor stator interaction compared to previous simulation. The velocity field in figure 5.4 revealed that the extension allowed the flow to enter the guide vane channels without being squeezed between its leading edge and the inlet boundary condition. There was still a difference between measured and simulated velocity distribution, which could help explain some of the difference to experimental pressure amplitudes. Unexpectedly, the readings for the amplitude at the guide vane passing frequency, Hz, were almost cancelled out on pressure side sensor one, while the second harmony was strong. 36

45 CHAPTER 5. RESULTS 5.2. EXTENDED INLET MODEL (a) Figure a. (b) Figure b. Figure 5.3: Figure a: (Upper) simulated pressure fluctuations on the pressure side, leading edge. (Lower) waterfall plot of amplitudes on the pressure side. Figure b: (Upper) simulated pressure fluctuations on the suction side, leading edge. (Lower) waterfall plot of amplitudes on the suction side. 37

46 5.2. EXTENDED INLET MODEL CHAPTER 5. RESULTS (a) Figure a. (b) Figure b. Figure 5.4: Figure a: Simulated mean velocity in one guide vane channel. Figure b: Difference between simulated mean velocity and experimental mean velocity. 38

47 CHAPTER 5. RESULTS 5.3. STAY VANES MODEL 5.3 Stay vanes model In figure 5.5, the pressure distribution in a guide vane channel has been plotted together with the pressure readings from two sensors. Sensor 58 was located in the beginning of the guide vane channel and sensor 69 closer to the runner as can be seen in the figure. The figure show how the runner blades were aligned to the guide vanes at the moment when the pressure was low or when the channel was open. The pressure peaks at the two different measurement points have different phase compared to each other, where point 69 peaked during a short window as the leading edge of the runner blade passed by on close distance. Measurement point 58 peaked during a longer time interval describing the pressure situation in the guide vane channel more generally. 50 time steps after (10 degree rotation) the situation described in figure 5.5 (a), the channel will be closed, at this point the time line in figure (b) will have reached time step 200. In figure 5.7 two spikes can be seen after the trailing edges of the guide vanes due to slight differences in the position of the wakes between experimental and simulated results. The boundary layers are otherwise more evenly reproduced, without spikes as in figure 5.2. Figure 5.8 show the pressure amplitudes at the guide vane passing frequency compared to experimental results. The comparison was good for all pressure sensors except for sensor one on the pressure side, were the pressure still was above experimental predictions. Figure 5.9 shows similar magnitude of velocity fluctuations at the runner blade frequency as predicted in figure

48 5.3. STAY VANES MODEL CHAPTER 5. RESULTS (a) Figure a. (b) Figure b. Figure 5.5: Figure a: Low pressure in guide vane channel at time step 150. Figure b: Pressure transducers 58 (outer) and 69 (inner), the vertical line marks the moment for the pressure situation in figure a, each time step corresponds to a 0.2 rotation. 40