POLITECNICO DI MILANO
School of Industrial and Information Engineering
Master Degree in Mechanical Engineering
3D unsteady CFD simulations of an axial turbine stage through time-inclined model
Supervisor: Prof. Giacomo Persico
Master Thesis of:
Francesco Sinopoli Matr. 852553
Contents
Introduction 1
0.1 Basic principles . . . 2
0.1.1 The role of the efficiency . . . 2
0.1.2 The turbines . . . 2
0.2 Test rig . . . 2
0.3 Hardware and software . . . 4
1 Theoretical remarks 5 1.1 Secondary flows . . . 5 1.1.1 Passage vortex . . . 6 1.1.2 Horseshoe vortex . . . 7 1.1.3 Corner vortex . . . 8 1.1.4 Shed vortex . . . 8
1.1.5 Tip leakage vortex . . . 9
1.1.6 Scraping vortex . . . 10 1.2 Rotor-stator interaction . . . 11 1.2.1 Potential interaction . . . 11 1.2.2 Wake-blade interaction . . . 11 1.2.3 Vortex-blade interaction . . . 12 1.3 Annular cascades . . . 12 1.3.1 Blade-to-blade plane . . . 13 1.3.2 Radial direction . . . 13 2 Numerical Model 15 2.1 Consistency, stability and convergence . . . 15
2.2 Navier-Stokes equations . . . 17
2.3 Flow speed classification . . . 17
2.4 Turbulence . . . 18
2.4.1 Boussinesq’s hypothesis . . . 20
2.4.2 One equation models . . . 21
2.4.3 Two equations models . . . 22
2.4.4 Reynolds stress models . . . 23
2.5 Discretization of balance equations . . . 24
2.5.1 Advection term . . . 26
2.5.2 Diffusion term . . . 28
2.5.3 Gradient term . . . 28
2.5.4 Transient term . . . 28
2.6 The issue of time dependence . . . 29
2.7 Solution of the linear equation system . . . 30
2.8 Sliding interface and grid interaction . . . 30
2.8.1 Mixing plane . . . 31 2.8.2 Transient stator-rotor . . . 31 2.9 Averaging technique . . . 32 2.9.1 Area averaging . . . 32 2.9.2 Availability averaging . . . 32 2.9.3 Mass average . . . 33 3 Physical Model 35 3.1 Description of the test rig . . . 35
3.2 Stator blade geometry . . . 37
3.2.1 Leaning angle . . . 38
3.3 Boundary layer modelling . . . 39
3.4 Considerations on the blade channels . . . 41
3.4.1 Stator blade transformation . . . 42
3.4.2 Time-inclined model . . . 42
3.5 Grid generation . . . 46
3.6 Boundary conditions . . . 46
3.7 Experimental data . . . 47
4 Steady Simulations of 25x25 architecture 51 4.1 Grid dependence analysis . . . 51
4.2 Comparison between k − ω SST and EARSM . . . 53
4.3 Comparison between k − ω SST and RSM . . . . 58
4.4 Contours on the secondary planes . . . 58
4.5 Further considerations on hub-to-shroud curves . . . 63
5 Steady Simulations of 22x25 architecture 65 5.1 Comparison between k − ω SST and RSM . . . . 65
5.2 Comparison between 22x25 and 25x25 steady simulations . . . 68
6 Unsteady Simulation of 25x25 architecture 73 6.1 Methodology . . . 73
6.2 Comparison between 25x25 steady and unsteady simulations . . . . 74
7 Unsteady Simulation of 22x25 architecture 81 7.1 Methodology . . . 81
7.2 Comparison between 22x25 steady and unsteady simulations . . . . 82
8 Comparison between unsteady 22x25 and 25x25 simulations 89 8.1 Differences in the flow field . . . 89
8.1.1 Comparison of the contours . . . 93
8.2 Differences in the simulation process . . . 94
CONTENTS v
A Convergence history of the simulations 103
A.1 Steady simulations . . . 103
A.1.1 SST model . . . 104
A.1.2 RSM model . . . 104
A.2 Unsteady simulations . . . 107
A.2.1 25x25 architecture . . . 107
A.2.2 22x25 architecture . . . 108
List of Figures
1 Stage blades . . . 3
2 Definition of leaning angle . . . 3
1.1 Planes’ definition in a blade channel . . . 6
1.2 Representation of passage vortices . . . 7
1.3 Representation of horseshoe vortices . . . 8
1.4 Vortices’ pattern in a cascade . . . 9
1.5 Representation of shrouded and unshrouded blades . . . 10
1.6 Representation of scraping vortex and its interaction with other swirling structures . . . 10
1.7 Effect of radial equilibrium on blade-to-blade plane . . . 13
2.1 CFX methodology for constructing a control volume around a node 25 2.2 CFX methodology for constructing the integration points in an element 25 2.3 Representation of upwind scheme . . . 26
2.4 Sweby diagram . . . 28
3.1 Sketch of the test rig . . . 35
3.2 Sketch of the turbine represented on the meridional plane . . . 36
3.3 Stator and rotor blades geometry at hub, midspan and tip . . . 37
3.4 Hub clearance on stator blade . . . 38
3.5 Isobaric lines in a channel between two leaned blades . . . 39
3.6 Interaction between the leaning-caused flow and the passage vortexes 39 3.7 Law of the wall . . . 40
3.8 Simplified representation of a stage with equal stator and rotor blade number . . . 43
3.9 Simplified representation of a stage with unequal stator and rotor blade number . . . 44
3.10 Grid of the stator in 22x25 architecture, blade to blade plane at midspan . . . 48
3.11 Grid of the stator in 22x25 architecture, meridional plane at periodic surface . . . 48
3.12 Grid of the stator in 22x25 architecture, inlet surface . . . 48
3.13 Grid of the rotor, blade to blade plane at midspan . . . 49
3.14 Grid of the rotor, meridional plane at periodic surface . . . 49
3.15 Grid of the rotor, inlet surface . . . 49
3.16 3D geometry of the stator in 22x25 architecture . . . 50
3.17 Inlet total pressure profile . . . 50
4.1 Hub-to-shroud curves in the grid independence analysis . . . 53 4.2 Stator hub-to-shroud curves with k − ω SST and EARSM models in
25x25 architecture . . . 56 4.3 Rotor hub-to-shroud curves with k − ω SST and EARSM models in
25x25 architecture . . . 57 4.4 Stator hub-to-shroud curves with k − ω SST and RSM models in
25x25 architecture . . . 59 4.5 Rotor hub-to-shroud curves with k − ω SST and RSM models in
25x25 architecture . . . 60 4.6 Total pressure contours on the stator measurement plane with
differ-ent turbulence models in 25x25 architecture . . . 61 4.7 Absolute total pressure contours on the rotor measurement plane
with different turbulence models in 25x25 architecture . . . 62 4.8 Highlight on the flow behaviour on stator hub-to-shroud chart . . . 63 4.9 Highlight on the flow behaviour on rotor hub-to-shroud chart . . . . 64 5.1 Stator hub-to-shroud curves with k − ω SST and RSM models in
22x25 architecture . . . 66 5.2 Rotor hub-to-shroud curves with k − ω SST and RSM models in
22x25 architecture . . . 67 5.3 Comparison of 25x25 and 22x25 stator’s hub-to-shroud curves . . . 70 5.4 Comparison of 25x25 and 22x25 rotor’s hub-to-shroud curves . . . . 71 5.5 Comparison of 25x25 and 22x25 relative total pressure hub-to-shroud
curves at rotor exit . . . 71 5.6 Blade-to-blade contour of the absolute velocity at midspan in 22x25
case . . . 72 6.1 Stator hub-to-shroud curves in the comparison between 25x25 steady
and unsteady simulations . . . 76 6.2 Rotor hub-to-shroud curves in the comparison between 25x25 steady
and unsteady simulations . . . 77 6.3 Stator hub-to-shroud curves in the 25x25 unsteady simulation with
standard deviation . . . 78 6.4 Rotor hub-to-shroud curves in the 25x25 unsteady simulation with
standard deviation . . . 78 6.5 Pressure distribution on the stator blade surfaces in 25x25 unsteady
simulation . . . 79 6.6 Pressure distribution on the rotor blade surfaces in 25x25 unsteady
simulation . . . 80 7.1 Stator hub-to-shroud curves in the comparison between 22x25 steady
and unsteady simulations . . . 83 7.2 Rotor hub-to-shroud curves in the comparison between 22x25 steady
and unsteady simulations . . . 84 7.3 Stator hub-to-shroud curves in the 22x25 unsteady simulation with
LIST OF FIGURES ix
7.4 Rotor hub-to-shroud curves in the 22x25 unsteady simulation with standard deviation . . . 85 7.5 Pressure distribution on the stator blade surfaces in 22x25 unsteady
simulation . . . 86 7.6 Pressure distribution on the rotor blade surfaces in 22x25 unsteady
simulation . . . 87 8.1 Stator hub-to-shroud curves in the comparison between 22x25 and
25x25 unsteady simulations . . . 91 8.2 Rotor hub-to-shroud curves in the comparison between 22x25 and
25x25 unsteady simulations . . . 92 8.3 Contour of the absolute flow angle in the unsteady 22x25 simulation 96 8.4 Contour of the absolute flow angle in the unsteady 25x25 simulation 96 8.5 Contour of the relative flow angle in the unsteady 22x25 simulation 97 8.6 Contour of the relative flow angle in the unsteady 25x25 simulation 97 8.7 Contour of the relative total pressure in the unsteady 22x25 simulation 98 8.8 Contour of the relative total pressure in the unsteady 25x25 simulation 98 8.9 Contour of the entropy in the unsteady 22x25 simulation . . . 99 8.10 Contour of the entropy in the unsteady 25x25 simulation . . . 99 A.1 Trend of the residuals for the 25x25 steady simulation using SST
model . . . 105 A.2 Trend of the residuals for the 22x25 steady simulation using SST
model . . . 105 A.3 Trend of the residuals for the 25x25 steady simulation using RSM
model . . . 106 A.4 Trend of the residuals for the 22x25 steady simulation using RSM
model . . . 106 A.5 Trend of the residuals for the 25x25 unsteady simulation . . . 107 A.6 Efficiency plot of the final timestep of the 25x25 unsteady simulation 108 A.7 Trend of the residuals for the 22x25 unsteady simulation . . . 108 A.8 Trend of the entropy on the measurement points for the 22x25
List of Tables
3.1 Main geometrical and operational characteristics of the axial turbine 36 3.2 Main geometrical and operational characteristics of the axial turbine’s
cascades . . . 37 3.3 Main geometrical and operational characteristics of the axial turbine’s
cascades distinguishing among hub, midspan and tip regions . . . . 37 4.1 Losses and efficiency in the grid independence analysis . . . 52 4.2 EARSM cases . . . 54 4.3 Losses and efficiency in the comparison between different models in
the 25x25 architecture . . . 54 4.4 Losses and efficiency in the comparison with the RSM in 25x25
architecture . . . 58 5.1 Losses and efficiency in the comparison between different turbulence
models in the 22x25 architecture . . . 65 5.2 Losses and efficiency in the comparison between the 22x25 and 25x25
steady simulations . . . 68 6.1 Losses and efficiency in the comparison between 25x25 steady and
unsteady simulations . . . 74 7.1 Losses and efficiency in the comparison between 22x25 steady and
unsteady simulations . . . 82 8.1 Losses and efficiency in the comparison between 22x25 and 25x25
unsteady simulations . . . 89
Abstract
The single stage axial gas turbine, sited at Laboratorio di Fluidodinamica delle Macchine at Politecnico di Milano, is representative of present-day high pressure stages for industrial applications. It has been extensively studied by means of experimental analyses and CFD simulation, however the real geometry of the turbine has never been extensively simulated in an unsteady fashion.
The aim of the present thesis work is to use a method recently implemented in ANSYS CFX (called time-inclined method or time transformation) which is capable of handling the real geometry in a transient fashion, including a different pitch between the stator and the rotor domain. This problem was previously circumvented by means of a particular geometrical transformation called homothety.
Both steady and unsteady 3D simulations are carried out in order to assess the quality of the two geometrical configurations. The results show that a steady simulation is not representative of the rotor flow field both with the real geometry of the stator and with the transformed one.
It is evident that unsteady simulations are needed to better investigate the flow field in the stage. The transient cases using the two different geometries present little but important differences. The two simulations have also different approaches starting from the pre-processing up to the post-processing, as well as a different computational time.
As a result there is not an always valid method: it is up to the engineer to choose the method nearer to its needs.
Keywords: CFD; axial turbine; time-inclined; homothety; transient; 3D.
La turbina assiale a singolo stadio, ubicata presso il Laboratorio di Fluidodi-namica delle Macchine al Politecnico di Milano, è rappresentativa di uno stadio ad alta pressione per applicazioni industriali. È stata studiata estensivamente sia mediante analisi sperimentali sia mediante simulazioni CFD, tuttavia la geometria reale della turbina non è mai stata ampiamente simulata in instazionario.
Lo scopo di questa tesi è di usare un metodo recentemente implementato in ANSYS CFX (chiamato metodo time-inclined o time transformation)capace di simulare la geometria reale in condizioni instazionarie, che prevede un pitch differente tra statore e rotore. Tale problema era stato precedentemente evitato grazie ad una particolare trasformazione geometrica chiamata omotetia.
Sia simulazioni stazionarie che instazionarie sono state condotte, in modo tale da indagare la qualità delle due configurazioni geometriche. I risultati mostrano che le simulazioni stazionarie non sono rappresentative del flusso dentro al rotore sia con la reale geometria dello statore sia con quella trasformata.
È evidente che sono necessarie delle simulazioni instazionarie per investigare meglio i campi di moto nello stadio. I casi instazionari presentano piccole ma importanti differenze con le due differenti geometrie. Le due simulazioni inoltre hanno anche approcci differenti partendo dal pre-processing fino al post-processing, così come un tempo computazionale diverso.
Ne risulta che non esiste un metodo che è sempre valido: spetta all’ingegnere scegliere quello che più si avvicina ai suoi bisogni.
Parole chiave: CFD; turbina assiale; time-inclined; omotetia; instazionario; 3D.
Introduction
Computational Fluid Dynamics (CFD from here on) simulations are needed in a wide range of engineering applications. In the case of a complex 3D geometry such as turbomachinery, it is a useful tool for the analysis of the behaviour of the flow inside the machine. Indeed, even if experimental campaigns are carried out on a test case, it is difficult to measure the behaviour of the fluid inside the whole machine, in particular the evolution of the pressure field, wakes and vortices. In the present work the aim is to analyse different models applied on the axial gas turbine installed in Laboratorio di Fluidodinamica delle Macchine at Politecnico di Milano. It has been studied in the last 15 years and so a wide number of experimental campaigns are available, for instance in [9], [7] and [8].
Also CFD simulations have been already carried out on this machine, such as in [20]. The problem in these simulations is that it has not been possible to simulate the real geometry of the turbine considering only one blade channel. This is the common practice in turbomachinery simulations, used to well predict the flow field and minimizing the computational effort. Indeed, since the number of blades on stator and rotor is not the same, it was necessary to transform the stator blade in order to obtain an equivalent geometry with the same pitch.
In this work it has been investigated the opportunity to use a time transformation and to use the real geometry, simulating a single blade channel as well. This model is based on the work of Giles (see [12]). The core of this thesis work is to analyse the differences in choosing one of already cited two approaches: transforming the geometry or the time coordinate. The expectation is that there will be some differences in the prediction of the flow and unsteady interactions, but the overall quantities will be in good agreement between the different models, since the aim of the geometrical transformation is to preserve as much as possible the effects of the original blade, while the time transformation is an internal feature of the software which is needed only during the computation.
In order to correctly represent the physics of the problem, time marching simulations have been carried out. This choice leads to a huge computational effort, which can be quantified in three weeks for the most demanding cases: a powerful processing unit is fundamental.
0.1
Basic principles
0.1.1
The role of the efficiency
Understanding how the flow evolves inside the machine is useful not only on an academic level, but it can be a very powerful tool in the design phase as a part of the optimization procedure. It is well known that rotor-stator interaction and the generation and evolution of wake and vortices have an impact on the efficiency of the whole machine. The other sources of losses are the shock waves and the boundary layers along the blades and on the shroud and hub surfaces. Increasing the efficiency of a machine has an impact on two levels:
• economic: increasing the efficiency means reducing the energy need in order to carry out the same work and ultimately reducing costs. This makes the machine itself more attractive on the market;
• environmental: since the greatest part of the electric energy on the grid is produced burning fossil fuels, using less energy means burning less fuel, thus producing a lower environmental impact.
0.1.2
The turbines
A machine such as the one analysed in the present work, namely an axial gas turbine, is usually one of the most important component in a turbogas. It is mounted downstream of the combustor and its aim is to drive the compressor (they share the same shaft) and eventually to drive also an alternator and other auxiliary services. This is done extracting energy from the fluid by means of the deflection imposed on the flow by the blades.
It is a complex assembly composed by several stages. Each stage is composed by two cascades:
• a stator (also called nozzle guide vane), which is static; • a rotor, which is a rotating vane.
Each cascade is annular in the case of axial machines and it is composed by several blades mounted on a disc. They form numerous blade channels, in which the flow is expanded and its enthalpy is converted in mechanical energy at the shaft by means of a variation of the moment of momentum in the flow.
0.2
Test rig
The test rig as a whole is a closed loop which contains, among all the components, a centrifugal compressor and the studied axial turbine. The centrifugal compressor feeds the axial turbine with air. The compressor can be used in order to obtain different operating points of the turbine, ranging from cases when the flow can be considered as incompressible up to transonic cases.
The simulated single stage gas turbine is representative of an high pressure stage, which is found at the beginning of the turbine assembly and it is characterised
0.2. Test rig 3
by a low blade height since the pressure is high as well as the density and so the volumetric flow rate is low. In this case the 3D phenomena have a strong impact on the behaviour of the flow and so on the efficiency of the stage itself. It has a particular architecture since it allows to vary the axial gap, namely the distance between rotor’s leading edge and stator’s trailing edge. The axial gap can vary from few millimetres up to the length of the vane’s axial chord. However in this work only the last case is considered. Before the nozzle guide vane, a honeycomb structure ensures that the flow is fully axial, since it comes from a centripetal channel. After the rotor there is a diffuser. Even if it is not common in existing turbines, in this specific case the hub of the diffuser is rotating with the rotor. This specific stage is representative of an high pressure gas turbine stage. It has 22 blades on the stator and 25 blades on the rotor. A picture of the blades is reported in figure 1.
(a) Rotor blade (b) Sator blade
Figure 1: Stage blades
The stator blades are designed according to a technique called leaning. This means that the blades are not stacked on a line parallel to the radial coordinate, but on a line inclined of a constant angle with respect to the radial direction, as reported in figure 2, where the black and vertical line is the radial direction, while the red line is representative of the stacking direction.
Figure 2: Definition of leaning angle
The rotor blades are twisted according to the free vortex methodology. This is done to obtain a more uniform flow at the outlet on the point of view of the axial velocity.
0.3
Hardware and software
In order to perform the simulations, ANSYS CFX has been used. It is a tool of ANSYS package specifically developed for turbomachinery simulations. It offers a very powerful meshing tool called Turbogrid, which provides high quality structured meshes. The user can specify the refinement level in specific regions of the domain, such as in the tip clearance, where an high number of elements is needed in order to properly capture the leakage phenomenon and the consequent vortex. Furthermore, pre-processing and post-processing tools offer a variety of specific tools for turbomachinery, which will be widely used in the present work. In order to carry out the calculations, a cluster has been used. It has 32 processors which allow a partition of the whole domain of the simulation, granting a strong reduction in computational time. It must be said that it allows also to allocate a rather high memory to the whole process, which couldn’t be possible on common PCs.
Chapter 1
Theoretical remarks
The focus of this work is to try to capture the stator-rotor interaction and to simulate it with different models. In order to correctly analyse such a complex topic, all the flow features must be considered. Since the flow field is very complex, all the secondary flows will be listed in this chapter. After that, all the characteristics of the rotor-stator interaction will be remarked. Considering this kind of interactions, one must remember that the rotor and stator mutually influence the flow field of the other cascade. Indeed, the structures coming from the stator have an impact on the rotor and they interact with the flow structures generated in the rotor itself. On the other hand, the rotor influences the stator with its presence. The axial gap, namely the distance between the stator trailing edge and the rotor trailing edge in axial direction, has a fundamental role in this interaction. In the present study it is fixed, but in [9] and in [7] it is shown that the axial gap variation strongly influences the flow field between rotor and stator and also downstream of the rotor on the same machine.
1.1
Secondary flows
For a better investigation of the flow behaviour inside the blade channel, it is widely accepted to divide the analysis of the flow in three significant planes, as shown in figure 1.1. In particular, these planes are:
• blade-to-blade plane, where some important feature can be studied, such as the flow deflection and so the Euler work. At first approximation it is parallel to the hub curve;
• meridional plane, where the meridional component of the velocity (the one that discharges the flow rate) is investigated. At first approximation it is parallel to the machine axis of rotation;
• secondary plane, orthogonal to the other two, where the secondary structures can be highlighted.
Figure 1.1: Planes’ definition in a blade channel
Focusing on the last plane, it must be said that several class of vortices affect the blade channel:
• passage vortex; • horseshoe vortex; • corner vortex; • shed vortex; • tip leakage vortex; • scraping vortex.
The tip leakage vortex has a huge impact on losses especially in this turbine because of the high tip clearance on the rotor with respect to the blade height.
1.1.1
Passage vortex
In order to have a description of this phenomenon, it is important to introduce a quantity called vorticity. It is a vector field which represents the local spinning motion of a fluid (in other words, the tendency of the fluid to rotate). It is the curl of the velocity, as stated in [14]:
#
ω = ∇ ×V# (1.1)
Attention must be paid to the fact that the vorticity #ω must not be confused with
the angular velocity ω, which instead regards a single particle and it is relative to the point where it is computed.
Recalling Crocco’s equation neglecting the body forces: ∇ht= V × ## ω −
∂V# ∂t +
∇ • ¯τ¯
1.1. Secondary flows 7
where ht is te total enthalpy, ¯τ is the shear stress tensor, T is the temperature¯ and s is the entropy. If we consider a isoenergetic, inviscid, isoentropic and steady problem, equation 1.2 becomes:
#
V × #ω = 0 (1.3)
If we consider a flow distribution as the one represented in figure 1.2, the term on the left side in equation 1.3 becomes:
#
ω = dVx dz
#
j (1.4)
meaning that a non-zero vorticity is present.
This implies that the non-homogeneities at the inlet of the channel in terms of velocity, due to the presence of the boundary layers, interact inside the channel with a pressure gradient from pressure side to suction side of the blades. This gradient pushes the low momentum fluid (namely the one in the boundary layer) from pressure side to suction side, thus creating two counter rotating vortices called passage vortices.
The presence of these vortices causes an overturning of the flow in the tip and hub region, while at midspan an underturning is present.
Figure 1.2: Representation of passage vortices
1.1.2
Horseshoe vortex
The horseshoe vortex is generated by the interaction between the flow and the leading edge of the blade. In particular, since a boundary layer is present because of the casings ad hub and tip, there is a local velocity gradient in the fluid. Since the boundary layer can be considered isobaric, a total pressure gradient exists in
it. When it interacts with the leading edge of the blade, there the flow decelerates and a static pressure gradient, directed from the the free stream to the wall, is formed in the boundary layer. This generates two counter rotating vortices, as shown in figure 1.3. It is clear that the vortices inside the channel, which are a low
Figure 1.3: Representation of horseshoe vortices
momentum flow structure, are pushed towards the suction side and that suction side vortex on a blade interact with a vortex coming from the pressure side of the subsequent blade.
1.1.3
Corner vortex
As shown in [22] corner vortices are generated by the interaction between a large vortex (the passage one) and the boundary layer. In particular the passage vortex is able to put in rotation a low momentum flow such as the boundary layer. The corner vortex is very little if compared with other swirling structures.
Given the interactions among corner, horseshoe and passage vortices, a representa-tion of the cascade studied in [22] is shown in figure 1.4.
As said by the same authors, figure 1.4 is representative only of the studied cascade, since some leg of the horseshoe vortices is strongly dependant on geometry. However the basic principles still hold.
1.1.4
Shed vortex
The shed vortex is generated at the trailing edge of the blade. In particular, at the end of the blade there is an interaction between pressure side and suction side which causes a recirculation of the flow from pressure side to the other side. Concerning the vorticity, it changes in sign from hub to tip as well as the one of the passage vortices in the channel. This is due to the presence and the interaction of the opposite legs of two passage vortices coming from two adjacent channels. The
1.1. Secondary flows 9
Figure 1.4: Vortices’ pattern in a cascade
vorticity of the shed vortex is opposite with respect to the one of the corresponding passage vortex.
1.1.5
Tip leakage vortex
Across the blade there are two pressure gradients: • from pressure to suction side;
• from upstream to downstream on turbines (the contrary on compressors). The second one derives from the reaction degree of the stage. The reaction degree is defined as:
χ = ∆hs
∆hstage
(1.5) where ∆hsis the static enthalpy drop across the rotor, while ∆hstageis the enthalpy drop across the whole stage and it can be static-static or total-total. In particular, if χ = 0, it means that there is no static enthalpy drop in the rotor. This implies an important feature of the velocity triangle: the relative velocities at inlet and outlet of the rotor have the same intensity and so an equal angle but opposite in sign. This means that in this case the rotor has a symmetric blade geometry and all the pressure drop is obtained in the stator. Consequently the second pressure gradient previously listed is not present in a pure impulse stage.
There is a way to considerably reduce the pressure gradient from pressure to suction side of the blade: adopting a shrouded rotor, namely covering the rotor blade tip and the blade duct is all contoured. The shroud has the function of a sealing, thus
reducing the flow leakage across the blade tip. An example of shrouded blades is reported in figure 1.5. In the studied turbines there is no shroud on the rotor and
(a) Unshrouded blades (b) Shrouded blade
Figure 1.5: Representation of shrouded and unshrouded blades
so the tip leakage is one of the main sources of losses in the whole stage because: • a portion of the whole mass flow is not deflected by the blades;
• the blade is less loaded and so less work is extracted;
• a vortex is generated downstream of the rotor by the leaked flow and so it generates mixing losses also far away from the rotor.
1.1.6
Scraping vortex
Another important feature of the unshrouded rotor is the interaction between the blade tip and the annulus boundary layer. This creates another vortex called scraping vortex. It interacts with the tip leakage vortex, as shown in figure 1.6. In this figure + and - represent respectively the pressure and suction sides of the blades, P is the tip passage vortex, T is the tip leakage vortex and S is the scraping vortex. It is counter-rotating with respect to the tip clearance vortex. However tip clearance vortex is usually much stronger than the scraping one.
Figure 1.6: Representation of scraping vortex and its interaction with other swirling
1.2. Rotor-stator interaction 11
1.2
Rotor-stator interaction
Due to the non-stationary intrinsic nature of the problem, an unsteady interac-tion between rotor and stator exists and it is due by the different flow structures generated in the whole stage. These structures are:
• potential interaction; • wake-blade interaction; • vortex-blade interaction;
One must keep in mind that all these phenomena occur at the same time, meaning that the flow pattern inside the blade channel is particularly complex. Other works on the same facility have already been performed, such as [20], proving how much these phenomena are important in a turbine stage and their complexity.
1.2.1
Potential interaction
It is due to the static perturbation in the flow in the proximity of the blade caused by the presence of the blade itself. In particular in a turbine the rotor influences the pressure field at the stator trailing edge because of its presence. As stated in [19], this causes a spatial periodicity in the flow. Obviously the most affected regions are the rotor leading edge and the stator trailing edge, if we consider an isolated stage as the analysed one.
1.2.2
Wake-blade interaction
The wake at the end of the blade is produced by the presence of two boundary layers on the blade, one on its suction side and one on its pressure side. After the blade these two low velocity flows merge in a flow structure which is called dead water and its characterised by low velocity with respect to the free stream velocity, high recirculation and thus high pressure. Going far away from the blade, the wake decays and the flow quantities tend to uniform to the free stream, generating mixing losses.
The rate of decay of the wake is very low with respect to the potential field. This means that it causes a more complicated interaction with the incoming blade row. Some important features of this phenomenon must be pointed out:
• the interaction between incoming wake and airfoil causes a periodic trend of the lift;
• the variation in lift has the same order of magnitude of the one induced by the potential field;
• the wake is chopped by the blades’ leading edges and in the channel is distorted because of the velocity gradients given by the presence of the boundary layers; • the passage of the wake on the blades creates local pressure gradients which increase the lift with a non-zero average, meaning that this phenomenon cannot be neglected even in a steady analysis.
The presence of the aforementioned local pressure gradients causes two counter rotating vortices in the blade-to-blade plane.
Furthermore, when the wake is chopped by two adjacent blades, the two extremities have a drift and one of them leaves the channel with a certain delay with respect to the other one. This is due to the blade loading and so to the pressure distribution on the different sides of the blade.
1.2.3
Vortex-blade interaction
The vortex-blade interaction s a very complex phenomenon since it is strongly case-dependent. It includes different phenomena, especially if incoming the vortex interacts with a rotating cascade. The vortex can undergo a breakdown, especially if it is near the pressure side of the blade. Furthermore it can be distorted and it can interact with the secondary flows which are formed inside the channel itself. In this rather complex scenario, another important factor must be considered: the blades are leaned and this is done in order to reduce the secondary flows formation inside the blade channel. This causes another geometrical feature which sets up a different case. As written in [19] (page 37), the same authors found different results in cases with cylindrical and leaned blades.
In the end, some basic features can be highlighted:
• the vortex can undergo a breakdown or it can become unstable, especially if it is near the pressure side of the blade;
• it can be distorted and convected in the rotor channel according to the pressure gradients and the flow behaviour, to the point that some counter-rotating legs can be formed.
1.3
Annular cascades
The main difference of an annular cascade with respect to a linear cascade is the presence in the former of a curvature of the flow passage, which will imply some important features exposed in this section. This curvature can cause in some blade design a static pressure gradient: the static pressure is higher at the tip with respect to the hub and so the low momentum flow structures are pushed towards the hub.
An analysis of this feature is obtained through Crocco equation in case of an isoentropic, isoenergetic, non viscous and steady problem. Crocco equation with these hypotheses reduces to equation 1.3, which is satisfied only if the curl of the velocity ∇ × #v = #ω is null. In a cylindrical reference (axial, tangential and radial
coordinates) and for an axial turbomachinery with the hypothesis of null radial velocity (Vr = 0) it can be rewritten as:
Va r ∂Va ∂ϑ − ∂(rVt) ∂z ! # it− Vt r ∂Va ∂ϑ − ∂(rVt) ∂z ! # ia+ Va ∂Va ∂r + Vt r ∂(rVt) ∂z ! # ir = 0 (1.6) Where Va, Vt and Vt are respectively the axial and tangential velocity components,
1.3. Annular cascades 13
each cylindrical coordinate.
In order to be satisfied, all the components in equation 1.6 must be null. The relationships in tangential and axial coordinates (the blade-to-blade plane) produce the same results, while the equation in radial direction must be analysed separately.
1.3.1
Blade-to-blade plane
By imposing that te first two terms in equation 1.6 must be equal to zero and neglecting the trivial solutions of Va = 0 and Vt = 0, the following equation is obtained:
∂Va
∂ϑ =
∂(rVt)
∂z (1.7)
This means that a change of Va along the axial direction imposes a change of rVt along the axial direction. As showed in figure 1.7, the presence of the blade causes
Figure 1.7: Effect of radial equilibrium on blade-to-blade plane
a variation of axial velocity because of the pressure field around the blade itself (there is also the effect of the reduction of the cross-sectional area because of the
presence of the blade itself). This causes a variation of rVt along z.
1.3.2
Radial direction
By equating to zero the third term in equation 1.6, the following relationship can be obtained: ∂Va ∂r = − Vt Va 1 r ∂(rVt) ∂r (1.8)
in a channel without a blade, this means that:
• if Va has a proper dependence on r, Vt/Va is not constant along r and so the flow angle α(r) is a function of the radial coordinate;
• if Va is constant with the radial coordinate, this means that rVt is also constant on the radial direction. This implies that the tangential velocity has an hyperbolic trend on the radial coordinate.
These observations introduce the idea of the free-vortex design methodology, where the idea is to keep the axial velocity constant along the radius at the outlet of the stage by imposing a Vt profile hyperbolically varying along the radius and so a varying discharge flow angle.
Chapter 2
Numerical Model
The idea behind the CFD is to solve a complex set of equations (such as the one which describes the behaviour of the flow) with the aid of a computer. Indeed the Navier-Stokes equations can be solved analytically only in simplified cases, which are far from the engineering applications. In order to achieve a result in interesting cases from an engineering perspective, these equations must be rewritten in another form, which is successfully handled by the computer. Indeed these are partial differential equations, which must be in some way transformed in algebraic equations through some approximations. This leads to an important point: while the partial differential equations are meant for the continuum, the algebraic equations need a discretization for the space. Indeed, in order to obtain such a result, a mesh (or grid) is created in the integration domain and the values of the physical quantities in the other points, which don’t belong to the mesh, are obtained by means of an interpolation. Also the time coordinate is discretized. The creation and the quality of the mesh is of paramount importance in a CFD simulation. As a general rule, some criteria must be satisfied in order to have a numerical scheme which provides a solution that is not far from the analytical model. These requirements are defined as stability, convergence and consistency.
In the following sections, the problem of consistency, stability and convergence will be considered. Then the analytical model will be exposed, with a focus on turbulence and compressibility, which are the main complications in the solution of the Navier-Stokes equations. Finally, the numerical schemes will be analysed with a focus on the generation of the grid and on the averaging procedure of the physical quantities on a surface.
2.1
Consistency, stability and convergence
It is not guaranteed that the numerical solution is near the analytical one. Obviously, in applicative engineering circumstances, if the analytical solution is available, there is no need to waste computational and human resources in a simulation of the same phenomenon. When this is not possible, the already cited criteria must be ensured in order to avoid unacceptable errors or a not converged simulation. These criteria represent different aspects of the interrelation between the analytical model, the numerical scheme and their solutions. In particular:
• consistency is a condition on the numerical scheme: it must tend to the differential equation when space and time steps tend to zero;
• stability is a condition on the numerical scheme: all the errors due to the finite algebra must be bounded iteration after iteration;
• convergence is a condition on the numerical solution: it requires that the numerical solution must tend to the exact solution of the mathematical model, when time and space steps tend to zero.
If the error ¯εn is defined as the difference between the computed solution un and the exact solution of the discretized equation ¯un:
¯
εn = un− ¯un (2.1) The stability conditions prescribes that the error must be bounded for n → +∞ with a constant time step ∆t:
lim
n→+∞|¯εn| ≤ K (2.2) with K independent on n.
If the error ˜εn
i instead is defined as the difference between the numerical solution and the exact solution of the differential equation:
˜
εni = uni − ˜u(i∆x, n∆t) (2.3)
The convergence condition can be expressed as: lim
(∆x,∆t)→(0,0)|˜ε
n
i| = 0 (2.4)
Please note that, as expressed in [13], stability and convergence do not refer to the same error.
It is clear that the conditions of stability, consistency and convergence are related one each other and this relation is expressed by the Equivalence theorem of Lax, which states that: for a well-posed initial value problem and a consistent discretization scheme, stability is the necessary and sufficient condition for convergence. This ensures that, by guaranteeing two of the three conditions, the other one is implicitly respected.
In particular, the theorem of Lax ensures that, in order to analyse a time dependent or initial value problem, two tasks must be performed:
• analyse the consistency conditions, in order to determine the order of accuracy and the truncation error of the discretization scheme;
• analyse the stability.
2.2. Navier-Stokes equations 17
2.2
Navier-Stokes equations
Concerning the analytical model, the equations which describe the behaviour in time and space of a continous, non reacting, compressive, Newtonian flow are the Navier-Stokes equations: ∂ρ ∂t + ∇ • (ρ # V ) = 0 (2.5) ∂(ρV )# ∂t + ∇ • (ρ # V ⊗V ) − ∇# hµ∇V + (∇# V#T)i= ρ #g − ∇ P + 2 3µ∇ • # V ! (2.6) ∂(ρe) ∂t + ∇ • (ρe # V ) − ∇ • (λ∇T ) − ρ #g •V + ∇ • (P# V ) =# = −∇ • " 2 3µ(∇ • # V )V# # + ∇ • ∇hµ∇V + (∇# V#T)•V#i+ ρQ (2.7)
Where equation 2.5 is the balance equation of continuity, equation 2.6 is the the equation of the momentum in the three directions for a 3D problem and equation 2.7 is the balance equation of the energy. In these equations, the hypothesis of Newtonian fluid is expressed by the fact that the viscous stress tensor is proportional to the velocity gradients. The problem is closed by the equations of state:
P = P (ρ, T ); e = e(ρ, T ) (2.8)
and by the expressions of the transport coefficients:
µ = µ(ρ, T ); λ = λ(ρ, T ) (2.9)
These equations of state close the problem, since now we have seven equations in seven unknowns. This means that the problem can be solved if suitable boundary and initial conditions are provided (use [15] as a reference).
An important difference between equation 2.7 and the energy equation implemented in ANSYS CFX is that in the latter case it is expressed in terms of total enthalpy, defined as: hT = e + P ρ + 1 2V 2 (2.10)
This is done because the total enthalpy is a more representative energetic quantity in turbomachinery, since the Euler work (the power per unitary mass flow rate extracted by the turbine) can be seen as a variation of total enthalpy.
2.3
Flow speed classification
The previous partial differential equations are written for a compressible flow. In case of an incompressible flow, their are much simpler since constant density can be assumed. However, one must identify when the hypothesis of incompressible flow is satisfied in order to avoid totally misleading results. The flow is incompressible if:
• if the fluid is compressible, the flow can be considered incompressible when the variation of density with pressure is negligible.
The latter case is analysed through the Mach number:
M = V
a (2.11)
where a is the speed of sound, a local characteristic of the fluid which depends on its nature and on its temperature. The flow can be characterised as follows:
• incompressible flow: the pressure variations are very low and so the density variations can be neglected. This happens when M < 0.3;
• subsonic flow: the flow is now considered compressible. The Mach range is 0.3 < M < 0.9;
• transonic flow: the flow in the greatest part of the domain is subsonic, but with transonic pockets. In these pockets, the density variations are very strong. The flow is considered transonic when 0.9 < M < 1.1;
• supersonic flow: the flow is supersonic in the greatest part of the domain. It can be subsonic near walls or obstacles. The Mach number range in this case is 1.1 < M < 5;
• hypersonic flow: other phenomena are present in this case, such as molecular dissociation. Other equations are needed for a correct formulation of the problem. This happens when M > 5.
In the present study the flow cannot be considered compressible, since the fluid is air and it is usually subsonic, with pockets of transonic flow on the suction side of the blades.
2.4
Turbulence
Navier-Stokes equations are rather complex and their are strongly non-linear. This is due to the compressibility of the flow, but also to the convection term, which is responsible of turbulence. This is a great problem when Navier-Stokes are attacked analytically, since this strong non-linearity implies the non uniqueness of the solution.
It is important to distinguish between the laminar flow and the turbulent one. Laminar flow is characterised by a stable, parallel and well ordered movement with no strong interaction between the different layers of the flow, low momentum convection and, finally, pressure and velocity are independent from time.
Turbulent flow is a much more complicated phenomenon. According to [21], the turbulent flow is random, unsteady and chaotic. It can be observed in a wide cases related to the everyday experience such as a waterfall ore a smoke stream from a chimney. Turbulence is also common in many engineering applications, such as the boundary layer on an aircraft wing or on a turbine blade.
2.4. Turbulence 19
• randomness: the turbulent flows seem chaotic and unpredictable, in contrast with the laminar flow;
• non-linearity: it means that the flow is unstable. This causes two important features in a turbulent flow:
– small perturbations can grow up to finite amplitude disturbances. If
these disturbances exceed the stability criteria, the flow becomes more unstable, reaching a chaotic state;
– it causes vortex stretching, a process by which three-dimensional
turbu-lent flows keep their vorticity as constant.
• diffusivity: given the macroscopic mixing processes inside the turbulent flows, they are characterised by an higher rate of diffusion of momentum and heat; • vorticity: turbulence is characterised by high levels of fluctuations of all the physical quantities, included the vorticity. The swirling structures which can be identified in a turbulent flow are called eddies. Turbulent flow is characterised by a wide range of eddy sizes, from the width of the region occupied by the turbulent flow up very small eddies. The largest eddies contain most of the energy and it is handed down from the largest to the smallest scale by means of non-linear interactions.
• dissipation: the vortex stretching mechanism transfer energy to smaller and smaller scales of the vortices, up to a scale where the vortex itself is smeared out by the fact that the gradients are too strong. This is a dissipative phenomenon dominated by the viscosity.
Given the fluctuations of all the quantities in the turbulent flow, a generic quantity
ϕ can be split in a mean part ¯ϕ and a fluctuating part ϕ0:
ϕ = ¯ϕ + ϕ0 (2.12)
This is a key point in the following dissertation because it introduced the idea of the Reynolds averaged Navier-Stokes equations.
Before going through the different turbulence models, it is needed to clarify the idea of energy cascade. Starting from the largest eddies, they are unstable by its nature and so the are prone to be broken up, transferring the energy to smaller eddies. These smaller eddies undergo a similar process and so they transfer the energy to smaller and smaller eddies. In relative large eddies, this process is merely an exchange of kinetic energy. When the eddy scale reaches the smallest one (also called Kolmogorov scale) the kinetic energy is converted into internal energy by means of viscous dissipation. This is the mechanism of the energy cascade, which is based on the eddies’ length scale.
In this scenario, there are several models meant to handle the turbulence in different ways. This is due to the fact that a compromise exists between the accuracy of the simulation of all the length scales and the computational time. Indeed, if the simulation of smaller and smaller length scales is required, not only the turbulence model becomes more complex, but also a stronger refinement of the grid is imposed, which ultimately is traduced in a larger number of points where the equations must
be solved.
The turbulence models can simulate (namely, directly compute from the equations) some eddies’ length scales and use simplified models for other ones. The turbulence models are:
• DNS: the Direct Navier-Stokes model aims to simulate all the eddies’ length scales;
• LES: the Large Eddy Simulation models only the viscous length scale; • RANS: the Reynolds Averaged Navier-Stokes model simulates only the largest
eddies, modelling all the others (even if they are far from the dissipative scale).
Another model exists, which is called DES (Detached Eddy Simulation) which is in between LES and RANS.
Due to the computational effort required to run a DNS, a LES or a DES, in this work RANS have been used. RANS is a wide family of turbulence models, but all of them have in common the idea to split the flow into an average part and a fluctuating part. This is done through an averaging procedure but, given the non-linearity of the equations, some fluctuating quantities have a non-zero effect on the mean flow. The averaged balance equations are:
∂ρ ∂t + ∇ • (ρ # V ) = 0 (2.13) ∂ρV# ∂t + ∇ • (ρ # V ⊗V ) = ρ ## g − ∇P + µ∇2V +# 1 3µ∇(∇ • # V ) − ∇ • (ρ #v ⊗ #v ) (2.14) ∂(ρcT ) ∂t + ∇ • (ρc # V T = k∇2T − ∇ • (ρcT0#v ) (2.15)
Where V is the average velocity and v is the fluctuating velocity. It is clear that the averaged continuity equation 2.13 is formally identical to 2.5. The averaged momentum equation 2.14 has an additional term with respect to equation 2.6. It is ¯¯r = ∇ • (ρ #v ⊗ #v ), called Reynolds stress tensor. It represents the turbulent
momentum transport which enhances the diffusivity and dissipation in a turbulent flow. Unfortunately a direct expression of the Reynolds stress tensor’ components doesn’t exist: a closure problem arises and modelling is needed to obtain a solution. Also the additional term in equation 2.15 represents the enhanced diffusivity present in a turbulent flow and it is called turbulent heat flux. Also in this case a closure problem arises.
Different solutions have been proposed in order to model the Reynolds stress tensor.
2.4.1
Boussinesq’s hypothesis
The Reynolds stress tensor is symmetric and can be expressed as the sum of an isotropic and a deviatoric component:
r = −ρ2
2.4. Turbulence 21
where a is the deviatoric component and −ρ2
3k is the isotropic one, in which k is
the turbulent kinetic energy, defined as:
k = 1
2(u
02
+ v02+ w02) (2.17) where u0, v0 and w0 are the components of the fluctuating part of the velocity vector. Boussinesq proposed a formal expression of the deviatoric component equal to the Newton stress-strain rate law:
a = −2µTD (2.18)
where µT is the turbulent viscosity (called also eddy viscosity). A similar solution has been proposed for the turbulent heat flux kT.
This hypothesis however seems justified only for shear flows. Dimensional consider-ations show that µT is proportional to a turbulent length scale lT and a turbulent velocity uT:
µT ∝ lTuT (2.19)
This is an improvement, since it easier to make hypotheses on a length scale rather than on a viscosity term. However the determination of lT is rather complex and so there is the need of more elaborated models.
2.4.2
One equation models
Two one equation models have been developed. These two models share the same complexity but are based on different assumptions and procedure. The idea however is to remove the turbulent length concept and to introduce another balance equation (namely, another partial differential equation).
Turbulent kinetic energy equation Prandlt deduced the turbulent velocity scale from the turbulent kinetic energy:
µT = clTuT = c0lT √
k (2.20)
A rigorous balance equation for k is deduced working out the RANS equations:
ρ∂k ∂t + ρ # V • ∇k = r : ∇V − 2µ# d : d+ ∇ • − P0#v + µ∇k − 1 2v 2#v ! (2.21) The equation states that the turbulent kinetic energy is dissipated by the viscosity. Indeed the dissipation ε is defined as:
ε = 2µd : d (2.22)
With the approximation:
ε = CD
k32
lT
(2.23) By prescribing a proper relationship for lT the system is closed. The main problem is that it still needs a way to identify a length scale.
Spalart-Allmaras model This model proposes a balance equation directly for the turbulent viscosity, thus avoiding the a-priori knowledge of a turbulent length scale. The balance equation is:
ρ∂µT ∂t + ρ
#
V • ∇µT = ∇ • (µT∇µT) + S (2.24) It is tailored for aerodynamic applications, also in transonic regime. I works well also in turbomachinery applications. It exhibit a poor behaviour in presence of jets, strong adverse gradients and separated flows.
2.4.3
Two equations models
The idea behind the usage of two balance equations is to avoid the estimation of specific quantities’ fields. This is done remembering that:
µT = Cµρ
k
ε (2.25)
and recalling the definition of the dissipation rate ω:
ω = ε
k (2.26)
Since we have a rigorous expression of the balance equation of the turbulent kinetic energy (equation 2.21), it will always be used in all the two equations turbulence models. The other modelled balance equation depends on the model itself.
k − ε model This model solves the following equation for ε:
∂ε ∂t + # V • ∇ε = Cε1 ε kr : ∇ # V − Cε2 ε2 k + ∇ • µ ρ + µT ρσε ! ∇ε ! (2.27)
with Cε1 = 1.44,Cε2 = 1.92 and σε= 1.3 are derived from considerations on shear flows and near wall behaviour.
The main problem of this model is that the equations diverges at the wall, therefore proper wall functions must be used.
k − ω model This model solves a balance equation for the dissipation rate similar to equation 2.27: ∂ω ∂t + # V • ∇ω = Cω1 ω kr : ∇ # V − Cω2ω2+ ∇ • µ ρ + µT ρσω ! ∇ω ! (2.28)
On the point of view of the coefficient, this equation is less sensitive on their tuning. This equation diverges far away from the wall. At the wall the wall function can be used to save computational time (namely adopting a coarser grid), but their are not needed.
2.4. Turbulence 23
Shear Stress Transport To combine the advantages of k − ε and k − ω models, shear stress transport model (also known as k − ω SST) proposes a blending between the two models with a balance equation (apart from the one for the turbulent kinetic energy) where a blending factor F is used and it is proportional to the distance from the wall:
∂ω ∂t + # V • ∇ω = Cω1 ω kr : ∇ # V − Cω2ω2+ ∇ • µ ρ + µT ρσω3 ! ∇ω ! + (1 − F )2∇ω∇k σω2ω (2.29) Near the wall, the equation seems the one of the k − ω model, while it is similar to the one of the k − ε model in the free stream. This avoids the critical issues of the two separated models.
In practice SST has been proven to be the best choice of industrial turbomachinery calculation and it is the standard turbulence model in this field.
2.4.4
Reynolds stress models
Citing [21], in Reynolds stress models (RSM) transport equations are solved for each component of the Reynolds stress tensor and for any other turbulent quantity which provides a turbulent length or time scale. This helps in predicting strong anisotropic swirling structures, which are not well captured even by the two equations turbulence model. This is because, in the latter models, there is the assumption of uniform turbulent kinetic energy in the three direction, as stated by equation 2.17.
In particular ANSYS CFX provides a set of ε-based and ω-based Reynolds stress models and the choice to use one or another RSM model is mesh-dependent. The guide itself suggests the ω-based Reynolds stress models (in particular the more advanced Baseline RSM) since they are more accurate near the walls with respect
ε-based ones. The major difficulty in using such a complex model is to set the
boundary conditions of each term of the Reynolds stress tensor. Furthermore, simulations last longer using RSM since it has more transport equation to be solved with respect to two equations models.
ANSYS CFX uses also another type of RSM called Explicit Algebraic Reynolds Stress Model (EARSM), which represents an extension of the standard two-equation models.
They are derived from the Reynolds stress transport equations, indeed they can be based on k − ε or k − ω models, but in this work only the -˛ω model has been used. Reynolds stresses are computed from the anisotropy tensor according to its definition: uiuj = aij + 2 3δij (2.30) where aij is a term of the anisotropy tensor a. It is computed by means of this implicit algebraic equation (from which the name EARSM is derived):
S and Ω denote the non-dimensional strain-rate and vorticity tensors, respectively.
They are defined as:
Sij = 1 2τ ∂Vi ∂xj + ∂Vj ∂xi ! (2.32) Ωij = 1 2τ ∂Vi ∂xj − ∂Vj ∂xi ! (2.33) where τ is a time scale given by:
τ = k ε = 1 Cµω (2.34) with Cµ= 0.09.
This model offers the possibility to avoid to solve a single balance equation for each term of the Reynolds stress tensor by solving only an implicit algebraic model, thus being faster even if it introduces some approximations.
2.5
Discretization of balance equations
The most important idea behind the computational fluid dynamics is the discretization of the domain and of the balance equations. There are two ways to discretize these equations and they differ on how the physical quantities are defined: • finite differences method: the quantities are defined over the nodes of the grid. The solution in this case is optimized on the indexing method of the grid nodes;
• finite volume method: the quantities are defined over the volume of the cells and referred to the centroid of the cell itself. The indexing is more cumbersome, but they are valid for any cell shape.
ANSYS CFX uses the finite volume method. With this approach, a control volume is considered on which the balance equations are discretized after being rewritten in integral form (from [13]):
∂ ∂t Z Ω ρdΩ + I S ρV • d# S# (2.35) ∂ ∂t Z Ω ρV dΩ +# I S ρV (# V • d# #S ) = Z Ω ρ#fedΩ + I S (τ − P I) • dS# (2.36) ∂ ∂t Z Ω ρedΩ+ I S ρeV •d# #S = I S k∇T •dS +# Z Ω (ρ#fe•V +q# H)dΩ+ I S ((τ −P I)•V )•d# #S (2.37) As stated in [1], ANSYS CFX uses the mesh to construct finite volumes, which are used to conserve relevant quantities such as mass, momentum, and energy at the nodes. As shown in figure 2.1, the control volume is constructed around each mesh node using the median dual (defined by lines joining the centres of the edges and element centers surrounding the node). The volume integrals are accumulated on the node on which the control volume belongs, while the surfaces integrals are
2.5. Discretization of balance equations 25
Figure 2.1: CFX methodology for constructing a control volume around a node
discretized at the integration points (ipn, shown in figure 2.2) located at the center of each surface segment within an element and then distributed to the adjacent control volumes.
The surface integrals represent fluxes (which can be advective or diffusive ones)
Figure 2.2: CFX methodology for constructing the integration points in an element
and generally they require two approximation levels:
• the integrand is approximated on one or more locations of the surface; • the cell-face values are approximated in terms of cell center values
The first bullet point is solved by using the Midpoint rule, where the only repre-sentative point of the surface is its center. This scheme allows to achieve a good accuracy, even if the global accuracy is determined also by the discretization scheme of the second bullet point.
2.5.1
Advection term
The advective terms are discretized by ANSYS CFX with the following equation:
ϕ = ϕup+ β∇ϕ∆ #r (2.38)
where ϕup is the value at the upwind node, and ∆ #r is the vector from the upwind node. Particular choices for β and ∇ϕ yield different schemes as described below.
Upwind scheme If the settings are β = 0 the scheme is upwind. This means that the value of ϕup on point f is set as follows, referring to figure 2.3:
ϕup = ϕP if V • ## n > 0
ϕup= ϕN if V • ## n < 0
Figure 2.3: Representation of upwind scheme
This is done by considering just the first term of the Taylor expansion of ϕ around point P : ϕ = ϕP + (xf − xP) ∂ϕ ∂x ! P + (xf − xP)2 ∂2ϕ ∂x2 ! P + o(xf − xP)2 (2.39) This scheme is very robust since it is unconditionally bounded. However it introduces a source of artificial viscosity, since the truncation error is similar to a diffusive term and it is:
εt = (xf − xP) ∂ϕ ∂x ! P (2.40) This means that this scheme is stable but not accurate. It can be used as a starting point for more accurate (yet less precise) schemes. The effects of this additional diffusion are two:
• it smears the sharp variations of the quantities;
• if the mesh is not flow oriented, it introduces additional diffusion also in the direction orthogonal to the flow.
High resolution scheme High resolution scheme is used in order to improve the accuracy of the upwind scheme. This is done by varying the value of β for each node individually, while ∆ϕ is evaluated from the upwind node.
2.5. Discretization of balance equations 27
an high accuracy scheme. The values done on β follow the the TVD (total variation diminishing) schemes, which, citing [15], have been specifically developed to achieve oscillation-free solutions.
Indeed a desirable property for a stable, non-oscillatory, higher-order scheme is monotonicity preserving. For a scheme to preserve monotonicity, some requirements must be set:
• it must not create local extrema;
• the value of an existing local minimum must be non-decreasing and that of a local maximum must be non-increasing.
This means that monotonicity-preserving schemes do not create new undershoots and overshoots in the solution or accentuate existing extremes. Ultimately, if one defines the total variations on the nodes (indexed with i) at the n-th iteration as:
T Vn=X i
|ϕi+1− ϕi| (2.41) The monotonicity is preserved if:
T Vn+1 ≤ T Vn (2.42) The necessary and sufficient conditions for a scheme to be TVD in terms of the
r − ψ relationship are:
• if 0 < r < 1 the upper limit is ψ(r) = 2r, so for TVD schemes ψ(r) ≤ 2r; • if r ≥ 1 the upper limit is ψ(r) = 2, so for TVD schemes ψ(r) ≤ 2; where r is defined as:
r = ϕP − ϕW ϕE − ϕP
(2.43) and the r − ψ relationship is involved in the calculation of ϕ:
ϕ = ϕP + 1
2ψ(r)(ϕE − ϕP) (2.44)
Other two requirements must be set in order to have a second order scheme: • if 0 < r < 1 the lower limit is ψ(r) = r, the upper limit is ψ(r) = 1, so for
TVD schemes r ≤ ψ(r) ≤ 1;
• if r ≥ 1 the lower limit is ψ(r) = 1, the upper limit is ψ(r) = r, so for TVD schemes 1 ≤ ψ(r) ≤ r.
This creates a region on the ψ − r diagram called Sweby region, where a scheme is TVD and second order accurate. It is represented in figure 2.4.
In this study all the presented results will be obtained using a TVD scheme. However the cases are initialized with upwind scheme, in order to have a good set-up for the more accurate scheme.
Figure 2.4: Sweby diagram
2.5.2
Diffusion term
The diffusion term is rewritten as a volume integral by means of Gauss theorem:
Z V ∇ • (Φϕ∇ϕ)dΩ = X f (Φϕ)f(#S ∇ϕ)f (2.45) where f is the generic face of the control volume. In this formulation, the terms:
• (Φϕ)f is interpolated on the surface; • (#S ∇ϕ)f depends on the mesh considered.
The latter term can be discretized as (referring to figure 2.3): (#S ∇ϕ)f) = |#S |
ϕN − ϕP
|#d | (2.46)
2.5.3
Gradient term
Considering the gradient term rewritten as:
Z
V
∇ϕdΩ =X
f
|#S |fϕf (2.47) in this expression the term ϕf is is evaluated resorting to linear interpolation between the two adjacent nodes.
2.5.4
Transient term
This term is fundamental in the presented simulations since the aim is to capture the unsteady effects in a turbine stage. Its discretization is handled by discretizing the volume integral:
∂ ∂t Z V ρϕdΩ = V (ρϕ) n+12 − (ρϕ)n−12 δt (2.48)
where the apexes of (ρϕ) indicates the values at start and end of the time step. For the discretization of this term, ANSYS CFX adopts two solutions, described in the following paragraphs. In the present work the explicit methods have been excluded a priori.
