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Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and Off-Design Conditions
Filippo Valentini
Master of Science Thesis EGI_2016-094 MSC EKV1169
KTH School of Industrial Engineering and Management
Machine Design
SE-100 44 STOCKHOLM
Master of Science Thesis
EGI_2016-094 MSC EKV1169
Numerical Modelling of a Radial Inflow Turbine with and without Nozzle Ring at Design and Off-
Design Conditions
Filippo Valentini
Approved
Examiner
Paul Petrie-Repar
Supervisor
Jens Fridh
Commissioner
Contact person
Abstract
The design of a radial turbine working at peak efficiency over a wide range of operating
conditions is nowadays an active topic of research, as this constitutes a target feature for
applications on turbochargers. To this purpose many solutions have been suggested, including
the use of devices for better flow guidance, namely the nozzle ring, which are reported to boost
the performance of a radial turbine at both design and off-design points. However the majority of
performance evaluations available in literature are based on one-dimensional meanline analysis,
hence loss terms related to the three-dimensional nature of real flows inside a radial turbine are
either approximated through empirical relations or simply neglected.
In this thesis a three-dimensional approach to the design of a radial turbine is implemented, and
two configurations, with and without fixed nozzle ring, are generated. The turbine is designed for
a turbocharging system of a typical six-cylinder diesel truck engine, of which exhaust gas
thermodynamic properties are known. The models are studied by means of a CFD commercial
software, and their performance at steady design and off-design conditions are compared.
Results show that, at design point, the addition of a static nozzle ring leads to non negligible
increments, with respect to the vaneless case, of both efficiency and power output: such
increments are estimated in +1.5% and +3.5% respectively, despite these data should be
compared with the uncertainty of the numerical model. On the other hand both turbine
configurations are found to be very sensitive to variations of pressure and temperature of the
incoming fluid, hence off-design performances are dependent on the particular off-design point
considered and a “best” configuration within all the combustion cycle does not exist.
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor, Jens Fridh, and my examiner, Paul Petrie-
Repar, for their availability and support, and without whom this work would have never been
realised.
Sincere gratitude to my Italian examiner Alessandro Talamelli, who followed the project of Dual
Degree between UniBo and KTH since its inception: it is also thanks to him that I ended up at
KTH, a circumstance which I will never regret.
I would like to thank my parents, who believed in me and allowed me to complete my academic
iter, providing full moral and material support.
A special mention goes to my Italian friends: Pietro, who was always ready to support me with a
Skype call and welcome me back during my short visits in Italy, and Simone, who, no matter
where he is, always finds a way to be present in the most crucial moments.
Thank you Danilo, you proved to be not only a respectful roommate but also a real valuable
person from all sides. And many thanks to all friends that I met in Stockholm, who turned my
stay abroad into an exciting and enjoyable experience, because “there’s no point in living if you
can’t feel alive”.
iii
TABLE OF CONTENTS
List of Figures.................................................................................................................................v
List of Tables.................................................................................................................................vii
Nomenclature...............................................................................................................................viii
1. Introduction...............................................................................................................................1
1.1. The Radial Turbine.............................................................................................................2
1.2. Sources of Losses...............................................................................................................5
1.3. The Design Process............................................................................................................7
1.4. Design and Analysis: State of the Art...............................................................................8
1.4.1. Inverse Problem: the Volute...................................................................................8
1.4.2. Inverse Problem: the Rotor.....................................................................................9
1.4.3. Inverse Problem: the Remaining Components......................................................10
1.4.4. Direct Problem: Performance Analysis.................................................................10
2. Motivation and Objective........................................................................................................11
2.1. Motivation........................................................................................................................11
2.2. Objective..........................................................................................................................11
3. Methodology and Tools..........................................................................................................12
3.1. Methodology....................................................................................................................12
3.2. Tools-Software.................................................................................................................13
4. Limitations...............................................................................................................................15
5. Design of Components............................................................................................................16
5.1. Design of the Volute........................................................................................................16
5.1.1. Theoretical Procedure...........................................................................................17
5.1.2. Implementation of the Theoretical Procedure.......................................................19
5.1.3. Implementation Under Additional Constraints.....................................................21
5.2. Design of the Nozzle Ring...............................................................................................23
5.3. Design of the Rotor..........................................................................................................25
5.3.1. Preliminary Design................................................................................................25
5.3.2. The Bezier Curve...................................................................................................27
5.3.3. Implementation of the Design Strategy.................................................................28
5.3.4. Supplementary Issues on 3D Design of the Rotor.................................................31
5.4. Design of the Diffuser......................................................................................................33
6. Mesh Generation......................................................................................................................35
6.1. Choice of the Grid............................................................................................................35
iv
6.2. Meshing the Boundary Layer...........................................................................................36
6.3. Quality of the Mesh..........................................................................................................38
6.3.1. Skewness................................................................................................................38
6.3.2. Orthogonal Quality................................................................................................39
6.3.3. Jacobian Ratio........................................................................................................39
6.4. Meshing of Components...................................................................................................40
7. CFX® Setup..............................................................................................................................46
7.1. Mathematical Model for Turbulence................................................................................46
7.2. Near-Wall Treatment........................................................................................................47
7.3. Boundary Conditions and Interfaces...............................................................................47
7.4. Choice of Off-Design Points............................................................................................48
8. Results.....................................................................................................................................50
8.1. Design Point.....................................................................................................................50
8.2. Off-Design Points............................................................................................................54
8.2.1. Off-Design Point 1................................................................................................55
8.2.2. Off-Design Point 4................................................................................................56
8.2.3. Off-Design Point 5................................................................................................57
8.2.4. Off-Design Point 7................................................................................................58
9. Discussion...............................................................................................................................59
10. Conclusions and Future Works...............................................................................................61
11. Bibliography............................................................................................................................62
Appendix 1: Absolute Angle at Rotor Inlet..................................................................................65
v
LIST OF FIGURES
NUM. TITLE PAG.
Figure 1 Thermodynamics of a radial turbine (Nguyen-Schäfer, [16], adapted) 1
Figure 2 Scheme of a 90° IFR turbine. Left: frontal view. Right: side view (Ventura,
[13])
2
Figure 3 Flow velocity triangles within a radial turbine (Dixon, [6]) 3
Figure 4 Thermodynamic diagram of the process through a 90° IFR turbine (Dixon,
[6])
4
Figure 5 Nominal design configuration (Saravanamuttoo, [19], adapted) 5
Figure 6 Behaviour of loss terms as function of incidence angle (Yahya, [33]) 6
Figure 7 Secondary flow in a blade passage (Yahya, [33]) 7
Figure 8 Full process iter in turbine design (Khader, ,[9]) 8
Figure 9 Schematic diagram of a vaneless volute casing (Whitfield, [31]) 16
Figure 10 Left – velocity profile across the centre-line of the volute section. Right –
variation of centroid radius at two subsequent azimuth positions (Whitfield,
[31], adapted)
18
Figure 11 Theoretical distribution of centroid radius, cross section area and flow
angle with azimuth location
20
Figure 12 Distribution of centroid radius, cross section area and flow angle with
azimuth location. Comparison between theoretical and implemented
solution, vaneless case
21
Figure 13 Distribution of centroid radius, cross section area and flow angle with
azimuth location. Comparison between theoretical and implemented
solution, vaned case
22
Figure 14 3D geometrical model of the volute casing. Left: vaneless. Right: vaned (no
nozzle)
23
Figure 15 Nozzle vane geometry definition (Rajoo & Martinez-Botas, [18]) 24
Figure 16 Nozzle ring. Left: sketch in the frontal plane. Right: shape of the blade 24
Figure 17 3D geometrical model of the nozzle ring 25
Figure 18 Sketch of the velocity triangles at rotor inlet and outlet (Saravanamuttoo,
[19], adapted)
26
Figure 19 Rotor views. Left: r-z (or meridional) plane. Right: θ-z (or blade to blade)
plane
27
Figure 20 Bezier curve of degree 3. Left: basis of vector space. Right: control points
(Floater, [7], adapted)
28
Figure 21 Rotor: θ-distribution (top), β-distribution (middle), thickness distribution
(down)
29
Figure 22 Rotor: distribution of wrap angle (top left), flow angle (top right) and
thickness (bottom) in the meridional plane
30
Figure 23 Rotor: variation from inlet to outlet of channel cross-section area (left) and
lean angle (right)
31
Figure 24 3D geometrical model of the rotor 31
Figure 25 Geometrical definition of the problem 32
Figure 26 Pressure distribution in a crosswise section. Effect of streamwise pressure
gradient (left), effect of blade-to-blade pressure gradient (middle),
ensemble (right) (Van den Braembussche, [28])
33
Figure 27 Conical diffuser. Left: 2D sketch. Right: lines of appreciable stall for given
geometrical configuration (Blevins, [4], adapted)
34
Figure 28 3D geometrical model of the diffuser 34
vi
Figure 29 Elements of a 3D mesh - tetrahedron, hexahedron, prism, pyramid 35
Figure 30 Velocity profile in a turbulent boundary layer (Bakker, [3]) 36
Figure 31 Non-dimensional velocity as function of 𝑦+ in the inner region (Kundu,
[11])
37
Figure 32 Stretching of a quadrilateral element. Nominal shape (left), deformed
shape (right)
38
Figure 33 Orthogonal quality on a 2D quadrilateral cell 39
Figure 34 Mapping of an hexahedral element (Bucki, [5]) 39
Figure 35 Example of setup of 𝐴𝑁𝑆𝑌𝑆®Meshing (volute) 40
Figure 36 Mesh of the volute (section) 42
Figure 37 Mesh of the diffuser (ensemble) 42
Figure 38 Mesh of the nozzle ring (one blade) 43
Figure 39 Topology for the rotor blade. The blade (blue) is surrounded by meshing
blocks
43
Figure 40 Mesh of the rotor (portion) 44
Figure 41 Mesh statistics for the rotor 45
Figure 42 Illustration of interfaces and boundary conditions for vaned configuration 48
Figure 43 Choice of representative off-design points (Mora, [1], adapted)
49
Figure 44 Spanwise distribution of 𝛽2, comparison at design point 51
Figure 45 Spanwise distribution of 𝛼3, comparison at design point 51
Figure 46 Rotor blade loading comparison at 10% span (top, left), 50% span (top,
right), 90% span (bottom), design point
52
Figure 47 Static entropy around the blade at 90% span. Left: vaneless. Right: vaned 52
Figure 48 Mach distribution around the blade at 10% span. Top: vaneless.
Bottom:vaned
53
Figure 49 Mach distribution around the blade at 50% span. Left: vaneless. Right:
vaned
53
Figure 50 Mach distribution around the blade at 90% span. Left: vaneless. Right:
vaned
53
Figure 51 Velocity contour around nozzle ring blade, 50% span, design point 54
Figure 52 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 1 55
Figure 53 Velocity distribution around the blade at 50% span. Left: vaneless. Right:
vaned
55
Figure 54 Velocity distribution around the blade at 90% span. Left: vaneless. Right:
vaned
55
Figure 55 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 4 56
Figure 56 Static entropy in meridional lane, circumferential average, off-design point
4. Left: vaneless. Right: vaned
56
Figure 57 Velocity distribution around the blade (left) and blade loading (right) at
90% span, off-design point 4
56
Figure 58 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 5 57
Figure 59 Static entropy in meridional plane, circumferential average, off-design
point 5. Left: vaneless. Right: vaned
57
Figure 60 Velocity contour around nozzle ring blade, 50% span, off-design point 5 57
Figure 61 Spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 7 58
Figure 62 Rotor blade loading, comparison at 10% span (top, left), 50% span (top,
right), 90% span (bottom), off-design point 7
58
Figure 63 spanwise distribution of 𝛼2. From top to bottom: design point, off-design 1,
off-design 4, off-design 5, off-design 7
66
vii
LIST OF TABLES
NUMBER TITLE PAGE
Table 1 Meanline design parameters for nozzle ring 19
Table 2 Meanline design parameters for vaneless and vaned volutes 20
Table 3 Relative angle at rotor outlet under nominal design condition 26
Table 4 Meanline design parameters for the rotor 28
Table 5 Meanline design parameters for the diffuser 33
Table 6 Estimation of first layer thickness for turbine components 41
Table 7 Mesh statistics for turbine components 41
Table 8 Thermodynamic properties of the studied off-design points 49
Table 9 Performance comparison, design point 50
Table 10 Comparison between mean velocity triangles at rotor inlet (top) and at
rotor outlet (bottom), design point
50
Table 11 Comparison of performance (top) and mean flow angles (bottom) at off-
design points
54
viii
NOMENCLATURE
ABBREVIATIONS
ATM Automated Topology and Meshing
CAD Computer Aided Design
CFD Computational Fluid Dynamics
DNS Direct Numerical Simlation
FEA Finite Element Analysis
FEM Finite Element Method
IFR Inflow Radial
JR Jacobian Ratio
LE Leading Edge
PDE Partial Differential Equation
RANS Reynolds Averaged Navier-Stokes
RPM Revolutions per Minute
OQ Orthogonal Quality
SBP Single Blade Passage
SST Shear Stress Transport
TE Trailing Edge
T-T Total-to-Total
1-2-3D One-Two-Three Dimensional
GREEK SYMBOLS
α Absolute flow angle
Constant (k- ω model)
β Relative flow angle
Blade angle (rotor)
Constant (k- ω model)
γ Specific heat ratio
δ Boundary layer thickness
ε Dissipation of turbulent kinetic energy
η Efficiency
θ Wrap angle (rotor)
Non-dimensional mass flow rate (volute)
Deformation angle (mesh statistics)
λ Angle (general notation)
μ Dynamic viscosity
μ𝑡 Turbulent viscosity
ν Kinematic viscosity
ξ General coordinate
ρ Density
ζ Constant (k- ω model)
η Shear stress
Φ, θ Azimuth angle (volute)
ψ Diffusion angle (diffuser)
Ω, ω Angular velocity
Dissipation rate of turbulent kinetic energy (RANS model)
ROMAN SYMBOLS
ix
a Speed of sound
A Area
b Passage width (volute)
Vane axial chord (nozzle ring)
c Absolute flow velocity
Vane true chord (nozzle ring)
𝐶𝑓 Skin friction coefficient
g Gravitational acceleration
h Enthalpy
Blade height (volute)
i,j,k Indexes (general notation)
k Constant (volute)
Turbulent kinetic energy (RANS model)
𝑘𝑎 Axial length (rotor)
l Axial length (diffuser)
L Work exchange
m Meridional coordinate
Vortex exponent (volute)
Mass (general notation)
𝑚 Mass flow
M Mach number
Moment of external forces (general notation)
n Rotational speed
Coordinate normal to streamline (rotor)
o Nozzle throat
P, p Pressure
Constant (volute)
q Heat exchange
R, r Radius
𝑅𝑒 Reynolds number
s Entropy
Nozzle pitch (nozzle ring)
S Angular momentum ratio (volute)
T Temperature
u Velocity (general notation)
U Blade speed
Velocity (general notation)
𝑢∗ Friction velocity
w Relative flow velocity
W Work exchange
X, x General coordinate
y General coordinate
𝑦+ Non-dimensional coordinate normal to a wall
𝑍𝑣 Number of blades (nozzle ring)
𝑍𝑏 Number of blades (rotor)
SUBSCRIPTS (*)
0 Stagnation property
1 At volute inlet
2 At volute outlet, At rotor inlet
x
3 At rotor outlet, At diffuser inlet
4 At diffuser outlet
a Relative to axial coordinate
b Relative to the blade
e Relative to ideal case (mesh statistics)
h Relative to blade hub (rotor)
i,j,k Relative to indexes i,j,k
m Relative to meridional coordinate
n Relative to the coordinate n
nr At nozzle ring
r Relative to radial coordinate
R Recirculating at volute tongue (volute)
ref Relative to a reference quantity
s Along isentropic transformation, Static
t Stagnation property
Relative to blade tip (rotor)
Relative to the tangential coordinate (rotor)
ts Total-to-static
tt Total-to-total
x Relative to the coordinate x
y Relative to the coordinate y
θ Relative to circumferential coordinate
(*) Some subscripts used in the notation are self-explanatory (i.e. inlet, wall, max…) and are not listed here
1
1 – INTRODUCTION
In an internal combustion engine a turbocharger is a system composed by a radial turbine and a
centrifugal compressor mounted on a common shaft. The turbine converts part of the enthalpy of
the exhaust gases into kinetic energy delivered to the shaft. The shaft drives the compressor
which increases the air pressure sent to the combustion chamber at each piston cycle, allowing
an increase in power output: for this reason turbocharging is often considered in applications
where power demand is a priority, such as in race car, heavy duty vehicle and marine engines.
Turbocharging is also beneficial from the point of view of overall engine efficiency. As noted by
Mora [14], more power available per unit cycle leads to smaller engines, with consequent
decrease of mechanical losses, and the higher density of air at the combustion chamber improves
volumetric efficiency. Moreover the performance of a supercharged engine is only minimally
affected by variations in ambient pressure, since pressure is a design parameter of the
turbocharger itself at steady conditions: consequently the engine operates at more constant
regime.
The absolute performance of a turbocharger is limited by the energy content of the working
fluid, defined by its total enthalpy. As shown in Fig.1, the work that the turbine can extract is the
difference between the total enthalpy at rotor inlet and the total enthalpy at rotor outlet, i.e.
without taking into account the contribution of flow speed. Whatever velocity component the
flow still has at outlet, infact, is “wasted” from a thermodynamic point of view; in the ideal case
the turbine should be able to expand the flow so to reach outlet pressure at zero velocity, and
without introducing losses (i.e. isentropically).
Figure 1: thermodynamics of a radial turbine (Nguyen-Schäfer, [16], adapted)
The ratio between the actual work and the total available energy defines the efficiency of the
turbine, which is a key driver in the design of the turbomachinery.
In this introductory part a description of the radial turbine is given: focus is placed on the
thermodynamic process underwent by the flow through all components of the turbine and on
2
physical aspects of power generation. Then the main dissipative phenomena arising in a real-case
turbomachinery are presented, since their knowledge is crucial for design purpose. The
subsequent section illustrates the steps of the design process, which also the present work is
based on. The introduction is concluded by a literature review highlighting the results obtained
so far in the design of radial turbines.
1.1 The radial turbine
For turbocharger applications a 90° IFR turbine is generally employed, as it guarantees
compactness, good efficiency within a wide range of operating conditions and high structural
strength. A sketch of such a turbine is shown in Fig.2.
Inside the volute the flow is accelerated and at the outlet it is delivered uniformly with a desired
outflow angle. At this stage the flow can enter directly into the rotor or pass through a nozzle
ring (stator): the former configuration is called vaneless, the latter vaned. The nozzle ring, when
present, aims at guiding the flow into the rotor inlet, and since the passage between blades is
convergent it further increases its speed. Inside the rotor (also called impeller) the flow exerts an
aerodynamic force on the blades, thus transferring part of its energy to the rotating device: the
goal of the impeller is to extract the highest possible amount of work from the working fluid.
The flow exits the impeller with a mainly axial velocity component and passes through a
diffuser, where it is slowed down and part of its static pressure is recovered before being
discharged.
Figure 2: scheme of a 90° IFR turbine. Left: frontal view. Right: side view (Ventura, [13])
The process can be described from a thermodynamic point of view, which is useful to highlight
the physical quantities affecting the performance of the turbine and to deduce preliminary
considerations about its design.
Most turbomachinery flow processes are adiabatic (Dixon, [6]), and also gravitational effects are
negligible; the first equation of thermodynamics for an open system (Negri di Montenegro, [15])
may then be re-written in terms of total enthalpy 0 as shown in eqn.(1.1)
3
𝑑 + 𝑐𝑑𝑐 + 𝑔𝑑𝑧 = 𝑑𝑞 − 𝑑𝐿 𝑑𝑞=0; 𝑑𝑧=0; −𝑑𝐿=𝑑𝑊 𝑑 + 𝑐𝑑𝑐
𝑑 +𝑐2
2
= 𝑑𝑊 0≝+
𝑐2
2 𝑑0 = 𝑑𝑊 (1.1)
The work per unit mass exerted by the flow on the turbine shaft is linked to the rate of change of
angular momentum that the flow itself undergoes inside the rotor, and may be expressed by the
so called “Euler turbine equation”, here in differential form (for the exact derivation see Mora
[14])
𝑑𝑊 = 𝑑 𝑈𝑐𝜃 (1.2)
The combination of eqn.(1.1) and eqn.(1.2) yields to eqn.(1.3):
𝑑(0 − 𝑈𝑐𝜃) 𝑟𝑜𝑡𝑎𝑙𝑝𝑦
= 0 (1.3)
which states that in the thermodynamic process through the impeller the rotational total enthalpy
𝐼 ≝ 0 − 𝑈𝑐𝜃 (also called rothalpy) is constant. The same expression is re-arranged according
to the velocity triangle in Fig.3, which relates the velocity vectors in the fixed and the rotating
frame of reference:
𝐼 = +1
2𝑐2 −𝑈𝑐𝜃 = +
1
2 𝑤2 + 𝑈2 + 2𝑈𝑤𝜃 − 𝑈 𝑤𝜃 + 𝑈 = +
1
2𝑤2 −
1
2𝑈2 (1.4)
Figure 3: flow velocity triangles within a radial turbine (Dixon, [6])
Through eqn.(1.4) it is possible to express the variation of static enthalpy between inlet and
outlet of the rotor (denoted by indexes 2 and 3, Fig.4), thus from the combination of eqn.(1.1)
with eqn.(1.4) the total specific work on the turbine is:
∆𝑊2−3 = +𝑐2
2
2−3=
1
2[ 𝑈2
2 − 𝑈32 − 𝑤2
2 − 𝑤32 + 𝑐2
2 − 𝑐32 ] (1.5)
Eqn.(1.5) leads to the following considerations:
4
a) In the case of an IFR turbine the term 1
2 𝑈2
2 − 𝑈32 is always positive because 𝑈2 > 𝑈3; the
same does not hold for an axial turbine, where inlet and outlet radiuses are equal. As
direct consequence, the work per unit mass that can be extracted with a single turbine
stage is higher in the former case, and so the efficiency, which justifies the exclusive
employment of radial turbines in turbochargers.
b) A positive contribution to the specific work is obtained when 𝑤3 > 𝑤2. This is achieved
if the channels between blades are convergent.
Eqn.(1.4) shows that the higher is 𝑤3 the higher is the static enthalpy jump 2 − 3 across the rotor (and consequently the specific work) but the lower is the static pressure 𝑝3 at rotor outlet (see Fig.4). However 𝑝3 cannot be lower than the atmospheric pressure,
otherwise the discharge would not be possible. The diffuser is used to allow 𝑝3 < 𝑝𝑎𝑡𝑚 (with a benefit for efficiency) and to recover part of the static pressure afterwards so that
at turbine outlet the condition 𝑝4 ≥ 𝑝𝑎𝑡𝑚 is fulfilled.
Moreover “accelerating the relative velocity through the rotor is a most useful aim of the
designer as this is conducive to achieving a low loss flow” (Dixon, [6]).
c) Since the specific work is proportional to the quantity 1
2(𝑐2
2 − 𝑐32) the absolute velocity
should be large at impeller inlet, which is achieved by means of the volute.
The volute (as well as the diffuser) is a static component (𝑑𝑊 = 0) and from eqn. (1.1)
the total enthalpy is conserved (see also Fig.4) which implies that the higher is the drop in
static enthalpy 1 − 2 the higher is the absolute velocity seen by the rotor at its inlet.
Figure 4: thermodynamic diagram of the process through a 90° IFR turbine (Dixon, [6])
With reference to point c) above, a deeper analysis is needed. Eqn.(1.2) can be evaluated
between points 2 and 3, leading to eqn.(1.6)
∆𝑊2−3 = 𝑈2𝑐𝜃2 − 𝑈3𝑐𝜃3 (1.6)
which shows that:
5
a high value of 𝑐2 is not the only requirement because what matters is the projection of 𝑐2
along the circumferential direction, 𝑐𝜃2. The ideal condition would be 𝑐2 ≡ 𝑐𝜃2 but this is
not physical since there would be no inflow at rotor inlet: the absolute flow angle is
chosen such that the relative velocity has only radial component.
A good volute design must therefore take into account the orientation of the absolute
velocity at rotor inlet, which may be accomplished by using vaned stators.
the specific work is increased if the absolute velocity of the flow at rotor outlet is axial,
i.e. 𝑐𝜃3 = 0.
The two conditions mentioned above constitute the so called nominal design (Fig.5).
Figure 5: nominal design configuration (Saravanamuttoo, [19], adapted)
1.2 Sources of losses
Nominal design is a theoretical condition which sets an upper bound to the maximum specific
work that a radial turbine can extract.
In real cases, however, this limit is never reached because dissipative phenomena arise from the
interaction between flow and solid surfaces of the turbomachinery: these phenomena (take Fig.1
as reference) increase the level of entropy and hence lower the jump in total enthalpy across the
turbine, which is linked to the amount of work by eqn.(1.1).
Several loss models have been developed and are now available in literature (Ventura, [13])
since careful evaluation of losses is crucial for performance estimation: however most of them
are based on empirical relations. In this section a qualitative description of the main sources of
losses in radial turbines is presented, and observations are made on how to limit them.
Channel losses
Dissipative phenomenon due to skin friction. Skin friction depends on wet surface
(portion of the surface in contact with the flow), surface roughness and flow speed: a
way to limit channel losses is then to make compact turbines with surfaces as smooth as
possible (limiting the flow speed would infact be detrimental for the expansion process
inside the rotor)
6
Incidence losses
Here are considered, without further distinctions, all losses which arise from the non-
ideal orientation of the velocity vector with respect to the rotor blades (and blades of the
nozzle ring, when present). For a 90° IFR turbine incidence losses originate when the
flow does not enter the rotor radially or leave it axially, as stated in the nominal design.
Incidence losses (also called shock losses, even if they are not related to compressibility
effects) grow almost parabolically with incidence angle (Dixon, [6]): the specific
behaviour depends on the loss model adopted but more in general Fig.6 shows that this
term contributes heavily to overall losses at off-design incidence angles.
Figure 6: behaviour of loss terms as function of incidence angle (Yahya, [33])
A well known solution to reduce incidence losses is to achieve better flow guidance at the
exit of the volute by means of a nozzle ring.
Secondary losses
Secondary flow generally denotes a portion of fluid which does not follow the streamlines
of the main flow.
As an example let us consider a channel between rotor blades (Fig.7). Suction and
pressure sides of adjacent blades create a pressure gradient orthogonal to the streamlines:
this induces a crossflow which develops a boundary layer in the local cross section planes
of the channel. A direct consequence is the onset of shear stresses because of the velocity
gradient (recall 𝜏 = 𝜇 𝜕𝑈
𝜕𝑦 𝑤𝑎𝑙𝑙
); moreover the vorticity injected in the flow creates
eddies which dissipate kinetic energy through turbulent mechanism (energy cascade).
Another region in which the presence of secondary flows is not negligible is in the
neighbourhood of the volute tongue, i.e. the connection between the inlet and the end of
the volute casing, where the flow has a lower pressure than at inlet because it was
accelerated inside the volute itself. The resultant pressure difference drives part of the
flow from the inlet around the tongue, creating a zone of recirculation thus causing
losses.
7
Figure 7: secondary flow in a blade passage (Yahya, [33])
Tip clearance losses
Tip clearance is a gap between blade shroud and casing which is designed to allow the
expansion of the rotor under thermal effect and centrifugal force. Across this gap leakage
occurs between the pressure and the suction side of the blade, driven by the pressure
gradient: the behaviour is the same of a wing of finite span.
This secondary flow through the clearance does not contribute to the work done on the
rotor, causing a reduction of the work output compared to the condition of designed mass
flow. Moreover tip clearance flow is responsible for the formation of vortices which may
also interact with other secondary flows (Siggeirsson et al., [22]).
Tip clearance losses depend on the relative size of the gap compared to the blade height.
1.3 The design process
In section 1.1 an ideal design condition was derived from studying the thermodynamics of the
expansion process inside the radial turbine. As this approach does not require any knowledge
about the internal flow pattern in terms of velocity and pressure fields, but only the average
physical quantities at specific sections of the turbomachinery, it is a common tool at early stages
of the design process: such approach is called meanline design.
Meanline analysis consists of treating the working fluid as a one dimensional flow at the mean
radius of the turbine while the flow parameters are assumed as reasonable average values across
the full span (Wei, [29]).
The flow in a turbine is, however, fully three-dimensional. As it was shown in the previous
section, some dissipative phenomena, like losses due to secondary flows, have an inherent three-
dimensional nature and meanline design would not be able to detect them. The next step is then
to develop a complete geometrical model which should yield to an expected behavior of the
internal flow (taking into account secondary flows as well, if their modelization is available) and
which, on average, is expected to have a performance close to the one forecasted with the
meanline design. Due to high complexity of the internal flow and the amount of constraints to be
fulfilled, well established procedures for this phase of the design process are difficult to develop,
and few results are reported in literature: a key role is still played by experience and personal
know-how of the designer.
8
Finally an accurate performance estimation is done on the 3D geometrical configuration by
simulating the internal flow with CFD techniques. A feasibility check is needed as well in order
to assess structural robustness under operating loads: FEA is commonly employed in this case.
Figure 8: full process iter in turbine design (Khader, ,[9])
“Turbine design is an iterative process” (Khader, [9]). Fig.8 shows the correlation between
different steps: in particular the initial 3D geometry is modified according to the outcomes of the
analysis step. The latter are continuously compared with the results from the meanline design
(which is an idealized design, thus represents the target) and the final configuration is reached
when sufficient convergence exists between the two steps.
1.4 Design and analysis: state of the art
“There are two problems of interest to designers of turbomachinery” (Yang, [34]). In the so
called “inverse” (or design) problem the overall geometry is defined, based on assumptions
about the flow pattern and other existing specifications, to yield desired performance
characteristics, while in the “direct” (or analysis) problem the performance of a given
geometrical configuration is assessed.
The direct problem is commonly tackled by means of CFD tools: despite the approximations
linked to numerically solving NS equations this method allows the simulation of three-
dimensional flows on complex geometries, which explains the presence of a vaste literature on
the analysis of radial turbines including comparative studies among different geometrical
configurations. On the other hand “research activities pertaining to the inverse design problem
has not been extensive” (Tjokroaminata, [26]).
Since both three dimensional design and performance analysis (second and third step of the
design process, respectively) are topics of the present work, a review of the main results of
public domain in these fields is now presented.
1.4.1 Inverse problem: the volute
In three articles Whitfield et al. ([12], [31], [32]) give a comprehensive treatment of vaneless
volutes for radial inflow turbines, which is taken as main reference for volute design in this
9
thesis. Firstly they developed a practical method, suitable for implementation in a program code,
based on simple hypotheses of incompressible flow and free vortex law. Since the outflow angle
is specified as input parameter the aim of Whitfield’s method is to design distributions of
centroid radius and cross section area which ensure the flow to be delivered uniformly and at the
appropriate angle to the impeller. A subsequent experimental investigation on such obtained
geometry showed that the free vortex pattern is only a fair approximation over the first 180° of
azimuth angle, while as the tongue position is approached the variation of the tangential
component of velocity with radius reduces considerably (Whitfield et al., [32]). The free vortex
law was therefore modified accordingly by means of a so called vortex exponent.
The method gives satisfactory results, at least at first attempt, but relies on several purely
empirical considerations such as the shape of the vortex exponent or the variation of Mach
number with azimuth angle. Moreover no indications are given about the optimal shape of the
volute cross sections. The latter aspect is studied by Shah et al. [20] who, after the comparison
between trapezoidal, Bezier-trapezoidal and circular cross sections, suggested that the circular
cross section will give a better efficiency.
Abidat et al. [1] suggest to design the distribution of centroid radius by means of a Bezier
polynomial and the distribution of cross section area by assuming a linear reduction of mass flow
with azimuth angle. The employment of parametric curves makes this method flexible for the
designer and allows her to easily take into account other possible constraints (for example on the
radius at volute outlet) but has somehow an empirical basis. Moreover there is no careful design
of the critical region around the tongue.
1.4.2 Inverse problem: the rotor
Yang [34] pointed out that the complex nature of the flow through a real turbomachine would
make a three-dimensional design procedure difficult if not impossible: for this reason lot of
effort was put on the development of approximate two-dimensional methods in the hub-to-
shroud plane.
An example of such method is presented in the work by Smith & Hamrick [24]. They prescribed
as input parameters blade shape, hub shape and velocity distribution along the hub, then they
introduced an estimated streamline from inlet to outlet of the rotor and checked for continuity of
flow through the annular streamtube within the hub and the streamline. If continuity is not
established, the streamline spacing is adjusted accordingly and another annular streamtube is
constructed over the first, following the same criterion: in this way the final streamline
determines the shape of the impeller shroud.
This method relies on the assumptions of isentropic, steady and non-viscous flow, but its real
limitation is due to the arbitrary choice of the input parameters, which has no theoretical basis
although strongly affects the final solution.
A turbomachinery blading design method in three-dimensional invscid flow was suggested by
Yang [34]. Here the blade is represented by a sheet of bound vorticity, i.e. bound to the solid
surface of the blade. Under the assumptions of steady inviscid and irrotational flow the only
vorticity in the flow field is that generated on the blade surfaces, which is related to the
circulation on a closed path around the axis of rotation: by carefully specifying the mean swirl
distribution (which, if integrated, gives the total amount of circulation) the distribution of bound
vorticity is specified as well, and the blade surface geometry is obtained as that location of the
bound vortex sheet in which the normal velocity vanishes. The condition of non-penetrating flow
must infact hold on the solid surface of the blade.
Tjokroaminata [26] highlights two main drawbacks of Yang’s method:
10
“for a wide variety of swirl distributions there always exists a region of inviscid reversed
flow on the pressure surface of the blade [...] which may result in flow separation”
(Tjokroaminata, [26])
Blades obtained with such design tend to be highly twisted, which may lead to structural
problems especially in applications where the rotational speed of the impeller is high
1.4.3 Inverse problem: the remaining components
Diffuser and fixed stator are relatively simple components and their design is not investigated
extensively.
Siggeirsson & Gunnarsson [22] report that “when the diffusion angle [(defined as the slope of the
wall of the divergent pipe)] is large the diffusion rate is rapid and can cause boundary layer
separation resulting in flow mixing and stagnation pressure losses. On the other hand if the
diffusion rate is too low the required length of the diffuser will be very large and the fluid
friction losses increase”. This opinion is shared by other authors such as Dixon [6], who sets to
7° - 8° the value of the diffusion angle which gives an optimal rate of diffusion.
Khader [9] relied on a direct approach for the design of a fixed-geometry nozzle ring. Straight
symmetric blades with rounded LE and TE are chosen, and an initial 3D geometry of the
component is guessed. A following analysis with CFD is performed, and modifications on the
incidence angle of the blades are made until no separation of the flow is detected. The method
followed by Khader, i.e. the iteration of the analysis step in order to reach an optimized design,
may be rather time consuming, but allows to end up with a configuration in which incidence
losses are minimized.
1.4.4 Direct problem: performance analysis
Simpson [23] performed a CFD analysis of existing test turbine geometries in both vaned and
vaneless configurations. A total of six geometries were analysed and the results compared with
measured turbine performance data. “Steady state predictions showed good agreement with the
experimental trends confirming the vaneless stators to yield higher efficiencies across the full
operating range” [23]
According to the author, vaned stators lead to a higher level of losses because of the wake
detaching from vane trailing edges, boundary layer growth and secondary flows.
Spence et al. [25] tested three pairs of vaneless and corresponding vaned stators within a range of
pressure ratios and flow rates. For each pair of stators the rotor was the same and the operating
conditions were identical. “The vaneless volutes delivered consistent and significant efficiency
advantages over the vaned stators over the complete range of pressure ratios tested. At the
design operating conditions, the efficiency advantage was between 2% and 3.5%” (Spence et al,
[25]).
Padzillah et al. [17] compared nozzled and nozzleless turbines under pulsating flow and found
that “the differences in flow angle distribution between increasing and decreasing pressure
instances during pulsating flow operation is more prominent in the nozzleless volute than in its
nozzled counterpart”, suggesting that the addition of a nozzle ring leads the turbine to more
stable flow angle configurations at off-design points. On the other hand Baines & Lavy [2]
claimed that the advantage of the vaned configuration consists of its highest peak efficiency at
design point, as at off-design this efficiency drops dramatically.
11
2 – MOTIVATION AND OBJECTIVE
2.1 Motivation
It is estimated (ATRI, [27]) that fuel-related cost accounts for around 35% of overall operational
costs of trucking, which justifies the need to improve engine efficiency and nowadays’ extensive
research on turbochargers for applications on heavy-duty vehicles.
Despite attempts to find optimal solutions have been done (and have been summarized in the
literature review), the problem of designing the turbine of a turbocharger so to guarantee the best
achievable performance is still open. The reason is that the complexity of the flow within a
turbomachinery requires the introduction of simplifying assumptions about its mean motion,
which may lead the designer to neglect some dynamics in the flow pattern (typically secondary
flows) and misevaluate losses.
Moreover, even assuming an ideal geometrical configuration, this would be valid for just one
design point. During a combustion cycle, however, pressure and temperature of the exhaust
gases vary over a considerable range, thus the performance of that configuration should be
optimal also at off-design points. Variable angle stators try to achieve the latter condition by
continuously adapting the inflow angle at rotor inlet, hence reducing incidence losses, but the
implementation of such devices is by now considered unfeasible due to high mechanical stresses
and vibrations.
A comparative study performed by Mora [14] pointed out that a turbine with fixed nozzle ring
has both higher efficiency and power output with respect to a vaneless turbine throughout all the
combustion cycle. This statement is based on a meanline analysis, but in order to be tested a
three-dimensional design of the two turbine configurations should be developed and an
investigation should be performed by means of CFD techniques so to describe the internal flow
in details and identify sources of losses otherwise undetectable.
2.2 Objective
In light of the previous research carried out by Mora, and considering the existence of different
opinions reported in literature, the purpose of this paper is to present a comparative
performance assessment between two turbine configurations, vaneless and with static nozzle
ring, working under the same operating conditions at both design and off design points.
12
3 – METHODOLOGY AND TOOLS
3.1 Methodology
The strategy used in the present work is meant to be an implementation of the design process
which was described in the introductory section. In order to achieve the objective several steps
were followed
Literature review
A literature review was carried out firstly. The purpose was to collect background
information about existing configurations of radial turbines and studies of their
performance, as well as techniques for the design of specific components, when
available.
Analysis of the meanline design
The reference meanline design was analysed in details, with specific focus on identifying
the average flow parameters at inlet and outlet of all components of the turbine and
potential geometrical constraints. This phase defines the inputs which enter the
subsequent step.
3D design of components
A theory-based procedure was developed for the 3D design of the volute and the
impeller. Diffuser and static nozzle ring were not designed with an inverse approach but
as first attempt geometries due to their relatively simple shapes. All the components were
drawn by means of a CAD software (for the rotor a dedicated software, allowing a high
flexibility in the design, was employed).
CFD simulation setup
All components were meshed separately. Structured hexahedral mesh was used for the
rotor blade passage while coarser tetrahedral mesh for the remaining components (with a
refinement close to the walls, so to better describe the boundary layer). After meshing,
vaneless and vaned configurations were assembled: the connection of different meshes
was obtained by specifying suitable interfaces on the contact surfaces. Fluid
thermodynamic properties and boundary conditions were specified as well, and an
appropriate turbulence model was chosen for the solution of RANS. At the end of this
step the two turbine configurations were ready for a CFD steady simulation.
Final design
The results of CFD simulations on both vaneless and vaned cases were compared and the
original 3D design was modified until the average velocity vector of the flow at rotor
inlet was the same for both configurations (since rotor and diffuser are also the same, this
guarantees a “fair” comparison between the two turbines, which only differ by the
presence/absence of a static nozzle ring: every difference pointed out in a subsequent
performance analysis may thus be attributed to that component). At the end of this
iterative phase the final geometrical configurations were obtained and the design process
is concluded.
Analysis of results
Given the two final configurations, comparative performance analysis was done at design
and off-design points in order to identify the “best” one, not only in terms of peak
efficiency but within the whole operative range.
13
3.2 Tools – Software
The methodology contains several sub-objectives to be reached before the final comparative
analysis can be done: drawing the 3D model of a component, meshing a blade passage...
For each intermediate task a set of computer programs is employed.
𝑴𝑨𝑻𝑳𝑨𝑩®
MATLAB® is a numerical environment suitable for matrix computation and
implementation of algorithms. Here it is used to implement the 3D design procedure of
the volute and to generate plots from vectors of data.
𝑺𝑶𝑳𝑰𝑫𝑾𝑶𝑹𝑲𝑺®
SOLIDWORKS® is the CAD software which was used to build the complete geometrical
models of volute, nozzle ring and diffuser. It is flexible because it allows to specify
relative constraints between parts of a sketch and adapt it when a modification is done, so
that it is most suitable for the iterative phases of the design process.
Moreover files generated in SOLIDWORKS® can be easily transferred to ANSYS®
package for a subsequent analysis (CFD, FEM...).
𝑫𝑬𝑺𝑰𝑮𝑵𝑴𝑶𝑫𝑬𝑳𝑬𝑹®
DESIGNMODELER® is a CAD software within ANSYS® package. It mainly handles
external geometry models, usually created for manufacturing purpose, and allows
modifications (for example the suppression of non meaningful details) so that the model
is ready for meshing and simulation. DESIGNMODELER® may also be used to draw
geometries from scratch.
In this thesis the program imports geometry files from SOLIDWORKS® and edits them
when needed (by rotating or translating components with respect to the frame of
reference) so that the turbine model is assembled correctly after meshing.
𝑩𝑳𝑨𝑫𝑬𝑮𝑬𝑵®
BLADEGEN® is an ANSYS® software specific for the design of rotative machinery
blades. The designer is allowed to specify the evolution of some representative sections
of the blade in a cylindrical frame of reference and the thickness distribution: the
program automatically generates the CAD model of the machine and monitors several
key parameters such as the cross-section area of the flow channel, the flow angle
distribution...
The design of the impeller of the radial turbine was done with BLADEGEN®.
𝑻𝑼𝑹𝑩𝑶𝑮𝑹𝑰𝑫®
TURBOGRID® is an ANSYS® software, specific for rotating machinery, which creates
high quality hexahedral meshes. The program imports the model of the impeller from
BLADEGEN® and meshes the blade passage: when ATM default feature is enabled, the
program chooses the optimal topology for a given blade geometry and allows to create a
good quality mesh in a highly automated way and with minimal effort, with no need for
control point adjustment.
𝑴𝑬𝑺𝑯𝑰𝑵𝑮®
The tool ANSYS MESHING® imports the geometry of a component from
DESIGNMODELER® and allows the creation of a mesh in a guided and automated way.
The designer has control over width and shape of the cells, can make local refinements
close to sharp edges or narrow passages and can build inflation layers, i.e. structured
14
mesh close to a wall to capture the boundary layer. Once fixed the setup parameters,
mesh generation is program-controlled, thus the process is fast and robust, and
subsequent modifications are directly implementable.
Volute, diffuser and nozzle ring are meshed with this software.
𝑪𝑭𝑿®
CFX® is a general-purpose commercial CFD software which is commonly used in
problems involving rotating machinery (turbines, pumps, fans...). CFX® has a
preprocessor in which the setup for the simulation is defined: this includes the definition
of fluid properties, boundary conditions, turbulence model, target of accuracy... All
components of the turbine, once meshed, are imported in CFX® preprocessor and
assembled there to reach the final configuration (either vaneless or vaned).
The final solution is considered achieved when some monitoring parameters, typically
the residuals of mass and momentum equations, have converged below a minimal
threshold value. CFX® also includes a postprocessor which allows to analyse the solution
and obtain a physical interpretation of the result.
15
4 – LIMITATIONS
Uncertainties of CFD computation
The present work consists of a numerical performance evaluation of a radial turbine, as
the internal flow is modelled with a CFD commercial software. However, “due to
modelling, discretization, and computation errors, the results obtained from CFD
simulations have a certain level of uncertainty. It is important to understand the sources
of CFD simulation errors and their magnitudes to be able to assess the magnitude of the
uncertainty in the results” [35].
Such uncertainty is mainly linked to the mathematical model describing the flow
(accuracy of the turbulence model, model for Reynolds stress when solving RANS, wall
functions...), to the method for the discretization of PDE’s, to the quality of the
computational grid and computer round-off errors. Moreover the validity of the
numerical solution is jeopardized wherever large regions of separated flow exist in the
domain, which may be the case for certain turbine configurations, especially at off-design
points. These sources of uncertainty constitute a limitation for the study because a certain
set of data (for example a performance difference between vaneless and vaned turbine
configurations at given working condition) may fall inside the uncertainty range, and
since experimental results are not available and cannot be used to validate the numerical
model, those data would lead to misleading interpretations.
Simplifications in the computational domain
In the following chapters it will be seen that the mesh of a full radial turbine may be
rather “heavy”, especially if low values of 𝑦+ are set as requirement. A simplifying
approach present in literature [22] and also adopted in this work is to mesh a single rotor
blade passage (SBP) and assume symmetry of the flow around the rotational axis: in this
way, however, the non-uniformities of the flow within different rotor blade-to-blade
channels are neglected.
Simplifications in the 3D geometrical model
The design of simplified geometrical models may lead to neglecting or misevaluating
certain features of the flow pattern, therefore it constitutes a limitation to the validity of
the results obtained in this study. This is particularly true for the rotor, where tip
clearance, scalloping and other geometrical details associated with the onset of secondary
flows are not accurately modelled at a first-attempt design.
Moreover the design of all components of the turbine is based on modelling assumptions
which introduce further simplifications (details in the following chapter)
16
5 – DESIGN OF COMPONENTS
5.1 Design of the volute
In a radial inflow turbine the volute has the purpose to decrease the static pressure of the working
fluid and to increase its speed (conversion to kinetic energy) in order to reach desired values of
velocity and flow angle at rotor inlet. Moreover the volute must distribute the flow uniformly
along the azimuth direction and perform the energy conversion as efficiently as possible, that is
with a minimum loss in stagnation pressure.
With reference to Fig.9, the constraints on the volute design are the following:
Radius at outlet (𝑅2) and passage width (𝑏2). These are set by the rotor geometry and
define the volute discharge area (𝐴2).
Mach number (𝑀2) and absolute flow angle (𝛼2) at outlet. These requirements are
imposed by the designed performance of the rotor.
Thermodynamic flow conditions at volute inlet (𝑃01 , 𝑇01 , γ). These parameters are set by
the working point of the engine.
Mach number (𝑀1) at inlet, linked to the velocity of the exhaust gases.
The goal of the preliminary 3D design is to size the flow passage in terms of the variation, with
azimuth angle (φ), of cross-section area and centroid radius. The solving strategy illustrated in
the following section is a modified version of the procedure whose original development is due
to Whitfield [31], and which is based on the assumptions of adiabatic incompressible flow and
conservation of angular momentum. Notice that incompressibility is a rough assumption, as the
flow undergoes a variation of Mach number inside the volute: however, according to the design
parameters (Tab.2), the flow regime is low subsonic, moreover the resultant design turns out to
be acceptable at first attempt, thus the assumption is ultimately justified by experience.
Figure 9: schematic diagram of a vaneless volute casing (Whitfield, [31])
17
5.1.1 Theoretical procedure
Newton’s second law of motion applied to moment forces reads: “For a system of mass m, the
vector sum of the moments of all external forces acting on the system about some arbitrary axis
A–A fixed in space is equal to the time rate of change of angular momentum of the system about
that axis” (Dixon, [6])
𝑀 = 𝑑
𝑑𝑡(𝑅𝐶𝜃) (4.1)
In a purely ideal case no external forces, so moments, are applied on a fluid particle moving from
the volute inlet to the volute outlet. In practice viscous shear forces, even if weak (due to high
temperature of the exhaust gas its viscosity is low), are not absent, and the angular momentum is
not fully conserved. Eqn.(4.1), evaluated from inlet to outlet, then becomes
𝑅2𝐶𝜃2 = 𝑆𝑅1𝐶𝜃1 (4.2)
where S is the angular momentum ratio across the volute.
For an adiabatic flow eqn.(4.2) can be developed in terms of absolute Mach numbers and flow
angles to give the volute radius ratio as
𝑅1
𝑅2=
𝑀2 sin𝛼2
𝑆𝑀1 sin𝛼1 1 + 𝛾 − 1 /2 𝑀1
2
1 + 𝛾 − 1 /2 𝑀22
12
4.3
The area ratio is derived from the conservation of mass between sections 1 and 2 (Fig.9)
𝐴1
𝐴2=𝑃02
𝑃01
𝑀2 cos𝛼2
𝑀1 sin𝛼1 1 + 𝛾 − 1 /2 𝑀2
2
1 + 𝛾 − 1 /2 𝑀12
− 𝛾+1
2 𝛾−1
(4.4)
where the stagnation pressure at outlet 𝑃02 is a function of the target efficiency of the volute (in
ideal case 𝑃02 = 𝑃01).
It should be noticed that eqns.(4.3) and (4.4) depend on 𝛼1, which, in the original procedure
(Whitfield, [31]), is given as input parameter. However the inflow angle at inlet is unknown at
this stage, because it is linked to the orientation of section 1 (take Fig.9 as reference) which in
turn depends on the tangent to the centroid locus in section 1, not yet determined. Thus 𝛼1 must
be either entred as guess parameter or estimated.
Here 𝛼1 is derived by equating the expressions of non-dimensionalised mass flow rate 𝜃 =𝑚
𝜌0𝑎0𝐴
between sections 1 and 2, expressed in terms of Mach numbers
𝑀1 sin𝛼1 1 +𝛾 − 1
2𝑀1
2 − 𝛾+1
2 𝛾−1
= 𝑀2 cos𝛼2 1 +𝛾 − 1
2𝑀2
2 − 𝛾+1
2 𝛾−1
(4.5)
This approach is not rigorous because the quantity which is conserved is not 𝜃 but 𝑚 . However
the final design under such approximation will be shown, after CFD simulation, to fulfill the
requirements at rotor inlet, and the qualitative behavior of 𝑅, 𝐴 and 𝛼 with azimuth angle
forecasted by Whitfield is still respected.
18
Once the overall volute geometry is defined in terms of inlet-to-outlet radius and area ratio the
next step is to extend the analysis to the derivation of geometric parameters as function of the
azimuth angle. To this purpose the following hypotheses are introduced:
Mach number at passage centroid increases linearly with θ → 𝑀𝑦 = 𝑀1 + Φ
2𝜋 𝑀2 −
𝑀1
Mass flow rate decreases linearly with θ → 𝑚 𝑦
𝑚 = 1−
Φ
2𝜋 +
𝑚 𝑅
𝑚 , where 𝑚 𝑅 is the flow
rate recirculating and mixing below the volute tongue (𝑚 𝑅 ≈ 5% at first estimate)
Stagnation pressure decreases linearly with θ → 𝑃02 = 𝑃01 − Φ
2𝜋 𝑃01 − 𝑃02
The vortex exponent m, which takes into account the modification of the free vortex
condition, varies with θ → 𝑚 = 𝑚0 − 𝑘𝛷𝑝 , where 𝑚0 = 1 is the exponent at volute
inlet while k and p are experimental constants.
Figure 10: Left – velocity profile across the centre-line of the volute section. Right – variation of
centroid radius at two subsequent azimuth positions (Whitfield, [31], adapted)
The swirling flow is subjected to the free vortex relation (conservation of momentum), corrected
by the vortex exponent which models the presence of small tangential forces arising from the
non ideal volute design
𝐶𝜃𝑅𝑚 = 𝑐𝑜𝑛𝑠𝑡 4.6
Eqn.(4.6) allows to express the variation of tangential velocity between any two angular
locations X and Y separated by a small angle 𝛥𝛷 (see Fig.10 – left)
𝐶𝜃𝑦
𝐶𝜃𝑥= 1−
𝛥𝛷
2𝜋
𝐶1𝑅1 sin 𝛼1
𝐶𝑥𝑅𝑥 sin 𝛼𝑥 1 − 𝑆
𝑆𝑥𝑦
𝑅𝑥𝑅2 𝑚𝑥
𝑅2
𝑅𝑦
𝑚𝑦
(4.7)
where 𝑆𝑥𝑦 represents the local dissipation of angular momentum due to wall skin friction forces.
19
The flow angle at Y can now be derived from eqn.(4.7) in terms of the known Mach numbers
and 𝛼𝑥 (known as well): the centroid radius 𝑅𝑦 , unknown at this stage, is derived from
geometrical considerations (Fig.10 – right).
Finally the conservation of mass is applied between section 1 and the generic section Y in order
to find an expression for the area ratio
sin 𝛼𝑦 = 𝑆𝑥𝑦 sin 𝛼𝑥 𝑅𝑥𝑅2 𝑚𝑥
𝑅2
𝑅𝑦
𝑚𝑦
1 + 𝛾 − 1 /2 𝑀𝑦
2
1 + 𝛾 − 1 /2 𝑀𝑥2
12
(4.8)
𝑅𝑦 = 𝑅𝑥 − 𝛥𝑅𝑥𝑦 = 𝑅𝑥 −𝑅𝑦𝛥𝛷
tan𝛼𝑥 → 𝑅𝑦 =
𝑅𝑥
1 +𝛥𝛷
tan 𝛼𝑥
(4.9)
𝐴1
𝐴𝑦=𝑃0𝑦
𝑃01
𝑚
𝑚 𝑦
𝑀𝑦 sin𝛼𝑦
𝑀1 sin𝛼1 1 + 𝛾 − 1 /2 𝑀𝑦
2
1 + 𝛾 − 1 /2 𝑀12
− 𝛾+1
2 𝛾−1
(4.10)
5.1.2 Implementation of the theoretical procedure
The procedure described above was implemented in a MATLAB®code, with the input parameters
coming from the meanline design.
In order to compare vaneless and vaned volutes at equal working conditions, both configurations
are required, at design point, to deliver to the rotor a flow with the same average velocity vector.
Thus the design specifications at volute inlet and outlet must be the same as well. However in the
vaned case the nozzle ring behaves as a convergent duct, hence the total acceleration is
distributed between the two components.
PARAMETER VALUE
Radius at nozzle inlet 𝑟𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 = 52.3 [mm]
Radius at nozzle outlet 𝑟𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡 = 41.8 → 41 [mm] (*)
Blade height = 13 [mm]
Table 1: meanline design parameters for nozzle ring
(*) The radius at outlet was slightly changed from the meanline value so to have complete accordance between the
geometrical dimensions of the two volute configurations.
The acceleration that the flow undergoes inside the nozzle ring is derived from the conservation
of mass between inlet and outlet sections. This estimate is approximated, as density and
temperature of the flow are supposed not to change during the process.
𝑚 𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 = 𝑚 𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡 → 𝜌𝑀𝑎 𝑢
2𝜋𝑟 𝐴
𝑛𝑟−𝑖𝑛𝑙𝑒𝑡
= 𝜌𝑀𝑎 𝑢
2𝜋𝑟 𝐴
𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡
(4.11)
If the blade height is constant, a rearrangement of eqn.(4.11) leads to an expression for the Mach
number at nozzle inlet, which is the only unknown.
20
𝑀𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 = 𝑀𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡
𝑟𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡𝑟𝑛𝑟−𝑖𝑛𝑙𝑒𝑡
(4.12)
where 𝑟𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 and 𝑟𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡 are reported in Tab.1. The input parameters for volute design are
summarized in Tab.2. The solution is shown in Fig.11.
PARAMETER VANELESS VANED (no nozzle)
Mach at volute inlet 𝑀1 = 0.21 𝑀1 = 0.21 Mach at volute outlet 𝑀2 = 0.56 𝑀2 = 𝑀𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 = 0.44
Flow angle at volute outlet 𝛼2 = 68 [°] 𝛼2 = 68 [°] Radius at volute outlet 𝑅2 = 41 [𝑚𝑚] 𝑅2 = 𝑟𝑛𝑟−𝑖𝑛𝑙𝑒𝑡
= 52.3 [𝑚𝑚] Outlet passage width = 13 [𝑚𝑚] = 13 [𝑚𝑚]
Flow properties at inlet
𝑃01 = 2.56 [𝑏𝑎𝑟] 𝑇01 = 846 [𝐾]
𝑚 1 = 0.3322 [𝑘𝑔
𝑠]
𝛾 = 𝛾𝑎𝑖𝑟 |𝑇01= 1.34
𝜌1 = 1.037 [𝑘𝑔
𝑚3]
𝑃01 = 2.56 [𝑏𝑎𝑟] 𝑇01 = 846 [𝐾]
𝑚 1 = 0.3322 [𝑘𝑔
𝑠]
𝛾 = 𝛾𝑎𝑖𝑟 |𝑇01= 1.34
𝜌1 = 1.037 [𝑘𝑔
𝑚3]
Additional constraints 𝑅1 = 82.14 [𝑚𝑚] 𝐴1 = 2733 [𝑚𝑚^2]
𝑅1 = 82.14 [𝑚𝑚] 𝐴1 = 2733 [𝑚𝑚^2]
Table 2: meanline design parameters for vaneless and vaned volutes
Figure 11: theoretical distribution of centroid radius, cross section area and flow angle with
azimuth location
21
5.1.3 Implementation under additional constraints
The solution deriving from the application of the theoretical procedure may not be
implementable due to the presence of additional constraints on the design. In particular,
requirements on the maximum size of the turbine may limit radius or area at volute inlet. The
designer may then look for a sub-optimal solution, i.e. a solution which is as close as possible to
the ideal one yet fulfilling all the constraints.
For the present case the meanline design reports values of 𝑅1 and 𝐴1 which were interpreted as
additional geometrical constraints (Tab.2): however the theoretical solution does not match the
values expected at volute inlet, as seen in Fig.11.
Qualitatively, a constraint on 𝑅1 is reflected into a modification of the radius distribution with
respect to the ideal case. Following a strategy suggested by Abidat [1], the modified centroid
radius distribution is modeled with a third order Bezier polynomial, a parametric curve defined
by 4 monitoring points, conveniently chosen, which allow high control over the shape of the
curve and its derivative. Details about the Bezier approach will be given in Section 5.3 “Design
of the Rotor”. The polynomial is numerically derived in order to obtain the flow angle
distribution, which is geometrically represented by the local tangent to the centroid locus (see
Fig.10- right). Given 𝛼 at the generic azimuth position and provided that all the other parameters
remain constant, the distribution of cross section area is computed from eqn.(4.10).
The implementation of the solving procedure under constraints is done in a MATLAB® code and
results are reported in Fig.12 and Fig.13.
Figure 12: distribution of centroid radius, cross section area and flow angle with azimuth
location. Comparison between theoretical and implemented solution, vaneless case
22
Figure 13: distribution of centroid radius, cross section area and flow angle with azimuth
location. Comparison between theoretical and implemented solution, vaned case
Observations
Overall the implemented solution follows the theoretical one in the central range of
azimuth positions (from 𝜑 ≈ 100° to 𝜑 ≈ 300° ). Major deviations occur close to the
inlet, where the constraints are specified, but local modifications are done after a
subsequent CFD analysis as part of the iterative phase of the design process.
Fig.13 shows that a modified radius distribution may lead to a non negligible variation of
the outflow angle with respect to the designed value. One solution is to use a higher
degree Bezier polynomial, with more control points in order to model the slope of the
curve at the edges. However this example highlights the difficulty to fulfill all constraints
at once and the limits of inverse design methods.
The Bezier polynomial is transformed into a cartesian curve in space and imported in
SOLIDWORKS®. The curve represents the locus of the centroids of all the cross sections.
Circular cross section is chosen, as it is the simplest shape and leads to an efficient configuration
according to literature (Shah, [20]). The area of each section is sized according to the solution of
the 3D design procedure: slight modifications are made so that, for a correct matching of the
components, the outlet passage width is kept constant at all azimuth positions and equal to the
designed blade height of the impeller,
3D CAD models of the volute casing are shown below (Fig.14).
23
Figure 14: 3D geometrical model of the volute casing. Left: vaneless. Right: vaned (no nozzle)
5.2 Design of the nozzle ring
The vaned stator must guarantee the designed flow angle at rotor inlet, ideally without
introducing additional losses.
The constraints on the design of the nozzle ring are below (their values are reported in Tab.1)
Radius at inlet (𝑟𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 ) and radius at outlet (𝑟𝑛𝑟−𝑜𝑢𝑡𝑙𝑒𝑡 ). These must match the outlet
radius of the volute and the inlet radius of the impeller respectively
Blade height (). This requirement is set by the volute outlet passage width
Moreover, other parameters from the meanline design are specified: mass flow rate from the
volute outlet section (𝑚 = 0.3322 [𝑘𝑔
𝑠]), fluid density (𝜌 = 1.037 [
𝑘𝑔
𝑚3]), speed of sound (𝑎 =
564.3 [𝑚
𝑠]), required flow angle at rotor inlet (𝛼 = 68 [°]).
Given the set of design parameters, a final configuration which fulfills all of them may not be
implementable if chocking occurs inside the nozzle vanes. Thus the condition 𝑀 ≤ 1 must be
verified for the design to be meaningful.
From Fig.15 𝛼 is expressed in terms of nozzle pitch and nozzle throat length as
𝛼 = cos−1 𝑜
𝑠 = cos−1
𝑚
𝜌𝑀𝑎 𝑍𝑣2𝜋𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡
𝑍𝑣
(4.13)
where 𝑠 =2𝜋𝑟𝑛𝑟 −𝑖𝑛𝑙𝑒𝑡
𝑍𝑣 is derived from geometrical considerations while 𝑜 =
𝑚
𝜌𝑀𝑎 𝑍𝑣 expresses the
link between throat area and mass flow rate for a nozzle vane.
24
Figure 15: nozzle vane geometry definition (Rajoo & Martinez-Botas, [18])
Eqn.(4.13) allows to compute the Mach number. By substituting the design parameters it is
found 𝑀 ≈ 0.36, thus chocking is avoided at design point (notice there is no guarantee that
𝑀 < 1 is also verified at off-design conditions)
Moreover eqn.(4.13) shows that the number of vanes has no influence on the outflow angle:
according to Rajoo & Martinez-Botas 𝑍𝑣 is the value which optimizes the pitch/chord ratio so to
achieve “compromise between friction losses and good flow guidance” [18]. Meanline design
specifies 𝑍𝑣 = 14.
Figure 16: nozzle ring. Left: sketch in the frontal plane. Right: shape of the blade
For the 3D design of the nozzle ring iterative procedures based on CFD analysis have been found
in literature [9] and the same approach is followed here. The blade has a simple symmetric
profile with straight sides closed at LE and TE by circumference arcs (Fig.16, right). The
25
inclination angle (Fig.16, left) is set arbitrarily: as initial guess value the designer chooses the
outlet flow angle 𝛼, since the flow is expected to “follow the blade” in ideal design. This value is
modified iteratively until no separation is detected.
3D CAD model of the nozzle ring is generated in SOLIDWORKS® and is shown in Fig.17.
Figure 17: 3D geometrical model of the nozzle ring
5.3 Design of the rotor
The rotor is the main component of a radial turbine, being the only one which extracts work from
the flow. Apart from the requirement of matching with the volute, which sets the dimensions of
the diameter and the height of the blade at inlet, there are no constraints on the design of the
rotor: the goal is to minimize all sources of losses so to achieve the maximum power output
under designed flow conditions.
5.3.1 Preliminary design
As illustrated in the introductory chapter, nominal design condition is associated with a
minimum in incidence losses. Therefore the rotor blade must be designed so that for all spanwise
locations (from hub to shroud, see Fig.18) the flow enters radially (𝛽2 = 0) and leaves the rotor
axially (𝛼3 = 0).
In literature [21] it is reported that the condition 𝛽2 = 0 actually leads to flow recirculation at the
suction surface of the blade, and the optimal inlet flow angle is identified within the range
𝛽2 = −10° and 𝛽2 = −40°. Given the flow parameters at volute outlet (𝑀2 = 0.56, 𝛼2 = 68°)
and the designed rotational speed (𝑛 = 85000 rev/min) it is 𝛽2 ≈ −26°, which falls inside the
optimal interval. It should be noticed that in general 𝛽2 may differ from the physical angle of
incidence, and this occurs when the blade angle at rotor inlet, namely 𝛽2𝑏 , is non zero. However
strength limitations require the blade to be radial in the segment furthest from the rotational axis,
26
where centrifugal forces are dominant: this constraints the inlet blade angle to be zero. In this
configuration the angle of incidence coincides with 𝛽2 thus it is in the optimal range as well.
Figure 18: sketch of the velocity triangles at rotor inlet and outlet (Saravanamuttoo, [19],
adapted)
At rotor outlet the condition 𝛼3 = 0 implies different values of 𝛽3 spanwise along the blade,
because while 𝑤3 is supposed to be constant and equal to its meanline value 𝑤3 = 288 𝑚/𝑠 (this
assumption is only a rough approximation) 𝑈3 increases proportionally with the distance from
the axis of rotation.
SPANWISE POSITION [mm] BLADE ANGLE AT OUTLET [°] (*)
𝑟3 = 11.1 (hub) 𝛽3 = 20.1
𝑟3 = 16.3 𝛽3 = 30.2 𝑟3 = 21.6 𝛽3 = 41.9 𝑟3 = 26.8 𝛽3 = 55.9
𝑟3 = 32 (shroud) 𝛽3 = 81.4
Table 3: relative angle at rotor outlet under nominal design condition
(*) These values are not to be interpreted as hard constraints for the final design but as an approximate guideline,
since they are obtained from simplifying hypotheses based on meanline approach. This is particularly true for hub
and shroud, i.e. close to the walls, where 𝑤3 differs considerably from its meanline value.
Tab.3 reports values of 𝛽3 computed for 5 representative spanwise positions. In ideal case the
blade is designed to be a streamline: indeed if the flow follows the blade “smoothly” incidence
losses are low. This implies that also at rotor outlet 𝛽3 ≡ 𝛽3𝑏 .
So far the blade angles at inlet and outlet have been derived from preliminary design
considerations, but this is not sufficient for the 3D design of the rotor as no indications are given
about the blade angle distribution in the streamwise direction, i.e. along 𝑚 (see Fig.19).
Moreover the evolution of the blade in the θ-direction (namely the “wrap angle distribution”) is
27
also undetermined. In the present work the approach to the 3D design consists of modelling the
wrap angle distribution at several spanwise locations by means of parametric Bezier curves.
Figure 19: rotor views. Left: r-z (or meridional) plane. Right: θ-z (or blade to blade) plane
5.3.2 The Bezier curve
Given 𝑛 + 1 control points 𝑃0, 𝑃1,..., 𝑃𝑛 the Bezier curve of degree 𝑛 in space is defined as the
function F: [0,1] → ℝ3 such that
𝐹 𝑡 = 𝑛
𝑖
𝑛
𝑖=0
𝑃𝑖 1− 𝑡 𝑛−𝑖𝑡𝑖 , 𝑡 ∈ 0,1 (4.14)
Bezier curves are interpolation functions. Eqn.(4.14) can be re-arranged so to highlight the set of
functions 𝐵0,𝑛 𝑡 ,𝐵1,𝑛 𝑡 ,… ,𝐵𝑖 ,𝑛 𝑡 ,… ,𝐵𝑛 ,𝑛 𝑡 which represents a basis of the vector space of
all polynomials of degree ≤ 𝑛, also named Bernstein basis.
𝐹 𝑡 = 𝑃𝑖 𝐵𝑖,𝑛(𝑡)
𝑛
𝑖=0
, 𝐵𝑖,𝑛 𝑡 = 𝑛
𝑖 𝑃𝑖 1− 𝑡
𝑛−𝑖𝑡𝑖 (4.15)
Control points 𝑃𝑖 are also the coefficients multiplying each element of the basis; thus by varying
the set of control points the whole vector space can be spanned. Moreover the position of the
control points identify a region inside which the curve will develop. 𝑃0 and 𝑃𝑛 are respectively
the first and the last element of the curve, while the remaining points do no lay on the curve
(apart from the trivial case in which all points are aligned) but locally affect its slope and
curvature.
The advantage to use Bezier curves as design tools is not only due to their flexibility but also to
the fact that the interpolation by means of Bernstein’s polynomials, unlike linear interpolation,
generates “smooth” curves i.e. curves which are continuous also in the first derivative. This
property is important when modelling rotor blades, in which sharp edges must be avoided so to
have continuous variation of the flow angle and limit incidence losses and the risk of local flow
separation.
28
As an example Fig.20 shows a Bezier curve of 3rd
order. Bernstein basis is formed by 3rd
order
polynomials (Fig.20, left). As mentioned, the curve is confined within the region identified by
the control points, whose position affects the local slope (Fig.20, right).
Figure 20: Bezier curve of degree 3. Left: basis of vector space. Right: control points (Floater,
[7], adapted)
5.3.3 Implementation of the design strategy
Data relative to the general dimensions of the rotor are provided by the meanline design. The
value of these parameters is reported in Tab.4.
PARAMETER VALUE
Rotational speed (rpm) 𝑛 = 85000
Inlet tip radius 𝑟2 = 40 [𝑚𝑚] Inlet blade height 𝑏2 = 13 [𝑚𝑚]
Exit tip radius 𝑟3𝑡 = 32 [𝑚𝑚]
Exit hub radius 𝑟3 = 11.1 [𝑚𝑚] Axial length 𝑘𝑎 = 23.9 [𝑚𝑚]
Number of blades 𝑍𝑏 = 12
Table 4: meanline design parameters for the rotor
Five equidistant spanwise locations, namely layers, are identified (the 1st being the hub, the 5
th
being the shroud) and for each of them the wrap angle distribution is modelled by means of a
Bezier curve. This operation is done efficiently in BLADEGEN®, where it is possible to drag, add
or delete control points so to determine the Bezier curve in terms of order and shape.
For the specific case a number of control points between 26 and 31 was used for each layer. High
degree polynomials were chosen in order to locally have control over the curvature, which is not
constant streamwise (it was mentioned that for mechanical reasons the blade is required to
develop radially in its first segment).
Given a wrap angle distribution BLADEGEN® numerically evaluates the tangent at each point.
The resultant plot corresponds to the flow angle distribution, since 𝜃 and 𝛽 are connected by the
relation 𝛽 =𝑑𝜃
𝑑𝑚 (see Fig.19 – right): hence the 𝜃-distribution is designed so that the associated 𝛽
fulfils the condition of axial flow at rotor outlet (Tab.3).
29
Figure 21: Rotor: θ-distribution (top), β-distribution (middle), thickness distribution (down)
30
For each layer BLADEGEN® must create the blade profile. The latter is defined by a mean line,
which is the wrap angle line, and a streamwise distribution of thickness, which is modelled by
the designer with a Bezier curve. Profiles of adjacent layers are connected together by
streamwise lofting and the result is the generation of the blade surface.
Distributions of 𝜃, 𝛽 and thickness are shown in Fig.21, while the result of such design on the
meridional plane is reported in Fig.22.
Figure 22: Rotor: distribution of wrap angle (top left), flow angle (top right) and thickness
(bottom) in the meridional plane
As seen in the introductory chapter, in order to increase the work exchange the flow must
accelerate inside the rotor, i.e. 𝑤3 > 𝑤2, which implies that for subsonic flows the channel
between rotor blades is convergent. Fig.23 – left shows that this is the case for the present
design, as the cross section area of the channel reduces from inlet to outlet.
The 3D CAD model of the impeller, which results from the design above, is presented in Fig.24.
31
Figure 23: Rotor: variation from inlet to outlet of channel cross-section area (left) and lean
angle (right)
Figure 24: 3D geometrical model of the rotor
5.3.4 Supplementary issues on 3D design of the rotor
So far the design of the impeller was made from a geometrical point of view, without due
consideration of the influence on the internal flow pattern. This relationship is impossible to
investigate analytically because it will be seen that, no matter how “good” the design is, the
presence of secondary flows cannot be avoided, thus the problem is fully 3D. CFD analysis is the
most common tool to have an insight on the internal flow, because the only way to have a
comprehensive description of it is by numerically solving the full set of NS equations: however
in this section an analytical approach is presented, whose purpose is to highlight the influence of
some design choices on the development of secondary flows, and possibly suggest improvements
at subsequent phases of the deign process.
32
This analysis is an adaptation of the study presented by Van den Braembussche [28] and
originally performed on the impeller of a radial compressor.
Blade lean is defined as the variation of 𝜃 of the blade from hub to shroud. Under the assumption
that the blade lean is null, i.e. 𝜕𝜃
𝜕𝑛= 0 (this is almost satisfied, as the lean angle lies within the
interval [0°,4°] everywhere from inlet to outlet, see Fig.23 – right) the components of velocity
which induce centrifugal accelerations in the meridional plane are:
Meridional velocity 𝑊𝑚 , whose radius of curvature is the one of the streamline, namely 𝑅𝑛 (the
subscript 𝑛 denotes that this radius is aligned with the spanwise direction)
Tangential velocity 𝑉𝑡 = 𝑊𝑡 − 𝛺𝑅, whose radius of curvature is 𝑅. Notice that this vector is
orthogonal to the meridional plane, but the resultant acceleration is in plane
Figure 25: geometrical definition of the problem
At all streamwise sections the overall centrifugal acceleration is balanced by a pressure gradient
orthogonal to the streamsurface (Fig.25). The equilibrium is expressed by eqn.(4.16)
1
𝜌
𝜕𝑝
𝜕𝑛= 𝑊𝑢 − 𝛺𝑅
2
𝑅cos 𝜆 −
𝑊𝑚2
𝑅𝑛 (4.16)
where 𝜆 is the angle between the meridional component of the streamline and the axis of
rotation. At inlet cos λ = 0 and the acceleration due to the curvature of the streamline increases
from hub to shroud because of decreasing 𝑅𝑛 , hence the pressure gradient decreases. At outlet
the direction of the pressure gradient depends on the values of 𝑉𝑡 and 𝑊𝑚, but with increasing 𝑅
from hub to shroud the spanwise acceleration due to the tangential velocity component tends to increase
as well (𝑎𝑡 ∝ 𝛺2𝑅), thus the pressure gradient is still negative.
The above considerations suggest that, together with the main motion of the flow from inlet to
outlet, there exist also a secondary flow moving from hub to shroud, i.e. against the spanwise
pressure gradient.
Imposing 𝜕𝑝
𝜕𝑛= 0 eqn.(4.16) gives the value of 𝑅𝑛 for zero pressure gradient
33
𝑅𝑛 =𝑊𝑚
2𝑅
𝑉𝑡2 cos 𝜆
(4.17)
However 𝑊𝑚 is higher at the suction side of the blade (where the flow is accelerated) and lower
at the pressure side, and from eqn.(4.17) two different values of 𝑅𝑛 should exist at the same
point, thus there is no design which can eliminate the spanwise pressure gradient, and the
correspondent secondary flow, at both sides of the blade. The hub-to-shroud pressure gradient
can be reduced by increasing the curvature radius of the meridional contour or by reducing the
blade height (hub-to-shroud distance).
Moreover there also exists a pressure gradient in the blade-to-blade plane because of the flow
moving between pressure side and suction side of adjacent blades. Inside the channel the flow
feels the simultaneous effect of both pressure gradients. The situation is depicted in Fig.26.
Figure 26: pressure distribution in a crosswise section. Effect of spanwise pressure gradient
(left), effect of blade-to-blade pressure gradient (middle), ensemble (right) (Van den
Braembussche, [28])
A possible strategy to reduce the effect of the secondary flow is the employment of splitter
blades.
5.4 Design of the diffuser
The diffuser must convert part of the kinetic energy of the flow into pressure in order to reach
the condition 𝑃4 > 𝑃𝑎𝑡𝑚 at the outlet of the turbine and allow the flow to be discharged.
The constraints on the design of the diffuser are
Inner radius (𝑟3) at diffuser inlet, which must be equal to the hub radius at rotor outlet
Outer radius (𝑟3𝑡) at diffuser inlet, which must be equal to the shroud radius at rotor outlet
Values provided by the meanline design are reported in Tab.5
PARAMETER VALUE
Outer radius at diffuser outlet 𝑟4𝑡 = 38.6 [mm]
Inner radius at diffuser outlet 𝑟4 = 0 [mm]
Axial length 𝑙 = 38.6 [mm]
Table 5: meanline design parameters for the diffuser
The main problem when designing the diffusers is the risk of boundary layer separation at the
wall [8], which depends on the diffusion angle ψ (Fig.27 – left).
34
Figure 27: conical diffuser. Left: 2D sketch. Right: lines of appreciable stall for given
geometrical configuration (Blevins, [4], adapted)
From meanline parameters the non-dimensional length of the diffuser is 𝑙
𝑟3𝑡= 1.21 while the
designed diffusion angle is expressed from Fig.27 – left as tan𝜓 = 𝑟4𝑡−𝑟3𝑡
𝑙 𝑦𝑖𝑒𝑙𝑑𝑠 2𝜓 ≈ 19.4°.
An extrapolation from the graph in Fig.27 – right suggests that this point is below the limit of
appreciable stall for conical diffusers, hence the present design configuration can be
implemented.
The 3D CAD model of the diffuser is presented below (Fig.28)
Figure 28: 3D geometrical model of the diffuser
35
6 – MESH GENERATION
The problem of how to partition the flow domain in order to create a grid on which discretized
NS equations can be solved is of primary importance in CFD, because it directly affects the
quality of the simulation and the computational time. From a practical point of view the ideal
mesh should be able to capture the critical aspects of the flow (presence of regions of separated
flow, recirculation bubbles, evaluation of losses...) with simulations lasting at most a few hours,
which implies that the mesh must be refined only in regions in which it is needed (typically in
the boundary layer, or where high velocity gradients are expected).
The first part of this section focuses on theoretical aspects about mesh generation: how to choose
an appropriate mesh for the problem in exam, how to mesh regions of the domain close to a wall
and the techniques to check if the mesh is “good”. In the second part the solution which was
implemented is illustrated and the mesh quality is assessed.
6.1 Choice of the grid
Two different techniques are employed for the discretization of a geometric domain. In the so
called structured grid the cells are ordered in a I×J×K array so that given whatever grid point
inside the domain this is univocally identified by a set of coordinates, say 𝑃𝑖 ,𝑗 ,𝑘 , and the points in
its neighbourhood are implicitly known (they will be 𝑃𝑖 ,𝑗 ,𝑘−1, 𝑃𝑖 ,𝑗 ,𝑘+1...). This implies that there
exist a regular pattern of connections among grid points which are close to each other. On the
other hand in an unsctuctured grid the above regularity is not present hence neighbouring cells
cannot be directly accessed by their indexes. The different way to build the two grids also affects
the geometrical shape of their cells, i.e. the elements: hexahedra are usually employed in
structured meshes while unstructured meshes are formed by tetrahedral elements or
combinations of different solids (see Fig.29).
Figure 29: elements of a 3D mesh - tetrahedron, hexahedron, prism, pyramid
The choice of the type of mesh should be done according to the following considerations:
Unstructured meshes can easily model every kind of domain because the shape of the
elements which is employed is not constrained to hexahedra. For the same reason also the
element size can vary considerably between adjacent cells. This flexibility is needed
when the geometry to be meshed is complex or when fast variation in the grid spacing is
desirable (for example close to the walls).
For the same amount of cells structured grids based on hexahedra allow the highest
accuracy in the solution. On the contrary unstructured grids tend to generate more
skewed elements, with consequent numerical errors.
The generation of an unstructured grid is much faster than a structured one. The time
strongly depends on the complexity of the problem, but while for the former it is usually
in the order of hours for the latter it can take up to weeks (Khare et al., [10]).
36
Meshing a radial turbine is a challenging task because structured hexahedral mesh would be
desirable for high computational accuracy, especially in regions where losses most probably
occur, but the complexity of the 3D geometry makes the unstructured mesh approach more
suitable. In the present work a hybrid solution is suggested. The mesh is structured in the
impeller because it was seen that complex secondary flows occur inside the blade-to-blade
channels and their accurate evaluation is critical for a correct performance assessment of the
turbine. Achieving such a degree of precision is less crucial for other components, like the volute
or the diffuser, where instead the main source of losses is given by skin friction: for this reason
an unstructured mesh is used, which becomes more regular only close to the walls in order to
model accurately the boundary layer.
6.2 Meshing the boundary layer
Boundary layer is a thin region adjacent to a solid surface in which there exist a strong velocity
gradient orthogonal to the surface itself. The reason is that mechanical equilibrium is achieved
between molecules of the flow and the wall in the contact region so that as a macroscopic effect
the velocities of the flow and the solid surface are equal (this constraint is called no slip
condition). The existence of a sharp velocity gradient suggests that for a “good description” of
the flow the grid inside the boundary layer shall be refined. But what happens qualitatively with
increasing Reynolds number is that the region where viscous effects are relevant gets more and
more confined to the walls and the boundary layer becomes thinner, thus the grid resolution must
increase accordingly, leading to an increase of the computational effort.
A rough evaluation of Reynolds number can be done taking as reference values the ones at
turbine inlet: 𝑈𝑟𝑒𝑓 is the meanline velocity, 𝐿𝑟𝑒𝑓 is the radius of the duct at volute inlet and
𝜈|𝑇=𝑇𝑟𝑒𝑓 is the kinematic viscosity of air at 𝑇 = 𝑇𝑖𝑛𝑙𝑒𝑡 at design point.
𝑅𝑒 =𝑈𝑟𝑒𝑓 𝐿𝑟𝑒𝑓
𝜈|𝑇=𝑇𝑟𝑒𝑓
≈120
𝑚
𝑠 ∗ 30 ∗ 10−3[𝑚]
9.06 ∗ 10−5 𝑚2
𝑠
≈ 4 ∗ 104 (5.1)
Eqn.(5.1) shows that 𝑅𝑒 is high, in the order of 104. The flow regime is turbulent, and in the
boundary layer exchange of momentum takes place not only between adjacent layers, at
molecular scale, but together with an exchange of fluid particles.
Figure 30: velocity profile in a turbulent boundary layer (Bakker, [3])
37
As a consequence the region close to the wall is subject to steep gradients normal to the
boundary (Fig.30 illustrates a typical velocity profile) and directly resolving the flow with a
suitable mesh is computationally demanding.
Experimental investigation showed that the boundary layer can be divided into an inner and an
outer region. The first region is dominated by viscous effects, and the velocity is a function of
the coordinate 𝑦+ which represents the distance from the wall nondimensionalized by the viscous
scale 𝑢∗
𝜈 (𝑢∗ ≝
𝜏0
𝜌 is called friction velocity).
𝑈
𝑢∗= 𝑓
𝑦𝑢∗𝜈 = 𝑓 𝑦+ (5.2)
The second region is dominated by turbulent mixing and the flow seems not to feel the presence
of the wall. The difference of velocity with respect to the reference value (called velocity defect)
is function of the coordinate 𝜉 which represents the distance from the wall nondimensionalized
by the boundary layer thickness 𝛿.
𝑈 − 𝑈𝑟𝑒𝑓
𝑢∗= 𝑓
𝑦
𝛿 = 𝑓 𝜉 (5.3)
The distance from the wall, y, scales differently in the two regions. However there exist an
intermediate overlap region in which the two expressions (eqn.(5.2) & eqn.(5.3)) are both valid.
By equating their derivatives (for details see Kundu, [11]) it is proved that the overlap region is
described by a logarithmic law, and the so called logarithmic layer becomes wider with
increasing 𝑅𝑒. The situation is illustrated in Fig.31.
Figure 31: non-dimensional velocity as function of 𝑦+ in the inner region (Kundu, [11])
38
Under the assumption that the logarithmic behaviour can be used to model the velocity
distribution near the wall, this provides a suitable law to link distance from the wall to velocity,
hence allows the estimation of the flow shear stress. In this way it is not necessary to resolve the
boundary layer because the velocity in the inner region can be estimated through the log-law,
thus a coarser mesh can be used.
Summarizing, there are two approaches to model the flow in the near-wall region
Wall function method. It is based on empirical formulas which link the velocity of the
flow to the position in the inner region of the boundary layer, thus avoiding to resolve it
and saving computational time. However additional assumptions must be introduced in
order to justify the validity of the wall function in the whole inner region, and this may
lower accuracy of the results.
Low-Reynolds-Number method. This method resolves the details of the boundary layer
profile by using very fine meshes with the first node located at 𝑦+ ∼ 1 or even closer to
the wall.
6.3 Quality of the mesh
There exist many criteria for evaluating the quality of a mesh, and an in-depth discussion on this
topic is beyond the purpose of the present work. However from a practical point of view a mesh
is considered of good quality if its elements are not warped too much with respect to their
nominal shape (tetrahedron, pyramid...): for example if an element is too stretched in a certain
direction (see Fig.32) the variation of any flow characteristic in that direction will be detected
with less accuracy because the grid behaves as if it were coarser.
Figure 32: stretching of a quadrilateral element. Nominal shape (left), deformed shape (right)
In this section the following quality parameters are discussed:
6.3.1 Skewness
Skewness is the measure of how close the shape of a cell is from the ideal shape. A possible way
to calculate it is through the so called normalized angle deviation method
𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 ≝ 𝑚𝑎𝑥 𝜃𝑚𝑎𝑥 − 𝜃𝑒
180 − 𝜃𝑚𝑎𝑥,𝜃𝑒 − 𝜃𝑚𝑖𝑛
𝜃𝑒 (5.4)
Eqn.(5.4) evaluates the deviations of the maximum (𝜃𝑚𝑎𝑥 ) and the minimum (𝜃𝑚𝑖𝑛 ) angle with
respect to the angle relative to an equiangular cell (𝜃𝑒 , which represents the ideal case) and takes
as value for the skewness the maximum between both.
A value of 0 represents an equilateral cell while a value of 1 stands for a degenerated cell (for
example in the case of a 2D cell this would become a 1D segment). Skewness is considered good
for values up to 0.5.
39
6.3.2 Orthogonal quality
Orthogonal quality is another way to evaluate how a cell is close to its ideal shape. Taking the
2D cell in Fig.33 as reference, orthogonal quality is the minimum of eqn.(5.5) computed ∀i
𝑂𝑄 ≝𝐴𝑖 . 𝑒𝑖
𝐴𝑖 𝑒𝑖 (5.5)
where 𝑒𝑖 is the vector joining the centroid of the cell with the centroid of the edge and 𝐴𝑖 is the
vector normal to the edge. Eqn.(5.5) is a measure of how much 𝑒𝑖 and 𝐴𝑖 are aligned, because the
scalar product at the numerator depends on the cosine of the angle between the two vectors.
The range for the orthogonal quality is [0,1], and the closer to 1 the more equilateral is the cell
(in this case, infact, all the vectors are aligned).
Figure 33: orthogonal quality on a 2D quadrilateral cell
6.3.3 Jacobian ratio
The Jacobian matrix describes the properties of the mapping between the computational space
(𝜉1, 𝜉2, 𝜉3), where the NS equations are discretized and solved, and the real domain (𝑋1, 𝑋2, 𝑋3).
In an ideal situation the two domains would coincide, thus the computed solution could be
transferred to the real case without loss of accuracy. For each element of the mesh the
determinant of the Jacobian matrix is computed at some sampling points (for example the corner
nodes, the centroid...).
Figure 34: mapping of an hexahedral element (Bucki, [5])
40
From eqn.(5.6) JR is defined as the maximum to the minimum value among those determinants.
Other definitions presented in literature [5] consider the maximum determinant at denominator,
but here is reported the one used by ANSYS® Meshing.
𝐽𝑅 ≝𝑚𝑎𝑥 𝑑𝑒𝑡 𝐽 𝜉𝑖
𝑚𝑖𝑛 𝑑𝑒𝑡 𝐽 𝜉𝑖 (5.6)
where 𝜉𝑖 , 𝑖 ∈ 1,2,… ,𝑛 is the generic sampling point.
At the end JR is a measure of the maximum distorsion of each element of the mesh. The situation
is sketched if Fig.34. A value close to 1 indicates that the mapping does not lead to distorsion of
the elements, while the higher the Jacobian ratio the worse is the mesh.
6.4 Meshing of components
Volute, nozzle ring and diffuser were meshed using the software 𝐴𝑁𝑆𝑌𝑆®Meshing. The process
of mesh generation is highly automated: the designer specifies general sizing parameters such as
degree of fineness of the grid (relevance), rate at which adjacent elements are allowed to grow
(transition), control over the element quality (smoothing), and the software creates an
unstructured mesh based on those requirements and on constraints about the dimension of the
elements to be used (limits on the size of edges and faces), which are specified as defaults
(defaults may be changed if necessary). Fig.35 shows as an example the setup of
𝐴𝑁𝑆𝑌𝑆®Meshing for mesh generation on the volute.
Figure 35: example of setup of 𝐴𝑁𝑆𝑌𝑆®Meshing (volute)
Generation of unstructured mesh by means of 𝐴𝑁𝑆𝑌𝑆®Meshing is a fast and robust process,
however for a more accurate description of the flow in the boundary layer a structured mesh
close to the walls is needed. This is achieved in 𝐴𝑁𝑆𝑌𝑆®Meshing by using inflation layers: to
this purpose the program requires definition of the surface around which inflation must be
41
performed (named selection) and specifications about inflation option (thickness of the first
layer, total inflation thickness…), number of layers and growth rate between adjacent layers. An
example is reported in Fig.35.
The main parameter to be set at this stage is the first layer thickness, i.e. the distance of the first
node of the mesh from the wall expressed in terms of 𝑦+. A target 𝑦+ is set and the
correspondent 𝑦 is determined through eqn.(5.2), but in order to do so the friction velocity 𝑢∗ must be estimated. Literature (White, [30]) reports a graph, namely Moody’s diagram, which
relates 𝑅𝑒 to 𝐶𝑓 (skin friction coefficient) for circular pipes with smooth walls in turbulent
regime. Many empirical relations were also formulated, among which the 1/7th
law is mentioned.
Assuming the validity of such relation in more general cases of ducts with non-circular shapes
(volute, diffuser…), once 𝑅𝑒 is calculated from eqn.(5.1) it is possible to estimate 𝐶𝑓 from
eqn.(5.7)
𝐶𝑓 = 0.027𝑅𝑒𝑥−1
7 (5.7)
and 𝑢∗ from eqn.(5.8)
𝑢∗ ≝ 𝜏0
𝜌= 𝐶𝑓
𝑈𝑟𝑒𝑓2
2 (5.8)
In Tab.6 are reported the target 𝑦+ for each sub-domain of the turbine (except for the rotor,
which will be discussed later) and the correspondent 𝑦, which must be set as first layer thickness
in the setup for generating inflation layers with 𝐴𝑁𝑆𝑌𝑆®Meshing. The reference values for the
computation of 𝑅𝑒 in eqn.(5.1) are considered to vary among the components.
Domain 𝑈𝑟𝑒𝑓 [𝑚/𝑠] 𝐿𝑟𝑒𝑓 [𝑚] 𝑅𝑒 [−] 𝑦+ [−] 𝑦 [𝑚]
Volute 𝑈1 ∼ 120 (speed at inlet)
3 ∗ 10−2 (radius at inlet)
4 ∗ 104 30 4 ∗ 10−4
Nozzle ring 𝑈𝑛𝑟−𝑖𝑛𝑙𝑒𝑡 ∼ 250 (speed at inlet)
3.2 ∗ 10−2 (blade chord)
8.8 ∗ 104 10 6 ∗ 10−5
Diffuser 𝑈3 ∼ 210 (speed at inlet)
2.1 ∗ 10−2 (width at inlet)
4.9 ∗ 104 30 2 ∗ 10−4
Table 6: estimation of first layer thickness for turbine components
Notice that for volute and diffuser 𝑦+ is located in the logarithmic layer while for the nozzle ring
it is placed in the buffer layer (𝑦+ ∼ 10). The goal is to achieve higher resolution in the nozzle
ring where the blades may cause separation of the flow, especially when the turbine is working at
off-design points.
DOMAIN NUMBER
OF NODES
NUMBER OF
ELEMENTS
AVERAGE
SKEWNESS
AVERAGE
JR
AVERAGE
OQ
Volute-
vaneless
113587 338479 0.256 1.069 0.872
Volute-
vaned
111830 336893 0.255 1.074 0.871
Nozzle ring 262591 995715 0.310 1.078 0.807
Diffuser 42571 122799 0.202 1.019 0.899
Table 7: mesh statistics for turbine components
42
Once the inflation is defined the setup of 𝐴𝑁𝑆𝑌𝑆®Meshing is complete and meshes are
generated. The result is shown in Tab.7, where the statistics for the final meshes are reported. As
can be seen all meshes fulfil the quality criteria, thus they will be used for the subsequent CFD
analysis.
The implemented solution is illustrated in the figures below, whose goals are to show the effect
of inflation layers (Fig.36), an ensemble mesh (Fig.37) and the presence of local refinements
around “critical points” (Fig.38, at LE and TE of the nozzle ring blade).
Figure 36: mesh of the volute (section)
Figure 37: mesh of the diffuser (ensemble)
43
Figure 38: mesh of the nozzle ring (one blade)
For meshing the rotor TURBOGRID® was used, a dedicated software for rotating machines which
creates a high quality structured hexahedral mesh. The process of mesh generation proceeds
according to the following steps:
1. The geometry of the rotor is imported from a BLADEGEN® file.
2. Based on the geometry of the blade TURBOGRID® automatically chooses the most
suitable topology for the mesh using the function ATM optimized. Topology denotes the
pattern in which the region of space around the blade is divided. Topology is chosen
according to the blade profile, the local curvature, the shape of LE and TE (cut-off or
rounded) and other geometrical factors: in each block the elements of the mesh are
oriented along the local shape of the blade, and interfaces between blocks are “smooth”
so to avoid warped elements which lower the quality of the mesh.
The topology for the rotor is sketched in Fig.39.
Figure 39: topology for the rotor blade. The blade (blue) is surrounded by meshing blocks
44
3. At this step TURBOGRID® requires to specify the fineness of the overall mesh, through the
global size factor, and the target 𝑦+. For the rotor 𝑦+ = 1 is set: the purpose is to directly
resolve the boundary layer (without the use of wall functions), as higher accuracy is
needed in the rotor, where the flow is fully 3D.
4. The final mesh is automatically generated. The final mesh for the rotor has 613788 nodes
and 579535 elements.
Fig.40 shows a portion of the mesh around the LE of the blade. Progressive refinement can be
appreciated close to the wall, due to the specification on 𝑦+, while the topology is seen from the
orientation of the hexahedral elements.
Figure 40: mesh of the rotor (portion)
After mesh generation TURBOGRID® provides tools for assessing the quality of the mesh. The
most significant ones are listed below:
Minimum/Maximum face angle. For each face of an element, the angle between the two
edges that touch a node is calculated. The smallest/largest angle of all faces is returned as
the value for the minimum/maximum face angle. This parameter evaluates how warped
the faces are with respect to the ideal angle (90°), therefore it can be considered a
measure of skewness.
Minimum volume. This is the minimum volume among all the cells of the grid. Its value
must be always positive in order avoid numerical errors.
Maximum element volume ratio. For each node the volume of all the cells touching that
node is computed and the ratio between the maximum and the minimum volume is
returned. This is a measure of the local expansion factor, and it should be low especially
in regions where high gradients are expected in the flow quantities (velocity,
temperature...).
Maximum edge length ratio. For each face of an element, the ratio between the longest
and the shortest edge is computed and the maximum value is returned. This parameter
measures the aspect ratio (see Fig.32).
Mesh statistics for the rotor are reported in Fig.41: after computation TURBOGRID® checks if
each quality parameter lays within the acceptable range, and returns a feedback.
45
Figure 41: mesh statistics for the rotor
The overall mesh for the vaneless configuration contains 769946 nodes and 1040813 elements,
for the vaned configuration 1030780 nodes and 2034942 elements.
46
7 – CFX SETUP
Once generated and meshed, all components of the turbine are assembled in CFX® − Pre.
However, before running a CFD simulation, it is necessary to specify the mathematical model
that must be solved together with boundary conditions to the problem, initial conditions and
interfaces between various components. Such topics are treated in this chapter.
7.1 Mathematical model for turbulence
A turbulent flow is characterized by swirling structures (eddies) which span a wide spectrum of
scales (Johansson & Wallin, [8], 2012). The dimension of the largest eddies is set by the
reference geometrical size of the problem, while the length scale of the smallest eddies is
supposed to only depend on dissipation rate and viscosity: this is, infact, the scale at which the
flow dissipates kinetic energy through viscous mechanisms. Within the two limits there exist a
range of eddies of intermediate dimensions which transfer kinetic energy from the biggest to the
smallest scales in a process called turbulence cascade.
The ratio between the largest and the smallest scales of turbulence depends on the Reynolds
number and can be estimated (for details see reference [8]) as 𝑅𝑒3/4
. Thus, a grid which aims to
describe all the details in a 3D flow domain would have a resolution of 𝑅𝑒9/4
, and considering
that for the present application the Reynolds number is at least in the order of 104 the mesh for
DNS would have 109 nodes and the computational effort would be massive. Alternatively it is
possible to describe a turbulent flow by means of RANS, where it is assumed that turbulence can
be modelled as fluctuations within an average velocity field: however when this assumption is
introduced in the NS equations it originates an extra term, namely the Reynolds stress, which is
unknown hence it must be modelled through a turbulence model.
Common turbulence models are the so called 𝑘 − 휀 and 𝑘 − 𝜔: both solve 2 additional
equations, together with the set of RANS, which account for the transport of turbulent variables.
In the former case those variables are turbulent kinetic energy, k, (representing the variance of
the fluctuations in velocity) and turbulence dissipation, ε, (representing the rate at which velocity
fluctuations dissipate); in the latter case ε is replaced by the specific turbulence dissipation, ω,
(𝜔 ≝휀
𝑘). Both models rely on the assumption that the Reynolds stress term is related to the
gradient of the mean velocity (strain) through the turbulent viscosity, 𝜈𝑡 , according to the
Boussinesq hypothesis (notice that this hypothesis is purely empirical)
−𝑢𝑖′𝑢𝑗′ 𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑠𝑡𝑟𝑒𝑠𝑠
= 𝜈𝑡 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
∗ 𝑓 𝜕𝑈𝑖𝜕𝑥𝑗
𝑠𝑡𝑟𝑎𝑖𝑛
, 𝑖, 𝑗 ∈ 1,2,3 (6.1)
An in-depth description of 𝑘 − 휀 and 𝑘 − 𝜔 turbulence models is beyond the scope of this work,
however from the point of view of applications 𝑘 − 휀 is reported to have general good
performances but is not applicable to flows under adverse pressure gradients and is not accurate
in forecasting separation, while 𝑘 − 𝜔 has complementary characteristics and is mostly suitable
in the near-wall region. In this thesis a hybrid turbulence model, namely SST 𝑘 − 𝜔, is used, as it
combines the advantages of both models because it allows a shift from 𝑘 − 𝜔 to 𝑘 − 휀 depending
on the distance from the wall.
The SST 𝑘 − 𝜔 formulation is shown below:
47
k-equation: 𝜕 𝜌𝑘
𝜕𝑡+
𝜕 𝜌𝑈𝑗𝑘
𝜕𝑥𝑗=
𝜕
𝜕𝑥𝑗 𝜇 +
𝜇 𝑡
𝜍𝑘3 𝜕𝑘
𝜕𝑥𝑗 + 𝑃𝑘 − 𝛽
∗𝜌𝑘𝜔 (6.2)
ω-equation:
𝜕 𝜌𝜔
𝜕𝑡+
𝜕 𝜌𝑈𝑗𝜔
𝜕𝑥𝑗=
𝜕
𝜕𝑥𝑗 𝜇 +
𝜇 𝑡
𝜍𝜔3 𝜕𝜔
𝜕𝑥𝑗 + 1− 𝐹1 2𝜌
1
𝜍𝜔2𝜔
𝜕𝑘
𝜕𝑥𝑗
𝜕𝜔
𝜕𝑥𝑗+ 𝛼3
𝜔
𝑘𝑃𝑘 − 𝛽3𝜌𝜔
2 6.3
where 𝑃𝑘 is the term responsible for the production of turbulence and depends on the Reynolds
stress (which is modelled as function of 𝜈𝑡).
𝐹1 ∈ [0,1] is called blending function and accounts for the position with respect to the wall. By
tuning 𝐹1 the model shifts from the expression of the standard 𝑘 − 휀 (when 𝐹1 = 0) to the one of
𝑘 − 𝜔 (when 𝐹1 = 1). The constants of the model (𝛽∗, 𝛽3, 𝜍𝑘3, ...) are a linear combination of
the coefficients appearing in the formulations of 𝑘 − 휀 and 𝑘 − 𝜔 through the blending function,
i.e. taken ϕ as a general constant it holds 𝜙𝑆𝑆𝑇 = 𝐹1𝜙𝑘−𝜔 + (1− 𝐹1)𝜙𝑘−휀 . The blending function is formulated empirically.
7.2 Near-wall treatment
In the previous chapter it was seen that, depending on how close the first node of the mesh is
from the wall, the flow in the boundary layer can be either resolved or modelled through wall
functions, and grids for all components of the turbine were designed based on the target 𝑦+.
However an exact estimation of 𝑦+ is difficult to achieve because 𝑦+ depends on 𝑅𝑒 , which
varies inside the turbine. In particular inaccuracies may occur if the wall function method is used
in regions where the first node of the mesh is inside the viscous sublayer, while if the Low
Reynolds Number method were used with the first node of the grid being in the log-law region it
would not be possible at all to describe viscous and buffer layer.
If the wall function is set to automatic, the correct near-wall treatment is automatically chosen by
CFX®. The program calculates 𝑦+ and if the mesh has a local near-wall distance corresponding
to 𝑦+ < 11.06 (default value, [22]) the boundary layer is resolved, otherwise wall functions are
used. In this way automatic wall function allows the highest possible accuracy.
7.3 Boundary conditions and interfaces
Boundary conditions are relative to the design point and are common to both vaneless and vaned
configurations. Boundary conditions are derived from the meanline design and reported below
Inlet total pressure: 2.5636 [bar]
Inlet total temperature: 845 [K]
Outlet static pressure: 1.1277 [bar]
Flow direction at inlet: normal to inlet section
Other combinations of boundary conditions are possible in CFX® (mass flow rate, velocity...) but
their use turned out to give solutions non-monotonically convergent (the residuals of mass and
momentum showed undamped oscillations) or even non convergent at all, while with the
specification of total pressure at inlet and static pressure at outlet the solution was stable. This
choice is also supported by literature ([9], [22])
No slip condition was set at the walls. Walls are modelled as adiabatic and smooth.
48
Interfaces are used to model the contact region between components of the turbine: across them
there can be a discontinuity in the mesh pattern, a variation of frame of reference (for example in
the matching rotor-stator) or a variation of pitch (this is the case when two contact domains have
different angular width).
For the present study the stage model provided by CFX® is used. In the stage model, also called
mixing plane model, the flow properties (i.e. velocity, pressure, temperature...) are averaged
circumferentially upstream of the interface in order to obtain the boundary condition for the
downstream component. Since the flow is averaged, the inlet boundary condition for the
component downstream of the interface is steady, thus this method is suitable for steady state
simulations, which is the case here.
Another type of interface is the rotational periodicity in a rotor SBP, which allows to model the
whole rotor as just one blade passage, thus saving computational time. All interfaces and
boundary conditions are highlighted in the ensemble Fig.42 (only the vaned configuration is
illustrated, as it is the most complete case).
Figure 42: illustration of interfaces and boundary conditions for vaned configuration
7.4 Choice of off-design points
The design point represents average conditions of inlet pressure and temperature, but within the
combustion cycle these parameters vary over a considerable range, as seen from Fig.43.
Off-design points were determined [14] by measuring the flow properties of the exhausts at
constant intervals of 2.32 milliseconds after the opening of the discharge valve: among them
only the most representative are chosen for this study, and are listed in Tab.8
49
Figure 43: choice of representative off-design points (Mora, [14], adapted)
In particular points 1, 4, 5 are associated with the “furthest” thermodynamic conditions with
respect to the average, while point 7 is the closest to it and is chosen to represent the effect that
small variations around the mean inlet conditions may have on the performance of the turbine.
Notice that the highest temperature of the combustion cycle is not associated with the highest
pressure, hence both points are studied.
POINT TEMPERATURE (total) [K] PRESSURE (total) [bar]
1 762.9 2.0754
4 993.1 2.8438
5 935.1 3.2098
7 829.1 2.6339
Table 8: thermodynamic properties of the studied off-design points
50
8 – RESULTS
This section presents the main results of the steady simulations run with CFX® on the designed
CAD model. For each working condition (design and off-design) the turbine is analysed in both
configurations, with and without nozzle ring, and results are compared. Such results will then be
discussed in Chapter 8 in order to answer to the stated objective.
8.1 Design point
PARAMETER VANELESS VANED
Mass flow [kg/s] 0.396 0.417
Torque on rotor [Nm] (*) -5.11 -5.29
Power [kW] 45.47 47.07
Efficiency T-T [-] 0.751 0.762
Table 9: performance comparison, design point
(*) The negative sign is a convention, as the sense of rotation of the impeller is clockwise.
Performance data relative to the design condition are calculated and reported in Tab.9, while
Tab.10 illustrates the mean velocity triangles at rotor inlet and outlet.
PARAMETER
(rotor inlet,
average)
VANELESS
(red)
VANED
(blue)
(*)
𝑈2 [𝑚/𝑠] 365 358
𝑐2 [𝑚/𝑠] 333 324
𝑤2 [𝑚/𝑠] 176 161
𝛼2 [°] 61.3 63.2
𝛽2 [°] -24.6 -24.7
PARAMETER
(rotor outlet,
average)
VANELESS
(red)
VANED
(blue)
𝑈3 [𝑚/𝑠] 153 161
𝑤3 [𝑚/𝑠] 398 403
𝑐3 [𝑚/𝑠] 311 311
𝛽3 [°] -38.8 -39.9
𝛼3 [°] -15.8 -15.96
Table 10: comparison between mean velocity triangles at rotor inlet (top) and at rotor outlet
(bottom), design point
(*) Sketches represent the projection in the meridional plane (where α and β are defined) but c and w are general
3D vectors and also have an out-of-plane component.
51
The spanwise distribution of 𝛽2 is illustrated in Fig.44, while distribution of 𝛼3 in Fig.45. For
each spanwise position the correspondent value is obtained as a mass average over the
circumferential coordinate.
Figure 44: spanwise distribution of 𝛽2, comparison at design point
Figure 45: spanwise distribution of 𝛼3, comparison at design point
52
Figure 46: rotor blade loading, comparison at 10% span (top, left), 50% span (top, right), 90%
span (bottom), design point
Figure 47: static entropy around the rotor blade at 90% span. Left: vaneless. Right: vaned
53
Figure 48: Mach distribution around the rotor blade at 10% span. Top: vaneless. Bottom: vaned
Figure 49: Mach distribution around the rotor blade at 50% span. Left: vaneless. Right: vaned
Figure 50: Mach distribution around the rotor blade at 90% span. Left: vaneless. Right: vaned
54
Fig.46 shows the comparison of blade loading for three representative spanwise positions: close
to hub (10%), mid span (50%) and close to shroud (90%). Fig.48-49-50 allow to visualize the
blade-to-blade velocity distributions (in terms of Mach number) at the same spanwise locations
at which the blade loading was evaluated. Fig.47, instead, is useful to visualize the static entropy
at a particular section in which flow separation is detected.
Fig.51 shows the flow around a blade of the nozzle ring at design point. The blade in the picture
is the closest to the volute tongue.
Figure 51 : velocity contour around nozzle ring blade, 50% span, design point
8.2 Off-design points
POINT CONFIGURATION Mass flow
[kg/s]
Torque on
rotor [Nm]
Power
[kW]
Efficiency T-T
[-]
1 vaneless 0.312 -2.88 25.64 0.757
vaned 0.321 -2.93 26.06 0.769
4 vaneless 0.428 -6.56 58.43 0.707
vaned 0.438 -6.86 61.10 0.729
5 vaneless 0.500 -8.08 71.89 0.795
vaned 0.506 -8.03 71.46 0.712
7 vaneless 0.424 -5.51 49.03 0.825
vaned 0.430 -5.51 49.03 0.763
POINT CONFIGURATION 𝛽2 [°] 𝛼3 [°]
1 vaneless -41.85 -15.68
vaned -44.32 -12.14
4 vaneless -13.25 -20.96
vaned -10.29 -21.89
5 vaneless -13.78 -22.38
vaned -14.04 -22.37
7 vaneless -22.28 -18.01
vaned -25.51 -16.42
Table 11: comparison of performance (top) and mean flow angles (bottom) at off-design points
55
8.2.1 Off-design point 1
Figure 52: spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 1
Figure 53 : Velocity distribution around the rotor blade at 50% span. Left: vaneless. Right:
vaned
Figure 54: Velocity distribution around the rotor blade at 90% span. Left: vaneless. Right:
vaned
56
8.2.2 Off-design point 4
Figure 55: spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 4
Figure 56: static entropy in meridional plane, circumferential average, off-design point 4. Left:
vaneless. Right: vaned
Figure 57: Velocity distribution around the blade (left) and blade loading (right) at 90% span,
off-design point 4
57
8.2.3 Off-design point 5
Figure 58: spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 5
Figure 59: static entropy in meridional plane, circumferential average, off-design point 5. Left:
vaneless. Right: vaned
Figure 60: velocity contour around nozzle ring blade, 50% span, off-design point 5
58
8.2.4 Off-design point 7
Figure 61: spanwise distribution of 𝛽2 (left) and 𝛼3 (right), off-design point 7
Figure 62: rotor blade loading, comparison at 10% span (top, left), 50% span (top, right), 90%
span (bottom), off-design point 7
59
9 – DISCUSSION
In this section results are analysed: the goal is to investigate the effects of the static nozzle ring
on the performance of the radial turbine. In order to do a “fair” comparison, at design point both
vaned and vaneless configurations must have the same average inflow conditions at rotor inlet.
Mean velocity triangles are reported in Tab.10. It holds 𝑐2_𝑣𝑎𝑛𝑒𝑑
𝑐2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠= 0.973,
𝛽2_𝑣𝑎𝑛𝑒𝑑
𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠= 1.004,
hence between the two cases the absolute velocity varies by less than 3% and the relative inflow
angle by less than 1%. The condition is fulfilled. Similar situation is present at rotor outlet,
where again the mean velocity triangles show non-appreciable differences (Tab.10, bottom).
According to a meanline analysis there would be no difference in incidence losses or TE losses
between vaned and vaneless configurations.
Fig.44 shows the spanwise distribution of 𝛽2. In both cases the relative inflow angle drops near
hub and shroud, while it is above the average value around mid span. At each spanwise position,
a strong correlation is noticed between the local 𝛽2 and the flow velocity around the blade: this
was an expected behaviour, as the inflow angle affects the position of the stagnation point at the
LE of the local blade profile and consequently the whole pressure distribution. At 90% span it
holds 𝛽2_𝑣𝑎𝑛𝑒𝑑 > 𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 (Fig.44): in the vaned turbine the stagnation point is shifted towards
the pressure side of the blade (the flow is more radial, which represents the ideal nominal design
condition), and as can be seen from Fig.50 this causes the flow on the suction side to separate
closer to the TE. This is in accordance with the correspondent blade loading plot (Fig.46,
bottom), where pressure on the suction side keep decreasing in the vaned case (blue curve) while
in the vaneless case the blade profile has already stalled (red curve). From Fig.46 it can be seen
that the rotor blade is more loaded in the vaned turbine: as blade loading determines the moment
on the impeller about the axis of rotation, the higher blade loading justifies the higher power
output of the vaned case (Tab.9). The difference is about 3.5%. Separation not only affects the
blade loading but also the efficiency. Regions of separated flow are turbulent and strong
dissipation of kinetic energy occurs. This is visualized in Fig.47, which represents the entropy of
the flow in the blade-to-blade plane (90% span). Higher entropy in the vaneless case denotes
higher amount of losses with respect to the turbine with nozzle ring: this difference is indeed
confirmed by the performance calculations and is about 1.5% (Tab.9).
In the vaned configuration separation may occur at nozzle ring blades. This is not the case at
design point, except for the blade downstream of the volute tongue. Here the free-vortex law,
which was the underlying assumption for the design of the volute, is not valid [32] and the flow
angle at volute outlet is higher than the design value. However the region subject to separation is
very small (see Fig.51).
At off-design points both configurations undergo substantial modifications of average inflow and
outflow angles with respect to the design value (Tab.11, bottom). This first observation suggests
that the nozzle ring does not constrain 𝛽2 to the design value (or close values) also at off-design
points. Inspection of Fig.52, 55, 58, 61 shows that not only the mean value but also the spanwise
distribution of 𝛽2 (and 𝛼3) vary considerably. Another straightforward consideration comes from
Tab.11: it does not exist a turbine configuration which guarantees better performance (in terms
of efficiency and power output) at all working conditions.
Efficiency seems to be related to the spanwise distribution of 𝛽2. Let’s consider off-design point
4. The blue curve (vaned) is always above the red one (vaneless), which means that 𝛽2 is closer
to 0 (ideal case) through all the span. A plot in the meridional plane (Fig.56) shows that this
condition corresponds to lower amount of static entropy, especially close to hub and shroud
(where, according to Fig.55, left, the difference in inflow angle increases even more). As a
60
consequence the efficiency of the vaned turbine is higher than vaneless for this off-design point
(around 3.1%). The relashionship between rotor inflow angle and entropy should be investigated
in details, but the intuitive explanation is that the blade has been designed under the condition
𝛽2,𝑏 = 𝛽2 = 0 at all spanwise locations and the less this condition is satisfied the more the flow
is subject to detachment or recirculation, which leads to an increase of entropy. The situation is
visualized in Fig.57: close to the shroud (90% span) both configurations separate at TE (the
velocity around the blade drops at streamwise position near 0.7) but the vaned separates after,
hence not only flow entropy is lower (Fig.56, shroud) but also blade loading is higher (Fig.57,
right) and ultimately power output is higher.
The same argumentation can be followed for off-design point 5, but here 𝛽2_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 >𝛽2_𝑣𝑎𝑛𝑒𝑑 everywhere except close to hub and shroud (Fig.58, left). The meridional plot of entropy
shows the forecasted behaviour (Fig.59) and efficiency is higher for the vaneless configuration
(Tab.11). Moreover in this case the region of separated flow at the blade of the nozzle ring is
bigger than at design point (again the difference is more evident for the blade close to volute
tongue).
At off-design point 1 the inflow angle is large and separation occurs both at LE and TE (Fig.54).
The situation regarding 𝛽2 and consequent flow pattern can be interpreted as the previous cases,
however here the difference in 𝛼3 between vaned and vaneless turbine is not negligible and as
𝛼3_𝑣𝑎𝑛𝑒𝑙𝑒𝑠𝑠 < 𝛼3_𝑣𝑎𝑛𝑒𝑑 (mean value) a difference in TE losses must be considered together with
losses due to 𝛽2. Overall efficiency for the vaned case is slightly higher (+1.6%) than vaneless.
Off-design point 7 falls inside the frame used to describe the previous cases.
Regarding mass flow at design point it can be noticed that the value for both turbine
configurations is higher with respect to the meanline value that was used during the design
phase. This difference has already been pointed out in literature [22]. The most plausible reason
is that the meanline code over-predicts the effects of blockage inside the rotor blade channel. As
can be seen from blade-to-blade Mach plots (Fig.48, 49, 50), Mach can be locally 1 (or higher),
especially near the shroud, but the channel is never chocked at all spanwise positions, not even
for off-design points implying the highest values of inlet pressure and temperature (in this case
points 4 and 5). The reason why mass flow in the vaned configuration is slightly higher than
vaneless at all working conditions may be linked to the inflow angle 𝛽2 near the hub: in the
vaneless case this stays well below the average (while in the vaned it tends to increase more
rapidly) and such low values of the inflow angle locally cause recirculation. Somehow chaotic
streamlines can be seen near the hub, and this may support the statement, but as this cannot be
proved a deeper investigation should be done on the topic.
This analysis shows that the reason why a turbine configuration is “better” than another one is
mainly related to the inflow angle at rotor inlet, which should be somehow close to the design
value for the rotor blade (in this case 0°). If the presence of a static nozzle ring guarantees this
condition in a certain working point the vaned configuration is preferable to the vaneless,
otherwise not, but from this analysis it is possible to conclude that the addition of a static nozzle
ring in a 90° IFR turbine does not improve performance in all the combustion cycle.
61
10 – CONCLUSION AND FUTURE WORKS
In this thesis a comparison between vaneless and vaned turbine configurations has been made on
the basis of numerical simulations carried out in steady conditions with a CFD commercial
software. The use of a static nozzle ring increases both efficiency and power output at design
point (increment is roughly estimated in +1.5% and +3.5% respectively). On the other hand, off-
design performances strongly depend on the thermodynamic flow conditions which characterize
the specific point, and a general trend does not exist: however it is proved that the
implementation of a fixed nozzle ring in a radial turbine does not guarantee higher efficiency
through all the engine combustion cycle.
Inputs for possible future works are listed below:
Results presented in this thesis solely rely on a numerical model, whose uncertainties
have been highlighted in Chapter 4 – Limitations. In order to assess the validity of the
results the latter should be compared with experimental data
In this study only steady conditions have been considered, and it is implicitly assumed
that the flow has enough time to adapt to variations of pressure and temperature in the
exhaust gases. This may not be true in general, especially when the engine operates at
high rpm, hence a further study should model the unsteadiness of the flow
Future efforts could be made for improving the 3D geometrical model of the radial
turbine: for example the casing which surrounds the rotor is not modelled here, which
implies a poor evaluation of tip clearance losses
No structural considerations are made regarding the design of the turbine. The rotor, in
particular, is subject to high centrifugal forces, and despite the blade tip speed is limited
to 𝑈2 < 400 𝑚/𝑠 under design parameters (𝑈2 ~ 356 𝑚/𝑠), there is no guarantee that
the blade thickness distribution suggested in this thesis results in mechanical stresses
which do not lead to fracture or plastic deformation of the blade. For this reason a further
study is needed in order to check the feasibility of the present design.
62
11 – BIBLIOGRAPHY
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63
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64
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65
APPENDIX 1: ABSOLUTE ANGLE AT ROTOR INLET
66
Figure 63: spanwise distribution of 𝛼2. From top to bottom: design point, off-design 1, off-
design 4, off-design 5, off-design 7
Notice that the spanwise distributions of α2 and β2 are strongly related, since at rotor inlet the
blade speed is constant through all the span (points are equidistant from the rotational axis);
however plots of α2 are reported as well, for completeness.
