A comparison between steady and unsteady methods for turbomachinery design using scaling analysis

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Dipartimento di Ingegneria dell’Energia, dei Sistemi, del Territorio e delle Costruzioni Master of Science in Energy Engineering

A comparison between steady and unsteady

methods for turbomachinery design using scaling

analysis

by

Rosa D’Amato

Student number: 483270

Thesis committee: Prof. U. Desideri, Università di Pisa, relatore Prof. M. Pini, TU Delft, relatore

Dr. A. Rubino, TU Delft, correlatore

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Preface

Ai miei genitori, punti cardine di tutto il mio percorso.

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Acknowledgements

I would like to acknowledge Professor Matteo Pini for giving me the opportunity to collaborate with the team of ’Flight Performance and Propulsion’ of Aerospace at TU Delft, for his useful and important ad-vice for my thesis.

I would like to thank my PhD supervisor at the TU Delft Antonio Rubino for his constant guidance. I sincerely appreciate your enthusiasm and the time you dedicated to me even when you were busy with your academic and personal duties. Your critical feedback, constant help and encouragement was a driving force in the successful completion of this work.

I would also like to thank my supervisor in Italy, professor Umberto Desideri, and the professor Daniele Testi of the University of Pisa for giving me the chance to elaborate and write this thesis in the Netherlands at the TU Delft.

Rosa D’Amato Delft, February 2019

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List of Figures

2.1 Enthalpy-entropy diagram for one process in a axial turbine stage . . . 4

2.2 The three loss components and the overall loss (Mee, 1990) . . . 5

2.3 Schematic of wake interaction from an upstream stator with a downstream rotor (Tucker, 2011) . . . 7

2.4 The convection of a wake through a blade row; contours of entropy (Denton, 1993) . . . . 8

3.1 Smith’s Chart . . . 14

3.2 Test Case 1 - Geometry Cascade . . . 15

3.5 Computational mesh for the Test Case 1 (a), Test case 2 (b), Test case 3 (c) . . . 18

3.6 Total to total efficiency: HB solution obtained for different time instances . . . 19

3.7 Verification of the similitude - Entropy contours for case 1 (a) and case 3 (b) calculated with MP method and with the HB method at five different time instances . . . 23

3.8 Verification of the similitude - Total to total efficiency and work and load coefficient as a function of the total inlet pressure p01 . . . 23

4.1 Test case 1 - Total to total efficiency (a) and load and work coefficient (b) as a function ofβ 25 4.2 Test Case 1 - Entropy contours forβ = 1.6 calculated with MP method and with the HB method at five different time instances . . . 25

4.3 Test Case 1 - Absolute Mach number and Relative Mach number as a function of the pres-sure ratio . . . 26

4.4 Test Case 1 - Mach Relative contour forβ = 2.4 calculated with MP method and with the HB at five different time instances . . . 27

4.5 Test Case 1 - Velocity distributions at the trailing edge of stator and rotor forβ = 2.4 (a) and β = 3.2 (b) . . . 27

4.6 Test Case 1 - Mach Relative contours forβ = 3.2 calculated with MP method and with the HB method at five different time instances . . . 28

4.7 Test Case 1 - Contributes of loss as a function of the expansion ratio in steady and unsteady conditions . . . 28

4.8 Total to total efficiency as a function ofβ for Test case 2 (a) and Test Case 3 (b) . . . 29

4.9 Difference between steady and unsteady total to total efficiency as a function ofβ for the differet test cases . . . 30

4.10 Test Case 2 - Mach Relative contours forβ = 3 calculated with MP method and with the HB method at five different time instances . . . 30

4.11 Test Case 3 - Mach Relative contours forβ = 2.4 calculated with MP method and with the HB method at five different time instances . . . 31

4.12 Total to total efficiency (a) and load and flow coefficient as a function ofγ for Test case 1 . . 32

4.13 Test Case 1 - Mach Relative contours for MM calculated with MP method and with the HB method at five different time instances . . . 33

4.14 Test Case 1 - Contributes of loss as a function of the heat capacity ratio in steady and un-steady conditions . . . 34

4.15 Total to total efficiency (a) and load and flow coefficient as a function ofγ for Test case 3 . . 35

4.16 Test Case 3 - Mach Relative contours for MM calculated with MP method and with the HB method at five different time instances . . . 35

4.17 Difference between steady and unsteady total to total efficiency as a function ofγ for the differet test cases . . . 36

4.18 Total to total efficiency as a function of ND gap for Test case 1 (a) and Test Case 2 (b) . . . . 36

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List of Figures v

4.19 Difference between steady and unsteady total to total efficiency as a function of ND gap for the differet test cases . . . 37 4.20 Test Case 1 - Contributes of loss as a function of the ND gap in steady and unsteady

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List of Tables

3.1 Test Case 1: Specificarions of the stage design point . . . 15

3.2 Test Case 2: Specifications of the stage design point . . . 16

3.3 Test Case 3: Specifications of the stage design point . . . 17

3.4 Verification of similitude: Non-dimensional numbers . . . 21

3.5 Verification of similitude: Main simulation parameters . . . 22

4.1 Test case 1: Non-dimensional numbers for studying the effect ofβ . . . 24

4.2 Test case 2 and Test case 3: Non-dimensional numbers for studying the effect ofβ . . . 30

4.3 Working fluid proprierties . . . 32

4.4 Test case 1: Non-dimensional numbers for studying the effect ofγ . . . 32

4.5 Test case 3: Non-dimensional numbers for studying the effect of_{γ . . . 34}

4.6 Test case 1 and Test case 2: Non-dimensional numbers for studying the effect of ND axial gap . . . 36

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1

Introduction

Background

Turbomachines are an essential part of modern life [15]. Nowadays, turbines are used in all signifi-cant electricity production in power plants, turbomachines, wind farms. Turbomachinery operating with special working fluid are used in various engineering fields. The hydrogen turbomachineries are applied to the propulsion system; the organic working fluid are applied to the organic Rankine cycle (ORC). In order to satisfy the growing demand of high performance turbine for increasing the economy, safety and environment protection, technological improvements and greater efficiencies are necessary. In recent years, the development and application of computational fluid dynamics (CFD) has made a significant impact on design. The computational design tools offer a feasible approach to solve a very complex optimization problems involving a large number of design variables. The application of the computational design optimization methods to turbomachinery blade designs can reduce design cost as well as increase efficiency.

Flow fields in turbomachinery are inherently unsteady, due to the relative motion of adjacent blade rows (Giles, 1992) in ref. [10]. Two forms of periodic unsteadiness: the wake interaction from the up-stream blade rows and the potential flow interaction of blade rows. The potential field can propagate both upstream and downstream. The magnitude of this effect depends on the Mach number and the axial distance from the blade row [11]. In high Mach number flows, potential interactions tend to be stronger than at lower Mach numbers. Unlike the potential interaction, a blade wake is only convected downstream. A wake profile is characterized by a velocity deficit, and the static pressure in it does usu-ally not vary significantly. It can influence the surface pressure, heat transfer and boundary layer of the downstream blades. When a turbine operates in the transonic regime, shock waves occur. In addition to the losses produced by the periodic movement of the shock itself, the shock wave can cause unsteady effect. These effects can have also a beneficial impact on the performance such as the recovery of the energy in the wakes or the wake-boundary layer interaction like a calming effect [19]. However, all these phenomena are not predicted by steady state assumption.

Motivation

Unsteady flow phenomena significantly can affect the overall stage efficiency and have other important consequences such as blade flutter or forced response, as well as noise and thermal stresses. However, due to high computational cost, the optimization methods for turbomachinery are mostly based on the assumption of steady flow. Nonetheless, there are some cases in which the unsteady flow phenomena like effect of the wake and the wake-boundary layer interaction cannot be neglected. Therefore, for a more realistic representation of the actual performance it could be important to include unsteady effects in turbomachinery optimization phase.

Scope and Objective

The aim of this work is to evaluate the difference between steady and unsteady flow predictions in or-der to investigate the impact of unsteady effects on the performance of axial turbine stages. Moreover,

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1.4. Outline 2

when the influence of the unsteady phenomena is not relevant, the use of steady-state methods can be sufficiently accurate to compute the fluid dynamic performance. Additional, the low impact of the unsteadiness on the performance could lead to the use of steady methods also for the aerodynamic de-sign of turbomachinery. The steady calculations are performed by using a mixing plane method (MP), whereas the unsteady calculations are solved with the harmonic balance method (HB).

A primary goal of this work is to verify the similitude in the CFD simulations in unsteady and steady conditions. By dimensional analysis, the meaningful dimensionless numbers based on input parame-ters for the two-dimensional (2D) axial cascade, are derived as a similarity criterion. The blade designs are studied by varying the expansion ratio, the working fluid and the non dimensional axial gap. The ef-fect of unsteady flow are quantified in terms of a degradation or enhancement of the unsteady efficiency in comparison to steady efficiency. To analyze and quantify the losses which arises as consequence of the unsteady flow, a model of losses estimation is taken into account. The loss generation in two dimen-sional axial cascade are produced from three sources: boundary layer losses, mixing and trailing edge losses and shock losses. Mee et al. [21] presented the kinetic energy loss coefficients as a measure for the different contributions of the profile losses of a single cascade. In this work they are calculated from the unsteady and steady numerical simulation results.

Outline

Chapter 2 presents the general concepts about the losses in the two-dimensional (2D) axial cascade, unsteady flows including the origins of unsteadiness and its effect on the performance and the theory of similitude. Chapter 3 describes the steady and unsteady methods used in this work and the test cases used for the simulations. This chapter also presents the verification of the similitude using the CFD Chapter 4 presents the analysis and the results of the study. Different cascades are examined varying the operating conditions. Three main non dimensional number are studied: the working fluid, the ex-pansion ratio and the non dimensional axial gap. The steady and unsteady performance prediction is discussed. This chapter the loss generated in the cascade are computed using the data from numer-ical simulations. The chapter 5 presents a discussion of the current work, the main conclusions and suggestions for future work.

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2

Theoretical Background

The aim of this chapter is to provide an overview of the losses developmemt in axial 2D turbine blade. Additional, it focuses on the sources of unsteady flow, unsteady loss generation mechanisms and pro-vides an insight into the mechanisms of interaction between the stator and the rotor of a turbine. Un-steady flow calculations form an important part of the current investigation. This chapter also discusses the use of similitude and the dimensionless analysis in designing turbine blades.

Losses in Axial flow turbine

There are three fluid dynamic processes that create entropy, and hence loss: viscous friction, heat trans-fer and non-equilibrium processes (Denton, 1993) in ref [1] . The total loss is computes as a sommation of the individual losses created in a cascade rows. The main loss mechanisms in two-dimensional cas-cades are:

• Boundary layer lossdue to shear stresses in the boundary layer. • Shock lossdue to the shock waves in the flow field.

• Mixing lossgenerated by mixing of the flow.

Several form of loss coefficient can be defined. If the losses are referred to the same energy, the total loss coefficient is the sum of boundary layer loss coefficient, shock loss coefficient and mixing loss or trailing edge coefficient as given below:

ξt ot= ξbl+ ξshock+ ξt e

In this work, the kinetic energy loss coefficients will be adopted to express each contribution of the total losses trough the cascade. The Figure 2.2 from Meein ref. [21] shows the individual contributions of boundary layer loss, shock loss and mixing loss in different Mach number regime for a single cascade. The primary loss coefficient is defined as :

ξt ot= 1 − η

Whereη is the efficiency, defined as the ratio of the actual kinetic energy of the flow at the mixed-out plane to that which would be present if the flow expanded isentropically from the inlet conditions to the measured mixed-out static pressure. This can be written in terms of enthalpy for stator and rotor respectively as: ηs= h01− h1 h00− h1s ηr= hr el 02− h2 hr el 01− h2s 3

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2.1. Losses in Axial flow turbine 4

Figure 2.1: Enthalpy-entropy diagram for one process in a axial turbine stage

The performance of the flow can be represented in an enthaly-entropy diagram in Figure 2.1. In this figure the ideal process through all the turbine stage is from the point 0 to 2s and a real process (with the losses) is from the point 0 to 2. The term h1is the static enthalpy at the stator outlet and term h2is the

static enthalpy at the rotor outlet. The h00is the total enthalpy at the stator inlet and the h01is the total

enthalpy at the stator outlet. Similar for the rotor, but considering the stagnation enthalpy of the relative flow, the hr el 01is the relative total enthalpy at the rotor inlet and the hr el 02is the relative total enthalpy

at the rotot outlet. The subscript ‘s’ refers to the isoentropic enthalpy at the outlet of both stator and rotor.

Boundary layers losses

Boundary layers are regions of steep velocity gradients and large shear stresses. These highly viscous regions are responsible for much of the loss created in a turbomachine. The flow over the blade sur-faces is decelerated by the viscous forces present inside the blade boundary layer. According to Denton (1993), the boundary layer loss depends upon the blade chord length, surface velocities and a dissipa-tion coefficient, funcdissipa-tion of the state of the boundary layer and the Reynolds number based on the local boundary layer thickness.

The method used in this work in order to determinate the boundary layer loss coefficient is pro-posed by Mee (1984) [21]. According to Mee, the boundary layer loss can be estimated by measuring the boundary layer profile at the trailing edge of the suction surface. He indicated that the pressure surface layer is an order of magnitude thiner than that on the suction surface and thus does not contribute sig-nificantly to the total boundary layer losses. The definition of loss regards the ratio of the kinetic energy dissipated in the boundary layer to the ideal kinetic energy at the mixed-out plane. The losses over the stator and the rotor can be separately calculated as shown below:

ξbl ,s= ∆ ˙ ek ˙ m(h00− h1s) ξbl ,r= ∆ ˙ ek ˙ m(hr el 01− h2s)

The kinetic energy dissipated in the boundary layer, considering also the pressure surface layer, can be estimated by the following equation:

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2.1. Losses in Axial flow turbine 5

Figure 2.2: The three loss components and the overall loss (Mee, 1990)

∆ek= 0.5[(ρeδew3e)ss+ (ρeδew3e)ps]

Where weis the flow speed andρeis the energy dissipation thickness of the boundary layer, defined by

Schlichting (1979) as δe= Zδ 0 ρw ρewe3 (w2_e− w2)d y

In this work , the measurement of the thickness of the boundary layerδ and the velocity profile at the trailing edge of the blade are computed by the output of the CFD simulations.

Shock losses

The shock losses are the losses due to viscous dissipation across the shock. Strong shock waves may cause complete boundary layer separation. This loss could be a substantial portion of the total profile losses, depending on Mach number and Reynolds number. When the flow passes through a shock wave its total pressure decreases. Shock losses can be estimated by examining the total pressure drop in the regions outside of the blade wakes. In this way it will be possible to exclude the effect of the losses of the boundary layer due to the blade and the losses of the wake generated at the trailing edge in order to consider only the losses due to the shock.

The formula proposed by Mee [21] for a single cascade and used in this work for the computation of the shock losses in the stator is:

ξsh,s= 1 − cpT02   1 − µ_P 2,now ake P02,now ake ¶ γ − 1 γ    cpT01   1 − µ_P 2,now ake ¯ P01 ¶γ − 1 γ   

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2.1. Losses in Axial flow turbine 6 becomes : ξsh,r= 1 − cpT02,r el   1 − µ _P 2,now ake P02,r el ,now ake ¶ γ − 1 γ    cpT01,r el    1 − µ_P ¯ 2,now ake ¯ P01,r el ¶ γ − 1 γ    

Trailing edge losses

The trailing edge losses are the losses due to the finite thickness of the blade trailing edge. This thickness gives rise to various loss generating mechanisms such as shock-wave interactions, boundary layer sep-aration and mixing of jets. Denton (1993) [1] considered that this loss is a function of the base pressure, flow angle at the outlet of the cascade, blade pitch, thickness of the trailing edge, boundary layer dis-placement and the momentum thickness at the trailing edge. For turbine cascades most of the increase of the losses can be attributed to the trailing edge loss. The mixing losses can be found by subtracting the boundary layer and shock loss components from the total losses. The trailing edge loss component is calculeted using the relation:

ξt e= ξt ot− ξbl− ξsh

This treatment is applicable for steady and unsteady data as well. For unsteady data the mean losses will be an average of the losses generated in different time-step.

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2.2. Source of unsteadiness 7

Figure 2.3: Schematic of wake interaction from an upstream stator with a downstream rotor (Tucker, 2011)

Source of unsteadiness

Flow in turbine blade rows is highly unsteady because of the periodically encountered flow distortions generated by the upstream and downstream blade rows [19]. These unsteady flow can have a substantial impact on the efficiency reducing also the operating life of a turbomachinery. Moreover, unsteady effects causes unsteady forces responsible of the mechanical fatigue and to noise generation in a engine. It is widely recognized that unsteady effects are the source of significant losses and therefore offer an obvious possibility to improve the efficiency of turbine stages in modern power plants. There are three principal two-dimensional sources of unsteadiness in a single stage of a turbomachine:

• Potential interactions of the upstream and the downstream blade rows..

• Interactions of the upstream wakes with the downstream blade rows.

• Shock wave interaction with the downstream blade rows.

The Figure 2.3 from Tucker (2011) [32] represents schematically these interaction.

It is useful to evaluate the strength of flow unsteadiness by the reduced frequency parameter as de-fined by Lighthill (1954). The reduced frequency is the ratio of time taken by the given particle for con-vection through the blade passage to the time taken for the rotor to sweep past one stator passage. It is expressed as ¯ f = f · yp va = C onvenc t i ont i me Di st ur bancet i me

where f is the blade passing frequency, yp is the blade pitch and vais the axial velocity at the blade

exit. The magnitude of the reduced frequency is a measure of the degree of unsteady effects compared to quasi-steady effects. If f À 1 , unsteady effects are significant and dominate the flow field, when f = 1 , unsteady and quasi-steady effects coexist. The reduced frequency ¯f also represents the number of wakes (or other upstream unsteady features) found in a single blade passage at any instant in time. Although the reduced frequency parameter characterizes the unsteadiness in a qualitative way, it does not shed much light on the magnitude of these effects. For this, the interactions them selves have to be considered.

Potential interactions

The potential interaction arises because the pressure in the region between the stator and rotor blade can be decomposed into a part that is steady and uniform and a part that is non-uniform but steady in the stator and rotor frame. As the rotor blades move, the stator trailing edges and the rotor loading edge experience an unsteady pressure due to the non-uniform part (Giles, 1993)[11]. The magnitude

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of this effect depends on a number of factors. The potential field varies with the axial distance from the blade row, the pitch of the blade row and the local Mach number. In high Mach number flows, potential interactions will tend to be stronger than at lower speeds. Confirmation of the downstream propagation potential fields, can be found in many experiments in literature. Parker and Watson (1972) [27] suggested that the effects of potential interactions will be insignificant for axial spacing greater than about 30 per cent of the blade pitch. Recently, Gaetani(2010) [9] confirmed that the maximum overall efficiency and the minimum unsteadiness are achieved for a vane-rotor axial gap equal to one third of the stator axial cord.

Wake blade interactions

Wake blade interaction causes unsteadiness because the stator wakes, which are approximately steady in the stator frame, are unsteady in the rotor frame of reference since the rotor is moving through the wakes and chopping them into pieces. This causes unsteady forces on the rotor blades and generates unsteady pressure waves (Giles, 1993 [11]). While the potential influence of the blade row extends up-stream and downup-stream, blade wake is only convected downup-stream.

The Figure 2.4 shows the convection of a wake through a blade row. The wake becomes highly dis-torted and stretched because the part adjacent to the suction surface convects more rapidly than that adjacent to the pressure surface. The velocity deficit of the center of the wake is reduced by this invis-cid effect as it also is by viscous effects. By the way, the convection of the upstream wake segment in the downstream blade row is characterized by bowing, reorientation, elongation, and stretching (Smith, 1966; Stieger and Hodson, 2005)[14]. This kinematic wake transport also effects the loss generated from mixing of these wakes in the downstream blade row. Moreover, there is a generic mechanism with sig-nificant influence of wake transport on performance: reversible recovery of the energy in the wakes (beneficial). [12]

The unsteady energy kinetic of the wake through a turbomachinery could be not necessarily lost. This energy could turn into work in the rotor domain and hence could be not equal to a loss. Mokulys defined an energy recover factor which is the ratio of the change of unsteady energy ratio over the rotor domain and the unsteady loss in the same region. In this way, he quantifies how much of the initial unsteady kinetic energy in the system is turned into energy in the rotor passage, and how much of it is turned into unsteady loss. [24]

Figure 2.4: The convection of a wake through a blade row; contours of entropy (Denton, 1993)

In additional, the wake interaction plays an important role in the development of the blade boundary layers. The wake induces a periodic-unsteady transition process that causes alterations in the boundary layer characteristic. As the wakes convect through the blade passage, induce boundary layer transition in a bypass mode. This effect can effectively inhibit the boundary layer separation that is periodically suppressed by turbulent and calming flow. This periodically changing of the boundary layer can have a positive impact on performance reducing the profile loss of the cascade. Unsteady wake-boundary layer interaction has been subject of interest for many researcher and a large number of author described the wake induced transition. [25] [30]

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Shock wave interactions

Transonic turbines are commonly used to obtain high-stage pressure ratios and so shock waves do oc-cur. The most serious consequence of the transonic flow in a turbine is a shock system at the trailing edge. The motion of the upstream periodic shock waves causes the surface of the downstream blade to be subjected to a significant unsteady pressure field [19]. Cause the entropy increase in a shock wave is such a nonlinear function of the preshock Mach number, any periodic motion of the shock will gener-ate increased loss. Larger shock amplitudes cause consequently larger increases in loss.[23] Effectively the increase in entropy generation when the shock is moving forward will be greater than the reduction when it is moving backward, Denton(1993) [1]. Moreover, the interaction between shocks and boundary layers can lead to unsteady boundary layer separations and so an increase in loss for transonic veloci-ties. These shock wave interaction can generate significant vibrations that cause high cycle fatigue and eventually, if not identified, to a catastrophic blade failure.

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2.3. Similarity Theory 10

Similarity Theory

Turbomachines can be compared with each other by dimensional analysis. This analysis gives various types of similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups that are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis can be used in turbomachines in order to compare data from various types of machines. This technique is used in the development of blade passages and blade profiles. To select various types of units based on maximum efficiency and pressure head required and for the prediction a prototype performance from tests conducted on a smaller scale model.

The fundamental basis of dimensional analysis includes the use of Buckingham’s Pi-theorem to de-rive dimensionless groups that affect the efficiency of an axial turbine stage. By the_{π Theorem, if a} problem involves in Nvrelevant variables and Ndindependent dimensions than it can be reduced to a

relationship of (Nv− Nd) non-dimensional groups formed from these variables. The elementary

quan-tities Nd used in the variables could be the dimension mass (M), length (L), and time (T). The variety of factors that involved in the evaluation of the performance of a cascade could be summarized as follow:

• Thermodynamic variables. The stage stagnation enthalpy drop∆h0 determines the specific

work output and signifies stage loading. The specific enthalpies h0, h1, and h2that represent the

progression in energy transfer through the stage.

• Kinematic variables. Velocity triangle at the output of the stator and rotor

• Geometric variables. The pitch (yp) that represents the distance the distance in the direction

of rotation between corresponding points on adjacent blades or also the gap (ga), the distance

between two blade rows

• Fluid Properties. The dynamic viscosity (_{µ) and the density (ρ)} • Fluid Characteristics. Gas constant (R) and heat capacity ratio(γ)

The major dimensionless groups that are used for each stage of the machines and that have the greatest impacts on the performance, according to Smith (1965), are the flow coefficient, the work coefficient and the degree of reaction. The Flow coefficient (ϕ)that is a measure of the flow capacity of the stage expressed in dimensionless form:

ϕ =Va

U

where Vais the axial flow velocity and U is the peripheral velocity of the rotor. The Load coefficientψ

that is a measure of the work capacity of the stage expressed in dimensionless form:

ψ = w U2

where w is the Eulerian Work that represents the change in total enthalpy or energy of the fluid. The degree of reaction is a measure of the static expansion that the flow undergoes in the rotor defined as

rx₌ h2− h1 h02− h00

Then the ratio between the static enthalpy rise in the rotor and the stagnation enthalpy rise in the stage. If the machines are geometrically similar and the flow remains dynamically similar, all the dimen-sionless parameters must remain constant. The Flow coefficients and the work coefficients are two of the most common means of classifying turbomachinery stages. They can be used to determine various off-design characteristics by using the so-called turbine maps. These maps, used in the preliminary de-sign study, predict the stage efficiency changes that are implied by the variation of these parameters. In this work, for the cascade analysis could be important to refers to these parameters.

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3

Methodology

This chapter presents an overview of the method used in this work. A brief summary of the steady and unsteady method used for the CFD simulations is given. This chapter presents the two dimensional axial turbine stages considered for the present study. The three test cases proposed in this Chapter will be simulate under steady and unsteady method. This chapter also describes the techniques used for the application of similitude to the CFD. Finally, the verification of similitude is discussed in this chapter.

Numerical Methods

The numerical simulations are performed using the open-source software SU2 [26]. It is a platform con-ceived for solving multi-physics Partial Differential Equation (PDE) and PDE-constrained optimization problems using general unstructured mesh. Unlike for structured meshes where a logical, ordered in-dexing can be assumed for neighboring nodes and their corresponding cells, for an unstructured mesh, a list of nodes that make up each element must be provided [26].

The RANS (Raynolds-Avarage Navier-Stokes) equations are discretized in SU2 using a finite volume method, with a standard edge-based structure on a dual grid with control volumes constructed using a median-dual vertex-based scheme. The PDE semi-discretized integral form is

Z

Ω

∂U

∂tdΩ + R(U) = 0 (3.1)

U is the vector of conservative variables, defined as follow U = (ρ,ρv1,ρv2,ρv3,ρE)

withρ the density, v = (v1, v2, v3) the velocity vector, and E the total specific energy.Ω and its boundary

∂Ω are assumed to vary their position in time,with velocity uΩ, without deforming. R(U) is the residual vector obtained by integrating the source term over the volumeΩ and summing up all the projected convective Fcand viscous fluxes Fvassociated with all the edges ofΩ. Convective and viscous fluxes are given by Fc=    ρ(v − uΩ) ρv × (v − uΩ) + p¯I ρE(v − uΩ) + pv    Fv=    − µ¯τ µ¯τ · v + κ∇T   

Where p is the static pressure, T is the static temperature,κ is the dynamic viscosity and τ is the viscous stress tensor. The turbulence modeling is considered, according to the Boussinesq hypothesis, by defin-ingµ = µl+ µtandκ = κl+ κt. The symbolsµlandµtare the laminar and the turbulent dynamic

vis-cosity, whereasκl andκtare the laminar and turbulent thermal conductivities. The time-discretization

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3.1. Numerical Methods 12

is performed using an implicit Euler scheme the first equation can be written as: µ ΩI ∆τ+ ∂R(Un₎ ∂Un ¶ ∆Un = −R(Un) (3.2)

In which∆Un = Un+1− Unandτ is the (pseudo) time-step which may be different in each cell by us-ing the local time-steppus-ing technique. This equation can be solved usus-ing different linear solver imple-mented in SU2.

Mixing Plane (MP)

Turbomachinery flow is inherently unsteady due to the relative motion of adjacent rotor and stator blade rows. However, at ordinary operation point, the time-averaged flow field in each blade passage in either rotor or stator blade row is almost the same in the relative frame of reference fixed to the corresponding blade row under consideration. Thus, quasi-steady calculation can be performed based on single blade passage, which can capture the major flow features but with much less computational effort than full unsteady simulation [6]. A cost effective way for quasi-steady calculation is the mixing-plane method which was first proposed in ref. [16]. The calculation of three-dimensional viscous flow through mul-tistage turbomachines In the mixing plane (MP) approach, each fluid zone is treated as a steady-state problem. Flow-field data from adjacent zones are passed as boundary conditions that are spatially av-eraged or "mixed” at the mixing plane interface. As pointed out by Denton, entropy will be generated by this mixing process and total pressure decreases abruptly through mixing plane. In general, there are two key steps to realize a mixing-plane method. First, an averaging technique is required to obtain the circumferentially averaged flow state at each side of the interface. Second, the circumferentially averaged flow state must be transferred through the interface. The most important issue is to prevent wave reflections from the interface. Non reflective boundary condition (NRBC), proposed by Giles in ref. [20], are adopted in order to prevent spurious, nonphysical reflection at inflow and outflow boundaries. However, this mixing removes any unsteadiness that would arise due to circumferential variations in the passage-to-passage flow field (wakes, shock waves, separated flow). So the mixing plane approach has two important consequence. One the one hand the mixing plane not allow to capture the unsteady interaction between the blade row, on the other and it introduces errors due to the mixing process of the flow.

Harmonic Balance (HB)

The harmonic balance methods is a technique for modeling unsteady nonlinear flows in turbomachin-ery at a reduced computational costs. Many unsteady flows of interest in turbomachinturbomachin-ery are periodic in time. Thus, the unsteady flow conservation variables may be represented by a Fourier series in time with spatially varying coefficients. This assumption leads to a harmonic balance form of the Euler or Navier–Stokes equations, which, in turn, can be solved efficiently as a steady problem using conven-tional computaconven-tional fluid dynamic (CFD) methods. The governing fluid equations of motion and the associated boundary conditions are then linearized about the mean flow solution to arrive at a set of lin-ear variable coefficient equations that describe the small disturbance flow . The time derivatives (∂/∂t) are replaced by jω , where ω is the frequency of the unsteady disturbance, so that time does not appear explicitly. The unsteady flow, in this work, is represented by a Fourier series in time with frequencies that are integer multiples of the fundamental excitation frequency, blade passing frequency in the case of wake/rotor interaction or the blade vibratory frequency in the case of flutter. Because the flow is temporally periodic, the conservation variables may be expressed as

U =X

n

ˆ uejωt where ˆu are the coefficients of the almost-periodic Fourier series.

The Fourier coefficients resulting from the application of the discrete Fourier transform (DFT) are calculated as follow [28] ˆ uk= 1 N N −1 X n Une− j ωt

where Un= [U0, U1, ..., UN −1] is the vector of the conservative variables evaluated at N time instances

t = [t0, t1, ..., tN −1] hence the corresponding Fourier Coefficients are

ˆ

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3.1. Numerical Methods 13

with N = 2k + 1, ωk = 2πk fk and k the number of the frequencies. The ensemble of the k resolved

frequencies is denoted by

ω = [ω0,ω1, ...,ωk,ω−k, ...,ω−1]

withω_−k_{= −ω}k. By defining the DFT matrix as

Ek,n=

1 Ne

− j ωktn

with n, k ∈ [0, N ] and the inverse discrete Fourier transform (IDFT) E_k,n−1 _{= e}jωktn

The Fourier coefficients are computed as ˆ uk= E ˆU

and the vector of the conservative variables

ˆ U = E−1uˆ

The linearized for of Navier-Stokes Equation 3.2, discretized both in space and time can be written as

µΩI ∆τ+ J

¶

∆U = − ˜R(Un₎ _(3.3)

in which∆U = Un− Un−1and ˜R(Un) = R(Un) + H∆U + HUn

H is the harmonic balance operator calculated for a set of k tonal frequencies, not necessarily integer multiple of a fundamental harmonic. It is the N ×N spectral operator matrix, with N = 2k+1 the number of resolved time instances. It is defined as follow

H = [E−1DE]

with D the diagonal matrix given by D = diag( j ω0, jω1, ..., jωk, jω−k, ..., jω−1). A more detailed

descrip-tion is given in given in the reference (Rubino at al,2017a) [28].

The Equation 3.3 is solved for each time instance in a segregated manner. Therefore, an unsteady flow problem characterize by k frequencies requieres that the solution of 2k + 1 non linear systems of equation need to be computed. Moreover, this method is computationally efficient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations [18]. The har-monic balance approach is implemented in the open-source SU2 code.

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3.2. Axial Turbine Stage 14

Axial Turbine Stage

Test Cases

In order to investigate the impact of the unsteady effect on the fluid dynamic performance, three axial turbine cascade with different design conditions are analyzed.

The preliminary mean-line design process typically begins with the selection of stage loading and flow coefficient, defined in the section 2.3. In addition, it is also necessary to select an appropriate degree of reaction and rotational speed. For a single stage design, the selection of the stage loading and flow coefficients will have a major impact on the stage performance (Smith, 1965). The figure Figure 3.1 shows the Smith’s Chart and the red points represent the selected conditions for the designed stages in this work.

Figure 3.1: Smith’s Chart

The three dimensionless parameters mentioned (stage loading, flow coefficient and reaction) are used for setting up the velocity triangles. The flow angles can be chosen for obtainig the maximum stage efficiency. Indeed, considering that the peripheral speed is constant, the outlet relative flow angleβ1and

β2and the inlet absolute flow angleα1andα2of the stator and of the rotor respectively, can be given as

a function of these non-dimensional numbers by [29]

t an_α1= t anβ1+ 1 ϕ rx_{= 1 − ϕt anα}1+ 1 4λ λ = 2(ϕtanα1− ϕt anβ2− 1) t anβ2= t anα2− 1 ϕ

The first test case is two-dimensional axial turbine stage, adapted from the 1.5 stage experimental setup of the Institute of Jet Propulsion and Turbomachinery at RWTH Aachen, Germany. The experimental

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Figure 3.2: Test Case 1 - Geometry Cascade

Parameter Unit Value

Gamma [-] 1.4

Inlet Total Pressure [bar ] 400000 Inlet Total Temperature [K ] 308.5 Outlet Static Pressure [bar ] 266667 Viscosity [k g /ms] 3.38 · 105

Pitch [mm] 44.64

Axial chord lenght [mm] 53.97

Axial gap lenght [mm] 16.42

Speed [m/s] 120

λ [-] 4.4

ϕ [-] 0.55

rx [-] 0.35

Table 3.1: Test Case 1: Specificarions of the stage design point

data are published by Stephan (2000) in ref. [31]. The other two dimensional blade geometry are gen-erated using Meangen, the mean-line program in Multall. Meangen is designed to use the minimum possible input data and so many of the parameters needed, such as grid point numbers, gas properties, blade thicknesses, blade row, and stage spacings are set by default. It predicts the blading parameters on a mean stream surface and writes an input file for Stagen. Thus, choosing three parameters,λ, ϕ and rx,all parameters for the velocity diagram are calculated and the blade geometry of the axial stage is generated. Stagen is a blade geometry manipulation program which generates and stacks the blading (Multall, Denton [2]) . The Stagen output data of the geometry are then imported as point in x and y coordinate in an open source mesh generator capable to create a mesh file for SU2 code.

Test case 1

The proposed test case, at the design point, presents a stage pressure ratio of 1.5 The design total inlet temperature is of 308.5 K and a rotational speed of 3500 rpm. These conditions result in a peripheral Mach number (M au) of about 0.354 and a peripheral Reynolds number (Reu) of approximately 7 · 105.

The Figure 3.2 shows the profile of the stator and rotor. The turbine blade pitch is 44,64 mm for both stator and rotor. The axial distance between each blade rows is 16.42 mm and the axial chord of the stator is 53,97 mm and the gap to axial chord ratio 0.3.The Table 3.1 shows the main specification parameters of the stage at the design point.

Test case 2

The second blade geometry is generated in Meangen maintaining constant the flow coefficient and the reaction of the previous geometry and using a lower value of the work coefficient. Low values of stage loading and flow coefficient should be led to the best stage efficiencies due to the low flow velocities and reduced friction losses. Indeed, highly loaded stages require greater fluid deflections and consequently generate higher losses. Modifying only the work coefficient, all the absolute and the relative flow angles change. The stage loading and flow coefficient are chosen to be 1.1 and 0.55, respectively. The reaction

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Gamma [-] 1.4

Inlet Total Pressure [bar ] 400000 Inlet Total Temperature [K ] 306 Outlet Static Pressure [bar ] 333333 Viscosity [k g /ms] 3.38 · 105

Pitch [mm] 44.64

Speed [m/s] 150

λ [-] 1.1

ϕ [-] 0.55

rx [-] 0.35

Table 3.2: Test Case 2: Specifications of the stage design point

was set slightly at 0.35, the inlet flow angle is assumed to be 0 degree. The outlet absolute flow angle α1= 63.4 degrees and the outlet relative flow angle of the rotor β2= −58.6 degrees. The Figure 4.10

shows the profile of the blade rows.

The Table 3.2 presents the main specification parameters of the stage at the design point.The test case 2 is designed with a total inlet pressure of 4 MPa and with a static outlet pressure is 3.3 MPa, which is equivalent to a stage pressure ratio of 1.2. The design total inlet temperature is of 306 K with a rotational speed of 150 m/s. These conditions result in a peripheral Mach number of about 0.36 and a peripheral Reynolds number of approximately 7 · 105. The blade pitch and the axial gap are the same of the first axial stage. The axial chord of the stator is 44.01 mm and the gap to axial chord ratio is 0.37.

Test case 3

The third blade geometry is generated keeping constant the work coefficient and the reaction of the previous geometry and changing only the flow coefficient. High flow coefficient (higher mass flow rate) results in higher pressure drop and corresponding losses also increase. The pitch and the axial gap between the blade row are the same of the previous stages. The Figure 4.11 shows the profile of the stator and rotor. The stage loading and flow coefficient are chosen to be 4.4 and 1.1, respectively. The reaction is 0.35, the inlet flow angle is assumed to be 0 degree. The outlet absolute flow angleα1= 57.8

degrees and the outlet relative flow angle of the rotorβ2= −52.8 degrees. The Table 3.2 presents the main

specification parameters of the stage at the design point. The test case 2 is simulated with a total inlet pressure of 4 MPa and with a static outlet pressure is 2.37 MPa, which is equivalent to a stage pressure ratio of 1.8. The design total inlet temperature is of 306 K.The axial gap to axial chord ratio is 0.29.

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Gamma [-] 1.4

Inlet Total Pressure [bar ] 400000 Inlet Total Temperature [K ] 306 Outlet Static Pressure [bar ] 266667 Viscosity [k g /ms] 3.38 · 105

Pitch [mm] 44.64

Speed [m/s] 125

λ [-] 4.4

ϕ [-] 1.1

rx [-] 0.35

Table 3.3: Test Case 3: Specifications of the stage design point

Computatutional Details

The steady state calculations are performed by the mixing plane (MP) approach while the unsteady analysis is solved with a harmonic balance (HB) method. The simulations are performed using the JST scheme for the discretization of the convective flux; second order accuracy is achieved by MUSCL recon-struction. For both steady and unsteady simulations, non-reflecting boundary conditions are imposed (Giles, 1990) [20] at the stator inlet and at the rotor outlet sections. The stator- rotor interface is resolved for the unsteady simulation using a sliding mesh approach, properly developed for general unstructured grids. The steady simulation are based on a conservative mixing-plane (MP) method.

The Spalart-Alarm turbolent model is adapted in this study, because it was developed primarily for aerodynamic flow (Javaerchy, 2010) and it presents good performance in adverse pressure regions (Menter, 2003)[22]. This model requires moderate computational cost to solve the viscous sublayer (Eu-litz and Engel,1997). The convergence of the solutions is monitored by checking the residuals of the numerically solved governing equations. The selected convergence criteria is the residual reduction. A decrease in residuals by 4 oreders of magnitude with respect to the initial value is adopted for all the simulations, steady and unsteady The number of iterations is typically 12,000 with a Courant-Friedrichs-Lewy (CFL) number of 6. An unstructured grid is used to discretize the 2D computational domain with 40000 triangular elements for each blade row and 15000 quad elements over each blade surface in order to ensure y+∼_{= 1. The Figure 3.5 shows the computational mesh of the three cascades.}

The HB solutions for all the cases are obtained for 3 frequencies, i.e.,5 time instances. The selected time instances correspond to the solution for the frequency vector

¯

ω = [0,±ω0, ±2ω0]

The resolved frequencies are multiples of the fundamental blade passing frequency calculated by

fb=

U yp

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Figure 3.5: Computational mesh for the Test Case 1 (a), Test case 2 (b), Test case 3 (c)

and

ω0= 2π fb=

2_πU yp

Where ypis the blade pitch and U the blade speed.

In order to verify the HB solution , the HB total to total efficiency of one test case found with 5 time instance is compared with the HB solution obtained with 3, 7, 9 and 11 time instances. The re-spectively frequency vectors for each time instances are ¯ωN3= [0, ±ω0], ¯ωN7= [0, ±ω0, ±2ω0, ±3ω0] ,

¯

ωN9= [0, ±ω0, ±2ω0, ±3ω0, ±4ω0] and ¯ωN11= [0, ±ω0, ±2ω0, ±3ω0, ±4ω0, ±5sω0].

The total to total efficiency , shown in the Figure 3.6 is obtained by spectra interpolation of the har-monic balance results. More details are explained in the reference (Rubino, 2017a [28]) in the reference. However, the cost to compute the unsteady flow using the HB approach increases with number of the harmonics. As a good compromise between accuracy and computational cost, 5 time instances are used for the all the simulations present in this work.

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3.3. Application of the similitude theory to CFD 19

Figure 3.6: Total to total efficiency: HB solution obtained for different time instances

Application of the similitude theory to CFD

The application of similarity laws to the aerodynamic performance is a standard procedure in turboma-chinery in ref. [4]. The Buckingham Theorem is used to obtain the correlations between the variables that are the inputs parameter of SU2. These correlations are used to estimate the measurement field of the performance of the axial casade. For finding the necessary and sufficient condition of similarity for the flow in the cascade, the parameters may be determined. There are nine characteristic parameters that completely describe the condition of a stage. These parameters include the inlet total pressure p01,

the inlet total temperature T01, the heat capacity ratioγ,the gas constant R, the output static pressure

p2, the dynamic viscosityµ, peripheral speed U and the characteristic dimension, the pitch ypand the

axial gap g , which determine all of the characteristic parameters, such as efficiencyη. Four basic dimen-sions are selected: Length L, Mass M, Temperatureθ and Time T. There are nine main parameters and four dimensions, so according to similarity theory there are 9 − 4 = 5 dimensionless groups. The object need to be studied is the efficiencyη which is the function of other parameters, as is shown in function Equation 3.4.

ηT T= f (p01, T01, p2,γ,R,µ,U, yp, g ) (3.4)

The equation can be described as dimensionless

[ηT T] =

h

pα₀₁, T₀₁β, p₂ε,γϑ, Rι,µκ,U%, yυ_p, gτi The dimensional equation is

·· _M LT2 ¸α [θ]β · _M LT2 ¸ε [1]ϑ ·_L θ ¸ι·_M LT ¸κ· _L T2 ¸% [L]υ[L]τ ¸ = [M0L0θ0T0]

The exponents must be determined by equating the exponents for each of the terms M,L,T andθ. Finally, the result of dimensional analysis is expressed among the pi terms as

ηT T= " p01 p2 ,γ, U p γRT01 ,ρ01U yp µ , g yp #

whereρ01is the density at the inlet defined as p01/RT01.

The non dimensional group p01 p2

is the expansion ratio over the stage, denotedβ, and γ is the heat capacity ratio, that is already a non dimensional numbers and it represents the characteristic of the fluid.

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The group_pU γRT01

represents the peripheral Mach and it expresses the compressibility effects. The

non dimensional groupρ01U yp

µ is a form of Reynolds number. Physically, Reynolds number expresses the ratio of inertia force to viscous force and reflects the properties of the fluid and the speed of the cascade U. Since fluid meridional velocity or any other flow velocity is proportional to U, then any such fluid velocity can be replaced by U. The last group g

yp

is a shape parameter and represent the non-dimensional gap express as the ratio between the gap and the pitch of the blade. Thus the efficiency of the stage can be expressed as follow

ηT T= f (β, γ, M au, Reu, gnd) (3.5)

One important point is that the efficiency of a turbine stage has been shown in the section 2.3 de-pends by the work coefficientλ and the flow coefficient ϕ and by the degree of reaction rx. These three represent the meaningful non dimensional parameters used in the preliminary design. They can be var-ied independently by a designer. Once a designer has selected the stage duty (λ and ϕ) and the reaction R, the losses and the efficiency will be determined. However, by a simple analysis, it is possible to find the correlation between the calculated non-dimensional parameters and the meaningful dimensionless parameters. The following equations refer to an isentropic process across the axial turbine in which evolves an ideal gas.

ψi s=λi s 2 = 1 − µ ₁ βT s ¶ γ − 1 γ M a2u(γ − 1) ϕi s= ψi s+ 1 t an_α1− t anβ2 rx=1 − ϕi s(t anα1+ t anβ2 2

The stage isotropic duty coefficientsλ and ϕ and the degree of reaction rxare determined by the non dimensional parametersβT s, M au,γ and by the general blade shapes expressed in terms of the angles

α1andβ2. The angleα1is the flow angle at the exit of the stator and the angleβ2is flow angle at the

exit of the rotor. The work coefficientλ, the flow coefficient ϕ and the degree of reaction rxare strictly correlated: changing one of them implies that also the other two change. Moreover, from the equations below it is possible to notice that, for a constant value ofβT s, M au, Reuandγ and for a constant profile

geometry (same angles), the coefficientsλ and ϕ and the degree of reaction rxare kept costant too. Sim-ilarly, just varying one non dimensional parameter betweenβT s, M au, Reuandγ, all the three change.

In additional, this analysis could be extended also to the real gases. To predict the behavior of the gas better than the ideal gas laws, one of the real gas law used is the Van der Waals equation:

µ P + a

Vm2

¶

(Vm− b) = RT

Where p is the pressure, Vmis the molar volume of the gas, R is the ideal gas constant and T the

temper-ature. The coefficients a and b are determined empirically for each individual compound or estimated from the relations. These coefficients depend on the critical temperature and the critical pressure. The amount by which a real gas deviates from the ideal gas can be expressed in terms of the flow non-ideality Z. This latest is called compressibility factor and is given by the following equation

Z = v vi d

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The thermodynamic state variables of a real gas are often expressed in the form of reduced pressure Prand reduced temperature Tr as shown below:

Pr= P Pc and Tr= T Tc

where Pcand Tcare the critical pressure and the critical temperature respectively. For the

compress-ibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature and reduced pressure should have the same compressibility factor

Z = f (Tr, Pr)

Recall the principle of similarity, it is necessary to add to the test case new input. The phenomena is expressed not only by the parameters described above but there are other two important parameters that are the critical pressure Pc and the critical temperature Tc. The performance of the axial stage is expressed as follow

[ηT T] =

h

pα₀₁, T₀₁β, pε₂,γϑ, Rι,µκ,U%, y_pυ, gτ, T_cχ, P_cωi

There are eleven main parameters and four dimensions , so according to similarity theory there are 11 − 4 = 7 dimensionless groups. The new non dimensional groups that complete the problem are:

pr,01= p01 Pc and Tr,01= T01 Tc

These non-dimensional groups represent the reduced pressure and the reduced temperature at the inlet of the cascade.

Verification of similitude theory

The first important goal of this work is the verification of the similitude in the SU2 code. In order to verify the similiarity the Test case 1 is taken into account. To achieve dynamic similarity in the CFD, all five of the dependent non-dimensional parameters in Equation 3.5 match between the reference and the other cases. The Table 3.4 shows the numerical values of the non-dimensional numbers of the reference. If the

Parameter Value

βR 1.8

M au,R 0.423

Reu,R 744910

γR 1.4

Table 3.4: Verification of similitude: Non-dimensional numbers

reference cascade is tested at some value of the the expansion ratio, gamma, Mach number, Reynolds number and non dimensional gap, the measured efficiency has to be guaranteed to equal that of the other cases if operated at the same dimensionless numbers. The reference case is denoted by ’R’ and the other case by ’O’. From the following equations, it is possible to calculate all the necessary input variables. βR= β0⇒ p01,R p2,R = p01,O p2,O γR= γO M au,R= M au,O⇒ Ã U p γRT01 ! R = Ã U p γRT01 ! O

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3.3. Application of the similitude theory to CFD 22 p01 [MPa] T01 [K] p2 [MPa] R [J/kg K] ρ [kg/m3] µ [kg/m s2] U [m/s] Reference 2 450 1.11 287.058 1.55 1.67·10−5 180 Case 1 3 530 1.67 287.058 1.97 2.31·10−5 195.3 Case 2 4 800 2.22 287.058 1.74 2.51·10−5 240 Case 3 5.4 620 3.11 287.058 3.98 4.31·10−5 211.3 Case 4 7.5 980 4.17 287.058 2.67 4.24·10−5 265.6 Case 5 9.2 1300 5.11 287.058 2.46 4.52·10−5 305.9

Table 3.5: Verification of similitude: Main simulation parameters

Reu,R= Reu,O⇒ µρ 01U yp µ ¶ R = µρ 01U yp µ ¶ O gnd ,R= gnd ,O⇒ gR yp,R = gO yp,O

The operating conditions for any dynamically similar stage can be found by changing two of the dimensional variables. For example, considering the same cascade, or similar geometry, known the inlet total pressure p01, the inlet total temperature T01and the heat capacity ratioγ, the relations become

p2,O= p01,O βR UO= M au,R p γROT01,O µO= ρ01,OUOyp Reu,R

In Table 3.5 there are the numerical values of some case studied. The same grid, numerical scheme, turbulence model, relaxation parameter, and so on, of the reference are used for all the cases in the steady and unsteady numerical simulations. The convergence of the solutions is monitored by checking the residuals of the numerically solved governing equations. Lower is the numerical value of residuals, more accurate is the the solution. In order to compare the different cases, the residual is reduced a 4 of orders of magnitude for all the simulations, steady and unsteady.

The Figure 3.8 shows that the similitude is verified for both steady and unsteady. Changing two of the dimensional number but keeping constant the dimensionless numbers that describe the test case, the efficiency is the same for each case. The figure shows the efficiency as a function of the total inlet pressure p01. If the pressure at the inlet increases, and the total temperature is not constant, the

effi-ciency remains the same. Dynamic similarity is verified this means that the load coefficient of the stage is conserved. Moreover, kinematic similarity of the inlet and outlet velocity diagrams is a necessary con-dition for dynamic similarity. This means that also the flow coefficient is kept constant and that the velocity angles are also conserved. The velocities in different direction are scaled by the same factors. Furthermore, the Figure 3.7 shows the entropy distibution for the case 1 (a) and the case 3 (b) reported in the Table 3.5. The difference of the efficiency between steady and unsteady is discussed in the next chapter.

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Figure 3.7: Verification of the similitude - Entropy contours for case 1 (a) and case 3 (b) calculated with MP method and with the HB method at five different time instances

Figure 3.8: Verification of the similitude - Total to total efficiency and work and load coefficient as a function of the total inlet pressure p01

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4

Results and Analysis

In this chapter the bi-dimensional axial stages are studied varying the operating conditions over a range of one dimensionless numbers, while the other parameters are kept constant. This non-dimensional number is used to estimate the field measurement of turbine performance. The cascades presented in the Chapter 3 are analyzed by varying the expansion ratio, the working fluid, and the non-dimensional axial gap. The steady calculations are performed by using a mixing plane approach (MP), whereas the unsteady calculations are solved with the harmonic balance approach (HB). This chapter also presents a method of identifying and quantifying the loss generated in unsteady conditions using the data from numerical simulations.

Variation of the expansion ratio

The aim of this section is to investigate the effect of expansion ratio (β) on the performance of the cas-cades. For the Test case 1 , the calculation have been performed with 8 expantion ratios, fixing the peripheral Mach number ( M au) , the Raynolds number (Reu) and the heat capacity ratio (γ) of the

fluid. The Table 4.1 reports the numerical value of these non-dimensional numbers.

Parameter Value

M au 0.449

Reu 838024

γ 1.4

Table 4.1: Test case 1: Non-dimensional numbers for studying the effect ofβ

The Figure 4.1(a) shows the total to total efficiency as a function of the pressure ratio for steady and unsteady method. The unsteady CFD results are denoted as symbols with dashed line, while steady CFD results are denoted as symbol with line. It is showed that the mixing plane(MP) approach gives results very similar to the harmonic balance approach. For both, as the pressure ratio increase, the efficiency decreases due to the change of the flow angles and the flow velocities. Since the velocity rises by increasing the expansion ratio, the shock losses gradually affect the performance. Additionally, a further increase in expansion ratio causes greater flow deflection and consequently generates higher losses. Therefore, increasing the pressure ratio from 1.4 to 3.2 causes a reduction in the efficiency of 3.5 per cent. The results of all conditions are summarized in term of flow coefficient and load coefficient in Figure 4.1(b). The performance maps provide by Smith (1965) shows that the efficiency of a turbine increases by decreasing the flow coefficient and the load coefficient. Changing the loading causes a variation in shock mechanism, which affects directly the losses in transonic axial turbine [17]. Moreover, the flow coefficient remains constant an expansion ratio of about 3.2, which correspond to the rotor chocking.

By comparing the steady and unsteady CFD performance results, it can be seen that the results of the two methods are quite close to each other. The RMSE (Root Mean Square Errors) is 0.33 per cent. It is observed that a low pressure ratio the unsteady total to total efficiency is higher than the steady

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4.1. Variation of the expansion ratio 25

Figure 4.1: Test case 1 - Total to total efficiency (a) and load and work coefficient (b) as a function ofβ

Figure 4.2: Test Case 1 - Entropy contours forβ = 1.6 calculated with MP method and with the HB method at five different time

instances

total to total efficiency. Forβ greater than 2, the relative error between steady and unsteady predicted is positive. It is worth notice that forβ = 2.4 , the unsteady calculation predicts the maximum efficiency penalty of 0.53 percent. After this value of beta, the discrepancy between the two results decrease with the expansion ratio. The periodically interaction of a upstream blade wake with the moving downstream blade row affects the predicted performance of the axial turbine stage in unsteady solutions.

The Figure 4.2 shows the difference between the harmonic balance solution and the mixing plane forβ = 1.6 in entropy distributions. The unsteady entropy reduction is of 0.6 kJ/kg K that means the 24 per cent in comparison with the entropy generation in the steady solution. In the HB results , the wake transport in the downstream blade row is evident. The red contours correspond to high entropy values,

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Figure 4.3: Test Case 1 - Absolute Mach number and Relative Mach number as a function of the pressure ratio

the light contours represent low values. When the stator wake enters the rotor passage, it is accelerated in axial direction in the mid-pitch region between neighboring blades. The wake, is transported from the pressure to the suction side in the rotor passage, so that the highest entropy values arise close to the suction side near the rotor blade surface. On the pressure side of the rotor blade the wake structure decreases in entropy intensity due to transient effects. The wake passing throughout the rotor cause transition of boundary layer from laminar to turbulent. After the passage of the wake, the boundary layer starts to relax back to its pre-transitional state. This relaxation period is called calmed period (Schulte, 1996)[3]. The trailing edge energy thickness (effectively loss of efficiency) and the velocity distribution profile change during the same wake-passing events. The increse of entropy during the turbulent period and the reduced entropy during calmed period are visible 4.2. In time avarage, the generated losses are less than the losses predicted by the mixing plane approach. The loss reduction is of about 0.4 per cent for values ofβ of 1.6, 1.8, 2. Overall, the unsteady Total to total efficiency is higher than the steady total to total efficiency.

The Figure 4.3 presents the numerical values of the Mach number as a function of the pressure ratio, respectively for the stator and for the rotor. For the rotor is considered both the inlet relative Mach num-ber and the outlet relative Mach numnum-ber. Once the inlet Mach numnum-ber exceeds about 0.9, atβ = 2.4, the flow on the blade will become transonic, leading to performance deterioration. The absolute Mach number at the exit of the stator in the unsteady solution is significantly higher than the exit Mach num-ber computed with the steady results. Thus, additional losses arise due to the presence of shock waves. The Figure 4.4 shows the relative Mach number distribution of steady and unsteady results forβ = 2.4. The shock wave generated at the trailing edge of the stator moves toward the right and it approaches the rotor leading edge. Any motion of the shock causes an increase in losses. This movement of the shock is not predicted in the mixing plane. As a result, the Mach number calculated at the interface between stator and rotor is different between steady and unsteady and the expected penalty in unsteady total to total efficiency of 0.51 per cent. The Figure 4.5 presents the boundary layer velocity profile at the trailing edge of the suction surface of the stator and of the rotor at different time instance in comparison with the steady profile. The measurements of the boundary layer velocity are non-dimensionalised by the isentropic exit velocity and the energy thickness of the boundary layer by the 99 per cent boundary layer thickness. The interaction between the shock and boundary layers lead to unsteady boundary layer separations for the t = 4/5T periodically suppressed by the turbulent flow at other times. The changing of the velocity distribution compared to the steady state solution is evident. The presence of unsteady velocity profiles in the boundary layer at the trailing edge of the rotor is due to wake passage.

Differently, the steady and unsteady calculations predict a similar shock system at the stator and rotor trailing edge at high expansion ratios. As a consequence, same efficiency are predicted with the two different methods. The Figure 4.6 shows the Relative Mach distribution forβ = 3.2 with the mixing plane and with the HB approach. The shock is strong enough to cause separation and this is predicted

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Figure 4.4: Test Case 1 - Mach Relative contour forβ = 2.4 calculated with MP method and with the HB at five different time

instances

A comparison between steady and unsteady methods for turbomachinery design using scaling analysis