Aerodynamic Inverse Shape Design of Compressor and Turbine

Aerodynamic Inverse Shape Design of Compressor and

Turbine Stages Using ANSYS-CFX

Araz Arbabi

A Thesis

in

The Department

of

Mechanical and Industrial Engineering

Presented in Partial Fulfillment of the Requirements

For the Degree of Master of Applied Science (Mechanical Engineering) at

Concordia University

Montreal, Quebec, Canada

December 2012

iv

January 15, 2013

iii

Abstract

Aerodynamic Inverse Shape Design of Compressor and Turbine Stages Using ANSYS-CFX

Araz Arbabi

An aerodynamic inverse shape design method is implemented into ANSYS-CFX using a User

Defined Function. The implementation is validated first; the method is then assessed on a

subsonic axial compressor stage and a turbine stage. The design method is based on specifying a

target pressure distribution over the blade suction surface (or a target pressure loading) and a

blade thickness distribution as the design variables. The blade wall moves with a fictitious

velocity, which is derived from a balance of design and target momentum fluxes, in order to

reach a blade shape that would produce the prescribed target pressure distribution. The wall

movement obtained from the design method is computed in a User Defined Function through the

CFX Expression Language; it is then communicated to ANSYS-CFX at each time step. In

ANSYS-CFX, a cell-centered finite volume formulation is used for space discretization. The

time accurate Reynolds-Averaged-Navier-Stokes (URANS) equations are written in an arbitrary

Lagrangian-Eulerian (ALE) form so as to account for the wall and mesh movement. The k-

turbulence model is used for both compressor and turbine stages. Once the UDF is validated,

ANSYS-CFX is used to redesign the E/CO-3 axial compressor stage and the E/TU-3 axial

turbine stage so as to improve the stage aerodynamic performance.

iv

Acknowledgement

I would like to thank all the great people who helped me so much throughout this research work.

I express my deepest appreciation to my professor Dr. Wahid Ghaly who trusted me and offered

me this great opportunity, motivated me, supported me continuously and provided me with his

exceptional guidance during the course of this work.

I sincerely thank my wonderful parents, without whom it was impossible for me to complete this

journey, for their financial and mental support, sympathy and care.

I will never forget the support of my senior colleague and my friend, Mr. Raja Ramamurthy who

was always helpful and kind to me. Thank you very much Raja for all your help.

v

Contents Abstract ........................................................................................................................................................ iii

Acknowledgement ....................................................................................................................................... iv

List of Figures .............................................................................................................................................. vii

List of Tables ................................................................................................................................................ ix

Nomenclature ............................................................................................................................................... x

Chapter 1 ....................................................................................................................................................... 1

Introduction .............................................................................................................................................. 1

1.1. Previous investigations ............................................................................................................. 2

1.2. Present investigations ............................................................................................................... 4

1.2.1. Mesh consideration .......................................................................................................... 4

1.3. Thesis outline ............................................................................................................................ 5

Chapter 2 ....................................................................................................................................................... 7

Flow Governing Equations ........................................................................................................................ 7

2.1. Mesh deformation .................................................................................................................. 10

2.1.1. Regions of motion specified ............................................................................................ 11

2.1.1.1. Mesh stiffness ............................................................................................................. 11

Chapter 3 ..................................................................................................................................................... 14

Inverse Design Methodology .................................................................................................................. 14

3.1. Inverse design formulation ..................................................................................................... 15

3.2. Inverse design variables .......................................................................................................... 18

3.2.1. Target pressure distribution on the blade pressure and suction surface ....................... 19

3.2.2. Target pressure loading and blade thickness distribution .............................................. 19

3.2.3. Target suction surface pressure and thickness distribution ........................................... 21

3.3. Design constraints ................................................................................................................... 22

Chapter 4 ..................................................................................................................................................... 24

Algorithm, Validation and Redesign Cases in ANSYS-CFX ....................................................................... 24

4.1. Implementation of inverse design methodology in ANSYS-CFX ............................................. 24

4.2. The UDF attached to ANSYS-CFX............................................................................................. 26

4.3. Analysis of the E/CO-3 compressor stage ............................................................................... 30

4.4. Validation of the inverse design implementation into ANSYS-CFX ......................................... 34

vi

4.5. Redesign of the E/CO-3 compressor stage at maximum flow ................................................ 37

4.6. Redesign of the E/CO-3 compressor stage at near surge ....................................................... 48

4.7. Assessment of the Analysis Scheme on the E/TU-3 turbine stage ......................................... 52

4.8. Redesigning of the E/TU-3 turbine stage ................................................................................ 54

Chapter 5 ..................................................................................................................................................... 62

Conclusion ............................................................................................................................................... 62

5.1. Summary ................................................................................................................................. 62

5.1. Future Work ............................................................................................................................ 64

References .................................................................................................................................................. 65

vii

List of Figures

Figure 1.1. E/CO-3 Stator: Mesh close-up near LE (left) and TE (right) ........................................................ 5

Figure 2.1. A typical two dimensional control volume in ANSYS-CFX ........................................................... 7

Figure 2.2. A schematic movement of the blade wall ................................................................................ 17

Figure 4.1. Computational algorithm for inverse design ............................................................................ 24

Figure 4.2. Building the structure data in UDF............................................................................................ 26

Figure 4.3. Algorithm for virtual velocity computation .............................................................................. 27

Figure 4.4. Algorithm for nodes displacement computation and updating the blade profile .................... 28

Figure 4.5. Functionality of the UDF attached to ANSYS-CFX .................................................................... 29

Figure 4.6. L2–norm of airfoil displacement ................................................................................................ 35

Figure 4.7. L2–norm of .......................................................................................................................... 35

Figure 4.8. E/CO-3 Rotor blade – Validation of inverse method using Design and Design .......... 36

Figure 4.9. E/CO-3 Rotor pressure distribution – Validation of inverse method using Design and

Design .................................................................................................................................................. 36

Figure 4.10. E/CO-3 Rotor: Convergence history for the Design ......................................................... 38

Figure 4.11. E/CO-3 rotor: Original and redesigned blade geometry for the Design .......................... 38

Figure 4.13. E/CO-3 Rotor: Convergence history for the Design ......................................................... 41

Figure 4.15. E/CO-3 Rotor: Design pressure distribution at maximum flow for and DP Design ......... 42

Figure 4.16. E/CO-3 Rotor: Blade profile at maximum flow for and DP Design .................................. 43

Figure 4.17. E/CO-3 Rotor: Pressure distribution close-up at maximum flow for and DP Design ...... 45

Figure 4.18. E/CO-3 Rotor – Pressure distribution of the original and redesigned blade (designed at

maximum flow) at design point .......................................................................................................... 46

viii

Figure 4.19. E/CO-3 Rotor: Pressure distribution of the original and redesigned blade (designed at

maximum flow) at Near Surge ............................................................................................................ 47

Figure 4.20. E/CO-3 Rotor Redesign in Near Surge condition, L2–norm of . Design

Variable: DP & thickness distribution ................................................................................................. 49

Figure 4.21. E/CO-3 Rotor: Original, target and design loading distributions at near surge ...................... 50

Figure 4.22. E/CO-3 Rotor: Original and redesigned blade geometry at near surge .................................. 50

Figure 4.23. E/CO-3 Rotor: Original and redesigned pressure distributions at near surge ........................ 51

Figure 4.24. E/TU-3 Stator: Convergence history ....................................................................................... 54

Figure 4.25. E/TU-3 Stator: Design, target and original loading distributions for DP Design ..................... 55

Figure 4.26. E/TU-3 Stator: Design and original pressure distribution for DP Design ................................ 55

Figure 4.27. E/TU-3 Stator: Design and original blade profiles for DP Design ............................................ 56

Figure 4.28. E/TU-3 rotor: Convergence history ......................................................................................... 59

Figure 4.29. E/TU-3 rotor: Design, target and original pressure distributions ........................................... 59

Figure 4.30. E/TU-3 rotor: Original and design blade profiles for Design ........................................... 60

ix

List of Tables

Table 4.1. E/CO-3 Stage geometric characteristics ..................................................................................... 30

Table 4.2. E/CO-3 Compressor stage results at maximum flow ................................................................. 31

Table 4.3. E/CO-3 Compressor stage results at design point ...................................................................... 32

Table 4.4. E/CO-3 Compressor stage results at near surge ........................................................................ 33

Table 4.5. Validation of the E/CO-3 compressor stage redesign: flow simulations obtained for two design

variables .............................................................................................................................................. 43

Table 4.6. E/CO-3 compressor stage results for analysis of the designed (designed in maximum flow) and

original blades at design point ............................................................................................................ 47

Table 4.7. E/CO-3 compressor stage results for analysis of the designed (designed at maximum flow) and

original blades at near surge ............................................................................................................... 48

Table 4.8. E/CO-3 Compressor stage results for the designed and original blade at near surge ............... 51

Table 4.9. Geometric characteristics of the E/TU-3 stage .......................................................................... 53

Table 4.10. E/TU-3 Assessment of the E/TU-3 turbine stage ..................................................................... 53

Table 4.11. E/TU-3 turbine stage results for the redesign of the stator blade at design operating point in

ANSYS-CFX ........................................................................................................................................... 56

Table 4.12. Redesign of E/TU-3 turbine rotor ............................................................................................. 60

x

Nomenclature

C Speed of sound

C Stiffness model exponent

D Distance

F Blade camber line

F Conservative flux vector, virtual momentum flux

G Viscous flux vector

M Mach number

N Normal vector

P Pressure

S Wall displacement

S Control surface, Source term

T Fictitious or physical time

T Thickness, Temperature

U Velocity component in x- direction

U Primitive variable vector

V Velocity component in y- direction

V Control volume

X x- coordinate

Y y- coordinate

xi

Greek Symbols

Relative flow angle

Under – relaxation factor for wall movement

𝛤 Diffusivity, Mesh stiffness

along the blade surface

𝜌 Density

Dynamic Viscosity

Node displacement

Total energy per unit of mass

Relaxation factor

Subscripts

0 Total (or stagnation)

1,2 Rotor inlet, outlet

Eff effective

i,j counter

New Current time step

Old Previous time step

Stiff stiffness

Tgt Target

X In the x- direction

Y In the y- direction

xii

Superscripts

Suction side

Pressure side

Acronyms

ALE Arbitrary Lagrangian–Eulerian

CEL CFX expression language

CFD Computational fluid dynamics

DP Blade pressure loading

LE Leading edge

PR Stage pressure ratio

PS,SS Blade pressure side, suction side

RANS Reynolds-avereaged Navier Stokes

TE Trailing edge

TRR Temperature rise ratio

UDF User defined function

1

Chapter 1

Introduction

Nowadays, Reynolds-averaged Navier Stokes (RANS) equations are used worldwide in

simulating the flow field in different industrial applications including gas turbine industry.

On the other hand, design methods, coupled with CFD techniques are used to improve the

performance of a compressor or turbine stage(s). One of these design methods is Automatic

Numerical Optimization [1-4] where the blade geometry is modified to satisfy a certain design

objective(s) subject to some constraints. The Numerical Optimization approach allows for

specifying the design objectives (e.g. turbine performance) and constraints (e.g. geometric

features) explicitly and exactly, then scanning the design space automatically and providing the

designer with the target blade profile. However, it is computationally expensive as it usually

requires a large number of flow simulations to compute the optimization objectives and

constraints.

Another design approach that is much less computationally intensive is the aerodynamic inverse

shape design. In that approach, the blade profile that satisfies a detailed flow performance is

targeted e.g., the static pressure distribution over the blade surfaces or the blade pressure loading

and thickness distribution. However, this approach is not as mature as the CFD analysis methods

are.

In this work, a recently developed inverse blade design approach is implemented into a

commercial CFD simulation package namely, ANSYS-CFX.

2

1.1. Previous investigations

Historically, inverse design method was implemented on two dimensional potential flow, then

being used for inviscid flow and finally viscous flow. These methods assumed a pressure

distribution over the blade surfaces [5-9], or Mach number [10], or velocity [11] or the pressure

loading and blade thickness distribution [12-15] as the target function. The design process started

from an initial guess for the blade geometry, then using the difference between the design and

target distributions, the blade shape deformed repeatedly in order to finally produce the

prescribed target function. Although it has been shown in different works that the inverse design

is efficient for internal flows [7,8,9,13,15] most of them still have some traces of the inviscid

flow.

Some approaches [7] make use of viscous-inviscid interaction, some use the tangency condition

to compute the designed blade camber-line [13], and in some other methods [8], the transpiration

condition has been used where the tangential and normal components of the velocity over the

blade surfaces are computed in order to find the blade’s new shape. One of the recent approaches

[9], uses both Navier-Stokes and Euler solver for the flow analysis and inverse design

respectively.

In all of these methods it has been assumed that the flow is attached to the blade and the

boundary layer is well behaved so that the blade’s new profile can be computed using the

velocity at the edge of the boundary layer.

Moreover, in most of these methods, the mesh movement that results from modifying the blade

geometry is ignored and the problem is solved as a quasi-steady problem [16] where the

governing equations do not account for the blade movement. This assumption raises a problem

3

as it afflicts the designed blade shape and therefore causes inaccurate pressure distribution in the

next iteration. Consequently the inaccurate pressure field leads to inaccurate blade shape and the

error is accumulated as time goes on and may lead to divergence of the iterative process. An

example of this situation is the work done by Yang and Ntone [17] who obtained a rather wavy

blade profile.

The above mentioned error due to the quasi-steady assumption can be removed by using a time

accurate formulation and modifying the governing equations to account for the mesh movement.

Using a time accurate formulation will improve the convergence even in difficult cases such as

transonic design cases. The improvement in the convergence was partly demonstrated by

Demeulenaere et al. [8]where they accounted for the mesh movement in the governing

equations, while still using time marching scheme. Daneshkhah and Ghaly [18] showed that by

using a time accurate formulation the problem converges in transonic cases while the quasi-

steady approach fails to converge in these cases.

The method developed by Daneshkhah and Ghaly [18,19] is compatible with the viscous flow

assumption where the fictitious velocity of the nodes located on the blade wall is computed from

the difference between current and target pressure distribution. The method makes use of the

time accurate formulation of the moving mesh into the RANS equations. The design approach

starts with an initial blade profile that evolves in time to reach asymptotically a profile that

would satisfy the target pressure distributions along the blade surfaces.

4

1.2. Present investigations

In this work, the method developed by Daneshkhah and Ghaly [18,19] has been implemented

into ANSYS-CFX where a Fortran subroutine which contains the inverse design shape

functionality, is compiled and linked to ANSYS-CFX to simulate the inverse design approach.

The method is consistent with the viscous flow assumption and the blade wall moves with a

virtual velocity computed from a balance of the current and target momentum fluxes.

The Reynolds-averaged Navier Stokes (RANS) equations are used to compute the flow field in

analysis mode while the unsteady Reynolds-Averaged Navier Stokes (URANS) equations, which

are written for the moving and deforming mesh using Arbitrary Lagrangian–Eulerian (ALE)

formulation, are used in design (unsteady) mode.

The method is validated and applied to a compressor stage and a turbine stage. The goal is to

demonstrate the robustness and generality of the inverse method for flow fields and conditions of

different nature and in the framework of a commercial software ANSYS-CFX. At the rotor-stator

interface, a mixing model using flux averaging is used [20].

1.2.1. Mesh consideration

An O-mesh is generated around the blade wall and care has been taken to resolve the boundary

layer near the wall in order to have y+<1 and the rest of the computational domain is filled with a

structured mesh.

Moreover, as this work focuses on 2D flow analysis and as ANSYS-CFX is a 3D flow analyzer,

the actual radius of the 2D section of the blade which is under question, is increased so that the

5

effects of the flow parameters in the 3rd

direction, specifically in rotor cases, could be ignored.

Care has been taken to adjust the blade rotational speed with the new radius in order to have the

same blade speed so as to respect the boundary conditions. A mesh close-up near LE and TE

regions of the E/CO-3 stator blade are shown in Figure 1.1.

Figure 1.1. E/CO-3 Stator: Mesh close-up near LE (left) and TE (right)

1.3. Thesis outline

This work consists of five chapters, including the introduction. The second chapter introduces

the governing equations being used for the steady and unsteady flow simulations. The third

chapter describes the inverse design methodology and formulations and discusses 3 different

choices of design variables. The forth chapter explains first the computational algorithm of the

inverse design in ANSYS-CFX and discusses the contribution of the subroutine compiled and

linked to ANSYS-CFX in redesigning the blade geometry. Later on in this chapter the validation

of the inverse method, performed on the E/CO-3 compressor rotor blade, is presented and then it

6

focuses on the implementation of the inverse method on a single stage subsonic compressor and

a single stage turbine. The method is applied first to the redesign of the E/CO-3 compressor rotor

blade at maximum flow conditions and by using 2 different choice of design variables in order to

increase the stage total to total efficiency while keeping the same overall loading and

aerodynamic characteristics. Afterwards in this chapter, the redesigned blade is analyzed at the

design point and near surge conditions, and the stage performance in terms of the total to total

efficiency is compared with that of the original blade. The E/CO-3 compressor rotor is then

redesigned at near surge conditions and the stage performance is measured and compared with

the original blade. Finally the E/TU-3 turbine stator and rotor blade rows are redesigned

respectively to improve the stage performance. The last chapter includes some concluding

remarks and recommendations for future work.

7

Chapter 2

Flow Governing Equations

In ANSYS-CFX a cell-centered finite volume method is used for space discretization.

Figure 2.1. A typical two dimensional control volume in ANSYS-CFX (reprinted from Ref. [20])

Figure 2.1 illustrates a two dimensional mesh in ANSYS-CFX where each node is surrounded by

a finite control volume which is identified by connecting the edge and element centers around

every single node. All the flow variables in ANSYS-CFX are stored at the nodes [20].

The conservation form of the two dimensional URANS equations accounting for mesh

movement which is written in Arbitrary Lagrangian–Eulerian (ALE) formulation is as follows:

8

(2-1)

Where ‘U’ is the solution vector which contains dependant flow variables, ‘F-Fg’ and ‘G-Gg’ are

the flux vectors relative to the moving grids, and Fv and Gv stand for the viscous flux terms. In

cases where there is no mesh movement, the terms Fg and Gg are zero.

The integral, conservation form of Eq.2-1 when there is no mesh deformation i.e., Fg and Gg are

zero, are [20]:

∫𝜌

∫𝜌

(2-2)

∫𝜌

∫𝜌

∫

∫

∫

(2-3)

∫𝜌

∫𝜌

∫𝛤

∫

(2-4)

Equations 2.2, 2.3 and 2.4 represent the conservation of mass, momentum and energy,

respectively. ‘V’ and ‘S’ indicate the volume and surface integration regions, respectively, and

‘dnj’ is the differential component of the vector normal to the control surface. ‘ ’ and ‘ ’ are

momentum and energy source terms, which are zero in the scope of this work since there is no

body forces nor heat generation in the computational domain. ‘eff’ is the effective or total

viscosity, which is the sum of molecular and turbulent eddy viscosity. ‘𝛤 ’ is the effective

thermal diffusivity which is the summation of molecular and turbulent diffusivity and is total

energy per unit of mass [20].

9

In the cases where the mesh or control volume deformation occurs, Equations 2-2, 2-3 and 2-4

should be modified to account for the mesh movement. In ANSYS-CFX this modification is

performed by making use of the Leibnitz Rule [20]:

∫

∫

∫

(2- 5)

By combining Equations 2-2, 2-3, 2-4 and 2-5 the URANS equations written in Arbitrary

Lagrangian–Eulerian (ALE) form is achieved [20]:

∫ 𝜌

∫𝜌

(2-6)

∫ 𝜌

∫𝜌

∫

∫

∫

(2-7)

∫ 𝜌

∫𝜌

∫𝛤

∫

(2-8)

The Reynolds-averaged Navier Stokes (RANS) equations are used to simulate the flow field

around a given blade profile (i.e., in analysis mode) while the unsteady Reynolds-averaged

Navier Stokes (URANS) equations, which are written for the moving and deforming mesh using

the Arbitrary Lagrangian–Eulerian (ALE) formulation, are used in simulating the flow around a

10

yet unknown blade profile that would produce a given, e.g., pressure distribution along that blade

(i.e., in design mode) where the flow is assumed unsteady [20].

In the time accurate simulations, a second order accurate backward Euler scheme is used for time

integration. This scheme is an implicit time stepping scheme, it can be used with constant or

varying time step sizes and is the default in ANSYS-CFX [20]. A high resolution scheme, which

is recommended by ANSYS-CFX for turbine and compressor stages, is used for the advection

terms and a first order scheme is used for the turbulence model for both steady and time accurate

computations. Moreover, the industry-standard two-equation k-omega turbulence model is used

for both compressor and turbine stages in all computations due to its highly accurate prediction

of flow separation in regions having adverse pressure gradient [20].

2.1. Mesh deformation

Since the blade profile moves at each time step, the mesh deformation has to be accounted for in

formulating and solving the flow governing equations. In ANSYS-CFX [20], different options

are available to account for mesh deformation:

None: is used when there is no mesh movement

Junction Box Routine: is used when the coordinates of all nodes in the domain, after

displacement, is predefined.

Regions of Motion Specified: is used when the motion of a boundary or a sub-domain is

specified.

11

In this work, the third option i.e., “Regions of Motion Specified” is chosen since the motion of

the blade boundary is specified while the mesh motion of the rest of the domain is unspecified.

2.1.1. Regions of motion specified

The movement applied to the nodes located on the blade walls is explicitly computed inside the

Fortran subroutine compiled to CFX (to be elaborated in Chapters 3,4) while for the remaining

nodes of the domain there is a mesh deformation model available in ANSYS-CFX called

“Displacement Diffusion” [20]. The idea of using this model is that, the displacement imposed

on the wall boundary is diffused to the rest of the domain by solving the following equation [20]:

( ) (2-9)

Where is the node displacement and is the “mesh stiffness” which determines how close

the mesh regions displace together. In the transient runs, this equation is solved at the start of

each time step. The merit of using this model is to retain the original relative mesh distribution

through the entire domain [20].

2.1.1.1. Mesh stiffness

The mesh stiffness could be a constant value or a varying one. By using a constant value, the

mesh displacement in the specified regions, here the blade wall, will diffuse evenly throughout

the domain while a varying value will make the mesh regions to have a smaller relative

12

displacement in the regions having higher stiffness and vice versa [20]. Varying mesh stiffness

is useful in the fine mesh regions where preserving the structure of mesh distribution and also the

mesh quality is of high importance e.g. the boundary layer around a blade, sharp corners, etc.

There are two options for the varying mesh stiffness in ANSYS-CFX:

Increase near small volumes: where the mesh stiffness will increase in the regions having

smaller control volumes. In this option the mesh stiffness is computed as follows [20]:

(

)

(2-10)

Where is the control volume size and is the “stiffness model exponent”. According to

Eq. 2-10, as the size of the control volume decreases, the mesh stiffness increases exponentially

and the value of indicates the degree to which the stiffness increases [20].

Increase near boundaries: where the mesh stiffness will increase in the regions near the

boundaries such as inlet, outlet, wall, etc. and is computed as follows [20]:

(

)

(2-11)

Where is the distance from a boundary. In this model, the mesh stiffness will increase

exponentially as the distance to a boundary decreases. Again indicates how fast the mesh

stiffness increases [20].

13

In this work, the second option i.e., “increase near boundaries” is used in order to preserve the

mesh quality and distribution near the blade wall, inlet, outlet and interfaces.

14

Chapter 3

Inverse Design Methodology

In this chapter, the aerodynamic inverse shape design methodology developed by Daneshkhah

and Ghaly [18,19] is summarized. The basic idea is that, the blade surface moves with a fictitious

velocity as to satisfy the prescribed target pressure distribution. The virtual velocity of the blade

surface is computed based on (and is proportional to) the difference between the current (or

designed) and the target pressure distribution which means that as the current pressure along the

blade surface gets closer to the target, the virtual velocity gets closer to zero as well. The

difference between the momentum fluxes of the current and target pressure distributions is the

source of virtual velocity computation. This virtual velocity moves the nodes to their new

position so that a new blade shape is designed which produces the target pressure.

The above explained process is performed in the unsteady mode using a time accurate scheme.

The RANS equations are written in the Arbitrary Lagrangian–Eulerian (ALE) form in order to

take the movement and deformation of the mesh into account. As pointed out in chapter 1, the

target pressure distribution depends on the choice of the design variable. It can be a target

pressure distribution on the blade suction surface or the target pressure loading or even target

pressure distribution over both suction and pressure surfaces of the blade.

15

3.1. Inverse design formulation

The fictitious velocity of the blade profile is derived from the difference between current and

target pressure. This is achieved by performing a balance between the transient momentum

fluxes of the current blade ‘ ’ and the fixed momentum fluxes of the designed blade ‘ ’ to be

obtained.

The momentum flux of the moving and deforming blade in the Navier-Stokes equations yields:

[ 𝜌 𝜌

𝜌 𝜌 ]

(3-1)

Where is the outward vector normal to the blade surfaces. The fictitious velocity of

the nodes located on the blade wall is computed by equating the momentum flux of the moving

wall ((3-1) with the momentum flux that is assumed to exist on the target blade shape. As the

blade reaches the shape that would satisfy the target pressure profile, the virtual velocities will

vanish and the design momentum flux reads:

[

]

(3-2)

The x- and y- components of the blade’s virtual velocity obtained by equating Equations 3-1 and

3-2 are:

16

(

| |

𝜌)

(3-3)

For the positive difference between the target and actual pressure on the suction surface of the

blade, the sign of the y component of the virtual velocity is negative and vice versa on the

pressure surface.

Having the x- and y- component of the virtual velocity, the virtual velocity of the blade in the

direction normal to the wall may be computed as follows:

. N (3-4)

The corresponding wall displacement is directly computed from Eq.3-4, However, in order to

respect the impermeability of the blade wall, the blade displacement should be in opposite

direction of the virtual velocity.

The blade displacement is illustrated in Figure 2.2. A heavy relaxation factor needs to be applied

to the computed virtual velocity in order to ensure the stability of the unsteady simulation[5].

The relaxation factor has the following form:

⁄ √| | 𝜌 (3-5)

17

Figure 2.2. A schematic movement of the blade wall

Where ‘ ’ is the relaxation factor, ‘ ’ is the speed of sound and ‘ ’ is the difference between

the current and target pressure distribution. Tong and Thompkins (21) had suggested a value of

0.01 – 0.02 for , however the current method allows for higher values for namely 0.1 – 0.2.

Having the above mentioned relaxation factor applied to the virtual velocity, the blade

displacement can then be written as:

(3-6)

Where ‘ ’ is the relaxation factor, is the velocity component normal to the blade wall and

‘ ’ is the user introduced transient time step size. The negative sign, as explained earlier,

implies the opposite direction for the blade displacement.

18

Where ‘ ’ is the introduced relaxation factor, is the velocity component normal to the blade

wall and ‘ ’ is the user introduced transient time step size. The negative sign, as explained

earlier, imposes the opposite direction for the blade displacement.

After imposing the computed displacement to the blade and update the blade profile coordinates,

the geometry is scaled back to the original axial chord length and then the new tangential camber

line is derived out of the new geometry. It is recommended to apply one or two loops of elliptic

smoothing over the new camber line in order to ensure the smoothness of the designed camber

line.

It should be noted that the method works well for both inviscid and viscous flows although the

viscous terms in (3-1) are neglected, and the convective terms only are involved in balancing the

flux and driving blade to a shape which asymptotically satisfies the target pressure [22].

3.2. Inverse design variables

There is a considerable difference between implementation of the inverse design method on

single or multistage blade rows with a single blade row [23]. In the case of a single blade row,

the designer can easily impose a fixed inlet and outlet boundary conditions such as inlet total

pressure, flow angles and outlet static pressure or mass flow rate without expecting variations in

the imposed values. However, in the case of a single stage or more considerably in the case of a

multistage blade rows, the flow conditions after the first blade row may vary, e.g., due to the

transient pressure loading variations which leads to the fluctuation of the pressure level in the

blade row downstream of the flow. In such cases, an absolute target pressure distribution for the

19

suction or pressure surface of the downstream blade row may not be achieved due to the

mentioned variations and it will impose to the designer to choose another target function i.e.,

target pressure loading which is the pressure difference between the pressure and suction

surfaces of the blade. Moreover, a designer may also tend to prescribe the target blade thickness

distribution as the structural and manufacturing constraints. However, in the case of a single

stage configuration, the mentioned transient pressure variation may be small and hence

ignorable.

Totally there are three choices of design variables available for the method:

3.2.1. Target pressure distribution on the blade pressure and suction surface

In this choice of design variable, target pressure distribution is prescribed for both pressure and

suction sides of the blade. Then the virtual velocity is directly computed from Equations 3-3 and

3-4 From the aerodynamic point of view, this choice of design variable works very well;

however, since the target blade thickness distribution is not prescribed and it is left to be a part of

the design solution, structural problems may rise. This is remedied by having the blade LE and

TE shapes be specified by excluding the first and last 2% from inverse computations as

discussed later.

3.2.2. Target pressure loading and blade thickness distribution

In this choice of design variable, a target pressure loading and a target thickness distribution is

prescribed. Here the virtual velocity may not be computed directly from the difference between

the current and target pressure loadings since the term in Equations 3-3 and 3-5 refers to the

20

difference between target and current static pressure of the suction or pressure surfaces of the

blade. Hence, it is first required to derive the target static pressure from the target loading.

Translation of the target loading to the static pressure of the suction and pressure surface is as

follows:

(3-7)

Where and respectively are the current static pressure distributions of the blade pressure

and suction surfaces obtained from the time accurate solution of the URANS equations at each

time step.

Moreover, in order to reach the imposed target thickness distribution, the following process is

performed. New camber line of the designed blade is computed by adding the average blade

displacement on both surfaces to the original camber line:

(3-8)

Then the discrete points on the camber line are brought back to their original x- location so that

the movement occurs only in the y direction. As explained in section 3.1, it is recommended to

perform one or two elliptic smoothing loops on the camber line to ensure the smoothness of the

blade profile. Care needs to be taken not to use too many smoothing loops as it drives the camber

line towards a straight line and consequently the convergence to a blade shape which satisfies the

target function may fail. The camber line is smoothed using the following formulation:

21

| |( ) | |( ) (3-9)

Where j refers to the position of the discrete points on the blade camber line which are sorted in

an ascending order from minimum x- to maximum x- coordinate.

The typical range of value for the smoothing factor ‘ ’ is between 0 and 0.2 and is highly case

dependant. Using this smoothing factor helps in eliminating possible small oscillations in the

blade geometry although it may delay convergence. More details are provided in Chapter 4.

Finally, the blade profile is updated by applying the imposed blade thickness to the designed

camber line:

(3- 10)

It should be mentioned that there is always a chance that the design process fails. This is usually

an indication that the chosen design variable(s) is not a physical one. It is notable that, smoothing

of the camber line does not apply exclusively to this choice of design variables and it is

recommended for the other two choices as well.

3.2.3. Target suction surface pressure and thickness distribution

The third choice of design variables is prescribing a target pressure distribution on the blade

suction surface and the thickness distribution.

Since the blade performance is mainly affected by the suction surface, the inverse design is only

applied on this surface. On the other hand, as the pressure distribution on the blade pressure

22

surface does not have a strong impact on the blade performance, the pressure obtained from the

URANS solution is imposed as target for the pressure surface at every time step which means no

virtual velocity is computed however this surface will change to satisfy the thickness constraint.

Moreover, working only on the suction surface gives the designer more control on the flow over

this surface and using a proper target pressure the blade performance may be improved by

decreasing e.g., the peak Mach number and the adverse pressure gradient and/or weakening the

shock, etc.

3.3. Design constraints

For every case, there are geometrical and non-geometrical constraints to be respected. The non-

geometric constraints such as mass flow rate, inlet flow angles, inlet Mach number, stage

reaction and rotational speed are mostly respected by the proper selection of the boundary

conditions and setup configurations. However, for external flow as shown by Mangler [24],

Lighthill [25] and later on by Volpe et al. [26], an arbitrary choice of the target pressure

distribution may lead to crucial geometrical problems especially near the LE/TE regions of the

blade where if proper care is not taken of, the design process may lead to an open leading edge or

a trailing edge crossover. In the current method, in order to avoid facing this problem, the inverse

design is implemented between 0.5%-3% and 97%-99.5% of the axial chord while the remaining

parts which fall near the LE/TE regions of the blade are going under analysis only and no design

is done in these regions. Then the slope of the camber line and blade thickness in these two

23

regions, in order to ensure the camber line smoothness at the transient points, is matched with

that prevailing from the design region [19].

24

Chapter 4

Algorithm, Validation and Redesign Cases in ANSYS-CFX

4.1. Implementation of inverse design methodology in ANSYS-CFX

To assess the CFD model that was chosen in ANSYS-CFX, it was used to simulate the flow

through a compressor stage and the results thus obtained were compared with the available

experimental data [27].

Figure 4.1. Computational algorithm for inverse design

25

Following that assessment, the inverse design method was implemented into the time accurate

Eulerian-Lagrangian formulation of ANSYS-CFX. At the initial time step, the steady state

converged solution obtained on the original geometry was taken as the prevailing flow field at

that initial time. Figure 4.1 shows the inverse design algorithm where the block on the right

constitutes the inverse design module.

The design iterations start from an initial guess of the blade shape that is moved in time based on

the difference between the instantaneous and target pressure distributions. The design iterations

continue until this difference is asymptotically driven to zero. This is similar to simulating the

flow around an airfoil that is executing a prescribed motion; in inverse design this motion is

calculated based on the difference between the instantaneous and the target pressure distributions

along the blade surfaces.

At the beginning of each time step, CFX computes the stationary problem up to a predefined

convergence level or number of iterations, then it passes to the User Defined Function (UDF) the

flow variables such as pressure, temperature, density and geometry at all the nodes located on the

blade walls. The UDF extracts these variables from CFX, it reads the target pressure distribution

and computes the fictitious wall velocity based on the difference between the current and the

target pressure distributions. The nodes displacements are computed and sent back to CFX to

modify the blade geometry. The grid velocities are then computed and added to the governing

equations, the computational domain is re-meshed to match the new blade shape and the solution

is advanced to the next time level. The remeshing occurs by moving the mesh points while

keeping the same mesh topology. The time accurate computations continue until either the L2-

norm of blade displacement or the difference in pressure between the target and the current

26

values falls within an acceptable tolerance where it can be said that the current blade shape has

asymptotically reached the target pressure distribution.

4.2. The UDF attached to ANSYS-CFX

In order to implement inverse design method in CFX, a FORTRAN subroutine which contains

the inverse design functionality is compiled with CFX, it is then attached to the blade boundary

conditions so that it can receive the nodal values of the blade shape at each time step.

Figure 4.2. Building the structure data in UDF

This is achieved by using a user “CFX Expression Language” CEL function. The user CEL

function is attached to the domain where the inverse design is carried out. Since CFX gives the

27

nodal values in a random order, after reading the user data and getting the nodal values they are

sorted in ascending order based on the x-coordinate, the values corresponding to the blade

pressure and suction surfaces are stored in separate arrays and the Design subroutine is then

called (Figure 4.2).

Since initially the nodes located on the blade walls do not have a one to one correspondence nor

do they fall at the same ‘x’ location, by passing a spline over the pressure surface, the nodes

move along the same curve to the same ‘x’ coordinate as the suction surface (this is required for

computing the tangential thickness and camber line).

Figure 4.3. Algorithm for virtual velocity computation

28

Then, based on the choice of the design variable, the target pressure distribution is computed and

based on the difference between current and target pressure distributions the virtual velocity is

calculated from Equations 3-3 and 3-4.

The node velocities are then smoothed by setting the averaged velocity of two neighboring nodes

as the velocity of the node between them and then the ‘Remesh’ subroutine is called (Figure 4.3)

where the nodes displacement is first computed then the tangential thickness and camber line for

the current and previous time steps are computed. The next step is the treatment of the blade

Figure 4.4. Algorithm for nodes displacement computation and updating the blade profile

29

leading and trailing edges where a small % of the blade LE and TE are preserved; this may be

done by replacing the current camber line in these preserved portions with that of the original

blade or by extrapolating the camber line in these regions.

Figure 4.5. Functionality of the UDF attached to ANSYS-CFX

30

Then the blade geometry is updated from Eq.3-10 and the L2-norm of the difference between

current and target pressure distributions is computed. As mentioned earlier, if this value is within

an acceptable tolerance, the process will stop and if not, the new blade profile will be sent to

CFX to update the geometry, remesh the domain and the whole process repeats until the L2-norm

of DP reaches the tolerance (Figure 4.4).

Figure 4.5 illustrates the whole process and the connection between CFX and the attached UDF.

4.3. Analysis of the E/CO-3 compressor stage

The mid-span section of the single stage subsonic compressor, called E/CO-3, is first analyzed in

ANSYS-CFX at three points on the design speed line (of 9,262.5 rpm) namely Maximum Flow,

Design Point and Near Surge. The geometrical characteristic of the stage is shown in the Table

4.1.

Table 4.1. E/CO-3 Stage geometric characteristics (Ref. [27])

Rotor Stator

Inlet blade angle 57.79˚ 36.64˚

Exit blade angle 43.03˚ -9.23˚

Number of blades 41 73

Stagger angle 49˚ 14˚

Space to chord ratio 0.9 0.7

Reynolds number 0.7 106 0.6 106

Axial chord length (cm) 2.68 2.85

31

The numerical results of the analysis at maximum flow, design point and near surge conditions

are provided and compared with the experimental data [27] in Tables 4.2, 4.3 and 4.4

respectively.

Table 4.2. E/CO-3 Compressor stage results at maximum flow

Measured Computed

Stage PR 1.196 1.196

Efficiency 85.7 85.76

TRR 0.0612 0.06114

Rotor

Inlet P0 (psi) 13.99 13.99

Inlet T0 (K) 296 296

Inlet flow angle 0.28˚ 0.28˚

Exit P0 (psi) 16.98 17.16

Exit T0 (K) 313 314.1

Exit flow angle 26.21˚ 26.8˚

Stator

Exit P0 (psi) 16.91 16.74

Exit T0 (K) 313 314.1

Exit flow angle -1.52˚ -0.64˚

32

It can be seen in the tables that the computed and measured data match almost perfectly. There is

one exception and it is the exit absolute flow angle of the rotor near surge where there is a

difference of about 3˚ between computed and experimental data. Here the predominant reason of

such a large difference between measured and computed value might be due to the separated

flow region close to the rotor blade trailing edge.

Table 4.3. E/CO-3 Compressor stage results at design point

Measured Computed

Stage PR 1.236 1.236


TRR 0.0707 0.0706

Rotor

Inlet P0 (psi)

13.97 13.97

Inlet T0 (K) 296 296


Exit P0 (psi) 17.43 17.65

Exit T0 (K) 315.54 316.85


Stator

Exit P0 (psi) 17.39 17.27

Exit T0 (K) 315.54 316.85


33

Another important reason is due to the fact that the experimental data are measured on a 3D

geometry while in this work only the mid-span of the entire blade row is analyzed. This, in some

cases will force the designer to manipulate the outlet static pressure, which is usually the

imposed outlet boundary condition, to a small extension in order to match the main flow

parameters with the experimental data hence may lose in some other flow variables.

Table 4.4. E/CO-3 Compressor stage results at near surge

Measured Computed

Stage PR 1.267 1.267


TRR 0.0822 0.0823

Rotor

Inlet P0 (psi) 13.95 13.95

Inlet T0 (K) 296 296


Exit P0 (psi) 17.81 18.05

Exit T0 (K) 319.12 320.29


Stator

Exit P0 (psi) 17.56 17.68

Exit T0 (K) 319.12 320.29


34

There are also other factors affecting the numerical results such as different inlet and outlet

radius of the mid-span section in real 3D blade geometry, flared gas path and etc.

4.4. Validation of the inverse design implementation into ANSYS-CFX

The mid-span of the E/CO-3 rotor, running at maximum flow conditions, is chosen to validate

the implementation of the inverse design method in CFX. For this purpose, the pressure

distribution obtained from analyzing the original blade profile has been set as the target pressure

distribution. Since the target pressure is the same as the original pressure, the target blade profile

should also be the same as the original blade geometry. Hence, the L2-norm of and blade

displacement are supposed to be zero. This is in fact observed with CFX running in the design

mode for two different design variables, i.e.,

1. SS pressure distribution and airfoil thickness distribution.

2. Dp distribution and airfoil thickness distribution.

Figure 4.6 shows that, regardless of choice of the design variable, the L2-norm of displacement

remains within machine accuracy and in the order of 10-5

(single precision is used); while Figure

4.7 shows that the L2-norm of fluctuates around 10-4

, Since pressure is related to velocity

squared rather than velocity.

Figure 4.8 shows that the obtained blade profiles for two design variables still fall on the original

blade profile and L2-norm of the blade airfoil error indicates that after 450 design steps, there is

almost no change in the blade shape.

35

Figure 4.6. L2–norm of airfoil displacement

Figure 4.7. L2–norm of

Time Step

L2

no

rmo

fa

irfo

ild

isp

lace

me

nt

0 100 200 300 400

10-8

10-7

10-6

10-5

10-4

10-3

10-2

DP Design

P-Design

Time Step

L2

no

rmo

fD

elta

(DP

)a

nd

De

lta

(P- )

0 100 200 300 400 500

10-7

10-6

10-5

10-4

10-3

10-2

10-1

P-Design

DP Design

36

Figure 4.8. E/CO-3 Rotor blade – Validation of inverse method using Design and Design

Figure 4.9. E/CO-3 Rotor pressure distribution – Validation of inverse method using Design and Design

x(m)

y(m

)

-0.005 0 0.005 0.01 0.015 0.02 0.025

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

Original

Dp Design

P-Design

L2

norm of airfoil error:

DP Design : 2.5 E -08

P-Design : 2.4 E -07

x (m)

P(k

Pa

)

-0.005 0 0.005 0.01 0.015 0.0230

40

50

60

70

80

90

100

110

120

130

140

Original

DP Design

P-Design

37

Figure 4.9 shows the corresponding pressure distribution over the original and designed blade

profiles which again fall on the top of each other. There are also some tiny spikes (Figure 4.7)

occurring every 50 time steps which will be addressed later in this chapter.

It is notable that the time step size and values of relaxation factors used for the above validation

case are all the same as the values being used in actual design cases. Here, the smoothing factor

is an exception which has been removed from the design computations in order to not to affect

the blade profile.

4.5. Redesign of the E/CO-3 compressor stage at maximum flow

The inverse design method is first implemented on the E/CO-3 compressor rotor blade in the

maximum flow while the stator blade is untouched. The design intent is to improve the stage

performance in terms of total to total efficiency by introducing a target pressure distribution

which lowers the negative incidence at the rotor inlet, decreases the peak Mach number over the

suction surface of the blade and lowers the adverse pressure gradient (i.e., decrease the diffusion

and pressure loss across the stage). Care was taken to maintain the original pressure loading.

The first design variable being used is the suction surface pressure distribution and thickness

distribution of the blade while the first and last 2% of the blade are running in the analysis mode

in order to ensure that the leading and trailing edges of the blade are smooth and consistent. L2-

norm of ) is a measure of the convergence of the current pressure distribution to the target

in a least-square sense. The problem converges after 64 time steps (Figure 4.10).

38

Figure 4.10. E/CO-3 Rotor: Convergence history for the Design

Figure 4.11. E/CO-3 rotor: Original and redesigned blade geometry for the Design

Time Step

L2

no

rmo

fD

elta

(P- )

0 10 20 30 40 50 60 70

0.05

0.1

0.15

0.2

0.25

P-Design

x(m)

y(m

)

-0.01 0 0.01 0.02-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

Original

P-Design

39

The designed and original blade profiles are shown in Figure 4.12 and Figure 4.12 shows and

compares the designed, target and original pressure distributions.

Figure 4.12. E/CO-3 rotor: Original, target and design pressure distributions for the Design

By looking at Figure 4.12, it can be seen that the peak Mach number over the suction surface as

well as the adverse pressure gradient are lowered which lead to the reduction in diffusion and

consequently reduction of pressure loss in the stage. Moreover the negative incidence at rotor

inlet has also been reduced and all of these factors together led to the improvement of the total to

total efficiency by 0.56 %.

It can be seen from Figure 4.10, that there is a tiny spike in the L2-norm of ) after 44 time

steps. The reason is because of the fact that CFX solver, while solving time accurate URANS

equations written in Eulerian-Lagrngian formulation, is stopped by the user after 44 time steps.

x(m)

P(k

Pa

)

0 0.01 0.02 0.03 0.04 0.05 0.06

40

60

80

100

120

P-Target

Rotor - Original

Rotor - P-Design

Stator - Redesigned Rotor

Stator - Original

Total to total Efficiency (%):Original: 85.76Design : 86.32

40

Then the obtained geometry up to that time step has been analyzed in the steady state mode to

remove the accumulated unsteadiness in the design mode to stabilize the flow field. Then again,

using the converged steady state, stable solution as the initial values, the design is continued until

64th

time steps when the final designed blade geometry, which satisfies the target pressure

distribution, is obtained.

A second validation case for the inverse methodology was performed where the obtained

pressure distribution for the designed blade in design case was used to compute the pressure

loading. Then it was set as the target loading distribution and the inverse design was performed

again with loading and blade thickness distribution as the design variable. It was expected to

reach the same blade profile and pressure distribution as those obtained in the first case (i.e., SS

pressure and blade thickness distribution). In fact, this expectation was fulfilled as the obtained

blade geometry and pressure distribution asymptotically fell on those of the design case.

Figure 4.13 shows the L2-norm of ) for the run. For this case, the design converges after

101 time steps and it was stopped 3 times at 13th, 33th and 73th time step to do the analysis on

the obtained blade profile and again continue the design process from a stable flow field.

Figure 4.14 shows the target, design and original pressure loading. Figure 4.15 shows the design

pressure distribution at maximum flow for and DP design, the corresponding blade profiles

are shown in Figure 4.16.

Table 4.5 gives the flow parameters obtained for two design variables and the original blade

geometry.

One difference between these two cases is that for the DP design, the CFX solver while running

in the design mode, has been stopped 3 times instead of 1. The reason might be related to the

41

choice of the design variable, pressure distribution and the values of relaxation and smoothing

factor or the time step size. For the stability reasons, the chosen time step size for the DP design

was smaller than that for the design and it more time steps were needed to reach the target.

Figure 4.13. E/CO-3 Rotor: Convergence history for the Design

Moreover, the smoothing factor for the DP design was greater than the design. The high

value of the smoothing factor moves the nodes away from the target but it is sometimes

necessary to use a high value in order to have a smooth, consistent blade shape.

The choice of values for smoothing factor is highly case dependent as even for the same blade

profile but running in different operating conditions or even the same conditions but different

design variable, it may not be possible to use the same value and actually this fact was observed

in ANSYS-CFX.

Time Step

L2n

orm

of

De

lta

(DP

)

0 20 40 60 80 100 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

DP Design

42

Figure 4.14. E/CO-3 rotor: Original, target and design loading distributions at maximum flow for the DP Design

Figure 4.15. E/CO-3 Rotor: Design pressure distribution at maximum flow for and DP Design

x(m)

DP

(kP

a)

-0.005 0 0.005 0.01 0.015 0.02

-60

-40

-20

0

20

Original

DP Target

DP Design

x(m)

P(k

Pa

)

-0.005 0 0.005 0.01 0.015 0.02

40

60

80

100

120

DP Design

P-

Design

43

Figure 4.16. E/CO-3 Rotor: Blade profile at maximum flow for and DP Design

Table 4.5. Validation of the E/CO-3 compressor stage redesign: flow simulations obtained for two design variables

Original P

- Design DP Design

Efficiency (%) 85.76 86.32 86.42

Stage PR 1.196 1.198 1.197

TRR 0.0611 0.0612 0.0610

Exit P0 (kPa) 115.4 115.6 115.5

Exit T0 (K) 314.10 314.12 314.10

Rel. flow angle at rotor exit 44.31˚ 44.11˚ 44.26˚

Rel. Mach no. at rotor exit 0.635 0.637 0.636

Rel. flow angle at rotor Inlet 54.87˚ 54.60˚ 54.69˚

x(m)

y(m

)

0 0.01 0.02-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

DP Design

P-Design

44

However, in the cases where the elimination of the smoothing factor does not endanger the

smoothness and the consistence of the blade profile, it has been observed that the design can

converge and reach the target without the need to stop the solver and stabilize the flow field in

the middle of the design process. One example is design of the E/CO-3 rotor blade at near surge

which will be presented later in this chapter.

From Table 4.5, one can see a 0.1% of difference between the total to total efficiency of the two

design cases. The reason may be found in the Figure 4.17 which focuses on a portion of the

suction surface of the blade in Figure 4.15, which has the maximum Mach number.

It can be seen from Figure 4.17 that the pressure distribution obtained from DP design (i.e. blade

pressure loading and thickness distribution) has a slightly lower slope and maximum Mach

number over the suction surface compared to the design case that could be the reason of this

small difference.

It should be also emphasized that 98.5% of the original loading has been retained for the

design, while for the DP design 99.1% of the target loading has been retained.

After redesigning the blade profile at maximum flow, the designed blade was analyzed at design

point and then near surge to evaluate the performance of the designed blade in all three

conditions and to compare it with that of the original blade.

First the designed blade was analyzed at design point. Figure 4.18 compares the obtained

pressure distribution for the designed blade with that of the original blade analyzed at design

point.

45

Figure 4.17. E/CO-3 Rotor: Pressure distribution close-up at maximum flow for and DP Design

It can be seen from the Figure 4.18 that the designed blade has a better performance than the

original blade as the peak Mach number over the suction surface as well as the adverse pressure

gradient have been lowered. Moreover, the negative incidence has decreased and all these

parameters lead to an improvement of about %0.3 in the total to total compressor stage efficiency

at design point.

Table 4.6 shows the flow parameters obtained from analysis of the designed (designed at

maximum flow) and original blades at design point.

However, when the performance of the designed blade was compared with that of the original

blade profile at near surge, it was observed that the original blade produces better performance.

x(m)

P(k

Pa

)

0 0.005 0.01

45

50

55

60

65

70

75

80

85

90

DP Design

P-

Design

46

Figure 4.19 shows the pressure distribution obtained from the analysis of the designed (designed

at maximum flow) and original blades at near surge.

Figure 4.18. E/CO-3 Rotor – Pressure distribution of the original and redesigned blade (designed at maximum flow)

at design point

Table 4.7 shows the flow parameters obtained from the analysis of the designed (designed at

maximum flow) and original blades at near surge.

From Figure 4.19 it can be understood that the adverse pressure gradient and peak Mach number

over the suction surface compared to the original blade have increased.

Moreover, as in the maximum flow the goal was to decrease the negative incidence, here not

surprisingly, it can be seen that a reduction of the negative incidence at maximum flow has led to

an increase of the positive incidence at near surge.

x(m)

P(k

Pa

)

0 0.01 0.02 0.03 0.04 0.05 0.06

70

80

90

100

110

120

130

Rotor - Original

Stator - Original

Rotor - P-Design (maximum flow)

Stator - Redesigned Rotor

Adiabatic Efficiency (%):Original: 88.60Design: 88.88

47

Table 4.6. E/CO-3 compressor stage results for analysis of the designed (designed in maximum flow) and original

blades at design point

Original P- Design

Efficiency (%) 88.60 86.88

Stage PR 1.236 1.238

TRR 0.0706 0.0707

Exit P0 (kPa) 119.1 119.2

Exit T0 (K) 316.89 316.93

Rel. flow angle at rotor exit 44.42˚ 44.24˚

Rel. Mach no. at rotor exit 0.6 0.6

Rel. flow angle at rotor Inlet 56.17˚ 55.96˚

Figure 4.19. E/CO-3 Rotor: Pressure distribution of the original and redesigned blade (designed at maximum flow)

at Near Surge

x(m)

P(k

Pa

)

0 0.01 0.02

40

60

80

100

120

Rotor - Original

Rotor - P-Design (maximum flow)

Adiabatic Efficiency (%):Original: 85.16Design: 84.60

48

Table 4.7. E/CO-3 compressor stage results for analysis of the designed (designed at maximum flow) and original

blades at near surge

Original P- Design

Efficiency (%) 85.16 84.60

Stage PR 1.267 1.267

TRR 0.0823 0.0826

Exit P0 (kPa) 121.9 121.9

Exit T0 (K) 320.29 320.563




Generally, increase in the positive (or negative) incidence is one of the reasons of the reduction

in the stage performance. All these reasons together caused a reduction of 0.56% in the total to

total efficiency of the stage compared to the original blade at near surge, as shown in Table 4.7.

4.6. Redesign of the E/CO-3 compressor stage at near surge

The inverse methodology was then implemented on the E/CO-3 compressor rotor blade at near

surge. The design variable is DP and thickness distribution and the goal is to improve the total to

total stage efficiency by lowering the positive incidence at the rotor inlet.

Figure 4.20 shows the L2-norm of ) for the run, where it can be seen that the problem

converges after 101 time steps. Figure 4.21 shows the target, design and original pressure loading

49

at near surge conditions. Figure 4.22 compares the original and designed blade profiles and

Figure 4.23 illustrates the original and designed pressure distributions.

Flow parameters obtained from the design along with those of the original blade are provided in

Table 4.8.

From Figures 4.21 and 4.23 it can be seen that the positive incidence has been significantly

lowered which accounts for the huge amount of improvement of about 1.9% in total to total stage

efficiency. This improvement is due to the considerable decrease of the pressure loss across the

stage (Table 4.8).

Figure 4.20. E/CO-3 Rotor Redesign in Near Surge condition, L2–norm of . Design Variable:

DP & thickness distribution

Moreover, as detailed earlier, for the design at near surge conditions the smoothing factor has

been removed and there was no need to stop the run in the mid way to stabilize the flow field.

Time Step

L2

no

rmo

fD

elta

(DP

)

0 20 40 60 80 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

DP Design

50

Figure 4.21. E/CO-3 Rotor: Original, target and design loading distributions at near surge

Figure 4.22. E/CO-3 Rotor: Original and redesigned blade geometry at near surge

x(m)

DP

(kP

a)

-0.005 0 0.005 0.01 0.015 0.02

0

20

40

60

80

Original

DP Target

DP Design

x(m)

y(m

)

-0.01 0 0.01 0.02

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

Original

DP Design

51

Figure 4.23. E/CO-3 Rotor: Original and redesigned pressure distributions at near surge

Table 4.8. E/CO-3 Compressor stage results for the designed and original blade at near surge

Original DP Design

Efficiency (%) 85.16 87.07

Stage PR 1.267 1.271

TRR 0.0823 0.0816

Exit P0 (kPa) 121.9 122.3

Exit T0 (K) 320.29 320.14




x(m)

P(k

Pa

)

-0.005 0 0.005 0.01 0.015 0.02

40

60

80

100

120

Original

DP Design

52

The reason, in the author’s opinion, could be related to the original and target pressure

distributions, since for the same blade profile but in different conditions (i.e., maximum flow) a

non-zero value for the smoothing factor was necessary.

4.7. Assessment of the Analysis Scheme on the E/TU-3 turbine stage

The mid-span section of the low speed axial turbine stage E/TU-3 is analyzed in ANSYS-CFX at

design point on the design speed line of 7800 rpm [27].

Table 4.9 shows the geometric characteristics of the stage. The numerical results of the analysis

at the design point are provided and compared with the experimental data [27] in Table 4.10.

A structured O-mesh is generated around the blade walls to ensure that and the rest of

the domain is filled with a structured H-mesh. The stator and rotor domains consisted of 25,160

and 30,886 cells, respectively.

Table 4.10 shows that the computed flow angle at stator exit angle matches rather well with the

measured value while relative flow angles at rotor exit and inlet off by 4.5˚ and 1.5˚ respectively.

This is probably due to the fact that the change in the streamtube height in flow direction has

been ignored, an effect that can not be accounted for when using ANSYS-CFX to simulate 2D

flow regions.

53

Table 4.9. Geometric characteristics of the E/TU-3 stage (Ref. [27])

Stator Rotor

Inlet blade angle β1 0˚ 45˚

Exit blade angle β2 67˚ 54.5˚

Number of blades 20 31

Stagger angle 45˚ 33˚

Pitch to chord ratio 0.65 0.66

Aspect ratio 1.92 1.22

Axial gap to stator chord 0.57

Nominal inlet Mabs 0.20 0.84

Nominal exit Mabs 0.84 0.4

Nominal inlet Mrel - 0.45

Nominal exit Mrel 0.45 0.69

Axial chord length (cm) 6.7 3.68

Table 4.10. E/TU-3 Assessment of the E/TU-3 turbine stage

Measured Computed

Stage PR 1.77 1.85

Efficiency (%) 89.9 90.44

Rotor

Inlet Rel. flow angle 44.7˚ 49.4˚

Exit Rel. flow angle -55.2˚ -53.81˚

Stator


54

4.8. Redesigning of the E/TU-3 turbine stage

The inverse method was applied to the redesign of the subsonic E/TU-3 turbine stage. First the

method is applied on the stator blade row while the rotor blade is untouched and is going under

analysis only.

Again the goal is to improve the total to total stage efficiency by reducing the peak Mach number

over the suction surface of the stator blade. The design variable is loading and blade thickness

distribution (DP Design).

Figure 4.24 shows the L2–norm of ; the problem converges after 44 time steps and the L2–

norm of decreases by about one order of magnitude.

Figure 4.24. E/TU-3 Stator: Convergence history

Time Step

L2

no

rmo

fD

elta

(DP

)

0 10 20 30 40 50

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

DP Design

55

Figure 4.25. E/TU-3 Stator: Design, target and original loading distributions for DP Design

Figure 4.26. E/TU-3 Stator: Design and original pressure distribution for DP Design

x (m)

DP

(kP

a)

-0.14 -0.12 -0.1 -0.08

0

10

20

30

40

50

60

Original

DP Target

DP Design

x (m)

P(k

Pa

)

-0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -0.0880

100

120

140

160

180

200

DP Design

Original

56

Figure 4.27. E/TU-3 Stator: Design and original blade profiles for DP Design

Table 4.11. E/TU-3 turbine stage results for the redesign of the stator blade at design operating point in ANSYS-

CFX

Original DP Design

Efficiency (%) 90.44 90.67

Stage PR 1.859 1.855

TRR 105.96 106.16

Exit P0 (kPa) 295.19 295.2

Exit T0 (K) -53.81˚ -53.81˚

Rel. flow angle at rotor exit -68.62˚ -68.41˚


Rel. flow angle at rotor Inlet 0.659 0.664

x (m)

y(m

)

-0.14 -0.12 -0.1 -0.08

-0.06

-0.04

-0.02

0

DP Design

Original

57

Moreover, as detailed earlier in sections 4.5 and 4.6, the fluctuations in the diagram represent the

time steps at which the design is stopped manually to solve the problem in the steady state mode

and then again restart the design process to finally reach a blade shape which satisfies the

prescribed target pressure distribution.

According to Figure 4.25, the peak loading is slightly reduced and the blade loading has moved a

little upstream the flow. Care has been taken to keep the overall loading fixed so that the specific

work and blade turning are least affected.

Figure 4.26 shows a decrease in the peak Mach number as well as the adverse pressure gradient

over the suction surface which leads to a lower diffusion. These effects combined result in an

improvement of about 0.23% in total to total stage efficiency.

Figure 4.27 represents the design and original blade profiles. Table 4.11 compares the flow

parameters obtained for the original and designed blade geometries.

A second design is implemented on the E/TU-3 rotor blade while the stator blade is untouched.

The design variable is suction surface pressure and blade thickness distribution ( Design). The

goal is to improve the stage performance, in terms of total to total efficiency, by reducing the

peak Mach number which is accomplished by increasing the minimum pressure over the suction

surface of the rotor blade.

Figure 4.28 shows the L2–norm of ; the problem converges after 64 time steps. Figure 4.29

shows a small gap between the designed and target pressure distributions near the point of

maximum turning. It was observed in ANSYS-CFX that getting closer to the target pressure at

this especial region will generate a small kink on the blade profile which is not preferred

although the effect on the stage performance is ignorable (i.e. 0.01%).

58

As it appears from Figure 4.30, the blade is highly cambered and it makes the problem more

complicated than the compressor case or even the E/TU-3 stator blade. An especial care should

be taken in order to retain the smoothness of the designed blade which may lead to small

deviations from the target pressure distributions in highly cambered regions of the blade.

Figure 4.28 shows that the CFX solver, while running in the design mode, has been stopped in

time steps 12, 32 and 53. However, after restarting the design process after the first stop i.e. time

step 13, a big jump occurs in the L2–norm of which demonstrates the huge amount of

accumulated unsteadiness in the first run. The reason lies in using a high value for the smoothing

factor i.e. 0.2.

As explained in sections 4.5 and 4.6, depending on the blade shape and pressure distribution, a

high smoothing factor, i.e. in Eq.3-9, might be required for the problem to converge. Here,

because of redesigning a highly cambered blade shape, a high value for the smoothing factor was

necessary to retain the smoothness of the camber-line. It should be noted that the amount of grid

displacement is directly affected by the user defined smoothing factor. That means higher values

of lead to a higher amount of grid displacement and consequently more unsteadiness at each

time step and in fact this behavior is the reason of having a big jump in the L2–norm of at

13th

time step compared to rest of the design steps where a lower smoothing factor has been

used.

Table 4.12 provides and compares the flow parameters obtained for the original and the designed

blade geometry which shows an improvement in the stage efficiency by 0.48%.

Moreover, 99.24% of the original pressure loading has been retained in the design process.

59

Figure 4.28. E/TU-3 rotor: Convergence history

Figure 4.29. E/TU-3 rotor: Design, target and original pressure distributions

Time Step

L2

no

rmo

fD

elta

(P- )

0 20 40 60

0.05

0.1

0.15

0.2

0.25

P-Design

x(m)

P(k

Pa

)

-0.02 -0.01 0 0.01

40

60

80

100

120

140

P-Target

Original

P-Design

Total to total Efficiency (%):Original: 90.44Design : 90.92

60

Figure 4.30. E/TU-3 rotor: Original and design blade profiles for Design

Table 4.12. Redesign of E/TU-3 turbine rotor

Original P

- Design

Efficiency (%) 90.44 90.92

Stage PR 1.859 1.861

Exit P0 (kPa) 105.96 105.86

Exit T0 (K) 295.19 294.85


Flow angle at stator exit -68.62˚ -68.57˚

Mach no. at stator exit 0.925 0.942


x(m)

y(m

)

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

Original

P-Design

61

The previously redesigned stator and rotor blades were analyzed as a stage and the results were

compared with those obtained for the original stage. Table 4.13 shows the flow parameters

obtained for the redesigned and original stage.

According to Table 4.13, the stage efficiency has increased by 0.71% which is exactly the

summation of the improvements accomplished when redesigning the stator and rotor blade rows

separately (Tables 4.11 and 4.12). This fact indicates that redesigning of a blade using inverse

methodology, does not affect the performance of downstream blade rows.

Table 4.13. E/TU-3 stage: Comparison of the original and redesigned stage

Original Design

Efficiency (%) 90.44 91.15

Stage PR 1.859 1.861

Exit P0 (kPa) 105.96 106.06

Exit T0 (K) 295.19 294.86


Flow angle at stator exit -68.62˚ -68.37˚

Mach no. at stator exit 0.925 0.937


62

Chapter 5

Conclusion

5.1. Summary

The inverse shape design method developed by Daneshkhah and Ghaly [18,19] has been

successfully implemented into ANSYS-CFX in the context of axial compressors and turbines in

two dimensional internal flows. The design variables used in this method were the pressure

distribution over the blade suction surface, or the blade pressure loading, and its thickness

distribution. The blade inverse design approach assumes that the blade surface moves with a

virtual velocity which is derived from a balance of transient (design) and fixed (target)

momentum fluxes, in order to fit a shape which satisfies the target pressure distribution.

The wall movement obtained from the design method is computed in a User Defined Function

(UDF) using the CFX Expression Language (CEL) and is communicated to ANSYS-CFX at

each time step.

In ANSYS-CFX, a cell-centered finite volume method is used for space discretization. The

URANS equations are written in an Arbitrary Lagrangian–Eulerian (ALE) form and are solved

in a time accurate fashion, in order to account for the mesh movement. The rotor-stator

63

interaction is based on a steady state circumferential averaging of the fluxes at the rotor-stator

interface, which is an option that is available in ANSYS-CFX.

The inverse method was first validated for the mid-span section of the E/CO-3 rotor, running at

maximum flow conditions, using two different sets of design variables. It was then applied to the

redesign of that axial compressor stage and then the redesign of an axial turbine stage.

First the E/CO-3 rotor blade was redesigned at maximum flow conditions; the target suction

surface pressure and the tangential blade thickness distributions were chosen as the design

variables. The stage efficiency was increased by reducing the peak Mach number and lowering

the adverse pressure gradient over the blade suction surface and also by reducing the negative

incidence at rotor inlet.

The designed blade was then analyzed at the design point and near surge. At the design point, it

was shown that the stage performance improved due to a decrease in negative incidence and

peak Mack number which resulted in a reduced diffusion region. Near surge, the performance

deteriorated since the decrease in negative incidence (at the design point) led to an increase in

positive incidence. A second compressor design case was performed on the compressor rotor

near surge conditions where the stage performance was highly improved by reducing the positive

incidence at rotor inlet.

Then the E/TU-3 stator blade was redesigned using the loading and thickness distributions as the

design variables where the performance was improved by reducing the diffusion and peak Mach

number over the suction side and finally the method was implemented onto the E/TU-3 rotor

blade where a reduced peak Mach number led to the improvement of the stage total to total

efficiency.

64

This design method was found to be useful in weakening or even removing diffusion regions

over the blade by lowering the adverse pressure gradient and peak Mach number over the blade

suction surface. However, it is highly dependent on the choice of an appropriate and realistic

target pressure distribution.

5.1. Future Work

The successful implementation of the design method into ANSYS-CFX opens the door for a

wide range of applications where the design method can be applied, namely any turbine blading

configuration that can be simulated using ANSYS-CFX. Most notably is to extend the current

method to blade design in three dimensional (3D) flow where the design issues that were

addressed in 2D flow and many new challenges associated with 3D flow will have to be

addressed. For example, what is a ‘good’ choice of the target suction side pressure distribution

(or pressure loading) in a 3D flow and how is it defined? The LE/TE regions, how would they be

defined?

Another important improvement that is needed is to replace the tangential thickness with a more

representative one i.e. the blade thickness normal to the camber line. This will help improving

the problem convergence especially for highly cambered blade shapes with high stagger angles

such as turbine rotor blades.

65

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Aerodynamic Inverse Shape Design of Compressor and Turbine