Jia Y U, Lucheng JI, Weiwei LI*, Weilin Y I

1 Copyright © 2015 by ISABE

ISABE-2015-20087

BLADE SHAPE OPTIMIZATION OF CENTRIFUGAL IMPELLER WITH ADJOINT METHOD

Jia YU, Lucheng JI, Weiwei LI*, Weilin YI

Beijing Institute of Technology, Beijing, China, 100081

Email: * [email protected]

Abstract Due to less computational cost, adjoint method has received considerable attention in recent

years. This paper focuses on continuous adjoint method, and couples with thin shear-layer N-S

equation to formulate an efficiency sensitivity analysis model for centrifugal impeller blades in

specified objective function. This model includes adjoint equation/boundary conditions, and the

sensitivity of objective function to design variable vector. Integrating a 3D blade perturbation pa-

rameterization and the simple steepest decent method, a frame of a gradient based aerodynamic

shape design system is constructed. Flow field of German and Swiss Turbo Compressor manufac-

turers a high pressure ratio centrifugal impeller aiming to increase efficiency and remain mass flow

rate and pressure ratio unchanged is taken.

Introduction As single-stage centrifugal compressors facilitate a high pressure ratio as small spatial require-

ments, they have been widely used in turbochargers and turboshaft engines. Extensive experimental

studies with state-of-the-art measurement techniques were conducted in centrifugal compressor

[1-6]. The trend in centrifugal compressor designs is to significantly reduce both size and weight for

cost reasons without efficiency penalties the aerodynamic loading of the blades is increasing. Opti-

mization is one of the important ways to improve the efficiency of the centrifugal compressor, keep

the pressure ratio and mass flow rate constant. However, increasing of the geometric complexity

makes the number of blade design variables numerous. The adjoint method used for gradient calcu-

lations has low computational cost, independent of the number of design variables involved. The

applications of the adjoint method in design optimization to both external and internal flows have

been widely used [7-22]. The gradient is calculated indirectly by solving an adjoint system, and all

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sensitivity derivatives are thus obtained with the cost of one flow solution and one adjoint solution

for each function. Hence, the total computing cost in one design cycle is roughly equivalent to twice

the cost of the iteration of the flow equations.

Compared with the axial compressor, centrifugal compressor is more complex because of the

complex internal flow field. The present paper attempts to develop the adjoint approach for the op-

timization of centrifugal compressor blades, and couples with thin shear-layer N-S equation to for-

mulate an efficiency sensitivity analysis model for centrifugal impeller blades in specified objective

function. This model includes adjoint equation, boundary conditions and the sensitivity of objective

function to design variable vector. Integrating a 3D blade perturbation parameterization and the

simple steepest decent method, a frame of a gradient based aerodynamic shape design system is

constructed. This new approach is applied to redesign of the German and Swiss Turbo Compressor

[6] aiming to increase efficiency and remain mass flow rate and pressure ratio unchanged.

Optimization Problem Objective function

In general, the increase of aerodynamic efficiency is obtained by means of minimizing the en-

tropy production rate. The entropy generation rate has a much simpler expression than that of the

isentropic efficiency as used by Oyama and Liou [23]. In this paper, the net mass-averaged entropy

production is chosen as the cost function. Constraint on the mass flow rate, pressure ratio and out-

flow angle is enforced by incorporating penalty functions in the cost function. The cost function of

centrifugal impellor design optimization is defined as

2 2 21 2 3( 1) ( 1) ( 1)

init init init init

s mIs m

π βσ σ σπ β

∆= + − + − + −∆

(1)

The definition for the entropy generation rate s∆ in the above expression is given by,

* *1 1

* *

out in

out in

T Tu ds u dsP P

suds uds

γ γγ γ

ρ ρ

ρ ρ

− −

∆ = −∫∫ ∫∫

∫∫ ∫∫

The subscript “init” denotes initial values; and m π are the ratios of mass flow rate and total pres-

sure ratio, respectively. 1 2,σ σ and 3σ are weighting factors taking different values for the centrifugal

impellor design. Different values of 1 2,σ σ and 3σ will usually lead to different designs, as it is also

indicated in [24]. A good choice of these values usually needs some trial and error.

Shape perturbation parameterization Hicks-Henne bump function [25] is used to explore the design space composed of blade cir-


cumferential bowing and blade suction surface. The advantage of the shape perturbation-based pa-

rameterization is that the original blade shape can be easily recovered by simply setting to zero the

perturbations. The overall net perturbations consist of the linear superposition of the bump function,

as it is shown in Eq. 2,

( ) ( )

( ) ( )

11ln 0.5 ln4

1242

ln 0.5 ln ln 0.5 ln2 2

1

, sin

sin sin

Ci

Ci Ci

Ri

i

X Ri

i

x r R

X R

δ α π

α π π

=

=

= ⋅

+ ⋅ ⋅

∑

∑ (2)

[ ] [ ]

[ ]

0,1 , ,

0,1 , ,

lele te

te le

hubhub tip

tip hub

x xX where x x x

x xr r

R where r r rr r

− = ∈ ∈ − − = ∈ ∈ −

,le tex x are the axial coordinates of the leading and trailing edges; hubr and tipr are radial coordinates of

hub and casing section.

The design variables are the coefficients of every basis function. The distribution of all design

variables can be marked by the bump centers of the corresponding basis functions. Once the values

of those design variables are known, the net perturbations are calculated through Eq. (2) and incor-

porated into the blade suction surface. The blade pressure surface is adjusted by keeping the blade

tangential thickness unchanged.

Adjoint equations and adjoint boundary conditions The implementation of the adjoint method based on thin layer RANS equations was described

previously [21]. Then the gradient vector of I toα can be written as,

( ) ( )

1

T

D D D

T T TT

D

T T

V V

T T TV Vx r

dI M Wfdv ds M ds dv W dvd

W D A B CU x r r

r B V P Udvr r r r r

MEn F F n G G n UUθ

λα α α

λ λ λλθ

λ λθ θ

λ λ λ

∂

−

∂ ∂ = + + + ∂ ∂

∂ ∂ ∂ ∂+ + + + + ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + ∂ ∂ ∂ ∂

∂ − + − + − − ∂

∫∫∫ ∫∫ ∫∫∫

∫∫∫

D

ds∂

∫∫

(3)

In order to eliminate the dependency of the domain integral toU in expression (3),λ should sat-

isfy,

1 0

T T TT

T T

V V

W D A B CU x r r

r B V Pr r r r r

λ λ λλθ

λ λθ θ

−

∂ ∂ ∂ ∂+ + + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + = ∂ ∂ ∂ ∂

(4)

Transpose equation (4), and the adjoint equations are,


1 0

TT T T T

TT T

V V

WA B C Dx r r U

r B V Pr r r r r

λ λ λ λθ

λ λθ θ

−

∂ ∂ ∂ ∂ + + + + ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + = ∂ ∂ ∂ ∂

(5)

There are also four kinds of boundary in adjoint field similar to flow field. In periodic boundary,

the boundary integral depending onU in expression (3) will be automatically cancelled due to the

symmetry of periodic boundary geometry. For another three kinds of boundary,λ should satisfy,

( ) ( ) 0T T TV Vx r

MEn F F n G G n UUθλ λ λ ∂

+ − + − − =∂

(6)

According to literature [11], the propagation directions of character information in adjoint field

are anti-physical, and the character velocity vector in adjoint field is opposite to that in flow field.

(1) Inlet boundary conditions

At inlet, 1, 0x rn n nθ= − = = , therefore equation (6) can be simplified,

0T ME UU

λ ∂− =∂

(7)

There is only one negative character velocity at the subsonic inlet boundary of adjoint field,

only one equation should be provided. ConsideringU is a function of static pressure p at inlet, we

have

,E p U pE Up pα α

∂ ∂ ∂ ∂= × = ×∂ ∂ ∂ ∂

(8)

Then, the inlet boundary condition is,

0T E Mp p

λ ∂ ∂+ =

∂ ∂ (9)

(2) Outlet boundary conditions

At outlet, 1, 0r xn n nθ= + = = . In a similar way, four equations should be provided at the sub-

sonic outlet boundary due to four negative character velocities, as follows,

0T VFF Mq q q

λ∂ ∂ ∂

+ − = ∂ ∂ ∂ (10)

where , , ,x rq V V Vθρ=

(3) Solid boundary conditions

At outlet, 0nV = , and only one equation should be specified. The solid boundary is slip, and

then the boundary condition should be,

2 3 4 5x rMn n rn wrnpθ θλ λ λ λ ∂

+ + + =∂

(11)

If the solid boundary is adiabatic, we also have,


5 0nλ∂

=∂

(12)

Where n indicates the normal direction.

The formulation procedure using continuous adjoint method coupled with the thin shear-layer

N-S equations in a cylindrical-coordinate system [22] is classified into three steps. The first step is

to linearize N-S equations, the second step is to formulate the adjoint partial differential equations,

and the third step is to discretize the adjoint equations for its numerical solving.

Case Description A high pressure ratio centrifugal compressor based on the experimental setup by Krain et al [26]

was chosen. The impeller design data are shown in Table 1. Further details can be taken from [6].

Table 1 Impeller design data SRV2-O

Inlet Total Pressure 0P 101325 Pa

Inlet Total Temperature 0T 288.15 K

Shaft Speed n 50000 rpm

Blade Count Full/Splitter f sz z 13/13 rpm

Mass Flow Rate m 2.55 /kg s

Blade Angle LE Tip 1tβ 26.5 deg

Impeller Tip Radius 2r 112 mm

Blade Angle TE 2β 52 deg

Impeller Press. Ratio 12π 6.1

Efficiency 12η 0.84

To make the Krain adapt our optimization solver, some modifications are applied to the impel-

ler. Firstly, the structure of leading and trailing edge were modified into rounded, as shown in Fig-

ure 1. Secondly, cancel the vaneless diffuser and reduce the exit area as shown in Figure 2. The flow

path inlet axial coordinate is 100mm. The outlet of the impeller radius coordinate is 130mm.

The performances of impeller with rounded leading and trailing edge in original flow path and

modified path are shown in Figure 3. The mass flow rate is normalized by its choked values. The

maximum stage pressure ratio is 6.8 with a corresponding isentropic efficiencyη of about 78% in

original path. The maximum stage pressure ratio is 7.3 with a corresponding isentropic efficien-

cyη of about 88% in modified path. The impeller in modified path has much shorter inlet domain

and without vaneless diffuser, so the isentropic efficiency is much higher than that in original path.


The inlet conditions are the same, but the inlet positions are different, so they have different in stage

pressure ratio.

Main Splitter Main Splitter

Figure 1 Modified structure of leading edge (left) and trailing edge (right)

Original Modified

Figure 2 Flow path of Krain

Figure 3 Performance of the original path (left) and modified path (b) impeller.


Results and Discussions The modified impeller is selected in this study. The design optimization is carried out at near

peak-efficiency operating point with its performance listed in Table 1. 242 design variables are used

to parameterize the blade shape perturbations, and 11 design variables are used to parameterize the

stacking line of the blade. The evolution of the entropy, isentropic efficiency and two constraints

with design cycles is showed in Figure 4. The reduction of entropy means the increasing of the is-

entropic efficiency. The isentropic efficiency is increased by 0.4%, with the total pressure ratio de-

creased by 0.2% and mass flow rate increased by 0.8% respectively.

The change of blade geometry is revealed in Figure 5. The profiles of main blade near un-

changed at tip region, but other spanwise move towards to left side of picture, especially at

mid-span. The profiles of splitter move towards to right side of picture at whole spanwise. The

change of blade geometry makes the flow field better. Figure 6 presents the contours of

blade-to-blade relative Mach number at 50% span and 90% span. As Figure 6 shows, the strength of

shock at leading edge region in main blade is reduced. At 90% span, the low speed region between

pressure surface of main blade and suction surface of splitter is decreased. Though the improvement

of flow field is not such significant, it still makes the isentropic efficiency increased by 0.4%.

The comparison is with the impeller global data, i.e. pressure ratio and efficiency vs. mass flow

rate, as shown in Figure 7. The comparison of the pressure ratio shows a very good agreement at

three different rotational speeds over the whole mass flow range. At the same time, the efficiency

has a 0.4% improvement by redesigned at three different rotational speeds. This shows that the re-

designed impeller has 0.4% improvement in modified flow path.

Considering the original flow path is much different to the modified one. The redesigned blade

was calculated in original flow path, the results are shown in Figure 8. From the comparison at

50,000 rpm, the efficiency has a nearly 0.4% improvement and the pressure ratio keeps unchanged.

This is consistent with the calculation in modified flow path. In other word, the optimization of the

impeller is effective, and the result is reliable.

Table 1 Performance comparison between the original and optimized blades

η initπ π initm m Original 0.877 1.000 1.000

Optimal 0.881 0.998 1.008


(a)

(b)

Figure 4 Evolution of the entropy, isentropic efficiency (a) and two constraints (b) with design cy-

cles

Figure 5 Comparison of blade geometry between original and optimal


Original Optimal

(a)

Original Optimal

(b)

Figure 6 Contours of relative Mach number on blade to blade: (a) at 50% span, (b) at 90% span


Figure 7 Efficiency and pressure ratio over mass flow rate for three rotational speeds

Figure 8 Efficiency and pressure ratio over mass flow rate in original flow path (50000rpm)

Conclusions In this paper, a constrained gradient evaluation is used for centrifugal impeller, and a continu-

ous adjoint method for the thin shear-layer N-S equations in a cylindrical-coordinate system is de-

veloped. With the adjoint-based gradient evaluation, perturbation-based shape representation, and

the steepest descent method, an efficient integrated shape optimization loop is constructed.

The proposed method is used to optimize the shape of the Krain impeller. The optimization is

able to raise the isentropic efficiency of the blade by 0.4% while maintaining the same mass flow

rate and total pressure ratio. Though the improvement of the isentropic efficiency is not such signif-

icant, it provides a new way to improve the centrifugal impeller.

Acknowledgments The authors would like to express their deep appreciations to the National Natural Science


Foundation of China for funding this work, Project No. 51176012.

Nomenclature Symbols

, ,E F G Axial, circumferential, and radial flux vectors

I Objective

N Residual of flow equations

nQ Normal flow flux vector

S∆ Net mass-averaged entropy production rate from the domain inlet to the domain outlet

,U U ′ Conservative and primitive flow variables

, ,u v w Axial, circumferential, and radial velocities

m Inlet and outlet average mass flow rate

, ,x r θ Cylindrical coordinates

, ,x rn n nθ Components of normal direction

α Design variable

1 2 3, ,σ σ σ Weighting factors

π Total pressure Ratio

λ Adjoint variable vector

γ Specific heat ratio

Superscripts and subscripts

( )*

Stagnation parameter ( ) Partial derivatives of flow terms to

design variables

( )init Baseline value ( ) ( ),

hub tip

Blade hub and tip

( ) Average value ( ) ( ),le te Leading and trailing edge

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Documents

Jia Y U, Lucheng JI, Weiwei LI*, Weilin Y I