Statistical evaluation of performance impact of flow variations for a transonic compressor rotor blade

Journal Pre-proof Statistical evaluation of performance impact of flow variations for a transonic compressor rotor blade Zhiheng Xia, Jiaqi Luo, Feng Liu PII:

S0360-5442(19)31980-2

DOI:

https://doi.org/10.1016/j.energy.2019.116285

Reference:

EGY 116285

To appear in:

Energy

Received Date: 23 April 2019 Revised Date:

31 August 2019

Accepted Date: 2 October 2019

Please cite this article as: Xia Z, Luo J, Liu F, Statistical evaluation of performance impact of flow variations for a transonic compressor rotor blade, Energy (2019), doi: https://doi.org/10.1016/ j.energy.2019.116285. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Statistical Evaluation of Performance Impact of Flow Variations for a Transonic Compressor Rotor Blade Zhiheng Xia

a,b

, Jiaqi Luo

a,1,*

, Feng Liu

c

a

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China Department of Aeronautics and Astronautics, Peking University, Beijing 100871, China c Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA92697-3975, United States b

Abstract The effects of flow variations to the aerodynamic performance of turbomachinery blades are considerable in the real world. Uncertainty quantification of aerodynamic performance is useful for evaluating the mean performance change, robust design, etc. The paper studies the performance impact of inlet and outlet flow variations for transonic compressor rotor blades using polynomial chaos. An adaptive sparse grid technique is employed to construct the model of adaptive non-intrusive polynomial chaos (ANIPC). Through statistical evaluation of performance changes for NASA Rotor 67, the response performance of ANIPC is firstly verified. Then the ANIPC is used to evaluate the changes of adiabatic efficiency and mass flow rate of Rotor 67 considering the variations of inlet total pressure and outlet back pressure at different operation conditions. The results reveal that the performance changes exhibit evident nonlinear dependence on the inlet and outlet pressure variations. Moreover, performance changes of the rotor blade in the whole operation range are evaluated and illustrated. Finally, by Monte Carlo simulation, the flow solutions along span and in the blade passage are statistically analyzed to demonstrate the impact mechanisms of inlet and outlet pressure variations to the performance changes. Keywords: uncertainty quantification, turbomachinery, non-intrusive polynomial chaos, adaptive sparse grid, flow variation, Monte Carlo simulation

*Corresponding author. 1

Email address: [email protected] (J. Luo)

1

Nomenclature CDF

cumulative distribution function

m

normalized mass flow rate

MCS

Monte Carlo simulation

N

number of sparse grids

NIPC

non-intrusive polynomial chaos

PDF

probability distribution function

P

pressure; order of polynomial chaos

Pb

UQ with outlet back pressure variation

Pi

UQ with inlet total pressure variation

PP

UQ with simultaneous pressure variations

STLI

shock/tip-leakage interaction

uj

coefficient of random basis function

UQ

uncertainty quantification

X

random process

Α

interpolation process

∆

interpolation difference, ∆i = Α i – Α i-1

εavr

mean of response error

εstd

standard deviation of response error

η

adiabatic efficiency

µ

statistical mean value

Ω

set of sparse grid points

Ψ

random basis function

σ

statistical standard deviation

θ

random event; total temperature ratio

Ξ

sparse grid points for Α

2

1. Introduction Traditionally, turbomachinery blades are designed at the specific operational conditions. However, in the real working environment the flow conditions at the inlet and outlet of turbomachinery blades cannot be always maintained as the nominal ones. For example, circumferential pressure distortion is commonly encountered at the inlet of multi-stage transonic compressor, the influence of which to the downstream flow and compressor surge margin have been investigated [1-3]. The high turbulence and spanwise non-uniformity at the outlet of upstream combustor can also bring about significant inlet flow condition variations to the downstream turbine [4,5]. Such kinds of flow variations at the inlet and outlet inevitably affect the aerodynamic performance of turbomachinery blades. In such cases, the effects of flow variations are necessary to be taken into account for evaluating the mean performance of turbomachinery blades in the real world, in which uncertainty quantification (UQ) of is one important module. Moreover, as robust design of turbomachinery blades has become more and more relevant to engineering community, studies on performance impact due to inlet and outlet flow variations have attracted more and more attentions. In last century, performance impact of turbomachinery blades was evaluated by the expensive and time-consuming experimental measurements [6-8]. At the beginning of this century, with the help of rapidly increased computing capabilities and high-performance numerical simulation, studies on performance impact for turbomachinery blades utilizing computational fluid dynamics (CFD) have become available [9-18]. After obtaining the numerical flow solutions, statistical evaluations of performance changes can then be accomplished by different UQ methods including the direct Monte Carlo simulation (MCS) [9], probabilistic collocation [10-13], polynomial chaos (PC) [14-17] and sensitivity-based method [18,19]. For example, Garzon et al. [9] successfully evaluated the performance change of manufacturing variability for turbomachinery blades using direct MCS and the converged

statistics

were

obtained

with

two

thousand

samples.

Obviously,

time-consuming CFD evaluations would be required if the direct MCS were employed in 3

the UQ studies involving a large number of uncertain parameters. Loeven et al. [11] investigated the performance impact of uncertain inlet total pressure profile for a transonic compressor rotor, NASA Rotor 37, by using the probabilistic collocation method. Pecnik et al. [15] studied the variations of boundary layer transition considering the turbulence inlet condition variations for a transonic turbine guide vane. Luo et al. [19] evaluated the performance impact of manufacturing variations for a turbine blade with sufficient accuracy and significantly reduced computational cost by the second order sensitivities. However, the sensitivity-based statistical evaluation method can only be applied to the small-scale UQ problems, such as performance impact of manufacturing variations. For the large-scale UQ problems, such as performance impact of considerable flow variations at the inlet and outlet of turbomachinery blades, PC methods perform better than the sensitivity-based method to evaluate the strong nonlinear dependent performance changes. PC involves a spectral representation for the uncertainty, which is decomposed into deterministic and stochastic components. Once the spectral representation is constructed with a series of training samples, function response can be obtained immediately and then MCS can be utilized to determine the statistics. Due to its easy implementation and reduced computational cost compared with direct MCS, PC methods have been widely used [14-17,20-24]. Generally, PC consists of intrusive polynomial chaos and non-intrusive polynomial chaos (NIPC). For the former, PC representation of interested outputs can be obtained directly by a one-time shot solution. However, the governing flow equations and the corresponding CFD codes have to be reformulated, which should be tiresome and time-consuming. On the contrary, no additional changes to governing flow equations are necessary for NIPC; moreover, the original CFD codes can be regarded as a black box [20]. Thus, NIPC can be used in UQ studies as a surrogate model, the function responses from which are used for MCS to obtain the statistics. Nowadays, there are some open literatures working on NIPC-based performance UQ for turbomachinery blades considering only the effects of inlet flow variations [11,15,21]. The present study will investigate the performance impact of flow variations at both the

4

inlet and outlet of a transonic compressor rotor blade, NASA Rotor 67, using NIPC. The effects of different kinds of flow variations to performance changes will be illustrated and compared in detail. Furthermore, mentioned that NIPC is used as a surrogate model in UQ, if the adaptive sampling technique can be used, the computational cost of UQ can be significantly reduced. Ghisu et al. [22] and Lucor et al. [23] have already introduced an adaptive NIPC (ANIPC) method through recursive approximation based on an iterative adaptation of polynomial basis to the solution. In the present study, an adaptive sparse grid technique will be employed to construct the model of ANIPC with as few training samples as possible. The deterministic sampling method along with Galerkin projection is used to calculate the model coefficients. ANIPC is firstly verified through statistical evaluations of performance changes for Rotor 67 at the operation condition near stall to demonstrate the response performance improvements. Then performance changes due to the variations of inlet total pressure and outlet back pressure for Rotor 67 are evaluated at different operation conditions and the results are exhibited in detail. Moreover, it is well known that the typical flow in blade passage including shock wave, flow separation, tip-leakage flow, etc., have dominant influence on the aerodynamic performance of transonic compressors. Statistical flow solutions obtained from direct MCS are finally presented to demonstrate the effects of pressure variations at the inlet and outlet to the typical flow in blade passage.

2. Flow Validation NASA Rotor 67 is a low-aspect-ratio transonic compressor rotor blade. It was designed and experimentally studied by Strazisar et al. [25]. It has 22 blades with the design pressure ratio of 1.63 and mass flow rate of 33.25 kg/s. The design rotational speed is 16,043 rpm with a tip speed of 429 m/s. The height of tip clearance is approximately 1.0 millimeter (mm). Figure 1 gives the configuration of Rotor 67. Steady-state Reynolds-averaged Navier-Stokes (RANS) equations are solved on a single-block H-type mesh by finite volume method and multi-grid acceleration technique

5

Figure 1: Configuration of NASA Rotor 67

[26]. The Spalart-Allmaras one-equation turbulence model [27] is also solved. At the inlet of Rotor 67, the given total temperature, total pressure and incidence flow angle are 288.15 K, 101,325 Pa and zero degree, respectively. At the outlet, the back pressure is given on the hub and the spanwise distribution of static pressure is determined by the radial equilibrium method. The fluid density and dynamic viscosity are 1.226 kg/m3 and 1.79E-05 N ⋅ s / m2 , respectively. Figure 2 illustrates the convergence histories of density,

ρ and eddy viscosity, µt /µl. The root mean square of the norm of density and the maximum eddy viscosity converge after about 500 and 150 iterations, respectively at the operation condition near peak efficiency.

Figure 2: Convergence history of flow computation at the operation condition near peak efficiency

At 100% design rotational speed of Rotor 67, the operation characteristic maps are obtained by adjusting the back pressure. Figure 3 illustrates the operation characteristics 6

Figure 3: Operation characteristics of Rotor 67: a) adiabatic efficiency; b) total pressure ratio

including adiabatic efficiency η and total pressure ratio π, which are compared with the experimental ones. The mass flow rate is normalized by the choked one. It can be found that the CFD solutions match well with the experiment ones. The two red solid points in the figures correspond to the operation conditions, near peak efficiency and near stall, involved in the following UQ studies. To further verify the CFD results for the whole operation ranges, Figure 4 and Figure 5 illustrate the outlet spanwise distributions of circumferentially mass-averaged flow turning β, total pressure ratio π and total temperature ratio θ at the operation conditions with 98.5% and 93% normalized mass flow rate, respectively. At the former operation condition with 98.5% normalized mass flow rate, the CFD-based total temperature ratio o

n

Figure 4: Spanwise distributions at the operation condition with 98.5% normalized mass flow rate: a) 7

flow turning; b) total pressure ratio; c) total temperature ratio

Figure 5: Spanwise distributions at the operation condition with 93% normalized mass flow rate: a) flow turning; b) total pressure ratio; c) total temperature ratio

only the outer spans is over predicted, resulting in decreased adiabatic efficiency. At the latter operation condition, the CFD-based total pressure ratio on only the outer spans is lower. Although the deviations on the outer spans is visible, the CFD-based spanwise distributions are close enough to the experimental ones.

3. Uncertainty Quantification of Flow Variations To precisely describe the uncertain flow variations at inlet and outlet, one has to take a large number of measurements, which is time-consuming and costly. In academic research, researchers usually focus on developing efficient algorithms to propagate flow uncertainties by simply assuming that flow variations meet one given distribution, such as Gaussian, uniform and other ones [10,11,21,28]. Once the description of flow variations is given, the proposed algorithms can be used to propagate uncertainties by the same UQ procedure. In the present study, a simple but widely used zero-mean Gaussian distribution is used to describe the pressure variations at inlet and outlet of Rotor 67.  1 ξ2 exp(− ), ξ ∈ [ − E , E ]  f (ξ ) =  2π 2  0, otherwise 

where the random variable is defined as

8

(1)

ξ=

p − pnomial

(2)

σp

where E=1.0, p and pnomial are the generic perturbed and reference pressure, respectively,

σp is the standard deviation of generic pressure variation. It should be clarified that the variations of inlet total pressure and outlet back pressure are assumed to be non-correlated random variables in the study.

4. Adaptive Non-Intrusive Polynomial Chaos 4.1

Adaptive Sparse Grid The Smoljak formula proposed by Smoljak [29] can be written as i

å

A (q, d )( f ) =

i

(D 1 Ä L Ä D d )

(3)

i£q

where i = i1 + L + id with d indicating the dimension, D is the interpolation difference calculated by D i = U i - U i- 1 with U representing univariate interpolation formula. Eq. (3) can also be written in another form for the k-level sparse grid [30,31]. A (q, d )( f ) =

q- i

å

(- 1)

k + 1£ i £ q

æd - 1 ÷ ö ç ÷×( U i1 Ä L Ä U id )( f ) ×çç ÷ çèq - i ÷ ÷ ø

(4)

where k = i - d . One only needs to evaluate the function value on the sparse grid

U k + 1£ i £ q

(X i ´ L ´ X i ) 1

(5)

d

where X i = {x 1i , L , x mi } denotes the grid points for U i . Instead of the exponential degrees i

of freedom O (N d ) of conventional approaches, only O (N ×(log N )d - 1 ) dependence gets involved for the discretization on sparse grids, which overcomes the curse of dimensionality to some extent [32]. N is the number of points used in each dimension. Besides, in the maximum-norm sense the sparse grid interpolation accuracy is O (N - 2 ×(log N )d - 1 ) .

9

To construct the adaptive sparse grid, a new variable named hierarchical surplus (HS) has to be defined first [33,34]. w kj ,i = f (x ij ) - A (k + d - 1, d )(x ji )

(6)

where j represents the number of basis nodes on the k-th grid level. w kj tends to zero for continuous functions when k increases to infinity. However, if the function has steep gradients, w kj should be large. Therefore, HS can be used as a suitable candidate for error estimation and tolerance control in the adaptive sparse grid. Following is the procedure for the adaptive sparse grid. Step 1: Define the maximum grid level, kmax; produce a number of initial nodes on the grid level k=1. Step 2: Produce a number of actnum nodes on the grid level k=2, which are stored in Ωa. Step 3: If k0, move all the nodes in Ωa to Ωo and calculate HS for each element in Ωo; or else end the interpolation process. Step 4: If HS is larger than the error tolerance ε, produce 2d additional nodes, which are then stored in Ωa. Step 5: k=k+1, go to Step 3. The additional nodes added in Step 4 are closest to the corresponding basis nodes and located on the grid level k+1. It should be noticed that no repetitive nodes appear in the function evaluation process. Moreover, to keep the adaptive sparse grid balanced, the nodes whose HS are lower than tolerance are maintained. 4.2

Adaptive Non-Intrusive Polynomial Chaos A general second-order random process X(θ), described as a function of a random

event θ, can be given as following [35]. ∞

X (θ ) = ∑ u j Ψ j (ξ (θ ))

(7)

j =0

where Ψj(ξ) is the random basis function corresponding to the j-th mode, the coefficients 10

uj are the amplitude of the j-th fluctuation. Different random distributions correspond to different polynomial chaos [35]. As the pressure variations are assumed to meet Gaussian distribution, Hermite polynomials are used as bases functions in the study. In practice, X(θ) is usually described in a summation form of p-order basis functions by the truncation theory. P

X (θ ) ≈ ∑ u j Ψ j (ξ ), j =0

P=

(n + p)! −1 n! p !

(8)

If the random process X and random variable ξ share the same density, exponential convergence of Eq. (8) versus p can be achieved [39]. Since the PC forms a complete orthogonal basis in the L2 space of Gaussian random variables, the coefficients uj can be determined by projecting the random process onto the PC basis. uj =

XΨj Ψ 2j

(9)

In the study, the Galerkin projection combined with the adaptive sparse grid is used to calculate the coefficients. Once the model of adaptive NIPC is constructed, the statistics can be obtained rapidly by MCS. 4.3

UQ Validation In order to investigate the response performance including model construction

efficiency and response accuracy of ANIPC, adiabatic efficiency change at the operation condition near stall of Rotor 67 is statistically evaluated. The results are compared with those obtained from direct MCS and static NIPC (SNIPC). The direct MCS is used to determine the statistics with a number of three thousand samples, which are also used as test samples for NIPC. By using NIPC, adiabatic efficiency can be calculated by p

p

η = ∑∑ aij H i (ξ1 ) H j (ξ 2 )

(10)

i =0 j =0

where p=8, Hi and Hj are Hermite polynomial chaos. The error tolerance for ANIPC is 1.0E-04. The inlet total pressure and outlet back pressure are perturbed simultaneously with σp = 0.0075 and E=1.0. The mean εvar and standard deviation εstd of response error 11

for NIPC

are defined as ε avr = ε

2 std

1 M

M

∑

f NIPC − f CFD

i =1

1 M 2 = ( f NIPC − fCFD ) ∑ M − 1 i =1

(11)

where fNIPC and fCFD represent the performance parameters predicted by NIPC and direct flow computation, respectively; M is the number of test samples and it is three thousand in the study. For the present study on UQ validation, the MCS statistics of adiabatic efficiency change are regarded as the ones closest to the exact ones, with which the results obtained from NIPC are compared. Figure 6 gives the convergence history of MCS-based statistics. It can be found that the statistic mean and standard deviation of adiabatic efficiency change already converge with about one thousand and five hundred samples.

Figure 6: Convergence history of MCS-based statistics of adiabatic efficiency change

The statistics of absolute change in percentage of adiabatic efficiency, response error 12

and the number of training samples for NIPC are given in Table1. N is the number of training samples for NIPC, which is also the number of sparse grids. µ, σ and skew represent the statistic mean, standard deviation and skewness of adiabatic efficiency change, respectively.

Table 1: Statistics of adiabatic efficiency change and response error of NIPC

µη/% ση skewη

εvar εstd N

MCS -0.108 7.42E-03 -0.628 3000

ANIPC -0.108 7.41E-03 -0.632 1.89E-05 2.24E-05 81

SNIPC -0.112 7.44E-03 -0.601 2.16E-05 1.69E-05 105

The definition of skewness is given as 1 skew = M

 δηi − µ  ∑  σ  i =1  M

3

(12)

where δηi denotes the adiabatic efficiency change of the i-th test sample. Compared with the direct MCS, the necessary CFD evaluations for NIPC are significantly reduced, resulting in reduced computational cost for NIPC-based UQ. Besides, compared with SNIPC, the statistics of ANIPC are much closer to MCS ones. Moreover, ANIPC requires less CFD evaluations than SNIPC. The results demonstrate that UQ accuracy and UQ efficiency can be improved by ANIPC. In the following, the results obtained from NIPC are the ones obtained from ANIPC exactly. Probability distribution function (PDF) and cumulative distribution function (CDF) are widely used for in-depth statistical analysis. Figure 7 illustrates PDFs and CDFs of adiabatic efficiency change for both direct MCS and NIPC. The PDF and CDF of NIPC are almost the duplicates of MCS ones, further demonstrating the effectiveness of NIPC for the present UQ study. Moreover, the significantly left skewed distributions of probability density demonstrate strong nonlinear dependence of adiabatic efficiency change on the variations of inlet total pressure and outlet back pressure. 13

5. Statistical Evaluation As the operation condition of Rotor 67 changes, the typical flow in the blade passage exhibits dramatic difference. The effects of pressure variations at the inlet and outlet to aerodynamic performance changes require to be investigated at different operation

Figure 7: PDFs and CDFs for adiabatic efficiency change: (a) PDF of MCS; (b) PDF of adaptive NIPC; (c) CDFs

conditions. The corresponding performance impact on adiabatic efficiency and mass flow rate changes requires to be evaluated. In this section, performance changes at two different operation conditions, near peak efficiency and near stall of Rotor 67, are evaluated using NIPC. The separated variation of inlet total pressure, outlet back pressure and the simultaneous variations of the two are taken into account. σp = 0.0075 and E = 1.0 for all the UQ studies. Three thousand samples are used to calculate the response errors of NIPC, the tolerance of which is 1.0E-04.

14

5.1

Near Peak Efficiency At the operation condition near peak efficiency, NIPC-based statistics of adiabatic

efficiency change and mass flow rate change are shown in Table 2 and Table 3, respectively. In the tables, N and p are the number of sparse grids (training samples) and the order of PCs, respectively, Pi, Pb and PP correspond to the UQ studies with inlet total pressure variation, Table 2: Statistics of adiabatic efficiency change near peak efficiency

µη/% ση skewη

εvar εstd Ν p

Pi -1.26E-02 1.00E-03 -0.252 6.99E-06 7.76E-06 7 5

Pb -1.46E-02 1.03E-03 -0.199 3.94E-06 4.75E-06 7 5

PP -2.19E-02 1.38E-03 -0.664 1.00E-05 1.15E-05 36 5

Table 3: Statistics of mass flow rate change near peak efficiency

µη/% ση skewη

εvar εstd Ν p

Pi -1.96E-02 6.17E-03 -0.0744 2.73E-06 3.34E-06 15 5

Pb -1.14E-02 2.04E-03 -0.295 3.18E-06 3.52E-06 13 5

PP -4.74E-02 6.58E-03 -0.198 4.22E-06 4.78E-06 86 5

outlet back pressure variation and the simultaneous variations, respectively. The response errors are also given in the tables. Figure 8 illustrates the PDFs and CDFs for the changes of adiabatic efficiency and mass flow rate. From Table 2 it is known that for the UQ studies Pi and Pb, the statistic means and standard deviations of adiabatic efficiency change are small and close to each other. As shown in Figure 8(b), the CDFs for these two cases are almost overlapped, indicating that the separate variation of either inlet total pressure or outlet back pressure results in similar distribution of adiabatic efficiency change. The non-zero skew as shown 15

in Table 2 demonstrates that the adiabatic efficiency change exhibits nonlinear dependence on the separate pressure variation. For the UQ study PP, all the statistics as shown in Table 2 are significantly larger than those of Pi and Pb, demonstrating stronger nonlinear dependence of adiabatic efficiency change on the simultaneous pressure variations, which can also be confirmed by the dramatically left-skewed PDF as shown by Figure 8(a).

Figure 8: PDFs and CDFs at the operation condition near peak efficiency: (a) PDF of adiabatic efficiency change for UQ study PP; (b) CDF of adiabatic efficiency change for all UQ studies; (c) PDF of mass flow rate change for UQ study PP; (d) CDF of mass flow rate change for all UQ studies

From the statistics given in Table 3 and CDFs illustrated in Figure 8(d), it can be found that mass flow rate change for UQ studies Pi and Pb are quite different and it is more sensitive to the inlet total pressure variation. Furthermore, compared with adiabatic efficiency change, mass flow rate change is more sensitive to pressure variations because the standard deviations as shown in Table 3 are larger than those given in Table 2 for all UQ studies. From Table 2, Table 3, Figure 8(a) and Figure 8(c) it can be found that the PDF of mass flow rate change is less left-skewed compared with that of adiabatic efficiency change. The results demonstrate that the nonlinear dependence of mass flow 16

rate change is much weaker on the simultaneous variations of inlet total pressure and outlet back pressure, although it is more sensitive to pressure variations. 5.2

Near Stall As the operation condition of transonic compressor rotor approaches stall from peak

efficiency, the blade loading increases and thus the flow in blade passage is more complicated [38,39]. Performance impact at the operation condition near stall is investigated and compared with that at the operation condition near peak efficiency in the following. Statistics of adiabatic efficiency change and mass flow rate change at the operation condition near stall of Rotor 67 are shown in Table 4 and Table 5, respectively. Similar to the results given in Table 2 and Table 3, the statistics of performance changes for the UQ studies Pi and Pb are much less than those of PP. Besides, compared with the statistics at the operation condition near peak efficiency, the statistic means and standard deviations of performance changes near stall are significantly increased, demonstrating that the performance changes at the operation condition near stall are more sensitive to the variations of inlet total pressure and outlet back pressure.

Moreover, compared with

Table 2 and Table 3, increased numbers of sparse grids and higher order PCs are necessary for constructing the models of NIPC at the operation condition near stall. Since adiabatic efficiency and mass flow rate exhibit larger changes due to pressure variations at the inlet and outlet at the operation condition near stall, it is reasonable that more training samples

Table 4: Statistics of adiabatic efficiency change near stall

µη/% ση skewη

εvar εstd Ν p

Pi -9.00E-02 5.63E-03 -0.279 1.06E-05 1.14E-05 13 8

Pb -4.67E-02 5.15E-03 -0.229 9.50E-06 1.02E-05 13 8

17

PP -1.10E-01 7.42E-03 -0.632 1.89E-05 2.24E-05 81 8

Table 5: Statistics of mass flow rate change near stall

µη/% ση skewη

εvar εstd Ν p

Pi -7.67E-02 1.50E-02 -0.157 3.15E-06 4.09E-06 31 8

Pb -8.10E-02 1.05E-02 -0.124 4.63E-06 5.95E-06 25 8

PP -1.58E-01 1.79E-02 -0.388 6.64E-06 8.84E-06 203 8

are needed to construct the models of NIPC. Figure 9 illustrates the PDFs and CDFs for the changes of adiabatic efficiency and mass flow rate at the operation condition near stall. Generally, the differences between the CDFs of Pi, Pb and PP are quite similar to those at the operation condition near peak efficiency. Besides, the left-skewed PDFs demonstrate nonlinear dependence of performance changes on the simultaneous pressure variations. Moreover, compared with Figure 8, the maximum changes of both adiabatic efficiency and mass flow rate at the operation condition near stall as shown in Figure 9 are significantly increased. It is conceivable because at the operation condition near stall, the blade loading and adverse pressure gradient of transonic compressor rotor blade increases [38,39]. In such situations, pressure variations at the inlet and outlet result in more changes to adiabatic efficiency and mass flow rate.

18

Figure 9: PDFs and CDFs at the operation condition near stall: (a) PDF of adiabatic efficiency change for UQ study PP; (b) CDF of adiabatic efficiency change for all UQ studies; (c) PDF of mass flow rate change for UQ study PP; (d) CDF of mass flow rate change for all UQ studies

5.3

Whole Operation Range In order to investigate the performance changes in the whole operation range, UQ

studies are further carried out at a series of different operation conditions considering the simultaneous variations of inlet total pressure and outlet back pressure. Figure 10 compares the nominal and perturbed operation characteristics of adiabatic efficiency. Moreover, the maximum deviations of adiabatic efficiency and mass flow rate away from the means are also plotted in Figure 10(a) and Figure 10(b), respectively. It should be noticed that the maximum deviation is inherently the statistic standard deviation of performance change. In Figure 10(a), the mean changes of both adiabatic efficiency and mass flow rate decrease as the operation condition approaches choke, while they increase as the operation condition approaches stall. Moreover, the maximum deviation of adiabatic efficiency increases as the operation condition approaches either choke or stall from peak efficiency. The results demonstrate that at the operation conditions near peak efficiency, the adiabatic efficiency change is the least sensitive to the simultaneous pressure variations. In Figure 10(b), the maximum deviation of mass flow rate increases as the operation condition approaches stall from choke, demonstrate that the mass flow rate c h a n g e

i s

m o r e

a n d

m o r e

19

s e n s i t i v e

t o

t h e

Figure 10: Statistic mean values and standard deviations of performance changes on the operation characteristic of adiabatic efficiency: (a) adiabatic efficiency change; (b) mass flow rate change

simultaneous pressure variations as the outlet mass flow rate decreases. In order to further illustrate the increased sensitivity of performance changes to the simultaneous pressure variations as the operation condition approaches stall from peak efficiency, Figure 11 gives two-dimensional PDF contours of both adiabatic efficiency and mass flow rate changes at three different operation conditions. In the figures, the black and red solid points represent the nominal and statistical performance, respectively. Obviously, the performance parameters vary in a wider range when the operation condition is close to stall. At the operation condition with 95.3% choke mass flow rate, the normalized mass flow rate in a considerable range is below 92%, which is already lower than the stall one [25]. Hereby, if the transonic compressor works at the operation condition near stall, the pressure variations at the inlet and outlet have disastrous effects to the aerodynamic performance. Generally, performance changes of Rotor 67 are nonlinear dependent on the pressure variations and the nonlinear dependence of adiabatic efficiency change is much stronger

20

Figure 11: Two-dimensional PDF contours of performance changes at different normalized mass flow rate: (a) m = 0.985; (b) m = 0.966; (c) m = 0.953

than that of mass flow rate change. When considering the effects of inlet total pressure and outlet back pressure variations simultaneously, performance changes increased dramatically. As the operation condition approaches stall from peak efficiency, performance changes become more and more sensitive to the pressure variations and exhibit more and more stronger nonlinear dependence on the pressure variations.

6. Statistical Analysis of Flow Solutions In previous section, the performance changes under the influence of pressure variations are quantified by the NIPC method. However, how the pressure variations effect the performance is still not clear up to now. In the following, in order to explore the impact mechanisms of pressure variations on the performance changes of Rotor 67, three thousand flow solutions are statistically analyzed by using the direct MCS. Since the nonlinear dependence of performance changes on the simultaneous pressure variations is much stronger, only the statistical flow solutions for the UQ study PP are illustrated. 21

At the operation conditions near peak efficiency and near stall, the contours of nominal pressure ratio, statistical mean and standard deviation of pressure ratio change on the suction side are presented in Figure 12 and Figure 13, respectively. The pressure ratio on the suction side is defined as the ratio between static pressure and inlet total pressure. The pressure ratio change is calculated by the difference between the perturbed and n

o

m

i

n

a

l

p

r

e

s

s

u

r

e

Figure 12: Contours on the suction side near peak efficiency: (a) nominal pressure ratio: (b) statistic mean of pressure change; (c) statistic standard deviation of pressure change

Figure 13: Contours on the suction side near stall: (a) nominal pressure ratio: (b) statistic mean of pressure change; (c) statistic standard deviation of pressure change

ratios. As shown in Figure 12(a) and Figure 13(a), the pressure ratio on the outer spans abruptly increases near the trailing edge and near middle chord, respectively. The abrupt pressure increase is attributed to the shock wave, which has already been verified in previous study [39]. From Figure 12(b) and Figure 12(c) it is known that in the transonic 22

flow region where shock wave appears, the statistical pressure ratio exhibits significantly larger change and the standard deviation is also dramatically larger. Similar distributions of statistics of pressure ratio change can be found at the operation condition near stall, as shown in Figure 13(b) and Figure 13(c). All the results demonstrate that the flow solutions in the transonic flow region are more sensitive to the pressure variations at the inlet and outlet. Moreover, the flow solutions at the operation condition near stall are more sensitive to the pressure variations. The effects of pressure variations to the shock wave have already been illustrated by Figure 12 and Figure 13. As illustrated in previous work [39], shock/tip-leakage interaction (STLI) is one of the main stall triggers in transonic compressor rotors and the severe downstream flow diffusion significantly deteriorating the aerodynamic performance. The effects of pressure variations to STLI also require to be investigated. Figure 14 and Figure 15 present the contours of nominal relative Mach number and the statistics of relative Mach number change on the blade-to-blade streamsurfaces at three different spans, 100%, 98% and 90% blade height at the operation conditions near peak e

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Figure 14: Contours on the blade-to-blade streamsurface near peak efficiency: (a) nominal relative Mach number; (b) statistic mean of relative Mach number change; (c) statistic standard deviation of relative Mach number change

23

Figure 15: Contours on the blade-to-blade streamsurface near stall: (a) nominal relative Mach number; (b) statistic mean of relative Mach number change; (c) statistic standard deviation of relative Mach number change

respectively. At the operation condition near peak efficiency, the arresting statistic mean and standard deviation of relative Mach number change on the streamsurface at 100% span indicate that the pressure variations at the inlet and outlet have significant influence on STLI. Moreover, since the blade loading increases at the operation condition near stall, tip-leakage flow and thus STLI are more intensive. Even on the streamsurface at 98% span, the statistic mean and standard deviation of relative Mach number change are significant, as shown in Figure 15(b) and Figure 15(c). All the results demonstrate that the flow in the tip gap and on the spans near blade tip are also sensitive to the pressure variations at the inlet and outlet. Figure 16 presents the circumferentially mass-averaged statistic mean and standard deviation of adiabatic efficiency change along span. It can be found that on the inner spans, the statistic means and standard deviation of adiabatic efficiency change exhibit slight variations along span. Moreover, the statistics on the inner spans are much lower than those on the outer spans. Such kinds of variations of statistics along span are not strange because the results illustrated in Figure 12 and Figure 13 have demonstrated that the flow is subsonic on the inner spans and transonic on the outer spans and the transonic flow on the outer spans is more sensitive to the pressure variations at the inlet and outlet. Generally, 24

larger positive pressure ratio change results in more shock loss and thus more adiabatic efficiency degradation, while larger standard deviation of pressure ratio change results in increased standard deviation of adiabatic efficiency change.

Figure 16: Spanwise distributions of statistics of adiabatic efficiency change

However, from Figure 16 it is found that the adiabatic efficiency drop at blade tip is not the largest, which is supposed to be the results of STLI. From middle span to 98% span, the shock wave is more and more stronger, resulting in increasing adiabatic efficiency drop along span. On the spans near blade tip, STLI induces significant flow diffusion. In such situations, the nonlinear dependence of adiabatic efficiency change on the pressure variations at the inlet and outlet is subsequently weakened. Meanwhile, the statistic mean of adiabatic efficiency change near the hub exhibits an opposite variation at the operation condition near stall, compared with that at the operation condition near peak efficiency. One possible reason is that as the back pressure increases near stall, the corner flow separation which almost vanishes at the operation condition near peak efficiency has significant influence on the aerodynamic performance, which requires further investigation. Most of the aforementioned results are of the UQ study PP considering the effects of simultaneous pressure variations. The separate contributions of either inlet total pressure variation or outlet back pressure variation to the performance changes require to be further 25

discovered. Figure 17 illustrates the spanwise distributions of circumferentially mass-averaged statistic mean and standard deviation of adiabatic efficiency change for all the three UQ studies at the operation condition near stall. Besides, the distributions with the index Linear as shown in the figures are the ones determined by simply summating the corresponding distributions of Pi and Pb. Notice that the standard deviations of pressure variations are the same, σp=0.0075, the comparisons as shown in the figures demonstrate that adiabatic efficiency change is more sensitive to the variation of inlet total pressure since the statistic mean and standard deviation of Pi are all larger than those of Pb. M

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Figure 17: Spanwise statistics of adiabatic efficiency change for different UQ studies

through comparing the distributions of PP and Linear, it can be found that PP contributes more adiabatic efficiency drop than the linear summation of Pi and Pb, whereas the standard deviation of PP is significantly reduced. The results demonstrate that the effects of inlet total pressure variation and outlet back pressure variation to the adiabatic efficiency change are not linearly coupled. Generally, the transonic flow on the suction side of Rotor 67 and the tip-leakage flow are sensitive to the pressure variations, which result in most of the performance changes. Besides, the nonlinear dependence on the pressure variations is much stronger for STLI. 26

However, on the spans near blade tip, the effects of pressure variations to performance changes are weakened by STLI. Moreover, from the spanwise distribution of statistics of adiabatic efficiency change, it is found that the inlet total pressure variation is not linearly coupled with the outlet back pressure variation. On the outer spans, these two kinds of pressure variations are obviously nonlinearly coupled.

Conclusion Uncertainty quantifications of adiabatic efficiency change and mass flow rate change at different operation conditions for the transonic compressor rotor blade, NASA Rotor 67, using an ANIPC method are presented. The variations of inlet total pressure and outlet back pressure are assumed to meet the Gaussian distribution. An adaptive sparse grid technique is employed to produce the samples. Compared with the direct MCS, the number of CFD computations for NIPC is reduced to about one tenth. Moreover, compared with SNIPC, the statistics obtained from ANIPC are more accurate, while with less training samples. The improvements on response performance including response accuracy and model construction efficiency of the proposed ANIPC are verified. The performance changes nonlinearly depend on the separate and simultaneous pressure variations. Compared with individual pressure variation, the coupled pressure variations result in more performance changes. Moreover, at different operating conditions, the sensitivity of performance changes to the simultaneous pressure variations varies. It is found that as the operation condition approaches stall from peak efficiency, the changes of adiabatic efficiency and mass flow rate are increased. The changes of adiabatic efficiency and mass flow rate at the operation condition near stall is increased by more than three times compared with those at the operation condition near peak efficiency. Furthermore, a series of flow solutions are statistically analyzed to explore the impact mechanisms of pressure variations. It is found that the typical flow structures including shock wave, tip-leakage flow and shock/tip-leakage interaction, are susceptible to the pressure variations, which account for the main performance changes. 27

Acknowledgements The authors would like to thank The National Natural Science Foundation of China (Grant Nos. 51676003, 51976183) and the Fundamental Research Funds for the Central Universities of China (Grant No. 2019QNA4058) for supporting the research work. References [1] Stenning A. Inlet distortion effects in axial compressors. Journal of Fluids Engineering, 1980, 102(1): 7-13. [2] Bry P, Laval P, Billet G. Distorted flow field in compressor inlet channels. Journal of Engineering for Gas Turbines and Power, 1985, 107(3): 782-791. [3] Peters T, Burgener T, Fottner L. Effects of rotating inlet distortion on a 5-stage HP-compressor, In: ASME Turbo Expo 2009: Power for Land, Sea, and Air, June 4–7, 2001, New Orleans, Louisiana, USA; 2001: GT2001-0300. [4] Butler TL, Sharma OP, Joslny HD, Dring RP. Redistribution of an inlet temperature distortion in an axial flow turbine stage. Journal of Propulsion and Power, 1989, 5(1): 64-71. [5] Dorney DJ, Gundy-Burlet KL, Sondak DL. A survey of hot streak experiments and simulations. International Journal of Turbo and Jet-engines, 1999, 16(1): 1-15. [6] Bammert K, Sandstede H. Influence of manufacturing tolerances and surface roughness of blades on the performance of turbines. Journal of Engineering for Gas Turbines and Power, 1976 98(1): 29-36. [7] Roelke RJ, Haas JE. The effects of rotor blade thickness and surface finish on the performance of a small axial flow turbine. Journal of Engineering for Gas Turbines and Power, 1983 105(2): 377-382. [8] Suder KL, Chima RV, Strazisar AJ, Roberts WB. The effects of adding roughness and thickness to a transonic axial compressor rotor. Journal of Turbomachinery, 1995, 117(4): 491-505. [9] Garzon VE, Darmofal DL. Impact of geometric variability on axial compressor performance. Journal of Turbomachinery, 2003 125(4): 692-703. [10] Loeven GJA, Witteveen JAS, Bijl H. Probabilistic collocation: an efficient non-intrusive approach for arbitrayily distributed parametric uncertainties. In: AIAA Aerospace Sciences Meeting and Exhibit, 8-11 January 2007, Reno, Nevada, USA; 2007: AIAA 2007-317. [11] Loeven GJA, Witteveen JAS, Bijl H. The application of the probabilistic collocation method to a transonic axial flow compressor. In; AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 12-15 April 2010, Orlando, USA; 2010: AIAA 2010-2923. [12] Liu Z, Wang X, Kang S. Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations. Energy, 2014, 73: 126-136. [13] Rahimi-Gorji M, Pourmehran O, Gorji-Bandpy M, Ganji DD. An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by collocation method. Ain Shams Engineering Journal, 2015, 6(2): 531-540. [14] Xiu D, Karniadakis GE. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 2003, 187: 137-167. 28

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Highlights

An adaptive sparse grid based NIPC method is proposed for uncertainty quantification.

By the adaptive-NIPC-based method, computational cost is significantly reduced.

Performance changes increase as the operation condition approaches stall.

Uncertainty effects to the flow make the most contribution to performance changes.

The uncertainty quantification method potentially paves the way for robust design.