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Research Papers

# Turbomachinery Active Subspace Performance MapsOPEN ACCESS

[+] Author and Article Information

Department of Engineering,
University of Cambridge,
Cambridge CB2 1PZ, UK
e-mail: ps583@cam.ac.uk

Shahrokh Shahpar

CFD Methods,
Rolls-Royce plc.,
Derby DE24 8BJ, UK

Paul Constantine

Department of Computer Science,
Boulder, CO 80309

Geoffrey Parks

Department of Engineering,
University of Cambridge,
Cambridge CB2 1PZ, UK

Fan and Compressor Subsystems,
Rolls-Royce plc.,
Derby DE24 8BJ, UK

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received October 21, 2017; final manuscript received December 27, 2017; published online January 17, 2018. Editor: Kenneth Hall.

J. Turbomach 140(4), 041003 (Jan 17, 2018) (11 pages) Paper No: TURBO-17-1193; doi: 10.1115/1.4038839 History: Received October 21, 2017; Revised December 27, 2017

## Abstract

Turbomachinery active subspace performance maps are two-dimensional (2D) contour plots that illustrate the variation of key flow performance metrics with different blade designs. While such maps are easy to construct for design parameterizations with two variables, in this paper, maps will be generated for a fan blade with twenty-five design variables. Turbomachinery active subspace performance maps combine active subspaces—a new set of ideas for dimension reduction—with fundamental turbomachinery aerodynamics and design spaces. In this paper, contours of (i) cruise efficiency, (ii) cruise pressure ratio (PR), (iii) maximum climb flow capacity, and (iv) sensitivity to manufacturing variations are plotted as objectives for the fan. These maps are then used to infer pedigree design rules: how best to increase fan efficiency; how best to desensitize blade aerodynamics to the impact of manufacturing variations? In the present study, the former required both a reduction in PR and flow capacity—leading to a reduction of the strength of the leading edge bow wave—while the latter required strictly a reduction in flow capacity. While such pedigree rules can be obtained from first principles, in this paper, these rules are derived from the active subspaces. This facilitates a more detailed quantification of the aerodynamic trade-offs. Thus, instead of simply stating that a particular design is more sensitive to manufacturing variations; or that it lies on a hypothetical “efficiency cliff,” this paper seeks to visualize, quantify, and make precise such notions of turbomachinery design.

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## Introduction

In this paper, turbomachinery active subspace performance maps are introduced as a new tool for blade design. These maps plot contours of efficiency, pressure ratio (PR), flow capacity, and sensitivity to manufacturing variations as objectives with respect to different blade designs; permitting an ameliorated exploration of the design space and complementing subsequent design optimization endeavors. Generating contour maps when the turbomachinery design space comprises solely of two design variables is intuitive and has been documented in the literature [1]. However, in this paper, such maps will be generated for a fan blade (see Fig. 1) with twenty-five design variables. To achieve this goal, active subspaces are utilized.

Active subspaces are a set of ideas that facilitate subspace-based dimension reduction for computational parameter studies. Using point-wise evaluations of a computational model, these ideas are used to determine dominant directions along which a chosen quantity of interest varies the most. It is useful to interpret active subspaces as an output-based principal component analysis. In that context, it is different from principal component analysis, which does not associate variability in the inputs to the corresponding model outputs. It is also different from functional analysis of variation decompositions [2] and the closely related Sobol' indices [3], which Constantine [4] classifies as methods for subset selection rather than subspace identification. Instead of determining which parameters are important, which Sobol' indices do by ranking them based on parameter-to-output variance, in subspace identification the objective is to identify which linear combination of all inputs drives the output. Active subspaces have previously been applied to investigate aerodynamic supersonic design [5], uncertainty quantification in hypersonics [6] and to study solar cell models [7], where dominant directions of quantities of interest in high-dimensional input parameter spaces were sought.

In this paper, active subspaces will be used to study the aerodynamics of a fan blade. Through this study, this paper will demonstrate how the mathematical definitions of the dominant directions—obtained via two active subspace strategies—are directly related to fundamental turbomachinery design principles. Then, this paper will illustrate how one may condense a multidimensional turbomachinery design space to a single two-dimensional (2D) contour map using these dominant directions. This facilitates a quantification of the trade-offs across various designs. In addition to flow quantities, this map will also convey the significance of manufacturing variations on the design process. Instead of simply stating that a particular design is more sensitive to manufacturing variations; or that it lies on a hypothetical efficiency cliff, this paper seeks to visualize, quantify, and make precise such notions of turbomachinery design.

The remainder of this paper is structured as follows: After a presentation of the underlying numerical tools used in this study in Sec. 2, Sec. 3 computes and explores the active subspaces for (i) maximum climb flow capacity, (ii) cruise efficiency, and (iii) cruise pressure ratio. These subspaces reveal the underlying physics governing these three performance metrics. In Sec. 4, these subspaces are exploited to generate a single map on which all objectives can be visualized. Finally, in Sec. 5, active subspace is computed using simulations on the manufactured geometry of the fan blade and the results are plotted on the same 2D contour map.

## Methods

###### Computational Tools.

All CFD results reported in this paper were obtained using the Rolls-Royce in-house flow solver, Hydra [9]. Hydra is an unstructured solver that uses a five-stage Runge–Kutta scheme with a block Jacobi preconditioner for steady-state Reynolds-averaged Navier–Stokes computations. In this paper, the Spalart–Almaras turbulence closure with wall functions is used within the flow solver. The computational domain for the CFD is a single fan blade with a downstream splitter (shown in Fig. 1(b)). Nonreflective exit capacity boundary conditions were imposed at both the engine sector stators (ESS) and the outlet guide vane (OGV) exits, and a standard total pressure, total temperature inlet profile was prescribed at the inlet. The fan rotor, hub, casing and the splitter utilized viscous wall boundary conditions, with the former two defined in the rotating frame. The entire domain was meshed using the Rolls-Royce geometry and meshing tool PADRAM [10]. The mesh is composed of downstream and upstream H blocks, upper and lower H blocks for the periodic boundaries, an O-mesh around the blade, and a C-mesh for the downstream splitter. In total, 1.75 × 106 cells were used after a mesh convergence study; the distribution of cells in the radial direction was found to be one of the important parameters in reducing the total temperature variation. A total of 129 points were used in the radial direction with increased clustering at the hub, tip, and those blade sections adjacent to the splitter. CFD-experimental validations of the PADRAM-HYDRA suite have previously been reported in the literature with a good match for moderate- to high-pressure fan and compressor flows [8,11].

In this study, PADRAM was also used for generating new blade designs by perturbing five spanwise airfoil sections—at 0, 25, 50, 75, and 100% span—and then radially interpolating (linearly) the sections to generate a new blade geometry. Individual sections were perturbed using the five degrees-of-freedom (5DOFs) illustrated in Fig. 2. These DOFs are dihedral (axial movement), sweep (tangential displacement), leading edge recambering, trailing edge recambering, and skew (rotation about the centroidal axis). Thus, in total, the design space consisted of 25 design variables (five spanwise stations with 5DOFs). It is important to note that these DOFs do not alter the thickness of the blade, but do change inlet angles, exit angles, and the camberline distributions. Such a design space has been previously used for design optimization studies [12]. Choices for the upper and lower bounds on each variable—indicated by the vectors $xU$ and $xL$, respectively—were determined such that (a) a subset of the design space could approximately emulate the effect of manufacturing variations on the blade and (b) a portion of the design space was suitable for design optimization. The resulting design envelope for the blade inlet and blade outlet angles, along with the camberline distributions for 75% and 95% span, is shown in Fig. 3.

###### Scope of Approaches.

For brevity, only a single turbomachinery blade case is explored in detail in this paper. However, it is emphasized that the methods used can be applied—using the same practices—to other compressor and turbine blades. Furthermore, the design space need not be restricted to three-dimensional (3D) blade parameters and may also include free form deformation or Hicks-Henne bump functions. Such are the intended scope and generalizability of turbomachinery active subspace performance maps.

## Active Subspace Performance Maps

In this study, there are three objectives that are of interest for which turbomachinery active performance maps will be generated:

1. (1)The first is maximum climb flow capacity, which is the flow capacity entering the OGV section at 105% shaft speed. It is given by $m˙OGVT0,OGV/P0,OGV$, where $m˙$ is the massflow rate, T0 is the total temperature, and P0 is the total pressure.
2. (2)The second performance metric is cruise adiabatic efficiency, which is computed using the total pressure ratio PR and the total temperature ratio (TR) at 95% shaft speed Display Formula
(1)$η=100×PR(γ−1)/γ−1TR−1$
where γ is the ratio of specific heats. The TR and PR ratios are mass-averaged over both the ESS and OGV exits.
3. (3)The third objective is the PR ratio, which is required to lie within a certain range for stable fan operation.

###### Maximum Climb Subspace: Flow Capacity.

To begin, an active subspace for the maximum climb flow capacity is sought with the aim of capturing its variation in 25-dimensional space with far fewer dimensions. The procedure detailed below seeks to construct an active subspace that reduces the dimensionality from 25 to 1, i.e., it finds a single variable—formed by a linear combination of the 25 design variables—along which capacity exhibits the greatest variability, on average. Details of the algorithm below can be found on page 5 of Ref. [4] with applications in Ref. [6].

###### A One-Dimensional Active Subspace Recipe.

First, N samples $x̂j$ were drawn uniformly from between $[xL,xU]$, where $j=1,…,N$. The number of elements (or rows) in vectors $xL$ and $xU$ is m = 25 and corresponds to the number of design variables. There are numerous heuristics for drawing these design of experiment samples; in this paper, optimal Latin hypercube sampling2 with N = 150 was chosen. From prior experience, N should be at least 4 to 5 times m. Next, for each sample Display Formula

(2)$xj=2x̂j−(xU−xL)(xU−xL)$
was computed. Equation (2) transforms all the dimensionalized samples $x̂j$ to lie between $xj∈[−1,1]m$. For each $xj$, a geometry can then be generated, meshed, and run through the flow simulation. Let the performance metric—the scalar output of the individual CFD simulations—be given by $fj=f(xj)$. The objective of this one-dimensional (1D) recipe is to fit a linear regression model to fjDisplay Formula
(3)$f(x)≈u0+u1x1+⋯+umxm$

where $x1,…,xm$ are the m components of $xj$ for some j. To compute the coefficients associated with this model, one seeks to solve the least-squares problem Display Formula

(4)$ũ=argminu‖Xu−f‖22$

where Display Formula

(5)$ũ=[u0⋮um], X=[1x̂1⋮⋮1x̂N], f=[f1⋮fN]$

where $||·‖2$ is the l2 norm.3 The 1D active subspace is then defined to be $w=u′/||u′||$, where $u′=(u1,…,um)$ is u without its first entry u0. The transform $yj=wTxj$ is referred to as a forward map, as this maps the 25-dimensional design vector to a 1D coordinate.

###### Flow Capacity Subspace.

The scatter plot in Fig. 4(a) plots the forward map for all $xj$ against fj, while the bar graph in (b) shows the components of the active subspace $wT$. In (b), each 3D blade design DOF has five bars associated with it, corresponding to the perturbation at the various spanwise locations. The 0% span perturbation is the first bar, followed by the remaining four bars for perturbations at 25, 50, 75, and 100% span, respectively. In plots like Fig. 4(a), one is ideally looking for the data to collapse into manifolds that may be used to build linear or quadratic response surfaces [5] and relate the shape of these manifolds to the governing physics. The y-axis in this plot has been nondimensionalized such that the variation between successive grid lines is 0.6% of the research blade's design intent flow capacity. For simplicity, throughout this paper, this will be expressed as research blade design flow capacity in imperial units.

The result in Fig. 4 shows that flow capacity varies linearly along a single vector in 25 dimensional space. This vector in turn is a linear combination of all design variables, but with greater contributions from leading edge recambering, trailing edge recambering, and skew. To examine the effect of moving along the capacity active variable, geometry profiles of designs at the lowest and highest capacity values in Fig. 4 are shown in Fig. 5. With increasing values of flow capacity, blade inlet angle is observed to increase.

From first principles, one knows that for compressible flows, capacity through a compressor will be set by the passage streamtube inlet area $A=s· cos(α1)$, where s is the blade pitch (which is fixed) and α1 is the flow angle. The streamtube inlet area varies along the span. Now, as the blade metal angle at the leading edge decreases,4α1 decreases, thereby increasing A, permitting more flow to pass through. Thus, the active subspace—the components of the vector w—has revealed the pedigree rule for changing fan capacity.

###### Cruise Subspaces: Efficiency and Pressure Ratio.

The 1D recipe described earlier may be used for computing a dominant direction for cruise efficiency. Recall that all the capacity simulations were run at 105% shaft speed. In this section, the goal is to obtain the active subspaces for two cruise flow metrics: cruise efficiency and cruise pressure ratio. At cruise, simulations need to be run at 95% shaft speed. There is a difference in geometry between the maximum climb and cruise conditions owing to blade untwisting due to different centrifugal forces. CFD boundary conditions are also different.

The cruise geometry was then perturbed using the 150 design vectors from earlier, meshed, and run through the flow solver. Then, using the 1D active subspaces recipe, an active subspace for efficiency was computed. The scatter plot is shown in Fig. 6. Such a plot is not too useful as it fails to reveal any trends in the data. However, this plot does suggest that a 1D model may be inadequate for capturing the variation in efficiency, motivating a higher-order approach.

###### A Quadratic Active Subspaces Recipe.

A global quadratic model was proposed for computing the active subspaces of efficiency. This model has the form Display Formula

(6)$f(x)=xTAx+cTx+d$

where A is a m × m symmetric matrix, c is a m-by-1 vector, and d is a constant. To determine the matrix-vector coefficients of the quadratic form, the least-squares problem Display Formula

(7)$ũ=argminu‖Mu−f‖22$
was solved. Here Display Formula
(8)$M=[1,D,DC]$

and $1∈ℝm$ is a vector of ones and Display Formula

(9)$D=[x1T,…,xNT], DC=[x1Tx1T,…,xiTxjT]$

where $i,j=1,…,N$. In Eq. (7), the total number of columns—i.e., polynomial basis terms—of M is $1+25+(25×26/2)=351$. This corresponds to constant, linear, and quadratic terms, respectively. To obtain a well-conditioned M for least squares, its number of rows should be at least a factor of 1.3–1.4 greater. Setting the total number of simulations N = 500, an additional 350 optimal Latin hypercube samples (recall that 150 samples were already evaluated for the 1D efficiency plot) were generated and used to compute u from Eq. (7). These coefficients were then used to determine A, c, and d.

Now, one can estimate the active subspaces with gradients of this quadratic model (see page 38 of Ref. [4]). To compute the active subspaces using this approach, the average outer product of the gradient of $f(x)$ must be computed as Display Formula

(10)$C=∫X∇xf(x)∇xf(x)Tρ(x)dx$

where $ρ(x)=2−25$ is simply the uniform density over $[−1,1]25$ and $X$ indicates the 25D domain. The physical interpretation of C is that of the covariance matrix of gradient samples. Plugging in the quadratic model, Eq. (10) becomes Display Formula

(11)$C=∫X(Ax+cT)(Ax+cT)Tρ(x)dx=23A2+ccT$

To find dominant directions along which $∇xf$ varies the most, the eigenvalue decomposition (using singular value decomposition) of the covariance matrix is computed as $C=WΛWT$ and can be partitioned as Display Formula

(12)$Λ=[Λ1Λ2], W=[W1W2], W1∈ℝ25×k$
such that the modes associated with the dominant eigenvalues are captured within the first k eigenvectors $W1$. Figure 7 plots the eigenvalues for the efficiency and pressure ratio objectives. In both cases, the first two eigenvectors of W, i.e., k = 2, were used to define their active subspaces. One can express each design vector as a linear combination of certain active variablesy and the remaining inactive variableszDisplay Formula
(13)$x=WWTx=W1W1Tx+W2W2Tx=W1y+W2z$

The intuition is that perturbations along the coordinates $y=W1Tx$ will dictate the value of f, while changes in $z=W2Tx$ should have a relatively smaller effect on f, on average.

###### Efficiency Subspace.

Figure 8(a) plots the efficiency values for N = 500 designs, where each design vector has been projected onto the 2D subspace $y=W1Tx$, while (b) plots the contributions of the 25 design variables to the two eigenvectors in $W1$. The efficiency values in (a) have been interpolated with a piecewise linear function to obtain the shaded contours. The design at an active variable coordinates of (0,0) corresponds to the design of the research blade. The same contours in Fig. 8(a) can be visualized as per Fig. 9(a), where the plot has simply been rotated for convenience. A quick visual inspection of the plot shows that along the vector $v=(0.4,−1.6)$, there lies a corridor of high efficiency designs. This is particularly interesting as all the designs lying to the right of (0,0) in this corridor have efficiencies higher than that of the research blade. In Fig. 9(b), radial profiles of the circumferentially mass-averaged values of efficiency (using total pressure and temperature ratios) for four designs along the vector are plotted. Trends for ten other designs on the vector were inspected and found to follow those shown in (b). The radial profiles reveal that the efficiency gains arise largely from the midspan to tip sections of the blade. In terms of the geometry, Fig. 10 shows the minimum and maximum camberline distributions at 95% span along with blade inlet angle profiles. Also shown are the tip profiles of the minimum and maximum efficiency designs compared with the research blade. These plots show that to obtain higher efficiency designs, the camberline must have reduced curvature and the blade inlet angle should be increased (closing the blade). The latter has the effect of reducing the flow capacity through the rotor; a point that is clear when analyzing the flow capacity subspace results in Fig. 5.

To understand the precise cause of the increase in efficiency, flow characteristics for the different designs are shown in Fig. 11, along with contours of the relative Mach number on the working line. From the data, one can infer that the increased efficiency is achieved by dropping the inlet relative Mach number by reducing the flow capacity. These designs are known as reduced Qa designs, where blade sections are closed relative to the research blade. A further inspection of the Mach number contours shows that design D itself is not quite at peak efficiency as the shock is swallowed (relative to the other blades), resulting in a slightly rear-ward shock position. The efficiency benefit seems to arise from the strength of the leading bow wave, as the relative Mach number drops; the bow wave will stand off the leading edge more, but will be weaker. This point is made clear by the contours of entropy around the leading edge for different designs, shown in Fig. 12.

The fact that the shock is swallowed for design D indicates that it is not quite at peak efficiency. This raises the question of whether there are designs beyond design D—lying on the vector $v=(0.4,−1.6)$—that have higher efficiencies. This point will be addressed in Sec. 4.

###### Pressure Ratio Subspace.

The quadratic active subspaces recipe discussed earlier was also applied to the cruise pressure ratio. Recall the eigenvalues of the outer product of the gradient of $f(x)$ shown in Fig. 7(b) which indicate that the first two eigenvectors are sufficient to capture the major trends. Figure 13 plots the pressure ratio on its active subspace and the contributions of the first two eigenvectors. In this case, the first eigenvector seems to be significantly more dominant than the second. Variations in the blade camberline distribution at 95% span, along with the blade outlet angle for the lowest and highest pressure ratios, are plotted in Fig. 14. The corresponding geometry changes are illustrated in Fig. 15. The results show that the first eigenvector increases the pressure ratio by increasing flow turning—which it does by incorporating more camber to the airfoil sections.

Figure 16(a) plots contours of the pressure ratio shown previously in Fig. 13(a). For a particular value of the pressure ratio, there exists a range of designs that lie on the quadratic manifold associated with that pressure ratio. Thus, while pressure rise contributions from the shock and from flow turning (subsequent diffusion) may vary with different designs, their combined effect is the same along constant pressure ratio manifolds. Going forward, it would be useful to visualize the change in efficiency along these manifolds.

In Fig. 16(b), a global quadratic polynomial is used to fit the data in (a). This figure shows that the active subspace for cruise pressure ratio can be well approximated by a quadratic response, which along with the linear response for capacity will be utilized in a subsequent section.

## Exploiting the Three-Dimensional Design Maps

As mentioned earlier, the efficiency contours in Fig. 9 indicate that there may be more efficient designs beyond its boundaries, toward the right-hand side of the black-dashed line. This raises the three key questions regarding this reduced representation of the efficiency data:

1. (1)What are the true boundaries of this surface given that the current boundaries are determined by the Latin hypercube samples?
2. (2)How do we compute the inverse map: the 25D vector required for CFD, given only the 2D coordinates from the efficiency active subspace?
3. (3)What is the error in this overall representation?

###### Zonotopes.

The reduced 2D range of the active variables for efficiency can be written in set notation as Display Formula

(14)$Y={y:y=W1Tx,−1≤x≤1}$

Here $Y$ is a convex polytope residing in 2D space, the vertices of which are a subset of the 225 vertices of a 25D hypercube projected on a 2D plane, essentially a silhouette of the 25D hypercube. Such projections are called zonotopes. The vertices of zonotopes can be computed using the search algorithm in Ref. [13], available in the Python Active-subspaces Utility Library [14]. The only requirement for determining these vertices is the eigenvectors in $W1$.

It is important to note that uniform sampling in 25D does not guarantee a uniform distribution of samples in the 2D zonotope. Recall that 500 optimal Latin hypercube samples were used in 25D. Their distribution in the 2D zonotope is determined by $W1$.

Figure 17(a) plots the boundaries of the 25D zonotope for efficiency; designs lying within demarcated black line segments are designs within the bounding box $−1≤x≤1$.

###### The Inverse Map.

Having computed the boundaries of the zonotope, one may wish to extend the boundaries of the efficiency contours, specifically along the vector $v=(−0.4,1.6)$ and its vicinity. A few candidate points are shown in Fig. 17(a). To compute the efficiency of the designs at these 2D coordinates, one needs to determine the 25D design vector required for CFD. Recall that the forward map defines the transformation from the fullspace to the reduced active subspace: for a given vector x, there is only one y in the reduced space. However, the converse is not true. Naïvely applying $x=W1y$ does not guarantee that x will lie within the original design space bounding box. Furthermore, for a given y, there are infinitely many x that will satisfy this linear equality. To mitigate this issue, regularization is necessary. Lukaczyk et al. [5] proposed the following linear program:

$minimizex 0Txsubject toy=W1Tx−1≤x≤1$

This minimum viable method for obtaining the inverse map uses a dummy objective function that is always 0. Essentially, the linear program requires that the forward map hold and that the design lie within the bounding box.

For each of the new designs in Fig. 17(a), the linear program is solved to determine the full 25D vector, x. The efficiency obtained from these designs—after generating the geometries, meshes and running CFD—is plotted in Fig. 17(b). Here, the new results have been interpolated with the prior results using a piecewise linear interpolant. It is clear from these new contours that higher efficiency designs are present well-beyond the prior efficiency contour boundaries. The efficiency peak in Fig. 17(b) in the vicinity of (0.5–1.5) corresponds to a 0.4% increase in efficiency relative to the research blade at coordinates (0,0).

###### Error in the Maps.

One of the drawbacks of the linear program is that, depending on the initial condition (initial design vector) used in the optimization, different design vectors may be rendered. However, if f truly admits an active subspace that is sufficiently well-defined by $W1$, then the differences in f among designs with the same 2D coordinates, but different 25D coordinates, should be small. To illustrate this point, consider the six 2D coordinates in Fig. 18(a). For each of these six designs, five 25D vectors were generated using different initial conditions in the linear program. The resulting efficiency values from CFD are shown in Fig. 18(b). These values—where the maximum error is within 0.18%—give confidence in the results that follow.

Similar numerical experiments were carried out on the flow capacity and pressure ratio subspaces, with maximum errors of 0.2% and 0.015%, respectively.

###### Active Performance Maps.

In Secs. 3.1.2 and 3.2.3, response surfaces for the maximum climb flow capacity and cruise pressure ratio were determined. Thus, for any 25D vector within the design space considered in this paper, an approximation to these two performance metrics could be obtained. Thus far, these performance metrics have been plotted in their own subspaces. However, it is useful to have their contours shown on the same axes as the efficiency.

To obtain this, the inverse map described in Sec. 4.2 was used for each grid point within the efficiency zonotope. This yielded a 25D vector for each red square in Fig. 19. Forward maps—using the linear and quadratic responses—for maximum climb flow capacity and pressure ratio were then used to determine these performance metrics for each design.

The final results are shown in Figs. 20(a) and 20(b). These plots should be compared with the efficiency contours in Fig. 18(a). These contour maps inform the fan designer that increases in efficiency—i.e., following the vector $v=(0.4,−1.6)$ toward the right half plane—will yield designs that have reduced pressure ratio and flow capacity. Furthermore, using these maps, the designer can get a quantitative assessment of the trade-offs, for example, the previously mentioned 0.4% increase in efficiency (see Sec. 4.2) yields a 0.041 decrease in the pressure ratio and a 2.02% decrease in flow capacity (relative to the research blade). These can be inferred from Fig. 20. Designs in the vicinity of (0.35, −0.85) on the other hand offer a 0.25% increase in efficiency with only a 1.27% decrease in flow capacity. The main point to note from this section is that using these turbomachinery active subspace performance maps such quantitative comparisons are easily made.

## An Active Map for Efficiency–Sensitivity

As mentioned in Sec. 1, this study was motivated by the discrepancy between the design intent CFD efficiency and the rig test obtained efficiency values—corroborated later by running CFD on the manufactured geometry. The distance between these two geometries is shown in Fig. 21; there are visible differences in the camberline distribution and in the skew, with the manufactured blade being slightly more open. The objective of this section is to use active subspaces to facilitate the design of blades that are desensitized to manufacturing variations. This goal rests on the assumption that all designed blades will, upon manufacturing, exhibit the same manufacturing excursions shown in Fig. 21. This perturbation form may be used to infer the effect of a bias in the manufacturing process. However, there is also a random component—i.e., even within a specific tolerance band (biased), one expects a range of blade profiles. Owing to the absence of additional manufacturing information, this paper does not account for the random component, although a few statements on how the methods below can be used with further manufacturing information are provided in Sec. 6. For the remainder of this section however, for every blade design, a geometry perturbation akin to Fig. 21 is applied.

###### An Insensitivity Subspace.

A total of 500 CFD simulations were carried out on these geometries simulating manufacturing at the cruise condition using the exact same design vectors as before. Thus, for each design vector, there are two geometries: the nominal design and the nominal design with the perturbation shown in Fig. 21. The quadratic active subspaces recipe was then applied using the difference between these efficiencies as the objective, i.e., $ηdesign−ηmanufactured$. This objective is termed efficiencysensitivity, and its scatter plot is shown in Fig. 22. A piecewise linear interpolant was applied on the active subspace to render the contour, where a moderate amount of noise is apparent. These contour lines are also shown in Fig. 23(a). It should be noted that designs having an efficiency–sensitivity value between −0.1 and 0.1 are extremely desirable compared to designs that may lie outside this range.

###### Mapping Insensitivity Onto the Efficiency Subspace.

In Sec. 4, flow capacity and pressure ratio contours were shown on the efficiency active subspace using both inverse and forward maps. The motivation was to visualize the trade-offs in efficiency, pressure ratio, and flow capacity on the same axes. The same process can be extended to efficiency–sensitivity. In Fig. 23(a), 1000 2D coordinates that yield an efficiency–sensitivity value between −0.1 and 0.1 were determined and an inverse map—using the linear program—was used to compute their corresponding 25D design vectors. These design vectors were then projected onto the 2D efficiency subspace, using the forward map for efficiency. The locations of all the designs are shown in Fig. 23(b). The results show a small region, to the right of the origin, that exhibits a significantly lower sensitivity to the manufacturing variations.

###### Extracting Pedigree Design Rules.

Figures 23(b) and 20 may be combined into a single plot: Fig. 24. The underlying contours are those of efficiency—but zoomed into a region of interest, in the vicinity of the research blade.

The translucent shaded black region indicates the bounds of the design pressure ratio, the striped region indicates the range of acceptable flow capacities, and the black contour indicates designs that are desensitized to manufacturing variations (to within $±0.1%$). The latter was obtained by approximating the outline of the samples in Fig. 23(b).

Consider the two designs indicated by the black and magenta squares. The magenta square at (0, 0) is the research blade, while the black square indicates the best approximation to the manufactured research blade, using the 3D design parameterization. In other words, the design parameters were varied to best match the manufactured geometry in Fig. 21, and the resulting design vector was projected onto the efficiency active subspace. This best approximation to the manufactured blade had an efficiency–sensitivity of −0.5%, slightly above the desired −0.78%. On the map, the manufacturing process has the effect of shifting the efficiency of the blade to the left of its design intent value. The research blade lies on the edge of an efficiency cliff (see contours in Fig. 24), and thus pays a significant penalty when manufactured (shifted left).

Now designs selected within the densitized to manufacturing variations region are less sensitive to manufacturing variations because the gradient of the efficiency contours is flatter compared to other regions. Thus, designs selected within this domain are more likely to be robust (in efficiency) to the leftward shift induced by manufacturing variations.

Two designs that lie within the desensitized to manufacturing variations region are shown in this figure: designs P (triangle symbol) and Q (diamond symbol). Their design intent geometries are colored in white, while their manufactured geometries are colored in gray (green in online version). The designs are selected, because their design intent efficiencies are comparable to that of the research blade. Their pressure ratios are also within the pressure ratio requirements. Now as the manufacturing process pushes designs to the left—on the efficiency active subspace map—these geometries incur a minor efficiency penalty compared to the research blade (see location of triangle and diamond green markers). Their flow characteristics are shown in Fig. 25. Designs P and Q have an efficiency–sensitivity value of approximately 0.2%.

While the pressure ratio for both manufactured P and Q designs is within the required range, their flow capacities are below the design requirement. Several other designs (not shown) at various locations were selected and analyzed in detail. The following three observations were made:

1. (1)It was easier to render high-efficiency blades that were insensitive to manufacturing variations at lower pressure ratios and capacities.
2. (2)At design capacity and design pressure ratio, the active subspace performance map indicates that the research blade design lies on a local efficiency peak.
3. (3)Desensitized blades that satisfied both flow capacity and pressure ratio requirements, yielded a minor (0.15%) drop in efficiency, however their efficiency values were less than that of the research blade.

In general, it was observed that blades that were desensitized to the manufacturing variations were more closed. This makes sense as the manufacturing variations effectively opened the blade up, thereby pushing the center of loading closer to the leading edge; increasing the inlet Mach number and loss.

Thus, the design intent of such desensitized blades operated at a lower flow capacity. Now, as the fan operates on a choked nozzle (which fixes the OGV exit flow function) to provide thrust, it would have to operate at a higher speed to pass the required flow. This may not be a viable solution due to stress considerations in the blade and disc. Thus, the other option to get the required flow capacity back would be to scale the fan up by 1–2% of area. This would also result in a minor increase in tip speed, but overall, because the fan will have an increased fan face area, the axial Mach number will reduce [15]. This should also permit higher efficiency designs to be achieved, as discussed previously.

## Conclusions

A new approach to visualizing and navigating the design space of a turbomachinery blade has been presented in this paper. The approach, turbomachinery active subspace performance maps, combines dimension reduction with fundamental turbomachinery aerodynamics. In this paper, these maps were generated for a high Mach number fan blade, where pedigree rules for efficiency, pressure ratio, flow capacity, and insensitivity to manufacturing variations were sought. From the computational studies in this paper, the following two pedigree rules were determined:

1. (1)There is a trade-off between increasing fan efficiency—by reducing the strength of the leading edge bow wave—and reducing flow capacity and the pressure ratio. This point is clear from the change in the geometries observed along the vector in the efficiency active subspace. The active subspace performance maps also suggest greater potential in finding higher efficiency designs at capacity values below that of the research blade.
2. (2)For desensitizing the blade to manufacturing variations, flow capacity must be reduced—relative to the research blade considered in this study. This is clear by inspecting the active subspace performance map, where the efficiency gradient in the desensitized regions is flatter compared to other areas of the map.

In isolation, these pedigree rules can be well established from first principles. However, in this paper, these rules are derived from the computation of the active subspaces. This facilitates a detailed quantitative assessment of the trade-offs in turbomachinery design.

One assumption made in this work—in the absence of additional manufacturing data—is that all new blade designs would have the same manufacturing excursions. While this may account for capturing the bias in the manufacturing process, it negates the random component. In practice, one can envision a range of different hypothetical manufactured designs based a single design intent geometry and known tolerances of the manufacturing process. On the performance map, this would then manifest as a ring of permissible designs around the design intent, instead of just one design (as considered in this work). This idea along with the application of turbomachinery active subspace performance maps to other compressor and turbine components is part of further work.

## Acknowledgements

This research was funded under an EPSRC Knowledge Transfer Fellowship between the University of Cambridge and Rolls-Royce plc. The authors would like to thank Nigel Smith, Mark Wilson, and Nick Crowder for their expertise in fan aerodynamics that was crucial to this project. The authors also thank Ralf Tilch, David Radford, Nick Cumpsty, and John Adamczyk. Finally, the authors would like to thank the anonymous reviewers for their comments and encouragement.

## Nomenclature

• A =

• c =

linear coefficients for 25D quadratic polynomial

• C =

covariance matrix

• d =

constant term for the 25D quadratic polynomial

• D =

dimension

• ESS =

engine sector stators

• $f(·)$ =

function of the argument in $(·)$

• N =

number of samples (CFD evaluations)

• OGV =

outlet guide vane

• PR =

pressure ratio

• TR =

temperature ratio

• u =

vector of least-squares coefficients

• W =

eigenvectors of the covariance matrix

• x =

full space (25D) coordinates

• y =

active coordinates

• z =

inactive coordinates

Greek Symbols
• $Λ$ =

eigenvalues of the covariance matrix

• $ρ$ =

the uniform density over the $[−1,1]25$ hypercube

Subscripts and Superscripts
• $0,OGV$ =

stagnation properties at the OGV inlet plane

• L =

lower bound of design variables

• U =

upper bound of design variables

## References

Taylor, J. V. , and Miller, R. J. , 2016, “Competing Three-Dimensional Mechanisms in Compressor Flows,” ASME J. Turbomach., 139(2), p. 021009.
Sobol, I. M. , 2001, “Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates,” Math. Comput. Simul., 55(1), pp. 271–280.
Owen, A. B. , 2013, “Variance Components and Generalized Sobol' Indices,” SIAM/ASA J. Uncertainty Quantif., 1(1), pp. 19–41.
Constantine, P. G. , 2015, Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies, Vol. 2, SIAM, Philadelphia, PA.
Lukaczyk, T. W. , Constantine, P. , Palacios, F. , and Alonso, J. J. , 2014, “Active Subspaces for Shape Optimization,” AIAA Paper No. AIAA-2014-1171.
Constantine, P. , Emory, M. , Larsson, J. , and Iaccarino, G. , 2015, “Exploiting Active Subspaces to Quantify Uncertainty in the Numerical Simulation of the Hyshot Ii Scramjet,” J. Comput. Phys., 302, pp. 1–20.
Constantine, P. G. , Zaharatos, B. , and Campanelli, M. , 2015, “Discovering an Active Subspace in a Single-Diode Solar Cell Model,” Stat. Anal. Data Min.: ASA Data Sci. J., 8(5–6), pp. 264–273.
Zamboni, G. , and Xu, L. , 2012, “Fan Root Aerodynamics for Large Bypass Gas Turbine Engines: Influence on the Engine Performance and 3D Design,” ASME J. Turbomach., 134(6), p. 061017.
Lapworth, L. , 2004, “Hydra-CFD: A Framework for Collaborative CFD Development,” International Conference on Scientific and Engineering Computation (IC-SEC), Singapore, July 5–8.
Milli, A. , and Shahpar, S. , 2012, “PADRAM: Parametric Design and Rapid Meshing System for Complex Turbomachinery Configurations,” ASME Paper No. GT2012-69030.
Seshadri, P. , Parks, G. T. , and Shahpar, S. , 2014, “Leakage Uncertainties in Compressors: The Case of Rotor 37,” J. Propul. Power, 31(1), pp. 456–466.
Seshadri, P. , Shahpar, S. , and Parks, G. T. , 2014, “Robust Compressor Blades for Desensitizing Operational Tip Clearance Variations,” ASME Paper No. GT2014-26624.
Fukuda, K. , 2004, “From the Zonotope Construction to the Minkowski Addition of Convex Polytopes,” J. Symbolic Comput., 38(4), pp. 1261–1272.
Constantine, P. , Howard, R. , Glaws, A. , Grey, Z. , Diaz, P. , and Fletcher, L. , 2016, “Python Active-Subspaces Utility Library,” J. Open Source Software, 1(5), p. 79.
Smith, N. , 2016, “Rolls-Royce Fan Design Specialist,” personal communication.
View article in PDF format.

## References

Taylor, J. V. , and Miller, R. J. , 2016, “Competing Three-Dimensional Mechanisms in Compressor Flows,” ASME J. Turbomach., 139(2), p. 021009.
Sobol, I. M. , 2001, “Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates,” Math. Comput. Simul., 55(1), pp. 271–280.
Owen, A. B. , 2013, “Variance Components and Generalized Sobol' Indices,” SIAM/ASA J. Uncertainty Quantif., 1(1), pp. 19–41.
Constantine, P. G. , 2015, Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies, Vol. 2, SIAM, Philadelphia, PA.
Lukaczyk, T. W. , Constantine, P. , Palacios, F. , and Alonso, J. J. , 2014, “Active Subspaces for Shape Optimization,” AIAA Paper No. AIAA-2014-1171.
Constantine, P. , Emory, M. , Larsson, J. , and Iaccarino, G. , 2015, “Exploiting Active Subspaces to Quantify Uncertainty in the Numerical Simulation of the Hyshot Ii Scramjet,” J. Comput. Phys., 302, pp. 1–20.
Constantine, P. G. , Zaharatos, B. , and Campanelli, M. , 2015, “Discovering an Active Subspace in a Single-Diode Solar Cell Model,” Stat. Anal. Data Min.: ASA Data Sci. J., 8(5–6), pp. 264–273.
Zamboni, G. , and Xu, L. , 2012, “Fan Root Aerodynamics for Large Bypass Gas Turbine Engines: Influence on the Engine Performance and 3D Design,” ASME J. Turbomach., 134(6), p. 061017.
Lapworth, L. , 2004, “Hydra-CFD: A Framework for Collaborative CFD Development,” International Conference on Scientific and Engineering Computation (IC-SEC), Singapore, July 5–8.
Milli, A. , and Shahpar, S. , 2012, “PADRAM: Parametric Design and Rapid Meshing System for Complex Turbomachinery Configurations,” ASME Paper No. GT2012-69030.
Seshadri, P. , Parks, G. T. , and Shahpar, S. , 2014, “Leakage Uncertainties in Compressors: The Case of Rotor 37,” J. Propul. Power, 31(1), pp. 456–466.
Seshadri, P. , Shahpar, S. , and Parks, G. T. , 2014, “Robust Compressor Blades for Desensitizing Operational Tip Clearance Variations,” ASME Paper No. GT2014-26624.
Fukuda, K. , 2004, “From the Zonotope Construction to the Minkowski Addition of Convex Polytopes,” J. Symbolic Comput., 38(4), pp. 1261–1272.
Constantine, P. , Howard, R. , Glaws, A. , Grey, Z. , Diaz, P. , and Fletcher, L. , 2016, “Python Active-Subspaces Utility Library,” J. Open Source Software, 1(5), p. 79.
Smith, N. , 2016, “Rolls-Royce Fan Design Specialist,” personal communication.

## Figures

Fig. 1

Research blade geometry (which has been scaled for proprietary reasons): (a) isometric view; (b) domain for simulations. OGV refers to the outlet guide vanes, and ESS to the engine sector stators.

Fig. 2

Blade 3D design parameterization showing the 5DOF (both positive and negative). Deflections are enlarged for clarity.

Fig. 3

Design space envelope for (a) blade inlet angle, (b) blade outlet angle, (c) camberline at 75% span, and (d) camberline at 95% span

Fig. 4

Scatter plot of maximum climb flow capacity using the 1D active subspaces recipe in (a) and components of the vector wT in (b)

Fig. 5

Perturbations along the capacity active subspace: (a) blade inlet angle profiles, (b) blade outlet angle profiles, (c) minimum capacity geometry at an active variable value of −1.2, and (d) maximum capacity geometry at an active variable value of 1.2; research blade is colored in a darker shade (orange in the online version)

Fig. 6

Scatter plot of efficiency using the 1D active subspaces recipe

Fig. 7

Eigenvalues obtained using the quadratic model where f(xj) is (a) efficiency and (b) pressure ratio

Fig. 8

Scatter plot of efficiency using the quadratic active subspaces recipe: (a) scatter plot; (b) components of the active subspace, WT, with two eigenvectors. A local piecewise linear interpolation has been applied on the scatter plot data.

Fig. 9

Contour map of efficiency (with a 0.2% difference between successive contour lines) in (a); radial efficiency profiles for designs A–D along the vector given by the dashed black line in (a) and (b)

Fig. 10

Perturbations along the vector in Fig. 9(a) in the efficiency active subspace: (a) camberline distribution at 95% span, (b) blade inlet angles, (c) minimum efficiency geometry, and (d) maximum efficiency geometry

Fig. 11

Characteristics of designs A–D at cruise: (a) efficiency, (b) pressure ratio, and (c) relative Mach number contours on working line at 90% span

Fig. 12

Entropy contours at the leading edge for Designs A–D at the cruise working line condition at 90% span. Lighter contours indicate more entropy.

Fig. 13

Scatter plot of pressure ratio using the quadratic active subspaces recipe: (a) scatter plot and (b) components of the active subspace, WT, with two eigenvectors. A local piecewise linear interpolation has been applied on the scatter plot data.

Fig. 14

Perturbations along the first eigenvector of the pressure ratio active subspace: (a) camberline distribution at 95% span and (b) blade outlet angles

Fig. 15

Perturbations along the first eigenvector of the pressure ratio active subspace: (a) minimum pressure ratio geometry and (b) maximum pressure ratio geometry

Fig. 16

Contours of cruise pressure ratio in its active subspace: (a) local linear interpolant response and (b) quadratic response. There is a 0.005 difference in pressure ratio between successive contours lines.

Fig. 17

Zonotope boundaries for efficiency: (a) new 2D coordinates selected and (b) CFD-based efficiency values of the new designs in (a)

Fig. 18

Error in the efficiency active subspace: (a) six 2D coordinates selected and (b) efficiency values of five different 25D vectors corresponding to each of the 2D coordinates

Fig. 19

Grid points in the efficiency active subspace used for approximating flow capacity and pressure ratio values

Fig. 20

Performance metrics on each grid point in Fig. 19 for (a) maximum climb flow capacity and (b) cruise pressure ratio. Successive contour lines in (a) are 0.8% research blade design flow capacity in imperial units, while contour lines in (b) are 0.005 apart.

Fig. 21

Geometry differences between design intent and manufactured blade. Positive values indicate an outward perturbation with respect to the surface normal of the design intent blade. (Geometries have been scaled for proprietary reasons.)

Fig. 22

Scatter plot of efficiency–sensitivity: (a) scatter plot and (b) components of the active subspace, WT, with two eigenvectors

Fig. 23

Contours of the efficiency–sensitivity subspace in (a); efficiency subspace in (b). A total of 1000 samples between –0.1 and 0.1 in (a) are mapped in (b).

Fig. 24

Fan active subspace performance map—combining pressure ratio, maximum climb flow capacity, efficiency, and efficiency–sensitivity. The underlying contours are those of efficiency with successive contours differing by 0.05%. Maximum and minimum contour levels have been changed to focus on the specific local region within the zonotope (see the x and y axis).

Fig. 25

Characteristics at cruise: (a) efficiency and (b) pressure ratio

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