Towards Robust Design of Axial Compressors with Uncertainty Quantification Pranay Seshadri∗ , Geoﬀrey Parks† and Jerome Jarrett‡ Engineering Design Centre, University of Cambridge, Cambridge, CB2 1PZ, U.K. Shahrokh Shahpar§ Rolls-Royce plc, Derby, DE24 8BJ, U.K. Operational uncertainties such as throttle excursions, varying inlet conditions and geometry changes lead to variability in compressor performance. In this work, the main operational uncertainties inherent in a transonic axial compressor are quantified to determine their eﬀect on performance. These uncertainties include the eﬀects of inlet distortion, metal expansion, flow leakages and blade roughness. A 3D, validated RANS model of the compressor is utilized to simulate these uncertainties and quantify their eﬀect on polytropic eﬃciency and pressure ratio. To propagate them, stochastic collocation and sparse pseudospectral approximations are used. We demonstrate that lower-order approximations are suﬃcient as these uncertainties are inherently linear. Results for epistemic uncertainties in the form of meshing methodologies are also presented. Finally, the uncertainties considered are ranked in order of their eﬀect on eﬃciency loss. Nomenclature HPC IPC PR P0 Pout,choke TR T0 Tout,choke √ m ˙ T0 /P0 γ η = (γ − 1)/γ · ln(P R)/ln(T R) ηa = (P R(γ−1)/γ − 1)/(T R − 1) fs (u) fp (u) m ˙ m ˙ leak /m ˙ choke ωleak θleak L(u) - high pressure compressor intermediate pressure compressor pressure ratio total pressure outlet total pressure at choke temperature ratio total temperature outlet total temperature at choke capacity ratio of specific heats polytropic eﬃciency adiabatic eﬃciency spectral collocation approximation of a function f (u) pseudospectral approximation of a function f (u) mass flow rate leakage mass flow as a fraction of choke mass flow leakage whirl velocity leakage meridional inflow angle Lagrange cardinal function ∗ PhD Candidate, Department of Engineering, Trumpington Street, AIAA Student Member University Lecturer, Department of Engineering, Trumpington Street ‡ University Lecturer, Department of Engineering, Trumpington Street, Senior Member AIAA § Royal Aeronautical Society Fellow, Rolls-Royce Engineering Associate Fellow, AIAA Associate Fellow † Senior 1 of 21 American Institute of Aeronautics and Astronautics I. Introduction The operating envelope of an axial compressor is vast. It must eﬃciently operate through a myriad of operating regimes and environments. This presents an arduous challenge to the compressor designer. Over the years a design methodology has evolved for such a complex machine. This process revolves around designing the compressor for a fixed operating point (chosen based on an assessment of key operating conditions), and then extrapolating that seed design to get the performance curves at all operating points. In addition to design point performance, surge margin and oﬀ-design performance of the compressor are extensively evaluated during the design phase. The aerodynamic workflow is to use historical data with mean-line and through-flow analysis, followed thereafter by 2D and 3D CFD computations. Here 3D CFD computations are particularly revealing, as they provide details on complex three-dimensional loss generating mechanisms within the compressor, that are diﬃcult to assess with 2D or mean line codes. Compressor Oﬀ-Design Performance The performance of an axial compressor can be determined by its location along its characteristic (see Figure 1). In steady state, the compressor will always operate on the working line – and thus if the shaft speed is known, the pressure ratio and eﬃciency can be determined. In service, excursions of the working line are expected, for instance, during accelerations and decelerations. For instance, in the HPC, the line falls during decelerations and rises during accelerations and power extractions. IPCs on the other hand see a rise during a deceleration. The line also deviates with downstream bleed movement and other control inputs.1 Thus, for a constant shaft speed, certain blade rows in the compressor may experience conditions varying from close-to-stall to close-to-choke. Figure 1. Compressor characteristic – eﬃciency and pressure ratio as functions of the flow function √ A deviation from the design point can also occur with shifts in the corrected speed (defined as N/ (T0 ), where N is the real shaft speed) curves (see Figure 1). These perturbations are caused by uncertainties in operation ranging from inlet distortions to axial clearances and blade erosion. As both the working line and constant velocity curve are prone to movement, it is critical to ensure that there is a suﬃcient stable margin above the working line for transient operations. For instance, Freeman2 shows that a 1% increase in the axial clearance during in-flight casing expansion, will result in a 1.4% drop in eﬃciency with a reduction in the surge margin. One approach to reduce the eﬀect of such operational uncertainties, may be to incorporate robust design methods with uncertainty quantification, into the design process. 2 of 21 American Institute of Aeronautics and Astronautics Uncertainty Quantification and Robust Design Recent work by Ghisu et al.3 and Kumar et al.4 has suggested that incorporating operational uncertainties within the optimization process can significantly reduce the performance variance of compressor components. In both cases, for each iteration of the optimizer, the new geometry is subjected to a sampling of uncertain flow conditions. These varying flow conditions reflect the probability distribution of the uncertainty. Once the flow conditions for a particular geometry have been evaluated, the mean and variance values of the objective function are fed back to the optimizer. Within such an architecture, the multiobjective optimizer maximizes the mean of the objective and minimizes its variance.4 Similar robust design studies for propagating uncertainty in blade manufacturing have been carried out by Bestle et al.5 Due to the large number of samples that need to be computed to obtain a realistic estimation of the uncertainty, the above works resort to using either two-dimensional CFD or lower fidelity aerodynamic codes. While such aerodynamic models provide a rough approximation of the flow field, they do not capture all the prevalent flow physics. Furthermore, in turbomachinery based design optimization, where the diﬀerence between datum and optimized components are of the order of a few percentage points, lower aerodynamic models may fail to yield truly optimal designs. As a result there is a need for 3D-RANS simulations to be the model of choice when carrying out robust design studies. This presents one of the key challenges in the field of turbomachinery design optimization – prohibitive sample sizes and long running times. Thus, there is a need for more eﬀective sampling techniques that can capture all the stochastic variation during optimization, whilst using 3D-RANS simulation models. In additon to the above, it is also critical to determine which uncertainties have the most deleterious eﬀects on performance. With tight constraints on the feasible number of RANS simulations, only a handful of uncertainties can be propagated, and thus it makes sense to propagate uncertainties with greatest eﬀect. These issues motivate the present work. Scope of the Paper There are two important questions this paper seeks to address. Firstly, what is the best approach to propagate axial compressor uncertainties within an optimization framework? Secondly, which operational uncertainties result in the greatest loss in axial compressor eﬃciency, and are thus primary candidates for robust design? To answer the first question, four major uncertainties are propagated. These include uncertainties in metal expansion (blade, hub, casing and both), boundary layer growth, inlet pressure distortions and leakage flows. Not covered here are the eﬀects of manufacturing errors and blade erosion, as they will be addressed in a forthcoming paper. Epistemic (model-form) uncertainties in the form of meshing methods for leakage flows are also considered. To propagate the uncertainties, several polynomial-based methods are used as they have recently shown considerable promise, and are more eﬃcient than sampling methods such as Monte Carlo and Latin hypercube. Principal component analysis methods have intentionally been avoided as they present the diﬃculty of parameterizing the uncertainty in terms of eigenmodes as opposed to blade and flow parameters. All uncertainties in this work are propagated from stall to choke, to account for shifts in the working line. A 3D-RANS model of an axial compressor blade is utilized for all computational studies. Here we use NASA Rotor 37.6 II. Computational Methods NASA Rotor 37 is a rotor blade for an axial core compressor that was developed at the National Aeronautics and Space Administration (NASA) Lewis Research Center. It was part of a broad program on axial flow fans and air-breathing compressors with the objective of attaining a high pressure ratio and eﬃciency well within stall margins, in minimal stages.7 It has been extensively tested, both as an isolated component and in tandem with a stator.6, 8 In this study, 3D RANS flow computations on a single Rotor 37 blade passage are carried out. Details on the meshing strategy and flow solver characteristics are provided below. A. Meshing Methodology The Rolls-Royce in-house mesh generator PADRAM9, 10 is used in the present study. PADRAM (or PArametric Design and RApid Meshing) is a multi-block, algebraic grid generator that uses a C-O-H grid topology for creating both multi-stage and multi-passage structured and unstructured grids. The code 3 of 21 American Institute of Aeronautics and Astronautics incorporates an elliptic partial diﬀerential equation based method for smoothing. In this work, three diﬀerent meshes for Rotor 37 were created and tested for grid independence: a baseline mesh and two other meshes to simulate hub-leakage flows. 1. Baseline mesh The meshing strategy employed consists of generating several 2D (along the streamline) airfoil grids at various heights (along the blade span), and then interpolating them to obtain the 3D grid. Each 2D airfoil mesh comprises of five sub-zones: two fore of the rotor leading edge, the rotor, and two aft of the rotor trailing edge. The two fore and aft sub-zones are H-meshes. These represent the stationary hub, as shown in Figure 2(a). The rotor sub-zone in the middle consists of an O-mesh around the airfoil, and an upper and lower H-mesh. This represents the rotating frame, which comprises of the blade and the rotating disk to which it is attached. A meridional view of the blade mesh is shown in Figure 2(b). A hub fillet mesh is incorporated at the base of the blade as shown in Figure 2(c). To model the tip gap flow physics 15 cells were used in the region from the tip of the blade to the casing. The total number of cells for the baseline mesh is 1.7 million. Z Z Y Z Y Y X X X Stationary zone Rotating zones Stationary zone (a) (b) (c) Figure 2. Baseline NASA Rotor 37 mesh: (a) Computational domain and zones; (b) Meridional view of the blade mesh; (c) Close-up of the hub fillet mesh 2. Surface Patch mesh Surface patches, with appropriate boundary conditions at the hub or casing, are often used in turbomachinery to simulate bleeds and injection jets. Meshes with surface patches converge far quicker than fully meshed cavities, and are thus quite advantageous. To simulate the eﬀect of hub-leakage flows, a patch 0.264 cm fore of the leading edge and 0.75 mm in width was created, as shown in Figure 3(a). A grid independence study revealed that 25 mesh points were required along the patch for convergence. The total number of cells for this mesh is 2.5 million. 3. Extruded Patch mesh While surface patches provide a fairly accurate representation of the passage flow, they often do not capture all the relevant flow physics. To explore the epistemic uncertainties associated with the diﬀerent meshing approaches, an extruded patch mesh was also created for the same width and location as the patch mesh. This permits the flow at the passage interface to be more accurately modeled. This mesh is shown in Figure 3(b). A total of 180 cells are located on all three walls of the rectangular cavity. The extrusion and its grid independent solution result in this mesh having 4.6 million cells. B. Flow Computations The 3D-RANS equations are solved using the Rolls-Royce solver HYDRA.11 HYDRA uses an edge-based data structure with the flow data stored at the cell vertices. A MUSCL-based flux-diﬀerencing algorithm is used to integrate the flow equations around median-dual control volumes. For the steady-state solution, 4 of 21 American Institute of Aeronautics and Astronautics Z Z Y Y X X Surface patch Cavity (a) (b) Figure 3. Meshes to simulate hub-leakage flows fore of the rotor leading edge: (a) Surface patch mesh; (b) Extruded surface patch mesh the pre-conditioning of the discrete flow equations is applied using a block Jacobi pre-conditioner and a fivestage Runge-Kutta scheme is employed for the convergence towards the steady-state solution. An elementcollapsing multi-grid algorithm is used to accelerate the convergence to steady state. In the present study both Spalart-Allmaras and k − ω turbulence models are used with fully turbulent boundary layers and wall functions. We prescribe radial distributions of the total temperature and total pressure at the rotor inlet. These values are obtained from Rotor 37 experiments at design speed.6 For the rotor exit, a fixed exit capacity boundary condition is enforced. Periodic boundaries are enforced at the upper and lower walls. For leakage simulations, the leakage surface patch/extruded surface is given a subsonic whirl mass inflow boundary condition. Mass inflow rate, inflow angle (in the meridional plane) and whirl velocity are prescribed for this patch, and the solver, in turn, determines the total pressure and temperature at the patch required for conservation of fluid properties. C. Validation with Experiment Overall performance metrics using the in-house codes with two diﬀerent turbulence models and their comparison with experiments (at design speed) and other computational studies are shown in Figure. 4. Results for the NASA SWIFT code are obtained from Chima12 and those for NASA H3D from Hah.8 The non-dimensional mass flow in these figures is defined to be the mass flow rate through the annulus at a given exit capacity (along the characteristic) divided by the mass flow rate at choke. For our computations the choke flow rate for Spalart-Allmaras was found to be 20.9402 kg/s, and for k − ω it was 20.9438 kg/s. We obtain acceptable agreement with experiments for pressure ratio and temperature ratio values. Radial distributions of the mass-mean total pressure and temperature ratios at 98% choke flow are plotted in Figure 5. These ratios correspond to the rotor exit stagnation pressure and temperature divided by the inlet static pressure and static temperature values respectively. Comparing the two turbulence models, good correlations between the total temperature ratio distribution are obtained. However, for the total pressure ratio, there are a few points to note. Firstly, the two turbulence models diﬀer in their prediction of the tip total pressure profile. Spalart-Allmaras predicts a marginal total pressure surplus from 90% span, while the surplus in k − ω is more dominant and begins at 70% span. Secondly, both turbulence models fail to capture the experimental hub deficit from 0% to 25% span. Other computational studies observe the same eﬀect (see Dunham6 ). This pressure deficit exists over a range of mass flows (at the design speed). While the tip pressure deficit smears out with increasing mass flow (towards choke), the hub deficit does not. Such observations were also noted by Strazisar,6 who concluded that these deficits were not an artifact of 5 of 21 American Institute of Aeronautics and Astronautics 1.32 2.25 2.2 1.3 2.15 Temperature ratio Pressure ratio 2.1 2.05 2 Experiments H3D−RANS (modified two equ.) H3D−LES SWIFT (BL) SWIFT(trans. stress lim.) HYDRA (SA) HYDRA (k−ω) 1.95 1.9 1.85 1.8 1.75 0.95 1.28 1.26 1.24 1.22 0.96 0.97 0.98 0.99 Non−dimensional mass flow 1.2 1 Experiments H3D−RANS (modified two equ.) H3D−LES SWIFT (BL) SWIFT(trans. stress lim.) HYDRA (SA) HYDRA (k−ω) 0.95 0.96 0.97 0.98 0.99 Non−dimensional mass flow (a) 1 (b) 100 100 90 90 80 80 Non−dimensional radius Non−dimensional radius Figure 4. Comparison of performance metrics in HYDRA using Spalart-Allmaras (SA) and k − ω turbulence models at 100% design speed: (a) Pressure ratio; (b) Temperature 70 60 50 40 30 20 Experiment k−ω Spalart−Allmaras 60 50 40 30 Experiment k−ω Spalart−Allmaras 20 10 0 1.9 70 10 1.95 2 2.05 2.1 2.15 Total pressure ratio 2.2 2.25 0 1.22 2.3 1.24 1.26 1.28 Total temperature ratio (a) 1.3 1.32 (b) Figure 5. Comparison of radial distributions with experiment at 98% choke mass flow: (a) Pressure ratio; (b) Temperature ratio 6 of 21 American Institute of Aeronautics and Astronautics downstream radial mixing. In fact, radial contours of the total pressure at the trailing edge also show the hub deficit. III. Uncertainty Quantification Methods Two polynomial-based, non-intrusive uncertainty quantification (UQ) techniques are pursued in this paper. These are stochastic collocation and pseudospectral approximations. A thorough analytical perspective on these two methods can be found in works by Xiu13 and Constantine et al.14 In both methods, a polynomial basis is used to generate a response surface from a given set of samples. This response surface can, in turn, be used to infer statistical data such as the mean, moment and kurtosis. The diﬀerence between the techniques is that while the collocation approach uses a Lagrange cardinal function basis, the pseudospectral method utilizes an orthogonal polynomial basis. The input samples in both cases are the nodes of quadrature rules. Generating these samples for a single variable is straightforward (see the paper by Trefethen15 for Clenshaw-Curtis and Gauss-Legendre rules). For multivariate problems several approaches exist including tensor grids (cross product of 1D quadrature rules), isotropic sparse grids16 (linear combination of 1D quadrature rules), anisotropic sparse grids17 (parameterweighted sparse grids) and adaptive sparse grids.18 Here we restrict ourselves to using isotropic tensor and sparse grids with the collocation and pseudospectral methods. An outline of these two methods follows. A. Spectral Collocation The spectral collocation approximation constructs an interpolating Lagrange cardinal function through points of a given stencil. These points may be from Clenshaw-Curtis, Gauss-Legendre or Newton-Cotes quadrature rules. The approximation is defined as: fs (u) = n−1 ∑ f (ξi ) Li (u) (1) i=0 where f (ξi ) corresponds to function evaluations at the quadrature points given by ξi . Here, the Lagrange ∏n−1 u−ξ cardinal function has the form Li (u) = j=0,j̸=i ξi −ξjj , for any i ≤ n − 1. Typically, Clenshaw-Curtis or Gauss rules are chosen as the stencil as they have non-equidistant points which are crucial when avoiding Runge-type spurious oscillations towards the ends of the domain. A more detailed study on stencil selection can be found in Trefethen’s book.19 In the same spirit as equation 1, the multivariate spectral collocation approximation can be written as: fs (u) = n1 ∑ n2 ∑ i1 =0 i2 =0 ... nd ∑ f (ξi1 , ξi2 , . . . , ξid ) Li1 (u1 ) Li2 (u2 ) . . . Lid (ud ) (2) id =0 ⇒ fs (u) = ∑ f (ξi ) Li (u) (3) i∈In where d is the number of variables, or the dimension of the problem, and In is the set of multi-indices of the quadrature points and polynomials, given to be { } In = i : i ∈ Nd , 1 ≤ ik ≤ nk ; k = 1, . . . , d (4) The multivariate polynomial basis here is eﬀectively a Kronecker product of univariate Lagrange cardinal functions. Once the spectral collocation approximation has been obtained, statistical data can be rendered by weighted sampling of the approximation, or by integrating the response surface with its input weights. B. Pseudospectral Approximation The pseudospectral approximation can be thought of as a truncated Fourier series approximation of a function. However, instead of using sines and cosines as the basis, orthogonal polynomials, πi (u), are used. The approximation can be written as: n−1 ∑ fp (u) = ϕi πi (u) (5) i=0 7 of 21 American Institute of Aeronautics and Astronautics where ϕi are the pseudospectral coeﬃcients. These coeﬃcients are obtained by projecting the function on a basis spanned by the normalized orthogonal (orthonormal) polynomials multiplied by their weighting function, w(u): ˆ ϕi = f (u) πi (u) w (u) du (6) U Using the quadrature rule corresponding to the chosen stencil, this integral can be expressed as: ϕi = n−1 ∑ f (ξj ) πi (ξj ) Wj (7) j=0 where Wj are the weights of the quadrature rule. As orthogonal polynomials have weights associated with them, the probability distributions of the input parameters typically dictate the choice of orthogonal polynomials. For instance, when using Hermite polynomials the quadrature weights in equation 7 are Gaussian, for Legendre polynomials they are uniform, and for a class of Jacobi polynomials they have a beta distribution, and so on. Using the apt orthogonal polynomials and their corresponding weights guarantees exponential convergence of the integral in equation 7 and the convergence of the pseudospectral coeﬃcients. The multivariate pseudospectral approximation is constructed via an index set as with collocation. It is defined as: nd n1 ∑ n2 ∑ ∑ fp (u) = ... ϕi1,2,...,d πi1 (u1 ) πi2 (u2 ) . . . πid (ud ) (8) i1 =0 i2 =0 id =0 ⇒ fp (u) = ∑ ϕi πi (u) (9) f (ξj ) πi (ξj ) Wj (10) i∈In ∑ where the coeﬃcients are: ϕi = j∈In Statistical moment computations for the pseudospectral method is particularly straightforward. The expectation or mean is simply the first pseudospectral coeﬃcient: E [f (u)] = ϕ0 (11) and the variance is the square of the mean subtracted from the sum of the squares of all the coeﬃcients: (n−1 ) ∑ (12) V [f (u)] = ϕ2i − ϕ20 i=0 1. Sparse Pseudospectral Approximation Method Sparse grid based pseudospectral approximations are a linear combination of tensor product pseudospectral operations. They are based on Smolyak’s algorithm, which can be expressed as: ∑ B= c (m) (U m1 ⊗ . . . U md ) (13) m∈I for a given univariate linear operator. Here |m| = m1 + m2 + . . . + md and the multi-index I corresponds to: { } In = m : m ∈ Nd , l + 1 ≤ |m| ≤ l + d (14) The coeﬃcients c (m) in equation 13 are defined as: c(m) = (−1) ( l+d−|m| d−1 l + d − |m| ) (15) where the parameter l is called the level, and governs the density of the sparse grid. The incremental number of points from one level to the next can be varied by using an appropriate growth rule – a linear growth rule, or an exponential growth rule. More details on sparse grid growth rules can be found in a paper by Eldred and Burkardt.20 The main advantage of sparse grids is that they do not compute points in the interior regions of the probability space, and thus save significant computational time by reducing the number of function evaluations. Details on the sparse pseudospectral method pursued here can be found in a paper by Constantine et al.14 8 of 21 American Institute of Aeronautics and Astronautics IV. Operational Uncertainties Stochastic Investigations We now apply the methods detailed above to quantify the eﬀect of operational uncertainties on axial compressor performance, through the metrics of eﬃciency and pressure ratio. Deterministic samples for the stochastic investigations here are computed by varying either the geometry or flow conditions. This results in a variation of the pressure ratio. Adiabatic eﬃciency comparisons for compressors with identical aerodynamic quality but diﬀerent pressure ratios are not valid, as the higher pressure ratio compressors will have lower adiabatic eﬃciencies. This penalization of higher pressure ratios is due to the divergence of constant pressure lines on a T-s diagram, which is a hidden eﬀect in the definition of the adiabatic eﬃciency.21 To circumvent this issue, we use the polytropic eﬃciencya which is the same for two compressors of identical aerodynamic quality but diﬀerent pressure ratios. A. Boundary Layer Thickness The eﬀect of inlet boundary layer thickness on compressor performance has been studied experimentally by Wagner22 and more recently through computations by Choi.23 In both works, the authors define a thin boundary layer to have a profile boundary layer thickness of δ ∗ = 0.88% span at the hub and δ ∗ = 1.45% span at the tip. For the thick inlet boundary layer case, these values were δ ∗ = 7.8% at the hub and δ ∗ = 7.9% at the tip. Using these values as a guideline, we distort the datum inlet pressure profile for Rotor 376 to obtain thick and thin boundary layer traverses. To ensure that the overall inlet flow conditions are the same for successive deterministic computations, we parameterize the curves in Figure 6(a), to ensure that the total inlet pressure (both radially and circumferentially mass-averaged) is constant. We assume that the actual profile is uniformly uncertain and lies somewhere between the thick and thin boundary layers. To propagate this boundary layer uncertainty a second-order, univariate, stochastic collocation approach is used with the objective of obtaining the mean and variance in the pressure ratio and eﬃciency. Mean eﬃciency values in the face of the uncertainty are shown in Figure 6(b) along with their standard deviations, which are plotted as error bars. Pressure ratio values showed negligible excursions and are therefore not presented. Full confidence intervals, which coincide with the supports of the output probability distribution of eﬃciency, are also shown. 88 87.8 0.24 87.6 Polytropic efficiency 0.25 Span, m 0.23 0.22 0.21 0.2 0.19 Baseline Thin boundary layer Thick boundary layer 87.2 87 86.8 86.6 Full confidence interval Standard deviation Baseline 86.4 86.2 0.18 0.96 87.4 0.98 1 1.02 Inlet total pressure, bars 86 17.5 1.04 18 18.5 19 Corrected exit massflow, kg/s (a) 19.5 (b) Figure 6. Eﬀect of uncertain inlet boundary layer thickness: (a) Inlet pressure profiles; (b) Eﬃciency characteristic The y-axis in Figure 6(b) is the polytropic eﬃciency and the x-axis is the corrected exit mass flow rate. On a characteristic, the x-axis is typically a formulation of the mass flow rate. When the focus is on the inlet conditions the non-dimensional mass flow or the corrected inlet mass flow rate are used. The latter is a By definition, the polytropic eﬃciency is always higher than the adiabatic eﬃciency for the same compressor. However, here we concern ourselves with the changes in polytropic eﬃciency as opposed to absolute values. 9 of 21 American Institute of Aeronautics and Astronautics √ √ given by m ˙ T0 /P0 · Pref / Tref , where Pref and Tref are the inlet reference conditions (typically, standard atmosphere). For the corrected exit mass flow rate here, we use Pref = Pout,choke and Tref = Tout,choke . Further details on this dimensionalization can be found in Cumpsty.21 The main point to note from this discussion is that all comparisons must be made at the same reference corrected mass flow rate as it represents operation at the same flow conditions. Returning to Figure 6, the mean values show a very small deviation from the baseline (no uncertainty) results. Standard deviation values are, however, far greater at stall, and virtually non-existent at choke. The full confidence intervals also follow the same trend. Overall, with thinner boundary layers the polytropic eﬃciency increases, while with thicker boundary layers, the eﬃciency drops. B. Metal Expansion Eﬀects On Axial Clearances Axial tip clearances have a profound impact on the mechanics of an aero-engine. These clearances exist to accommodate thermal and centrifugal growths of the disk, blades and casing. These allowances also take additional factors into account, such as production tolerances and the impact of gas loads. Excursions in geometry for a standard axial compressor are shown in Figure 7. Resultant values of these clearances can exhibit considerable variations in flight. These variations result in overall eﬃciency and surge margin reduction and are therefore important to quantify. In the investigation that follows, four scenarios based on the above discussion are considered: (1) uncertainty in blade centrifugal growth, (2) casing thermal expansion, (3) disk expansion, and (4) in both disk and casing expansion. In cases (1)–(3), the eﬀective tip gap uncertainty is maintained, while for case (4) a wider tip gap excursion is explored. For case (1) the blade span is varied, while for cases (2)–(4) the blade span is fixed. Deterministic computations for each uncertainty require a modification of the mesh of the geometry. In PADRAM, hub and casing lines can be raised or altered parametrically, and a new mesh with diﬀerent endwall distributions across the annulus can be generated automatically. Furthermore, blade span can be varied by specifying the desired tip gap using a rolling ball algorithm. A combination of these Figure 7. Change in clearances from build to cruise, approaches – altering the casing and hub endwalls adapted from Freeman2 and the tip gap – is used to generate the geometry (and the meshes) for the following simulations. 1. Blade Expansion-Contraction In this study the eﬀect of an uncertain blade expansion at steady state is examined, keeping both hub and casing fixed. This results in a varying tip gap. Tip gap measurements for NASA Rotor 37 taken by Suder,24 reveal that at 100% speed the tip gap was 0.356 mm, while at 80% speed, this increased to 0.5 mm and at 60% it increased further to 0.58 mm. This corresponds to a 0.5-0.8% span tip clearance. Here, we assume that at 100% speed, the blade may exhbit the same uncertainty range. Thus it may contract by 0.144 mm or expand marginally by 0.033 mm, leading to a tip gap uncertainty of 0.5 mm to 0.323 mm, as shown in the schematic in Figure 8. Here we consider a beta(2,1) and uniform distribution for the tip gap arising from blade expansion. The beta distribution is chosen to reflect the high probability that the measured tip gap will be around the datum value of 0.356 mm. The uniform distribution is an artifact of no a priori information on the problem. Stochastic collocation on a Chebyshev stencil is used to propagate the blade expansion/contraction uncertainty. The results for the peak eﬃciency condition are summarized in Tables 1 and 2. Owing to the linear relationship between the tip gap and performance metrics, acceptable convergence for the polytropic eﬃciency 10 of 21 American Institute of Aeronautics and Astronautics Figure 8. Eﬀect of blade expansion-contraction on eﬀective tip gap (η) and pressure ratio (PR) is observed even for a first-order expansion. It is also clear from the tables that the beta distributed blade expansion has higher mean PR and η values, with lower variances. Table 1. Summary of blade expansion UQ study for tip gap, close to peak eﬃciency for a uniform distribution Order 1 2 3 Mean PR 2.0945 2.0944 2.0943 Var. PR 5.9533e-5 5.9494e-5 5.9494e-5 Mean η 87.355 87.349 87.347 Var. η 0.04211 0.04234 0.04346 Table 2. Summary of blade expansion UQ study for tip gap, close to peak eﬃciency for a beta(2, 1) distribution Order 1 2 3 Mean PR 2.1002 2.1001 2.1002 Var. PR 1.8247e-5 1.8413e-5 1.8776e-5 Mean η 87.507 87.501 87.502 Var. η 0.01284 0.01328 0.01383 The blade growth study was extended to other points along the constant velocity characteristic. It was found that at high mass flow rates (close to choke), the response surfaces were almost constant. At lower mass flow rates, close to stall, the response surfaces were quadratic in form. It was observed that closer to stall, N + 1 polynomial orders were required for reasonable accuracy, compared to N for peak eﬃciency and N − 1 for choke. Complete characteristics are shown in Figure 9 with standard deviations of both distributions along with their full confidence intervals. The eﬀect of blade growth on pressure ratio, for the range of values selected, is rather limited. Variations in eﬃciency on the other hand are quite large. Close to stall, the standard deviation in eﬃciency, owing to a uniform uncertainty in blade growth is approximately 0.8%. At peak eﬃciency (corrected mass flow rate of 18.9 kg/s) this values reduces to 0.38%. Closer to choke, both the full confidence levels and standard deviations drop. 2. Disk Expansion-Contraction A disk expansion/contraction study was carried out by raising and lowering the hub endwall, whilst keeping the casing and blade span fixed. This resulted in the blade shifting radially outward during expansion and moving radially inward during contraction. Disk movement was attributed to both centrifugal and 11 of 21 American Institute of Aeronautics and Astronautics 88 87.8 Polytropic efficiency 87.6 87.4 87.2 87 86.8 86.6 Full confidence interval Standard deviation (uniform) Standard deviation (beta) Baseline 86.4 86.2 86 17.5 18 18.5 19 Corrected exit massflow, kg/s (a) 19.5 (b) Figure 9. Compressor characteristic for blade expansion-contraction uncertainty for: (a) Pressure ratio; (b) Polytropic eﬃciency (plotted are mean values with standard deviation error bars) thermal expansion and was assumed to be within the range from −0.144 mm to +0.033 mm. These bounds were chosen as they were in tune with the metrics in Figure 7 and because they resulted in an eﬀective tip gap uncertainty of 0.323 mm to 0.5 mm, just as in the prior study (see Figure 10). Once again we use stochastic collocation on a Chebyshev stencil to propagate this univariate uncertainty. Results for the characteristic are shown in Figure 11(a) and (b). Figure 10. Eﬀect of disk expansion-contraction on eﬀective tip gap As the disk contracts, the tip gap increases leading to pressure losses and reduced eﬃciency. As the disk expands, the tip gap is reduced and the eﬃciency rises. The trends shown in Figure 11 are quite similar to those in Figure 9, wherein close to choke there is hardly any variability in the pressure ratio and eﬃciency and the opposite close to stall. When plotted with the mean values of the blade expansion study (uniform distribution case), the mean values of the disk expansion polytropic eﬃciency are a few tenths of a percent lower. At peak eﬃciency, there is a 0.3% drop in eﬃciency due to the disk movement. 3. Casing Expansion-Contraction In real compressors, the casing undergoes both axial and circumferential variations in growth. Thus, if seen from the annulus line, at a fixed axial location the eﬀective tip gap across the annulus is not constant. Here, as in the prior metal expansion studies, we model coupled circumferential and axial variations in 12 of 21 American Institute of Aeronautics and Astronautics 2.2 87.8 87.6 87.4 Polytropic efficiency Pressure ratio 2.15 2.1 2.05 2 1.95 17.5 Full confidence interval (disk) Standard deviation (disk) Baseline Standard deviation (blade) 18 18.5 19 Corrected exit massflow, kg/s 87.2 87 86.8 86.6 Baseline Uniform blade expansion Uniform disk expansion 86.4 86.2 86 17.5 19.5 18 18.5 19 Corrected exit massflow, kg/s (a) 19.5 (b) Figure 11. Compressor characteristic for disk expansion-contraction uncertainty for: (a) Pressure ratio; (b) Polytropic eﬃciency (plotted are mean values with standard deviation error bars) growth using a single parameter to characterize the overall displacements. Uncertain casing thermal expansion and contraction was simulated with a fixed hub and blade span, as shown in Figure 12. The casing rise was restricted to 0.144 mm and its contraction to 0.033 mm, thereby ensuring the tip gap variation was the same as the prior two studies. Results for the casing shift are shown in Figure 13. Figure 12. Eﬀect of casing expansion-contraction on eﬀective tip gap As with the disk movement study, for the same eﬀective tip gap, the uncertainty in casing expansion yields lower mean polytropic eﬃciencies compared to the blade growth study. The full confidence level here is 0.7% of the eﬃciency, and the standard deviation is 0.6%. 4. Combined Disk and Casing Expansion-Contraction In the prior subsections dealing with metal expansion and contraction uncertainties, the stochastic variation in the tip gap was held constant. As stated earlier, this was done such that a fair comparison could be made between the eﬀects of hub, blade and casing displacement. In this subsection, both hub and casing displacements are varied simultaneously, using the same uncertainty bounds as in the previous studies. The tip gap thus varies from a lowest value of 0.29 mm to a maximum of 0.644 mm. For this bivariate uncertainty, tensor grids are utilized with the collocation method on Clenshaw-Curtis quadrature points. A summary of 13 of 21 American Institute of Aeronautics and Astronautics 2.2 87.8 87.6 87.4 Polytropic efficiency Pressure ratio 2.15 2.1 2.05 2 1.95 17.5 Full confidence interval (cas.) Standard deviation (cas.) Standard deviation (blade) Standard deviation (disk and cas.) Baseline 18 18.5 19 Corrected exit massflow, kg/s 87.2 87 86.8 86.6 86.4 86.2 86 19.5 17.5 18 18.5 19 Corrected exit massflow, kg/s (a) 19.5 (b) Figure 13. Compressor characteristic for casing expansion-contraction uncertainty for: (a) Pressure ratio; (b) Polytropic eﬃciency (plotted are mean values with standard deviation error bars) the results is shown in Table 3. Table 3. Summary of combined hub and casing movement results Tensor grid order [1, 1] [3, 3] Mean η 86.9472 86.9729 Var. η 0.1003 0.0900 Mean tip gap (mm) 0.467 0.467 Var. tip gap (mm) 0.00523 0.00524 Here the [1, 1] tensor grid has 4 points (all end points), while the [3, 3] tensor has 16 points. The firstorder approximation is able to capture 99% of the variability. Upon closer investigation, it was found that both polytropic eﬃciency and pressure ratio exhibited a linear response to changes in both casing and disk movements. One of the main findings of the metal expansion study was the highly linear response surfaces for pressure ratio and eﬃciency. Thus, for a given geometric variation, as long as the mass flow rate stays within the peak eﬃciency to choke envelope, domain end points may be suﬃcient to capture the mean and variance of the uncertainties in question. C. Hub-Leakage Flows Leakage flows have been shown to have detrimental eﬀects on axial compressor performance by significantly altering the blockage distribution along the main flow path and causing a pressure loss.25, 26 These flows are created by pressure diﬀerentials that exist across small clearances or channels, thus fostering fluid movement from high pressure regions to low pressure ones. Leakage flows can be extracted from the main power stream via bleeds, or injected into the main power stream in the form of jets. In both experimental test rigs and aero-engines, they commonly arise from gaps between rotating and stationary bodies.25 Leakage flows for NASA Rotors 37 and 35 have been investigated by Shabbir et al.26 In fact, Shabbir et al. postulated that in the actual experiments conducted by Reid and Moore,7 there was a leakage flow emanating fore of the leading edge, between the rotating center body and stationary hub. Their hypothesis was that the leakage flow triggered a pressure deficit at the hub, which is what was observed in the experiment. To simulate this leakage flow, they used a small surface patch with a mass flow inlet boundary condition. In their study, they assumed that the flow emanating from the patch had a whirl velocity equivalent to half the rotor speed, ωleak = 900 rad/s, and an inflow angle of θleak = 90◦ in the meridional plane. Inlet mass flow rates were varied from 0.10% to 0.333% of the choke flow rate. Here, we carry out a similar study, but from a stochastic perspective. We treat the leakage flow to be uncertain governed by the same three uncertain parameters: leakage mass flow rate, whirl velocity and inflow angle. Based on Shabbir’s work we 14 of 21 American Institute of Aeronautics and Astronautics 60 40 θ θ leak 80 60 leak 80 60 θ leak 80 40 0.3 0.3 1200 0.25 0.2 /m leak choke 0.3 1200 0.25 0.2 1000 0.15 m 40 0.15 800 0.1 m ωleak 600 /m leak (a) choke 1200 0.25 0.2 1000 0.15 800 0.1 600 1000 m ωleak /m leak choke 800 0.1 (b) 600 ωleak (c) Figure 14. Deterministic samples for hub-leakage uncertainty: (a) Tensor grid [2,2,2]; (b) l = 1 sparse grid; (c) l = 2 sparse grid define the bounds for each parameter in Table 4, along with its assumed probability distribution. To resolve the leakage flows, emanating from the small gap, we use an extruded patch, as shown in Figure 3(b). Initial leakage computations were carried out for a fixed exit corrected mass flow of 17.8 kg/s, located between peak eﬃciency and stall. To propagate the leakage uncertainty, both the stochastic collocation on tensor Table 4. Uncertain parameters, bounds and distribution for hub-leakage flow No. 1. 2. 3. Parameter m ˙ leak θleak ωleak Uncertainty bounds 0.10% − 0.333% choke 25◦ − 90◦ 600 − 1200 rad/s Uncertainty distribution uniform uniform uniform grids and the sparse pseudospectral approximation method14 are used. For the collocation approach, a Chebyshev stencil (with end points) of univariate order two (n = 2) is utilized. This corresponds to a total of N = (n + 1)3 = 27 deterministic samples. Higher-order tensor grid computations were not carried out, as for n = 3 the total number of samples becomes N = 64. Furthermore, due to a lack of a priori information regarding the sensitivities of the various parameters, anisotropic tensor grids could not be used. For the sparse pseudospectral approximation method, a sparse grid of levels 1 and 2 and a linear growth rule are used. Gauss-Legendre quadrature points are used for the stencil. The total number of samples required for the sparse grids are 7 and 25 respectively. In fact, all 7 points for level 1 are nested within the points in level 2. These stencils are shown in Figure 14. Computed mean and variance values for the polytropic eﬃciency are shown in Table 5. From the table it is clear that an acceptable level of numerical Table 5. Summary of hub-leakage results using an extruded surface patch Method Collocation Pseudospectral Pseudospectral Stencil Tensor - Chebyshev Sparse - Gauss-Legendre Sparse - Gauss-Legendre Order [2,2,2] l=1 l=2 Mean η 86.0273 86.0287 86.0442 Var. η 0.0178 0.0148 0.0225 accuracy is obtained with the level 1 sparse grid for the mean polytropic eﬃciency. Variance values for this low-order expansion seem less trustworthy. It is hypothesized that the wide range of variance values may be a facet of operation between stall and peak eﬃciency. In other words, samples suﬃcient for a linear approximation may not be enough for uncertainties in this region, due to a more non-linear response surface. To understand the contribution of the three unknown parameters to the overall pressure ratio and eﬃciency, samples from both sparse and tensor grids were plotted in parallel coordinates (see Figure 15). From the plots it is apparent that for low values of m ˙ leak /m ˙ choke , high pressure ratios and eﬃciencies are maintained. Thus, here the output is insensitive to changes in ωleak and θleak . For high values of m ˙ leak /m ˙ choke and moderately high values of ωleak , the eﬃciency drops along with the pressure ratio. Thus 15 of 21 American Institute of Aeronautics and Astronautics (a) (b) Figure 15. Parallel coordinate plots for hub-leakage study: uncertain inputs that yield: (a) A high polytropic eﬃciency; (b) A low polytropic eﬃciency for future leakage simulations, using anisotropic sparse or tensor grids, a higher weight could be given to the uncertainty in m ˙ leak /m ˙ choke . Such weighted approaches would secure quicker convergence. Radial distributions of the total pressure ratio were computed using both pseudospectral and collocation approaches. These are shown in Figure 16(a), along with the collocation (tensor) full confidence interval. 100 88 Full confidence interval Baseline Std.dev. sparse, l=2 90 80 87.5 3 Polytropic efficiency Percentage span Std.dev. tensor, 2 70 60 50 40 30 20 87 86.5 86 Standard deviation (leak) 10 0 1.95 Baseline 2 2.05 2.1 Total pressure ratio 2.15 85.5 17.5 2.2 18 18.5 19 Corrected exit massflow, kg/s (a) 19.5 (b) Figure 16. Hub-leakage study carried out with extruded surface patch: (a) Radially mass-averaged total pressure ratio; (b) Polytropic eﬃciency (plotted are mean values with standard deviation error bars) The large excursion from 10% to 40% span is attributed to high leakage flow rate and whirl velocity. There is a large discrepancy between full confidence levels and standard deviation error bars. This discrepancy reduces from 50% span onwards. Both full confidence intervals and the UQ results exhibit a small variance towards the tip. Overall, both sparse and tensor grids show extremely good correlations. The overall loss in polytropic eﬃciency can be quite significant for leakage flows. The study discussed herein was extended to another operating point to obtain the characteristic shown in Figure 16(b). We report a 1.1% loss in eﬃciency due to the leakage flow. Mean and standard deviation stagnation pressure flow fields were computed using the sparse pseudospectral approach. Figure 17 plots these values for an axial slice taken towards the trailing edge of the rotor blade. Also shown is the baseline, no leakage case for comparison. The left-hand side of each figure corresponds to the pressure side, while the right-hand side corresponds to the suction side. Compared to the peak eﬃciency contour plot, the mean leakage case has a larger low stagnation region at the hub and another one ranging from 60% span to the tip, both on the suction side. This explains the pressure deficit seen in these two regions in Figure 16. Interestingly, the leakage flow does not seem to perturb the stagnation pressure contours on the pressure side. The contour plot of the standard deviation of 16 of 21 American Institute of Aeronautics and Astronautics the stagnation pressure reveals that for a variation in the three leakage uncertainties, the suction side hub corner experiences the greatest sensitivity. There also seems to be a radial migration of the leakage flow from the hub to the core flow region, which also experiences some sensitivity to the three uncertainties. On the pressure side, a small amount of cross flow at the hub is observed. (a) (b) (c) Figure 17. Trailing edge stagnation pressure contours for: (a) Baseline case with no leakage; (b) Mean leakage condition; (c) Standard deviation leakage condition. Contours obtained through sparse pseudospectral approximation of 25 sparse grid samples for extruded suface patch D. Surface Roughness Numerous studies27, 28 have looked into compressor performance degradation with the introduction of surface roughness. Surface roughness is caused by the deposition of dust, sand and other particulates on blades. It may also be brought about by intrinsic factors such as a manufacturing roughness or a thermal barrier coating29 for turbine blades. With increased surface roughness, the onset of laminar-turbulent transition shifts upstream towards the leading edge. Also, the skin friction arising from the turbulent boundary layer increases, causing performance degradation. The surface roughness model used in HYDRA is based on the law of the wall for a rough surface (see Guo et al.29 ). It involves a shift in the velocity profile from the smooth law of the wall along with an integration of the momentum equations across the wall layer to reconcile the wall condition with the grid point directly above the wall. In this study, the surface roughness of the rotor blade was assumed to be uniformly uncertain, with centre line roughness values ranging from Ra = 6.45 to 9.97 microns. It should be noted that modern compressor blades have an order of magnitude lower surface roughness values. The roughness value of 6.45 microns was obtained from Morini et al.,28 but could not be substantiated. Nevertheless, we use this value bearing in mind that this is an academic exercise. We utilize stochastic collocation on a Chebyshev stencil. For this study, both baseline and rough simulations were carried out using the k − ω turbulence model. Representative results are shown in Figure 18. The inclusion of surface roughness leads to a consistent drop in both pressure ratio and polytropic eﬃciency. Variance changes from choke to stall are rather small in contrast to the characteristics obtained in the prior studies. V. Epistemic Stochastic Investigations Epistemic uncertainties arise due to the physical limitations of our computational models. In this section, we analyze the eﬀect cavity-based meshing on the performance metrics. A. Patch Modeling In the prior leakage studies, a single RANS computation on an 8-core, 3.5 GHz machine took approximately one day. This highlights the main problem when simulating fully meshed cavities: long running times. To circumvent this, it is a usual practice in turbomachinery to resort to using surface patches with mass inlet boundary conditions. Surface patch simulations on the same computer were found to reach convergence 17 of 21 American Institute of Aeronautics and Astronautics 2.14 87.4 Full confidence interval Std. dev. surface roughness (k−ω) Baseline (k−ω) 2.12 87.2 87 Polytropic efficiency Pressure ratio 2.1 2.08 2.06 2.04 86.8 86.6 86.4 2.02 86.2 2 86 1.98 18 18.2 18.4 18.6 18.8 19 19.2 Corrected exit massflow, kg/s 19.4 19.6 Full confidence interval Std. dev. surface roughness (k−ω) Baseline (k−ω) 18 18.2 18.4 18.6 18.8 19 19.2 Corrected exit massflow, kg/s (a) 19.4 19.6 (b) Figure 18. Eﬀect of surface roughness on compressor characteristic: (a) Pressure ratio; (b) Polytropic eﬃciency (plotted are mean values with standard deviation error bars) in approximately nine hours – thus highlighting their advantage. In this section, we seek to determine the epistemic uncertainties that arise when resorting to surface patches to simulate leakage flows. To quantify these uncertainties, a tensor grid of samples corresponding to a [2,2,2] order was evaluated at the same points shown in Figure 14(a) using a surface patch mesh (see Figure 3(a)). Radial distributions comparing the full confidence intervals are shown in Figure 19(a) and the standard deviations in Figure 19(b). The full confidence intervals show that the total pressure probability density function has a larger support 100 100 90 Cavity Extruded patch 80 Patch Percentage span Percentage span 80 Surface patch 60 40 70 60 50 40 30 20 20 10 0 1.95 2 2.05 2.1 2.15 Total pressure ratio 2.2 0 1.95 2.25 2 2.05 2.1 Total pressure ratio (a) 2.15 2.2 (b) Figure 19. Comparison of cavity modeling methods for radial distributions of mass-mean pressure ratio: (a) Full confidence intervals; (b) Standard deviation and mean values. Results were obtained from tensor grid based stochastic collocation using 27 samples when a leakage patch is used. This is particularly dominant from the hub to 20% span. Mean and standard deviation values from Figure 19(b) show remarkably good correlations between the two traverses. VI. Ranking of Uncertainties The aim of the present work has been two-fold. On the one hand, we have sought to apply a set of computationally inexpensive methods to quantify operational uncertainties in an axial compressor. On the other hand, we seek to determine which of these uncertainties is the most dominant, and thus a suitable 18 of 21 American Institute of Aeronautics and Astronautics candidate for robust optimization. In this section we address the latter objective. To carry out a fair comparison between the results from the various uncertainty studies, the mean polytropic eﬃciency values are compared with the polytropic eﬃciency obtained for the baseline (no uncertainty propagation) case at a corrected exit mass flow close to peak eﬃciency. The loss in mean polytropic eﬃciency (expressed as a percentage) from the baseline case was computed for the chief uncertainties presented in the prior sections. These include the eﬀect of disk and hub expansions, leakage flows, surface roughness and blade growth. The histogram in Figure 20 shows the relative eﬀect of the uncertainties. Inlet profile distortions and the epistemic uncertainties are not presented here, as depending on their uncertainty bounds, they were found to increase the polytropic eﬃciency. 1.4 Loss in mean efficiency 1.2 1 0.8 Disk and hub movement Leakage flows Surface roughness Blade growth Casing movement Disk movement 0.6 0.4 0.2 0 Figure 20. Loss in mean polytropic eﬃciency for uniformly distributed operational uncertainties close to peak eﬃciency As shown in the graph, leakage flows were found to be the most dominant uncertainty, resulting in an eﬀective loss of approximately 1.1%, followed by geometry changes associated with disk and hub expansions. Surface roughness, casing and disk movements had approximately the same contributions to overall loss at 0.4%. Blade centrifugal growth and contraction was found to have the least eﬀect at 0.34%. It should be noted that this histogram represents the eﬀect of these uncertainties at 100% design speed, and it is expected that losses resulting from geometric variations would be diﬀerent at lower speeds. Conclusions In this study, the major uncertainties (both operational and epistemic) inherent in a transonic axial compressor were propagated. The eﬀect of these uncertainties on the polytropic eﬃciency and pressure ratio were quantified, through a series of computationally aﬀordable techniques, for five aleatory uncertainties and one epistemic case. The uncertainties and their bounds were chosen from prior experimental and computational investigations and represent the uncertainties encountered by axial compressors. Stochastic collocation and pseudospectral approximations on both tensor and sparse grids were used to propagate these uncertainties. The following are some of the key conclusions of the present work: 1. Compressor characteristics for most uncertainties showed greater variation in eﬃciency and pressure ratio values close to stall than close to peak eﬃciency and choke. 2. The response surfaces generated for the operational uncertainties revealed that most of the uncertainties exhibit a linear behavior with respect to the global performance metrics. 3. Sparse grids were found to provide representations as accurate as those given by tensor grids, with far fewer RANS computations. Far coarser, but reasonable, estimates of multivariate uncertainties can also be obtained by domain end point sampling. 19 of 21 American Institute of Aeronautics and Astronautics 4. Hub-leakage flows were found to be the most dominant aleatory uncertainty followed by losses due to geometric uncertainties. Contours of mean stagnation pressure at the trailing edge revealed the greatest sensitivity to leakage flows was on the suction side at the hub corner. 5. For geometric uncertainties, the eﬀective tip gap was the most important length factor. For studies conducted with the same tip gap uncertainty, albeit triggered by blade growth, casing or hub expansion, the eﬀect on performance loss was almost identical. 6. Surface patch based cavities were found to represent leakage flows accurately when compared with extruded surface patches – for radial traverses of total pressure. Patch solutions converged in roughly a third of the time it took to run extruded patch cases. On the whole, the total number of 3D RANS computations performed was small as most of the uncertainties were accurately represented by second- and third-order polynomials. This small number and the design advantages than can be anticipated justifies housing these UQ methods within optimization loops to facilitate robust design. Acknowledgements The authors are grateful to Rolls-Royce plc for permission to publish this work. The authors would like to thank David Radford for his advice regarding some aspects of this research. The authors would also like to thank Richard Northall and Dario Bruna for their helpful discussions. This study was funded through a Dorothy Hodgkin Postgraduate Award, under a joint partnership between EPSRC and Rolls-Royce plc. References 1 Walsh, P. P. and Fletcher, P., Gas Turbine Performance, Blackwell Science, 1998. C., “Tip Clearance Eﬀects in Axial Turbomachines,” Von Karman Institute, Lecture Series 1985-05 , 1985. 3 Ghisu, T., Parks, G. T., Jarrett, J. P., and Clarkson, P. 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