Effect of Gradient Reconstruction Method on

Transonic Fan Performance

Forrest L. Carpenter∗ and Paul G. A. Cizmas†

Texas A&M University, College Station, TX, 77843, United States

This paper presents an investigation of the choice of gradient evaluation method onthe performance of a transonic fan, namely NASA Rotor 67. Turbulent solutions werecomputed using gradients found via the Green-Gauss, un-weighted least-squares with QRdecomposition (LSQR), weighted LSQR (WLSQR), and weighted essential non-oscillatory(WENO) methods. Partial speedlines at the design wheel speed were computed for eachgradient case, and the flow at the near peak efficient point was analyzed. The un-weightedLSQR gradient solutions were found to under-predict all performance quantities, while theother three methods showed good agreement with experiment. Increased flow separationwas found to be associated with the un-weighed LSQR solutions, leading to higher losses.

Nomenclature

E Total energy per unit massH Total enthalpyk Thermal conductivity coefficient

n Normal vector, n = nx, ny, nzTp Pressure

r Position vector, r = x, y, zTt TimeT TemperatureU Transport velocity vector, U = ω × rU Normal transport velocity component, U = U · nV Absolute velocity vector, V = vx, vy, vzTW Relative velocity vector, W = V −UW Relative contravariant velocity, W = W · n(x, y, z) Cartesian coordinatesηtt Adiabatic efficiencyκ Turbulent kinetic energyµ Dynamic viscosityµT Turbulent eddy viscosityπ∗ Total pressure ratio, π∗ = p02

/p01

ρ DensityΩ Cell volume

ω Angular velocity vector, ω = ωx, 0, 0Tω Rate of dissipation per unit turbulent kinetic energySubscript0 Total, or stagnation, quantity1 Inlet or upstream condition2 Outlet or downstream conditionEQ Equivalent conditioni, j Node indices

∗Graduate Research Assistant, Aerospace Engineering, College Station†Professor, Aerospace Engineering, College Station, Associate Fellow

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I. Introduction

The accuracy of a typical computational fluid dynamics (CFD) solver greatly depends on the fidelityof the gradient calculation. The gradient calculation is need for estimating viscous stresses found in themomentum and energy balance equations,1 as well as for obtaining higher-order spatial accuracy.2 Forturbulence modeling, the gradients of velocity are responsible for production,3,4 while the gradients of theturbulence variables are required for flux computations.

A number of methods are available for the evaluation of the gradients. The two most basic methods arebased on the Green-Gauss theorem,5 and on the least-squares method.5 A variation on the standard least-squares approach decomposes least-squares matrix into orthogonal matrix Q and an upper right triangularmatrix R to avoid solving the system using matrix inversion.6,7 Row weighting can be applied to theleast-squares system to improve the accuracy of the calculation.6

It has been shown that using the un-weighted least-squares approach on meshes with high aspect ratiosand curvature results in poor approximations of the gradients.8,9 Elements like these are commonly found inthe boundary layer region of meshes, a region where accurate estimation of the gradients are essential. Usingthe inverse distance between connected nodes as a weighting factor has shown to correct the deficiencies ofthe un-weighted least-squares.8,9

One alternative to the basic gradient evaluation methods previously mentioned is the weighted essentiallynon-oscillatory (WENO)10 scheme. This scheme is well suited for smooth flows which contain shocks.11 Onebenefit of this scheme is that it inherently supplies a measure of limiting to the reconstruction.

This paper explores the effect of the gradient evaluation method on the performance prediction of atransonic fan. The gradient evaluation methods used were the Green-Gauss, WENO, and the least-squareswith QR decomposition methods. Both an inverse-distance weighted and an un-weighted least-squaresformulation were considered.

The governing equations of the flow are presented first, followed by an overview of the numerical methodsused. The details of the transonic fan are then presented. Following this, a comparison is made of the overallperformance of the fan predicted by each gradient method, and a detailed look at the predicted flow at nearpeak efficiency is presented.

II. Flow Model

The flow of fluid within turbomachinery is governed by the equations for the conservation of mass,momentum, and energy. Collectively, these equations are known as the Navier–Stokes equations, and arealso referred to herein as the governing equations. It is convenient for rotating geometries to write thegoverning equations in a relative reference frame which is affixed to the rotating geometry. In doing so, theunsteady motion of the geometry can be solved in a steady fashion. The governing equations, written in therelative reference frame, are given by1

∂

∂t

∫Ω

QdΩ +

∮∂Ω

(Fc − Fv) dS =

∫Ω

GdΩ . (1)

The conservative state vector is given by Q = ρ, ρV , ρET , where the velocity has been cast in termsof absolute velocities.12,13 The source term in Eq. (1) arises from writing the governing equations in therelative reference frame, and accounts for both the Coriolis and centrifugal forces. The vector of sourceterms, written using absolute velocities, is given by G = 0, 0, ρvzωx, −ρvyωx, 0T .

The convective and viscous flux vectors found in Eq. (1) are given, respectively, by

Fc =

ρW

ρvxW + pnx

ρvyW + pny

ρvzW + pnz

ρHW + pU

Fv =

0

nxτxx + nyτxy + nzτxz

nxτyx + nyτyy + nzτyz

nxτzx + nyτzy + nzτzz

nxΘx + nyΘy + nzΘz

, (2)

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where the components of the shear stress tensor are given using Einstein notation by

τij = µ

(∂vi∂xj

+∂vj∂xi− 2

3

∂vk∂xk

δij

),

and the viscous work plus heat conduction term is

Θℵ = v1τℵx + v2τℵy + v3τℵz + k∂T

∂ℵ. (3)

The ℵ in Eq. (3) is a placeholder variable that can be interchanged with x, y, or z.Equation 1 was both Reynolds and Favre averaged so that the compressible turbulent flow could be

approximated. The Favre-averaged Reynolds stress tensor, which resulted from the averaging, was approx-imated using the Boussinesq hypothesis. The κ − ω SST turbulence model by Menter3,14 was used tocomputed the eddy viscosity. The frame invariant definition for the eddy viscosity is given by15

µT =a1κ

max (a1ω, |Sij |F2),

where |Sij | is the magnitude of the strain rate tensor, F2 is a blending function,14 and a1 is a model constant.The values of the model constants were taken from Ref. 14.

III. Numerical Method

The governing equations were solved using an in-house, unstructured flow solver initially developed forturbomachinery applications.16,17 Since then it has been applied to stationary internal flows,18,19 and toexternal aeroelastic cases.20

The flow solver uses the finite volume method (FVM) to discretize both the governing equations andthe two turbulence model equations. A cell-vertex spatial discretization is utilized by the solver, with thecontrol volumes being specified by a median-dual sub-mesh. Using the FVM approximations for the volumeand surface integrals found in Eq. (1) leads to the discrete form of the governing equations

Ωi

∆ti∆qi = −

NV∑j=1

(Fc − Fv)ij Sij − Ωigi

, (4)

where NV is the number of vertices connected via edge to node i, Sij is the edge based area for the edgeconnecting nodes i and j, and Ωi is the volume of the median-dual associated with node i. The vectors qand g in Eq. (4) are the volume averaged state and source vectors, respectively.

The convective fluxes in Eq. (4) were computed using Roe’s approximate Riemann solver21 with Harten’sentropy fix.22 Viscous fluxes were averaged at the dual-mesh faces. Directional derivatives were used whencomputing the average gradients at the faces to prevent solution decoupling.7

Piecewise linear reconstruction2 was used to obtain second-order spatial accuracy. The required gradientsof the flow variables were computed using one of four methods: (1) Green-Gauss,5 (2) un-weighted least-squares with QR decomposition (LSQR),7 (3) inverse-distance weighed LSQR (WLSQR),23 and (4) weightedessentially non-oscillatory (WENO).10 The computed gradients were stored at the vertices of the mesh andwere also used to calculate the viscous shear stresses. The Dervieux24,25 limiter was used to prevent thecreation of non-physical oscillations in regions of strong gradients.

Equation (4) was integrated in time using the General Minimal Residual Algorithm (GMRES).26 Thediscrete form of the turbulence model equations were integrated using a second-order accurate four-stageRunge-Kutta scheme.27 Multiple Runge-Kutta steps were taken during each GMRES step such that theturbulence equations were stepped forward in time by the same amount as the governing equations. Localtime-stepping28 was utilized for convergence acceleration.

IV. Results

This section explores the effect of the gradient reconstruction on the prediction of the performance of atransonic fan. The first part of the section describes the transonic fan and its operating conditions. This isfollowed by the prediction of the partial compressor map speedlines. The section ends with a comparisonbetween the predicted and measured flows at near peak efficiency.

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IV.A. Transonic Fan Description

The NASA Rotor 67 geometry was chosen as the test case due to its popularity with other researchers29–34

and the availability of experimental data.35,36 Rotor 67 is the first-stage rotor of a two-stage fan developedat NASA Lewis.37 Twenty-two multiple-circular-arc blades make up the full rotor. The rotor has a designpressure ratio and mass flow rate of 1.63 and 33.25 kg/s, respectively, at a wheel speed of 16,043 RPM. Theaxial hub and tip chords are 91.04 mm and 40.29 mm, respectively.

IV.A.1. Mesh Generation

The computational domain extended five axial hub chords upstream and downstream of the rotor to reducethe effect of reflected waves from the boundaries on the solution. The domain was discretized using 3,387,830nodes using structured grid blocks.1 The entire computational domain is shown in Fig. 1a. The mesh wasconstructed by stacking topologically identical layers along the rotor’s span. Figure 1b shows the structuredmesh on the tip grid layer. A total of 135 grid layers were used to define the rotor with 167 points definingthe rotor cross-section on each layer. The viscous spacing along the rotor varied with the chord to providea near constant nondimensional viscous spacing of y+ = 2.

The tip leakage region of the domain was meshed by adding two grid blocks within the interior of therotor’s internal volume.1 Forty spanwise layers were used to define the desired gap height. A constantviscous spacing of z1 = 1.2× 10−3 mm was applied normal to the rotor tip. Figure 1c shows the distributionof spanwise layers for the tip leakage grids.

IV.A.2. Boundary Conditions

Freestream conditions were defined using the standard conditions at sea level (p∞ = 101325 Pa, T∞ =288.3 K). Inlet total conditions were set isentropically for an absolute Mach number of 0.55. The computedtotal conditions were: p01

= 124452.6 Pa and T01= 305.7 K. Flow was assumed to enter the domain axially.

The turbulence length scale and intensity were used to define the inlet turbulence conditions. The valuesof the length scale and intensity were 0.2 mm and 0.64%, respectively. The turbulence boundary conditionsused resulted in an inlet eddy viscosity ratio of µT /µ = 1.5.

At the outlet, the static pressure was set at the hub casing and was then varied along the span using theaxisymmetric radial momentum equation.12 The value of the hub static pressure was raised or lowered froman initial value to defined different operating conditions along the speedline. The hub static pressure wasvaried between p2/p01 = 0.85 and p2/p01 = 1.0125.

An equivalent wheel speed was used such that the inlet conditions used for the current simulationsmatched the reported conditions from the literature. The equivalent wheel speed was defined as NEQ =

N/√θ, where θ = T01/Tstd (Tstd = 288.3 K). The equivalent wheel speed was computed to be 16,520 RPM.

IV.B. Overall Performance

The performance characteristics were found by extracting the solution at planes upstream and downstreamof the rotor. The axial positions of the extraction planes with respect to the rotor were the same as thoseused in Ref. 36. Appropriate averages of the flow variables were used to find the mass flow rate, adiabaticefficiency, and total pressure ratio.1

The simulations were run on an IBM/Lenovo x86 HPC cluster, using forty cores per case. Parallelcommunication was handled by Message Passage Interface (MPI). The time per iteration using Green-Gausswas 6.802× 10−5 s/iter/nodemax, where nodemax = 106, 257 was the size of the largest grid piece out of theforty grids created by the domain decomposition. Both LSQR simulations were 1% slower than Green-Gauss,and WENO was found to take 28% more time per iteration than Green-Gauss.

The maximum mass flow rates found using each gradient reconstruction method, except WENO, arepresented in Table 1, where they are compared against two experimental values. A value for WENO wasnot obtained because issues with solution divergence were encountered at the higher flow rates.

Using un-weighted LSQR resulted in the lowest predicted maximum flow rate. The maximum flow ratespredicted by Green-Gauss and WLSQR were quantitatively similar. The difference between the measuredvalues35,36 and the value predicted by Green-Gauss was 1.26%. The current difference between experimentand prediction was quantitatively similar to the one reported in Ref. 32.

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(a) Full mesh

(b) Tip structured grid layer (c) Tip leakage grid blocks

Figure 1: Rotor 67 computational mesh.

Table 1: Predicted and measured equivalent maximum mass flow rates, given in kg/s.

Green-Gauss LSQR WLSQR Pierzga & Wood35 Wood36

34.48 34.21 34.48 34.92 34.96

Partial compressor map speedlines were computed using each of the gradient construction methods.Operating points in the near stall region were not simulated as the point near peak efficiency was thedesired operating point. The mass flow rates for each gradient case have been normalized by their respectivemaximum values to allow for a better comparison with experiment. The maximum mass flow rate for Green-Gauss was used for the WENO speedline due to the overall similarity between the two sets of dimensionalspeedlines.

Figure 2 presents the predicted speedlines with a comparison to the measured speedlines. The speedlinesgenerated using Green-Gauss, WLSQR, and WENO were found to be in good agreement with the experi-mental data. While the un-weighted LSQR speedline showed qualitative agreement with the experimentaldata trend, the values of adiabatic efficiency and total pressure ratio were lower than that predicted by theother methods.

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84

86

88

90

92

94

0.9 0.95 1

Adiabatic Efficiency [%]

Normalized Flow Rate, m./m.max [-]

Green-GaussWENOLSQRWLSQRPierzga [1985]Wood [1990]

1.3

1.4

1.5

1.6

1.7

1.8

0.9 0.95 1

Total Pressure Ratio [-]

Normalized Flow Rate, m./m.max [-]

Figure 2: Rotor 67 compressor map for the design wheel speed. Solid symbols indicate the “near peakefficiency” operating point for each case.

IV.C. Flow at Near Peak Efficiency

The performance characteristics at near peak efficiency have been collected in Table 2. At near peak effi-ciency, the Green-Gauss, WENO, and WLSQR solutions predicted nearly identical performance parameters.Again, the un-weighted LSQR solution resulted in reduced performance in comparison to the other solutions.Ignoring the LSQR solution, the predicted mass flow rates were found to be 2% lower than the measuredvalues. However, the predicted values of adiabatic efficiency and total pressure ratio were in good agreementwith the experimental data. In particular, the predicted efficiencies and pressure ratios were within 0.1% ofthe measurements from Ref. 35.

Table 2: Predicted and measured performance characteristics at near peak efficiency.

Green-Gauss WENO LSQR WLSQR Pierzga & Wood35 Wood36

m [kg/s] 33.90 33.92 33.27 33.89 34.61 34.67

ηtt [%] 91.00 91.01 89.90 90.94 91.13 93.16

π∗ [–] 1.610 1.611 1.573 1.610 1.609 1.647

Spanwise profiles of static pressure ratio and absolute flow angle were computed at the aforementioneddownstream measurement plane for each case. Figure 3 compares the computed profiles against the mea-surements found in Ref. 36. In general, all of the methods showed good qualitative agreement with themeasured profiles of both variables. Quantitatively, LSQR under-predicted the static pressure along theentire span of the rotor, while the other methods showed very good agreement with the measured values.The predicted flow angles were largely unaffected by the choice of gradient evaluation method, and showedgood quantitative agreement with the data. LSQR slightly under-predicted the other methods for percentspans greater than 90%. LSQR suffers more in this region as gradients of the flow play a larger role here,where the secondary flows generated by the tip leakage flow exist.

Contours of the relative Mach number were extracted at 90% of the span from the hub for each gra-

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0

20

40

60

80

100

0.9 1 1.1 1.2 1.3

Percent Span [%]

Norm. Static Pressure, p2/p01 [-]

Wood [1990]

Green-Gauss

WENO

LSQR

WLSQR

0

20

40

60

80

100

20 40 60 80

Percent Span [%]

Abs. Flow Angle, α [deg]

Figure 3: Spanwise profiles of the predicted flow downstream of the rotor at the near peak efficiency comparedagainst experiment.36

dient case. The different predicted flows are shown in Fig. 4, where they are shown in comparison to theexperimentally determined flow field from Ref. 35. It was found that the flow fields predicted using Green-Gauss, WENO, and WLSQR were indistinguishable to the eye. All three solutions showed good qualitativeagreement to measurement.

The LSQR solution offered a noticeably different flow field. The point of impingement of the bow shockon the suction side of the rotor was moved forward, and accounted for more of the flow deceleration thanthe passage shock. The LSQR solution also exhibited enhanced flow separation in comparison to the otherflow solutions. The additional flow separation is due in part to the location of the bow shock impingement,and also in part to poor accuracy of the gradients resulting

The difference between the LSQR and the other gradient methods can be further illustrated by examiningthe surface pressure distributions. The pressure coefficient, computed using the relative dynamic pressure(qrel = 1

2ρ∞W2∞), is shown in Fig. 5 at 90% span. The pressure distributions indicate that the near wall

flow is slower on the pressure side of the rotor, just downstream of the leading edge. The difference in thelocation where the bow shock hits the rotor is also made clear by Fig. 5. The impingement point occurswhen the pressure is seen to increase near the trailing edge. For LSQR this pressure increase occurred atx/caxial ≈ 0.64, and at x/caxial ≈ 0.72 for the other methods. The increased pressure seen with LSQR alongthe aft 20% of the axial chord also indicated larger amounts of separation present in the flow.

V. Conclusions

This paper explored the effect that the gradient evaluation method had on the performance of a transonicfan. The gradient methods examined were the Green-Gauss, un-weighted least-squares with QR decomposi-tion (LSQR), inverse distance weighted LSQR (WLSQR), and weighted essentially non-oscillatory (WENO)methods. The Green-Gauss, WENO, and WLSQR methods provided good overall predictions for the massflow rate, total pressure ratio, and adiabatic efficiency of the rotor in comparison to experimental data.Using un-weighted LSQR resulted in the under-prediction of all three performance parameters.

At near peak efficiency, the total pressure ratios and adiabatic efficiencies predicted by Green-Gauss

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(a) Green-Gauss (b) WENO

(c) LSQR (d) WLSQR

(e) Experiment35

Figure 4: Near peak efficiency relative Mach number contours (∆Mrel = 0.05) at 90% of the span.

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-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Pressure Coefficient, cp [-]

Normalized Axial Chord, x/caxial [-]

Green-Gauss

WENO

LSQR

WLSQR

Figure 5: Pressure coefficient profiles at near peak efficiency and 90% span, measured from the hub.

and the two weighted gradient methods were in good agreement with the measured values. The flow fieldspredicted using these three gradient methods showed good qualitative agreement with experiment. Thematch between un-weighted LSQR and experiment was found to be poor at this operating point. Alteredshock structures and increased amounts of flow separation were both observed for this case. These re-sults demonstrate that un-weighted LSQR is not a suitable choice for the gradient evaluations required byturbomachinery simulations.

Acknowledgments

The authors would like to thank Neil Matula for his insight during the mesh generation process. Theauthors would also like to thank the Texas A&M High Performance Research Computing Center for providingthe computational resources that were needed by this work.

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