Aeroengine fans and compressors increasingly operate subject to inlet distortion in the transonic flow regime. In this paper, innovations to loworder numerical modeling of fans and compressors via volumetric source terms (body forces) are presented. The approach builds upon past work to accommodate any axial fan/compressor geometry and ensures accurate work input and efficiency prediction across a range of flow coefficients. In particular, the efficiency dropoff near choke is captured. The model for a particular blade row is calibrated using data from singlepassage bladed computations. Compared to fullwheel unsteady computations which include the fan/compressor blades, the source term model approach can reduce computational cost by at least two orders of magnitude through a combination of reducing grid resolution and, critically, eliminating the need for a timeresolved approach. The approach is applied to NASA stage 67. For uniform flow, at 90% corrected speed and peakefficiency, the body force model is able to predict the totaltototal pressure rise coefficient of the stage to within 1.43% and the isentropic efficiency to within 0.03%. With a
In the preliminary design of compressors, maintaining low computational cost for numerical simulation of the blade aerodynamics is paramount. Traditionally, various 1D and 2D mathematical models have been employed. 1D calculations (meanline codes) yield the shortest computational time but with a reduction in both accuracy and available flow details. 2D models (streamline curvature methods, throughflow methods) provide a greater degree of accuracy; however, the flow is typically assumed to be axisymmetric.
In this paper, a 3D distributed source term modeling approach, commonly and henceforth referred to as a body force model, is developed. The approach can be applied to any axial compressor and captures efficiency dropoff near choke. This model is used to capture distortion transfer and efficiency penalties associated with inflow distortion. A body force model replaces physical blades in a numerical simulation with a volumetric source term field. Doing so considerably reduces the grid density required to capture key phenomena such as loading distribution, overall pressure rise, and work input when compared to bladed unsteady Reynoldsaveraged NavierStokes (URANS) computations. More significantly, the source term model is a “smearedout” circumferential average of the flow field; even in the case of nonaxisymmetric inflow, source terms are timeinvariant, eliminating the need for an unsteady calculation. In circumferentially nonuniform inflow, there is no frame of reference in which the flow is steady. However, the smearedout forces are not framedependent and do not vary with time if two conditions are met regarding the flow [
The distributed source term approach has been extensively used in the literature to study fan and compressor response to inlet flow distortion. A few recent examples which do not form an exhaustive list include the following:
The methodology was first introduced by Marble [
Based on an adaptation from Gong’s distributed source term model, Peters developed a body force model to investigate fan inlet and nacelle design parameters for low pressure ratio fans with increased fan and inlet coupling [
More recently, an incompressible, inviscid, normal force model was developed by Hall, Greitzer, and Tan [
In this paper, an innovative methodology for calibrating a body force model of a transonic compressor is demonstrated. A compressible adaptation of the incompressibleflow normal force model developed by Hall et al. [
Computations in this work are carried out on a transonic compressor, NASA stage 67 [
Design characteristics for NASA stage 67 rotor at 90% speed and maximum efficiency [
Parameter  Value  Parameter  Value 

corrected angular speed 
1512  hub solidity 
3.11 

1.20  tip solidity 
1.29 
totaltototal pressure ratio 
1.48  
0.375 
corrected mass flow rate 
31.10  
0.478 
number of rotor blades 
22  maximum isentropic efficiency 
92.2 
aspect ratio 
1.56  
0.39 
Design characteristics for NASA stage 67 stator at 90% speed and maximum efficiency.
Parameter  Value  Parameter  Value 

number of stator blades 
36  tip solidity 
1.33 
aspect ratio 
2.16 

0.50 
hub solidity 
2.57 

0.53 
The normal force model in this work is based on the incompressible, inviscid turning force model developed by Hall et al. [
Body force model constrained relative flow angle creates a mismatch of relative swirl velocity due to the absence of blade metal blockage.
The formulation allows for the correct flow angle and thus correct loading at the blade leading edge, as well as correct swirl velocity and hence total enthalpy rise at the blade trailing edge. Ensuring the correct blade loading at the leading edge is critical for the model to be able to drive the upstream flow redistribution in nonuniform flow.
Peters developed a loss model for studying the effect of ultrashort nacelles on low pressure ratio fans [
For the calibration of the body force model, singlepassage bladed RANS calculations as well as previously published experimental data by Fidalgo et al. [
As discussed in Section
A highlevel overview of the body force model calibration process is illustrated in Figure
A highlevel overview of the distributed source term model calibration process.
Calculating the
The next step is to calculate the camber reduction,
The last step is to calibrate the parallel force model. To reduce the computational cost, educated guesses for the initial values of the model constants are important. Given that the flow coefficient at peakefficiency is known from experimental data, a simplification to (
The body force grid in this work is a
Meridional view of the body force computational domain with axial measurement stations.
NASA stage 67 contains sharp hub radius changes in the rotor and stator. Within the stator, these area contractions coincide with the adverse pressure gradient produced by flow straightening and endwall boundary layer development. Due to these conditions, nonphysical severe flow separation was observed. The separation is not consistent with the results observed in the singlepassage model. To avoid this flow separation, the noslip condition is not enforced on endwalls in the body force computations. Within the rotor and stator blade regions, the parallel force model accounts for the endwall losses. Using slipwall conditions leads to the absence of boundary layer development upstream of the rotor. While it is desired to be able to capture this effect, the use of slipwalls only affects the flow in the bottom 5% span and upper 5% span. For the purpose of this study, the overall performance trends are still adequately captured.
Without the physical presence of blades in the body force model, the pressure surface to suction surface flow over the rotor blade tips is not captured. Having no source terms in the tip gap region in the distributed source term model was attempted, but, without physical blades, nonphysical backflow occurs. Thus, in this work, blade forces are extended into the tip gap, which for this machine is less than 0.4% of the blade span, yielding a tip region swept volume that is only 0.64% of the entire swept volume of the rotor. A consequence of this approach is that work is done on the flow in the tip gap, meaning the total enthalpy rise is overestimated, and the tip leakage flow details cannot be observed. However, the relatively small tip gap regions mean that the model is still able to predict the overall stage work to an acceptable degree, as will be shown.
A series of five grids were tested for the body force model, with increasing refinement in the axial, radial, and circumferential directions. Grid independence was assumed as the rotor work at peakefficiency changed significantly less than 1% between each of the successive grids. Rotor efficiency is not a concern in determining grid independence as the parallel force model is calibrated based on the selected grid. Instead, the additional constraint placed on grid selection is the tradeoff between refinement and computational cost. As grid density increases, work input by the rotor changes at a rate that is acceptable, <0.5% between all cases. The efficiency, however, is largely dependent on the grid discretization. The selected grid has a fullwheel cell count of
The
Final
After seven iterations, the camber reduction constant converged to a value of
Normal force model rotor camber reduction.
Rotor work input characteristic before and after blade recambering, 90% rotational speed.
The parallel force coefficients found from calibration are shown in Table
Parallel force coefficient values for stage 67.
Rotor  Stator  


0.0145 

0.052 

650 

5 

1125  

1.007 

0.6199 

0.9870  

0.9868  
0.6045 
Body force model rotor isentropic efficiency characteristic and previously published results [
This section examines the body force model’s global and local performance metrics in both uniform and nonuniform flow, compared against fullannulus URANS computations.
To evaluate the model accuracy, a comparison is made against the uniform inflow singlepassage RANS calculation. At the peakefficiency flow coefficient, 0.505, the key performance metrics are shown in Table
Uniform inflow body force model overall performance versus single passage RANS at peakefficiency flow coefficient.
single passage  body force  % error  


1.48  1.49  1.43 

92.5  92.4  0.03 

0.143  0.148  3.37 
At flow coefficients away from peakefficiency, the model is compared against data from Fidalgo et al. [
Rotor isentropic efficiency, total temperature, and total pressure versus flow coefficient.
In observing the spanwise mass flux profile at rotor trailing edge, shown in Figure
Spanwise rotor trailing edge mass flux at peakefficiency, compared against singlepassage results.
Spanwise rotor trailing edge total temperature at peakefficiency, compared against singlepassage results.
Recall that the intended application of the body force model approach is prediction of compressor response to nonuniform inflow. To assess the model in such conditions, a comparison was made against a nonuniform inflow study performed by Fidalgo et al. [
Far upstream midspan circumferential traverse of total pressure.
The rotor work input and trailing edge absolute swirl angle of the body force model are well matched with URANS results, as shown in the middle part of Figure
Midspan circumferential traverse of absolute whirl angle, total temperature, and total pressure at the rotor trailing edge.
In addition to the circumferential flow characteristics, the body force model also captures blade trailing edge spanwise flow properties. As shown in Figure
Spanwise profiles of total pressure, total temperature, and absolute swirl angle at the rotor trailing edge,
Overall performance metrics for the rotor are listed in Table
Comparison of body force model rotor total pressure and isentropic efficiency changes from clean to distorted inflow versus Fidalgo et al. CFD [
URANS (clean)  URANS (distorted)  body force (clean)  body force (distorted)  


1.46  1.46  1.45  1.45 

93.5  92.0  91.9  90.4 

1.50  1.46 
Within the stator blade passage, the qualitative flow response to the distortion is in excellent agreement with the results by Fidalgo et al. [
Midspan circumferential traverses of absolute swirl angle, total temperature, and total pressure at the stator trailing edge.
To quantify the overall stage performance of the body force model in nonuniform inflow, the stator trailing edge total temperature RMS error is 0.31%. The flow field at this location is a result of distortion transfer through both blade rows, showing that distortion transfer and blade response to distortion are wellcaptured by model. In comparing the total pressure at stator exit of the body force model to the URANS results, an RMS error of 3.1% exists. This is largely due to excessive loss forces in both the rotor and stator. To isolate the model from this loss force overestimation, an alternative RMS error is calculated. Artificially shifting the body force model data such that the mean value is equal to that of the URANS data, an RMS error of 0.57% is found. This supports the postulation that the model is able to capture distortion transfer as well as blade row losses; the accuracy of the calibration of the loss model is the limiting factor.
In this paper, a transonic body force modeling approach is developed by expanding upon existing flow turning and loss models. The model requires calibration of a compressibility correction factor that is both machine and operating speed dependent. Additionally, a camber reduction term is implemented to account for the absence of mass flow blockage. A twosided loss model is implemented to allow for an enhanced ability to capture choking effects. The resulting model obtained for NASA stage 67 is compared against singlepassage uniform inflow and fullannulus nonuniform inflow.
The key outcomes from this work are the following:
At peakefficiency, in uniform flow, the body force model is able to predict the rotor total pressure ratio within 1.43% as well as the isentropic efficiency within 0.03%. In the stator domain, the entropybased loss coefficient is found to have an error of 3.37%. Considering the rotor total temperature ratio, total pressure ratio, and isentropic efficiency away from peakefficiency, the largest error is found within the total pressure ratio. Even still, the overestimation in total pressure ratio is limited to 4.1% compared to URANS results. In the case of a severe nonuniform inflow, the stator trailing edge total temperature ratio at midspan has an RMS error of 0.31%, showing that the distortion transfer and blade work input are wellpredicted throughout the entire stage.
Future work will involve an automated, optimized scheme to calibrate the normal and parallel force models with greatly reduced user effort. Additionally, a mass flow blockage source term could be implemented to allow for the removal of the
Reference area
Staggered blade spacing
Number of blades
Lift coefficient per unit span
Blade axial chord
Volumetric source term per unit mass
Local rotor reduced frequency
Enthalpy
Blade loading coefficient
Combined loss coefficient
Design loss coefficient
Offdesign loss coefficient
Corrected mass flow rate
Relative Mach number
Reference Mach number
Camber surface normal
Pressure
Radial coordinate
Entropy
Temperature
Velocity component in stationary frame
Velocity magnitude in stationary frame
Velocity magnitude in rotating frame
Cartesian coordinates
Absolute swirl angle
Relative swirl angle
Flow deviation
Compressibility correction
Entropybased loss coefficient
Isentropic efficiency
Circumferential coordinate
Local blade camber angle
Camber reduction constant
Density
Blade solidity
Flow coefficient
Work coefficient
Corrected rotational speed.
Axial measurement planes
Inlet
Isentropic
Leading edge
Normal to streamwise direction
Parallel to streamwise direction
Total/stagnation quantity
Trailing edge
Freestream.
Massaveraged quantity.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest regarding this publication of this article.
This work is funded by the NSERC Discovery Grants program. It is based upon the work completed in Hill’s thesis [