Considerations for High-Fidelity Modeling of Unsteady Flows in a Multistage Axial Compressor

Abstract

This paper presents the development and validation of a high-fidelity, unsteady, computational fluid dynamics (CFD) model of the Purdue 3-Stage Axial Research Compressor. A grid convergence study assesses the spatial discretization accuracy of the single-passage, steady-state computational model. Additionally, the periodic-unsteady convergence of the unsteady signals of a multiple-passage transient blade row model was explored. Computational predictions were compared with experimental measurements to evaluate the efficacy of the various modeling decisions. Notably, transient blade row model calculations employing the Scale-Adaptive Simulation (SAS) formulation of Menter’s Shear Stress Transport (SST) turbulence model exhibited a significantly improved agreement with experimental data compared to steady-state calculations. Particularly, in conjunction with the SAS-SST turbulence model, the transient calculations significantly improved the spanwise (radial) mixing characteristics of the transient-average stagewise total temperature profiles. Spectral analyses of the transient signals compared with unsteady pressure measurements showed fundamental and second harmonic blade-passing frequency amplitudes matching within 5–7% in the embedded stage. This research underscores the importance of including accurate geometry, practical minimization of modeling assumptions using higher-fidelity physics models, comprehensive convergence assessment, and the comparison and validation of computational predictions with experimental measurements.

1. Introduction

This paper introduces and comprehensively defines the core methodologies implemented for multistage compressor computational models derived from the Purdue 3-Stage Axial Compressor Research Facility (thoroughly detailed by Kormanik et al. [1]). The development of this model and derived models is approached from an experimentalist perspective by controlling what variables can be controlled and quantifying errors from those that cannot. The primary goal of this work is to help guide experimental planning and supplement the understanding of experimental results.

Although rapidly evolving, the current standard for multistage turbomachinery CFD models relies on steady-state Reynolds-averaged Navier–Stokes (RANS) solvers and representing full-annulus blade rows by single passages with periodic boundary interfaces. These single-passage domains are typically limited to the primary gas path between the hub and shroud, often neglecting the potentially significant effects of secondary flow paths, including stator cavity flows beneath shrouded stator platforms, bleed and leakage paths, and any other flows associated with geometries deemed too complex to model or assumed to have a minimal impact on the overall solution.

These steady-state, single-passage computational models must account for pitch differences and reference frame changes between adjacent rows, with one of the most widely used inter-row interfaces being the mixing plane method. One treatment of the mixing plane method [2] circumferentially averages the spanwise flow properties upstream of the domain interface, assuming flow properties mix out with pitchwise uniform enthalpy and entropy. This uniform flow is then imposed as the steady inflow to the next domain, still having uniform pitchwise stagnation enthalpy and entropy but generally pitchwise non-uniform static pressure and flow direction determined by the downstream domain.

While the mixing process is conservative (mass, momentum, and energy), it is inherently loss-generating, causing increased entropy and loss across the interface and decreased isentropic compression efficiency. Further manifestations occur as spanwise (or radial) mixing discrepancies in computational predictions compared with experimental measurements. Significant research efforts and debate in the 1980s [3,4,5,6] aimed to identify the mechanisms of multistage mixing and realistically model the turbulent processes.

Although the debate continued into the next decade, the closure of these discussions identified the importance of secondary flow and turbulent diffusion in axial flow compressor mixing processes [7]. Ironically, the mixing plane model is the principal reason for mixing deficiencies with steady-state RANS solver turbomachinery predictions. Recent efforts by Cozzi et al. [8] have continued pursuing the improvement of steady-state RANS computations. The authors explain the introduction of circumferential averaging, necessary for modeling the inherently unsteady processes as steady, disrupts the development of secondary and clearance flows, minimizes blade row interactions, and partially cancels the transport of streamwise vorticity.

In this paper, the computational methodology is first introduced, including the geometric scope of the computational domain, the CFD solver used for calculations, boundary conditions and modeling decisions, spatial discretization and grid sensitivity, an introduction of the transient blade row model, and finally an assessment of the periodic-steady signal convergence of the transient calculations. CFD model results are then compared with steady and unsteady experimental measurements to understand the strengths and limitations of the modeling approach. This paper demonstrates that the transient blade row model used in conjunction with the Scale-Adaptive Simulation formulation of Menter’s Shear Stress Transport model matches experimental results significantly better than steady-state RANS CFD methods.

2. Computational Methodology

2.1. Scope of the Computational Domain and Geometric Definitions

Figure 1 shows the two-dimensional cross-section of the Purdue 3-Stage PAX200 axial research compressor flow path. PAX200 models the rear stages of a modern high-pressure, small-core compressor comprising an inlet guide vane (IGV) followed by three stages of rotors and stators. The compressor, geometrically scaled up by a factor of four, matches engine-representative Mach and Reynolds numbers, thereby maintaining geometric and kinematic similitude to its full-engine counterpart and demonstrating appreciable compressibility effects. PAX200 is a component-level technology demonstrator with blading designed by the Rolls-Royce Corporation. The airfoils are designed using current three-dimensional methods, resulting in airfoil profiles with pronounced sweep and endwall dihedral. The annular flow path area decreases by about 22%, and airfoil aspect ratios (span-to-chord) are less than one. Further facility details are described in [1].

The flow path cross-section is superimposed with the computational domains of the numerical model in Figure 1. Stationary frame domains, including the Front Strut (FS) domain, are shown in blue, and relative frame domains are in red. Similarly, the dashed lines indicate the stationary endwalls and rotating endwalls in lighter shades of blue and red. Black dashed lines distinguish the axial domain interfaces. The IGV and S3 cavities have been simplified by defeaturing the variable IGV penny gaps and associated endwall clearances, as well as small areas near the centerline and rotor shaft. Despite these geometric simplifications, Figure 1 shows a stark departure from the primary gas-path-only modeling methodology.

Flow path geometry definitions, including the converging hub and spanwise sections of airfoil coordinates, have been provided based on “hot” geometry established at the aerodynamic design point. Projected close-down calculations set all static and running clearances at their design values. All other geometric definitions are derived directly from engineering drawings for part manufacture, matching design intent as closely as possible. In the S3 cavity computational geometry, small tubes simulate the experimental S3 cavity bleed, accounting for the mass flow rate across the S3 knife seals. The computational geometry models the circumferentially distributed, larger-diameter bleed tubes with smaller-diameter tubes matching the fractional flow area of a passage sector. Further details of all aspects of the computational model are extensively documented in [9].

2.2. Ansys CFX-5 Solver

The commercial Ansys CFX-5 solver 2023 R2 (CFX) [10] uses an unstructured, element-based, finite-volume method to discretize and solve the governing equations. Control volumes are constructed around each node of the three-dimensional grid using a median dual-discretization method. CFX primarily uses second-order accurate approximations, striking a balance between accuracy and numerical stability. The CFX solver is pressure-based with pressure–velocity coupling. The steady-state solver uses a pseudo-transient method to advance the solution.

Transient turbomachinery calculations have a few options in CFX, but the present work is limited to unsteady RANS (URANS) methods. URANS calculations can be conducted using a typical true-transient approach with full-annulus model geometries or partial-annulus geometries with symmetry. These models, particularly for multistage turbomachinery that commonly use prime number blade counts to reduce the aeromechanical response, quickly become prohibitively large for the available computational resources. The CFX solver offers Transient Blade Row (TBR) methods—models to reduce the size of the computational model while maintaining reasonably accurate solutions.

One available method is the Time Transformation (TT) method, based on Giles’s time inclination method [11], which handles pitch differences in adjacent rows by applying temporal transformation of the physical time coordinates of the rotor and stator in the circumferential direction to make the models periodic in transformed time coordinates. For single-stage models, the physical timestep is set to the stator passage, and a computed transformed timestep is set to the respective rotor passage.

The current solver guidelines emphasize restricting the Time Transformation method to single-pitch ratios with a different domain interface model between two time-transformed domains. However, Cornelius et al. [12] have demonstrated the validity of extending the Time Transformation method to all domain interfaces in a six-stage axial compressor. The current study will demonstrate that applying this method to the multistage environment results in multiple transformed timesteps to satisfy the model requirement that each blade row completes the same number of transformed timesteps per period.

2.3. Boundary Conditions and Modeling

The inflow and outflow boundary conditions are derived from experimental measurements acquired with 7-element Kiel-head rake probes. These measurements are scaled to sea-level static, dry air conditions for a standard day; the inlet total temperature profiles are scaled to 15 °C (59 °F), and the total pressure profiles are scaled to 101.325 kPa (1 atm, 14.696 psia). The working fluid is the CFX-defined Air Ideal Gas material. It is important to note that this definition includes thermally and calorically perfect gas assumptions, and, therefore, the heat capacities at constant pressure and volume are not temperature dependent. These assumptions are considered acceptable for this case since there is a relatively low temperature rise.

Primary path outflow conditions are applied as exit-corrected mass flow rates matching experimental conditions within about 0.15%, and the S3 cavity bleed is applied as the physical mass flow rate computed from experimental measurements. The endwalls are considered adiabatic and, following Shabbir and Turner [13], hydrodynamically smooth. The logarithmic comparison of cell Reynolds number as a function of the ratio of nondimensional surface roughness to grid cell height (k_s⁺/y⁺) averages less than 3.6, also less than the hydraulically smooth criteria presented in Reference [13], thus avoiding the introduction of additional modeling.

Heat transfer within the fluid domain is governed by the total energy equation, including the contribution from viscous work. This term accounts for the energy associated with viscous stresses, effectively modeling the work and energy associated with viscous shear forces within the fluid, such as windage.

The single-passage, steady-state model is solved using the mixing plane interfaces between all domains and periodic interfaces. The multi-passage, transient model is solved with Time Transformation interfaces between all rotors and stators. The mixing planes between the Front Strut and IGV and between the IGV and R1 domain interfaces are maintained.

The steady-state and transient models have 5% turbulence intensity applied at their inflow boundaries and are assumed to be fully turbulent. Menter’s Shear Stress Transport (SST) model, including the curvature correction production term, is used for turbulence closure [14,15]. In addition, the transient models are solved with the relatively advanced turbulence modeling approach using the Scale-Adaptive Simulation model (SAS) formulation of the SST turbulence model [16,17,18]. Introducing the von Kármán length scale allows the SAS model to adjust its length scale dynamically to resolve turbulent structures in unsteady regimes and produce LES-like (Large Eddy Simulation) results. When this unsteadiness is absent, the model reverts to the standard SST result. This turbulence model is compared with the standard URANS SST model, focusing on its effects on spanwise mixing. Resolving the large turbulence scales, rather than quickly damping them out as occurs in standard models, will improve the overall prediction of mixing through viscous interactions. A comparison of these different modeling techniques is shown in Figure 2, where contours of streamwise vorticity show considerable differences in turbulence scale resolution and propagation between reference frames.

2.4. Spatial Discretization and Grid Sensitivity

The entire computational domain, Stations 0–9 indicated in Figure 1, is spatially discretized using GridPro [19], a commercial multi-block, structured grid generation software. Each rotor domain has an identical grid topology and block structure. The primary flow path topologies of the stator domains are nearly identical, with variations caused by the integrally meshed cavity geometries beneath the stator shrouds; the stators and their shroud cavities are meshed as single domains with no interface between primary and secondary flow paths.

Aside from the previously mentioned simplifications in the geometric definition, the primary and secondary flow paths include endwall fillets on the airfoils, as well as all details of the hub shroud geometries and rotor assembly (i.e., the forward and aft compression plates, the rotor spacers including knife seals, and rotor disk geometries) in the stator cavities. The blocking ensures O-grid topology around all corners in the domains for orthogonal, flow-aligned cells throughout the model. The O-grid topology extends through the tip gaps, resulting in a conformal mesh through the tip clearance and eliminating interface modeling and unnecessary boundary layer clustering associated with H-grid topologies.

Staggered block topology features are introduced through the periodic surfaces to increase orthogonality and reduce cell skewness. The average first cell offset from every wall is consistent with an average and maximum y+ of less than one and five, respectively, ensuring a well-resolved viscous sublayer. With an expansion ratio of 1.1, more than ten elements are located within the boundary layer. The final cell counts for the computational domains are listed in Figure 3. The radial, pitchwise, and axial counts provided are the number of cells prior to boundary layer clustering, and the total cell count is summed in the Euler Grids column referencing the inviscid Euler equations. Once the specified boundary layer clustering is applied, the increased cell counts are summed in the Navier–Stokes (N–S) Grids column.

Examples of the final grids are shown in Figure 3 for the embedded stage Rotor 2 and Stator 2. Every other grid edge is removed in each direction, and the remaining line weights have been increased for clarity and visibility. Figure 3a and Figure 3d depict the primary flow path grids and airfoils of Rotor 2 and Stator 2, respectively. More detailed views of the rotor tip surface and a sheet through the conformal tip grid are shown in Figure 3b. An example of a corner fillet is shown in Figure 3c at the airfoil’s leading edge. Figure 3e shows a closer view of the embedded stage Stator 2 cavity, including the knife seals and seal land ring. Additionally, the leading edge fillet mesh and the airfoil O-grid topology are evident on the hub of the main flow passage.

A grid sensitivity study of the single-passage, steady-state model (also used to initialize the Time Transformation method models) was conducted at a nominal loading condition matching the fourth point of the experimental 100% corrected speedline. This loading condition, coincident with the aerodynamic design point, is denoted L4-NL. The grid sensitivity study follows the Grid Convergence Index (GCI) method proposed by Roache [20] and adopted by the AIAA standard guideline [21] and the ASME standard guideline by Celik et al. [22].

Figures of merit (ϕ) reported here are the embedded Stage 2 and overall total pressure ratios (TPR), total temperature ratios (TTR), and isentropic efficiencies (η_s). Initially, the base grid spacings and cell counts (h_base, N_base) listed in Figure 3 were halved (h_coarse = 2.0, N_coarse = 0.5 × N_base) and doubled (h_fine = 0.5, N_fine = 2.0 × N_base) in each direction; however, the coarse grid solutions were seemingly outside the asymptotic convergence range for TPR calculations. Therefore, another simulation was run, coarsened by a factor of 0.67 (h_coarse = 1.5, N_coarse = 0.67 × N_base) in all directions for TPR. All refinement factors, or the coarser-to-finer grid spacing ratios, exceed the empirical recommended minimum of 1.3.

The selected results of the grid sensitivity study are shown in Figure 4 for the overall quantities with GCI error bars and listed in

Table 1. Results of the grid sensitivity study.

$\begin{array}{c}{\mathit{r}}_{\mathit{b}/\mathit{f}}={\mathit{h}}_{\mathit{b}}/{\mathit{h}}_{\mathit{f}}\\ {\mathit{r}}_{\mathit{c}/\mathit{b}}={\mathit{h}}_{\mathit{c}}/{\mathit{h}}_{\mathit{b}}\end{array}$	TPR [–]		TTR [–]		ηs [–]
	Stg2	Ovr	Stg2	Ovr	Stg2	Ovr
$r_{b/f}\left(2.0\times N_b\right)$	2.0	2.0	2.0	2.0	2.0	2.0
$r_{c/b}\left(0.5\vee 0.67\times N_b\right)$	1.5	1.5	2.0	2.0	2.0	2.0
${\varphi }_{fine}^{*}$	1.000	0.998	1.000	1.000	1.001	1.000
${\varphi }_{base}^{*}$	0.999	0.993	0.999	0.998	1.004	0.997
${\varphi }_{coarse}^{*}$	0.995	0.984	0.999	0.998	0.993	0.983
$\Delta \left({\varphi }_{fine}-{\varphi }_{base}\right)$	0.001	0.010	0.001	0.002	−0.002	0.002
$\Delta \left({\varphi }_{fine}-{\varphi }_{coarse}\right)$	0.006	0.026	0.001	0.002	0.007	0.014
$\delta \left({\varphi }_{fine},{\varphi }_{base}\right)$	0.10%	0.52%	0.05%	0.13%	0.29%	0.21%
$\delta \left({\varphi }_{fine},{\varphi }_{coarse}\right)$	0.49%	1.46%	0.08%	0.15%	0.82%	1.63%
p	3.75	2.14	1.09	2.55	1.94	2.71
$GC{I}_{base}^{c/b}/r^{p}GC{I}_{fine}^{b/f}$	1.001	1.005	0.221	0.029	0.997	1.002
${\varphi }_{ext}^{b/f*}$	1.000	1.000	1.000	1.000	1.000	1.000
${\varphi }_{ext}^{c/b*}$	1.000	1.000	0.999	0.998	1.008	1.000
$e_{approx}^{b/f}$	0.10%	0.52%	0.05%	0.13%	0.29%	0.21%
$e_{approx}^{c/b}$	0.38%	0.93%	0.02%	0.02%	1.10%	1.40%
$e_{ext}^{b/f}$	0.01%	0.15%	0.05%	0.03%	0.10%	0.04%
$e_{ext}^{c/b}$	0.11%	0.67%	0.02%	0.00%	0.38%	0.25%
$GC{I}_{fine}^{b/f}$	0.01%	0.19%	0.06%	0.03%	0.13%	0.05%
$GC{I}_{base}^{c/b}$	0.14%	0.85%	0.03%	0.01%	0.48%	0.32%

following the ASME guidelines in Ref. [22], with all performance values normalized by the extrapolated value of the fine grid and denoted with an asterisk. Since the base grids are used for all calculations, GCI_base is the relevant metric of the conservative estimate of spatial discretization error for each quantity; it is subsequently referred to as simply GCI. The table organization follows definitions from Reference [22] with three exceptions. The absolute differences between the finest grid and the base and coarse grids are listed, denoted by uppercase delta (Δ), and the similar relative differences are represented by lowercase delta (δ). Ratios of GCI_base and GCI_fine, accounting for refinement factor, r, and apparent order of convergence, p, are also listed, and values close to unity indicate that the solutions are well within the asymptotic range of convergence.

The GCI ratios of TPR and isentropic efficiency are nearly unity, but the TTR values are closer to zero. However, the differences between the coarse and fine TTR values are small, indicating values have reached the asymptotic range ahead of the other figures of merit. When this occurs, it is noted that the slight changes in the values do not work well with this procedure and should not necessarily be taken as signs of an unsatisfactory grid.

While the selected quantities listed in Table 1 do not include Stage 1 and Stage 3 calculations, these were also performed and included in the following summaries. The stage total pressure ratio GCIs range between 0.14–0.44% and 0.85% for the overall total pressure ratio, still larger than expected but within the asymptotic range, demonstrated in Figure 4. The stage total temperature ratio GCIs are particularly small, between 0.0–0.03% and 0.01% for the overall total temperature ratio. Lastly, the isentropic stage efficiency GCIs are between 0.48 and 1.37%, and the overall isentropic efficiency GCI is 0.32%.

The total pressure and temperature profiles, normalized by their inlet area averages, are shown in Figure 5. The experimental data comprises a 20-point pitchwise passage traverse of 7-element Kiel-head rake probes at the station numbers indicated in Figure 1. These data are subsequently pitch-weight averaged, yielding the passage-average radial profiles. The profile data are radius-weight averaged, resulting in the entire area-averaged property of the station being used to normalize the experimental profiles and calculate performance parameters. Identical procedures are deployed to compute the results for the different grids in the sensitivity study. The computational radial profiles reiterate the results from Table 1. There are minor changes for most of the spanwise values between the different grid densities; however, there is a noticeable improvement between the coarse and base grids, while the base and fine grids are nearly superimposed.

The computational total pressure ratio profiles match the experimental results well, with a notable exception occurring at 80% span for several stations. This spanwise location coincides with the rotor tip leakage vortices as they migrate radially inward from the casing and convect downstream. Two possible explanations for this discrepancy may be the computational over-prediction or further losses incurred by interactions between the vortices and rake probes in the experiment. However, the considerably improved match between experimental measurements and the transient calculation results using the SAS-SST turbulence model, introduced in a subsequent section, indicates that the discrepancy is likely a further symptom of the steady flow assumption.

The steady-state computational total temperature ratio profiles initially agree with the experimental results at forward measurement stations through Station 4 at the S1 exit; however, this agreement deteriorates as discrepancies between the experimental profiles and computed predictions grow as the flow convects downstream. These discrepancies are consistent with the deficits of the computational mixing plane model—higher total temperature predictions near the adiabatic endwalls and lower total temperature predictions toward midspan, resulting in the parabolically shaped profile. By contrast, the experimental temperature profile is flatter; while the parabolic profile is still present through midspan, a curvature reduction (even inflection of the profiles) occurs near the endwalls.

While the computational model walls maintain the simplified adiabatic assumption, the shapes of the experimental total temperature profiles near the endwalls are the development of an internal flow thermal boundary layer when heated walls are present; total temperature increases toward the wall where the no-slip condition applies. The heat source in these walls occurs from conduction through the relatively thick aluminum casing from the hotter aft section to the cooler section upstream.

Neglecting these heat transfer effects is likely contributing to the observed temperature profile differences. Active, concurrent research in this facility is working to understand the influence of these heat transfer effects. Recent modifications to the current steady-state computational model incorporate solid structural domains for conjugate heat transfer analyses. Comparisons of computational solutions and experimental measurements have begun showing the importance of considering the true diabatic boundary conditions of the experimental test article and facility [23,24].

2.5. Transient Blade Row Model

Transient calculations were carried out using the Time Transformation transient blade row method, discussed previously. The pitch ratios between the Time Transformation interfaces are reduced to be as close to unity as practically possible by doubling the number of rotor domains and increasing the number of stator domains by a factor of three relative to the single-passage steady-state model, resulting in pitch ratios of adjacent rows ranging between 0.94 and 1.18. Additional domains would only improve the ratio for the largest pitch ratio, reducing 1.18 to 1.07; however, to do so would require more than doubling the number of domains with Time Transformation interfaces and increasing global cell count by nearly 88%. Since these pitch ratios satisfy stability limits, this trade-off is readily accepted. The multiple-passage transient blade row computational domain is shown in Figure 6 in a color scheme consistent with Figure 1; walls shaded blue are stationary, while walls shaded red are rotating.

When the Time Transformation method is applied at all interfaces, it was discovered that the adjacent row pitch ratio does not necessarily dictate the transformed timestep typically calculated by the method. Typically, the period is chosen as a blade-passing period of the Time Transformation interface. The only common period of the modeled machine is a complete revolution, and the transient blade row model was developed with this definition. The physical timestep chosen covers 0.1° of a full revolution at the compressor design speed, and therefore, there are 3600 timesteps per revolution (defining one period). The solution results are efficiently stored through Fourier compression; however, analysis of the transient solution requires Fourier reconstruction. This process effectively band-pass filters the solutions at the fundamental frequency associated with the adjacent rotor blade or stator vane passing frequencies and their harmonics. An important aspect of the present work is investigating the validity of this method in a multistage environment concerned with blade row interactions.

In physical time, this timestep results in approximately 80–100 timesteps per rotor blade-passing period in the stationary reference frame and 60–70 timesteps per vane-passing period in the rotating reference frame, depending on the blade row count. The computational model result in the present work represents the solution after 19 revolutions. Transformed time is less straightforward and is defined by the embedded stage blade counts. This global dependency on the embedded stage blade counts results in the expected transformed timesteps differing from the physical timesteps for the rotating reference frame of R1, R2, and R3, but also as an unexpected transformed timestep for the stationary reference frame of S2. An illustration of the difference between physical and transformed timesteps and a listing of the ratios of transformed timestep to physical timestep is shown in Figure 7.

2.6. Assessing Periodic-Unsteady Convergence

Iterative convergence, or residual accuracy, of steady-state solutions is relatively straightforward. Convergence is achieved by satisfying the following conditions:

(1)

Conservation equation residuals have reduced orders of magnitude (typically 4–6, highly dependent on model complexity), reaching a level below a predefined threshold.

(2)

Global and domain imbalances of mass, momentum, energy, and other scalar quantities are near zero (less than 1%).

(3)

The solution’s figures of merit no longer change with subsequent iterations (e.g., TPR, TTR, mass flow rate, torque, entropy).

However, the convergence of the periodic-unsteady solution of a transient simulation can be more challenging to define. Convergence may be qualitatively assessed by monitoring flow field solution quantities over a defined fundamental period, for example, a blade-passing, a rotor revolution, or a mass flow rate fluctuation.

A method proposed by Clark and Grover [25] provides a means of quantitatively evaluating the same criteria. Full details of the method and an example are available in [25], but it is briefly summarized here. Five parameters are computed over two successive signal cycles (or fundamental periods), defining a series of fuzzy sets representing distinct aspects of the degree of signal convergence. The fuzzy set intersection combines these sets, defined by standard fuzzy set operations as simply the minimum value of all fuzzy sets. The equations and further explanation are available in [25], and the fuzzy set parameters, their descriptions, and aspects of consistent signal representation are listed in

Table 2. Fuzzy set definitions representing aspects of periodic-unsteady convergence.

Fuzzy Set Parameter	Parameter Description	Consistent Signal Representation
$f_M$	Temporal History and Mean	Mean Level
$f_A$	Spectra Amplitude for Chosen Signal Components	Amplitude
$f_{\varphi }$	Spectra Phase for Chosen Signal Components	Phase Angle
$f_S$	Cross-Correlation of Cycles 1 and 2	Overall Signal Shape
$f_P$	Power Spectral Density Estimate	Fractional Signal Power
$f_C$	Overall Convergence Level

. Evaluating the parameters yields membership grades of the fuzzy sets. The maximum possible value of membership grade is unity, indicating perfect periodic convergence of the particular signal aspect.

While the first three parameters (f_M, f_A, f_ϕ) may be intuitive, the following two (f_S, f_P) need further explanation. The set parameter representing the convergence of the overall signal shape (f_S) is the cross-correlation value of the two signal periods at zero time lag. Complete convergence of the overall signal shape would produce a value of unity at zero time lag, implying that the first and second periodic signals are identical and the true period has been determined. The set parameter evaluating the fractional signal power of the chosen signal components (f_P) from the power spectral density (PSD) estimate represents how much of the overall signal power is contained within the expected dominant frequencies. The closer the value is to unity, the more the full signal power is captured. If this value is substantially below one, more of the full signal power is missing, implying the presence of inherent unsteadiness, significant contribution to the signal power from higher harmonics of expected frequencies, or a combination of both.

Before applying the method to the transient blade row model, the method was developed and validated by repeating the results of the example signal in the original work. These results matched between 0.0 and 6.3%, increasing with higher engine order (EO) signal components, but are quite small despite potential errors introduced by digitizing and interpolating the original signals.

After confirming the values in the original work, the method was applied to the transient monitor point data recorded at each timestep over the two final revolution periods, and it is noted that these are not the Fourier-reconstructed signals. The locations of the monitor points of the analysis are at the 50% span element of the 7-element Kiel-head rake probes in the stator computational domains, reporting non-reconstructed quantities in transformed time. While all domains were monitored qualitatively for convergence, this convergence analysis method is constrained to the first compressor stage for two reasons: (1) the transformed timestep affects the transformed time values from the S2 domain, and (2) the inherently greater unsteadiness present downstream of the S2 and S3 domains. Both aspects pose challenges for analysis using this convergence quantification method. Since the pressure field becomes periodic well before the entropy field from different propagation rates of finite pressure waves and viscous disturbances [25], the total pressure is used in this analysis case as it encompasses both aspects.

The results of these analyses are shown in Figure 8 for the first stage signal processing, and the resulting fuzzy set parameter membership grades are listed in

Table 3. Membership grades of the fuzzy sets.

Fuzzy Set		Membership Grade		Fuzzy Set		Membership Grade
R1 BPF		Station 3– R1 Exit	Station 4– S1 Exit	R2 BPF		Station 3– R1 Exit	Station 4– S1 Exit
fA	R1	0.983	0.989	fA	R2	0.925	0.987
fϕ	R1	0.995	0.997	fϕ	R2	0.999	0.989
fA	2 × R1	0.988	0.935	fA	2 × R2	0.898	0.887
fϕ	2 × R1	0.999	0.984	fϕ	2 × R2	0.976	0.994
fA	3 × R1	0.959	0.890	fA	3 × R2	0.996	0.983
fϕ	3 × R1	0.984	0.996	fϕ	3 × R2	0.989	0.933
fA	4 × R1	0.951	0.855
fϕ	4 × R1	0.984	0.890
fA	5 × R1	0.902	0.912	fA	R1 + R2	0.606	0.276
fϕ	5 × R1	0.888	0.968	fϕ	R1 + R2	0.993	0.903
		Station 3–R1 Exit				Station 4–S1 Exit
fM		1.000		fM		1.000
fS		0.945		fS		0.968
fP		0.904		fP		0.895

. The total pressure signals in transformed time are qualitatively similar for both monitor point locations, and the following conclusions quantitatively support the qualitative observations.

(1)

Each monitor point location has unchanging temporal means between periods and an associated membership grade of one (f_M = 1.00).

(2)

High membership grades are associated with the amplitude and phase of the fundamental through fifth harmonic of the upstream R1 blade-passing frequency (BPF) (f_A, f_ϕ > 0.85) and the fundamental through third harmonic of the downstream R2 blade-passing frequency (f_A, f_ϕ > 0.88).

(3)

The cross-correlation function (CCF) applied to the successive periods shows minimal time lag between the signals. The associated membership grade (f_S > 0.94) is the value of the cross-correlation function at zero time lag and indicates a consistent signal shape between periods.

(4)

Membership grades (f_S > 0.89) of the fractional signal power set parameter computed from the PSD estimate include the same frequencies chosen for the amplitude and phase spectra, indicating that most of the signal power is accounted for in the chosen frequencies.

A signal component associated with the blade row interactions of R1 and R2 presents as the fundamental frequency related to the sum of their blade counts. While the consistent amplitude set parameter is rather low (f_{A,R1 Exit} = 0.606 and f_{A,S1 Exit} = 0.276), the consistent phase membership grades are at similarly high levels as the rest of the parameters in Table 3. The lower values of indicated amplitude consistency may be associated with a periodic constructive and destructive interference of the total pressure signal at the monitor point location.

3. Comparing CFD Methods with Experimental Measurements

Computational predictions, including the steady-state RANS and transient-average URANS results, are contrasted with experimental measurements in Figure 9, comparing the passage-average radial profiles of total pressure and total temperature. There is a moderate improvement in the agreement between the experimental and URANS total pressure profiles compared with the RANS prediction, particularly near the endwalls. The total pressure discrepancy between the RANS calculations and experimental measurements is markedly reduced, exhibiting much closer agreement between the experimental and URANS transient-average results.

In contrast to this moderate improvement, the agreement between the experimental and computational total temperature profiles has improved considerably between the steady-state RANS calculations and those with URANS SST. The improved radial mixing reduces endwall temperature predictions while increasing midspan predictions, resulting in a flatter profile compared with the parabolically shaped profile in the steady-state RANS with mixing plane model interfaces. While the URANS method is responsible for most of this improvement, the SAS-SST turbulence model is also an important contributor.

A similar direct comparison of the URANS SST and SAS-SST turbulence models is shown in Figure 10. The differences between the models are shown with halved abscissa axis scales relative to Figure 9, underscoring the smaller differences between the URANS solutions than if either is compared with steady-state RANS. However, differences are apparent, highlighting the SAS-SST turbulence model’s ability to resolve the larger turbulent structures as the radial profile twists and turns in the radial direction. Both turbulence models show agreement with each other and experimental measurements; however, the SAS-SST turbulence model tends to match the experiment more closely in a general sense, particularly near the endwalls with increased viscous interaction and vorticity levels.

The passage-average radial profiles show good agreement with the experimental measurements; still, the averaging obscures important information like passage wake development and other flow structures. The circumferential flow field surveys comprising the passage-average radial profiles are compared in Figure 11 between the 20-point traverse of 7-element Kiel-head rakes and the corresponding computational predictions at the exit of the embedded stage, the S2 exit measurement plane.

The total pressure wake depth and width are well captured with both computational methods, with the profiles for each spanwise location showing a good comparison. The 80% span total pressure discrepancy is caused by a higher total pressure prediction of the midspan flow near the pressure side of the vane, between 50 and 100% vane passage, with the steady-state RANS calculations.

Similar conclusions are drawn from the total temperature wake profiles. The URANS SAS-SST calculations closely follow the experimental total temperature measurements over the entire passage at each spanwise location. The model effectively captures the intra-stator transport of the rotor wake and its tendency to migrate toward the stator pressure surface as it convects through the passage, consistent with the wake transport theory proposed by Kerrebrock and Mikolajczak [26] and further described by Montomoli et al. [27]. The high-enthalpy flow accumulates on the stator pressure surface caused by the negative jet developed from the slip velocity component of the relative frame rotor wake, corresponding with higher total temperature fluid. This rotor wake migration is evident in the circumferential total temperature profiles (Figure 11) for both the experimental data and the URANS SAS-SST calculations, appearing as circumferential peaks off the stator pressure surface at all spanwise measurement locations. The steady-state RANS solution matches similarly well at 20% span and 80% span, coinciding with the pitch-average profiles crossing at these locations. Other spanwise locations show pitchwise discrepancies corresponding with the differences in pitch average. Near the hub and shroud, 10% and 90% span, the shapes of the pitchwise profile are predicted with exaggeration; however, the midspan profiles, between 35 and 65% span, show flat total temperature wakes in contrast to the experimental values and URANS predictions.

So far, comparisons have been made with the experimental measurements representing steady-state operation, steady-state RANS results, and the transient-average results of URANS calculations. Comparisons with fast-response experimental measurements provide insight into the strengths and weaknesses of the transient blade row model and the Fourier compression of the solution. High-frequency response pressure transducers flush-mounted in the casing upstream of each rotor record the endwall pressure fluctuations of the rotor clearance gap.

Six circumferentially spaced sensors are installed upstream of each rotor row by about 20% of their axial chord lengths, typically used for tracking stall cells during intentional stall events. These sensors have protective screens, limiting their dynamic range to about 20 kHz. Difficulties with the sensors upstream of R3 limited this row to four sensors, and these sensors exhibited lower amplitudes than expected. The sensors were sampled at a rate of 100 kHz, and the analog signals were low-pass filtered at 40 kHz before conversion to digital signals.

These measurements are phase-locked to the shaft with a once-per-revolution laser optical tachometer. Unsteady pressure measurements were divided into ensemble revolutions in the time domain before averaging for an ensemble-averaged revolution of each sensor. The ensemble-averaged amplitude spectra for each sensor are then averaged in the frequency domain for the average amplitude spectra for each axial plane.

Similar methods were implemented in the computational model, with six transient monitor points spaced equally over each stator pitch at the same axial locations as the fast-response pressure transducers. Spectral amplitudes were computed from these transient monitor points and similarly averaged in the frequency domain for the experimental data. The computational spectra are computed for the transient monitor data recorded with each timestep and the Fourier reconstructed signals from the results. The time signals $\left({\psi }_{S-S}\right)$ and amplitude spectra $\left(\left|{\psi }_{S-S}\right|\right)$ of the upstream R2 and R3 static-to-static pressure rise coefficients are shown in Figure 12, and the five most significant responses for each signal are listed in

Table 4. Largest amplitudes of the upstream R2 and R3 static-to-static pressure rise spectra.

Upstream R2						Upstream R3
Unsteady Measurement		CFX Transient Monitor		CFX Fourier Reconstruction		Unsteady Measurement		CFX Transient Monitor		CFX Fourier Reconstruction
BPF	$\left|{\psi }_{S-S}\right|$	BPF	$\left|{\psi }_{S-S}\right|$	BPF	$\left|{\psi }_{S-S}\right|$	BPF	$\left|{\psi }_{S-S}\right|$	BPF	$\left|{\psi }_{S-S}\right|$	BPF	$\left|{\psi }_{S-S}\right|$
$\mathrm{R}2$	0.0771	$\mathrm{R}2$	0.0810	$\mathrm{R}2$	0.0798	$\mathrm{R}3$	0.0418	$\mathrm{R}2$	0.0688	$\mathrm{R}3$	0.0743
$2\times \mathrm{R}2$	0.0154	$2\times \left(\mathrm{R}2+\mathrm{R}3\right)$	0.0188	$4\times \mathrm{R}2$	0.0286	$\mathrm{R}2$	0.0085	$\mathrm{R}2\&\mathrm{R}3\mathrm{SW}$	0.0145	$2\times \mathrm{R}3$	0.0205
$\mathrm{R}1$	0.0117	$2\times \mathrm{R}2$	0.0131	$4\times \mathrm{R}1$	0.0130	$2\times \mathrm{R}3$	0.0082	$2\times \mathrm{R}3$	0.0130	$2\times \mathrm{R}2$	0.0184
$3\times \mathrm{R}2$	0.0064	$\mathrm{R}1$	0.0125	$3\times \mathrm{R}1$	0.0125	$\mathrm{R}1$	0.0067	$\mathrm{R}2$	0.0119	$\mathrm{R}2$	0.0150
$\mathrm{R}1+\mathrm{R}2$	0.0047	$\mathrm{R}2+\mathrm{R}3$	0.0069	$5\times \mathrm{R}2$	0.0116	$3\times \mathrm{R}3$	0.0053	$\mathrm{R}2\&\mathrm{R}3\mathrm{SW}$	0.0104	$3\times \mathrm{R}2$	0.0145

.

The reconstructed results show the band-pass filter effect from the Fourier compression. While the compression method is efficient, the harmonics beyond the fundamental frequency are not well predicted and skew between each fundamental frequency’s third and fifth harmonics. At the same time, the spinning wave modes, or Tyler–Sofrin modes [28,29,30], associated with blade row interactions are removed entirely.

The upstream R3 calculations have confirmed the suspected difficulties with the corresponding pressure transducer signals. The fundamental R3 blade-passing frequency amplitude from the experimental measurements differs from the transient monitor and reconstructed data by factors of about 1.6–1.8. Despite the discrepancies, the blade-passing frequencies associated with the five largest amplitudes compare favorably between the three datasets. After the dominant fundamental R3 response, each dataset contains the fundamental R2 and second harmonic responses in the top four amplitudes.

The R2 and R3 transient monitor points register spectral amplitudes at frequencies attributed to spinning wave modes. These modes indicate high wavenumber (high engine order) blade row interaction modes occurring in the computational flow field outside the dynamic range of the fast-response pressure transducers. For transient R2 monitor data, these occur as the fundamental frequency and harmonics of the sum of the R2 and R3 blade counts. By contrast, the R3 transient monitor data have relatively significant amplitudes associated with the spinning wave modes containing combinations of the R2 and R3 blade counts and their differences. However, these are not as straightforward to identify and, in these cases, are denoted by ‘SW’ in Figure 12 and Table 4.

The computational spectra feature harmonics of the R1 blade-passing frequency upstream of R2 and R3, exhibiting capabilities of the Time Transformation method for simulating blade row interaction in the multistage environment. However, if blade row interactions are of primary interest, these comparisons indicate that the Time Transformation transient blade row model combined with Fourier compression is not ideal. In these cases, further reduction in the multiple passage pitch ratios, ideally to unity values between each row, or forgoing transient blade row modeling for full-annulus calculations may be necessary while directly exporting transient data from the solver. Nevertheless, these analyses lend confidence to the Time Transformation transient blade row model as a sufficient and computationally efficient means of predicting a multistage flow field and determining compressor performance.

4. Conclusions

A multistage axial compressor representing modern, small-core, high-pressure compressors was numerically investigated, emphasizing higher-fidelity modeling when practically possible. Facilitating this investigation, extensive modeling of the machine’s geometry included aspects that had not traditionally been included due to computational constraints. For example, in the current investigation, the stator domains comprised nearly half of the discretized cells in the single-passage model and 66% in the multiple-passage model. The S3 domain alone claimed 20–28% of the discretized cell counts, and a significant reason for these cell distributions was the inclusion of the integrally meshed stator cavities and knife seals with the primary flow path.

Additionally, although one model was traded for another, the mixing plane interfaces between all rotors and stators were replaced with transient rotor-stator interfaces with the Time Transformation transient blade row model, and the turbulence closure of the unsteady Reynolds-averaged Navier–Stokes equations was achieved using the advanced Scale Adaptive Simulation formulation of the SST turbulence model.

The computational model was rigorously compared with available experimental data supporting the validation of modeling decisions. These efforts have proven favorable, with exceptional agreements between the computational results and experimental data. While the model development and validation process may be considered tedious, it is essential to instill confidence in the tools that are relied upon for informing important decisions. A thoroughly verified and well-validated model, acknowledging confidence in its strengths and knowledge of its limitations in predicting outcomes for specific research inquiries, renders a computational model an invaluable tool complementing experimental research. As the experimental database of these test articles grows, further validation of this and similar models becomes possible and should be an ongoing process. More importantly, further validation and confidence in these models help guide future experimental research, including supporting the development of novel experimental methods and probe design and placement.