Abstract
Stagnation pressure loss coefficient is still the most commonly used loss metric for performance evaluation and is routinely used to validate simulations. This is because it is easy to measure and readily available from the experiments. However, it was previously shown (TURBO-24-1006) that the stagnation pressure loss coefficient can become unreliable when high levels of inflow unsteadiness are present. As the current design trends are moving toward more compact machines and higher work coefficients, the levels of unsteadiness are likely to increase. It is therefore desirable to assess how higher inflow unsteadiness levels affect the performance of new blade designs and what loss metrics should be used to reliably estimate it. Motivated by the need to understand how performance prediction changes under highly unsteady inflow conditions, we perform a series of high-fidelity scale-resolving simulations. We use this data to construct energy budgets for a variety of mid-span compressor cases with varying Reynolds numbers and inflow turbulence intensities. This allows us to systematically assess the impact of inflow conditions on loss prediction when using stagnation pressure based loss metrics. Stagnation pressure loss coefficient was found to be least reliable for the high inflow turbulence intensities and at high Reynolds numbers.
Introduction
The ability to correctly measure losses and translate them into turbomachinery performance prediction is one of the most important tasks for the development of novel designs. Among the different loss metrics, the entropy generation by irreversibilities can be considered the most rational choice [1]. Despite some attempts (see e.g., Ref. [2]), entropy remains impractical to measure directly due to the inadequate accuracy of temperature measurements required for accurate entropy estimation. Instead, typically, the stagnation pressure loss is used as it can be measured very accurately with high frequency. This choice is motivated by the observation that for the adiabatic flow through a stationary blade row, stagnation temperature remains constant and as a result entropy changes depend only on stagnation pressure changes.
Given a turbomachinery cascade, loss can be estimated by employing a control volume approach. Then, the changes in stagnation pressure are measured between the inlet and outlet to estimate the cascade loss. However, when this approach is used experimentally the accurate prediction of loss is only warranted if there are no non-uniformities present at the inlet or outlet of the control volume. These non-uniformities may correspond to high level of inlet turbulence, unsteady interactions, and viscous effects within the turbomachinery blading and are practically impossible to avoid in a time-varying flow.
Two seminal papers (among others) discuss how the inlet and outlet averaging should be properly conducted. Namely, Prasad [3] discussed and demonstrated how to obtain a mixed-out condition, i.e., a state that carries the same properties of the initial state but averages flow non-uniformities into a fully mixed-out state. On the other hand, Cumpsty and Horlock [4] provided a rational guide to the different averaging methods, by evaluating several typical averaging procedure for non-uniform flows. They also concluded that, in most cases, the different averaging procedures may provide very similar results, given that the measurement errors often provide a larger inaccuracy than the averaging methods.
Nowadays, high-fidelity simulations give the possibility to fully explore the loss generation mechanisms (e.g., Ref. [5]). Several recent works [6–13], have directly explored the transport equations of entropy, stagnation pressure, or mechanical work potential by pursuing an integral approach within the control volume rather than the classical inlet to outlet loss estimation. In these approaches, irreversibilities are directly related to the volume integral of the viscous dissipation (e.g., Refs. [6,7,12]). Similarly, to account for the unresolved dissipation, it is possible to write the kinetic energy budget in the Navier–Stokes equations with the Reynolds decomposition. The total dissipation is mainly linked to the turbulent kinetic energy production, the mean viscous dissipation, the substantial variation of the kinetic energy, and the turbulent transport [8,10,13]. The contribution of this latter term has been shown as significant in compressor bladings under unsteady inflow conditions in Przytarski et al. [13], even though typically it has been considered negligible in the experimental practice since Moore et al. [14]. Particularly, it has been shown that the turbulent transport terms are the difference between the entropy changes in the control volume and the mass averaged stagnation pressure loss coefficient.
The present paper extends the previous work of Przytarski et al. [13] by considering several high-fidelity simulations of a compressor blade at different inflow conditions. The paper provides a rationale for loss estimation using typical averaging operations. The role of turbulent transport is explored in detail and is related to the variation of the turbulence within the blade passage. In particular, the paper will address the following questions:
What is the correct way to define a transport equation consistent loss coefficient?
What is the impact of flow averaging on the loss prediction?
What drives the difference between different stagnation pressure loss coefficients?
Is there a correlation that can be used to estimate the difference between the loss coefficients?
Computational Setup
Case Setup.
Motivated by the findings of Przytarski et al. [13], the cases considered here were chosen to cover a range of inflow conditions representative of compressor cascade experiments under steady turbulent inflow conditions. The Reynolds numbers, Re, were selected to vary between k and k, inflow turbulence intensities, , between 2% and 9%, while the turbulence lengthscales, , were set to either 4% or 8% of the axial chord. Both on- and off-design conditions were considered by varying the blade incidence between and . The geometry used for these simulations was the controlled-diffusion profile (CDA) representative of a modern high-pressure ratio compressor used previously [9]. This cascade is characterized by the solidity of 1.7, diffusion factor of 0.45, and nominal turning of . In addition, Fig. 1 shows its nominal loading distribution and skin friction coefficient under various on-design inflow conditions. In total, 21 on-design and 12 off-design large-eddy simulation (LES) cases were performed, along with three validation cases using a finer mesh. The summary of all the cases is shown in Table 1.

Loading coefficient and skin friction coefficient of the CDA blade at on-design conditions under different Re numbers and
Details of the simulation cases
Geometry | Operation | Resolution | Re | Ma | Incidence | # of cases | ||
---|---|---|---|---|---|---|---|---|
CDA | On-design | M | k, k, k | 0.20 | 2–9.2% | 4% | 15 | |
CDA | On-design | M | k, k, k | 0.20 | 3.5%, 7.3% | 8% | 6 | |
CDA | On-design | M | k, k, k | 0.20 | 7% | 4% | 3 | |
CDA | Off-design | M | k | to | 3.2–9.2% | 4% |
Geometry | Operation | Resolution | Re | Ma | Incidence | # of cases | ||
---|---|---|---|---|---|---|---|---|
CDA | On-design | M | k, k, k | 0.20 | 2–9.2% | 4% | 15 | |
CDA | On-design | M | k, k, k | 0.20 | 3.5%, 7.3% | 8% | 6 | |
CDA | On-design | M | k, k, k | 0.20 | 7% | 4% | 3 | |
CDA | Off-design | M | k | to | 3.2–9.2% | 4% |
Solver Details.
The LES cases were performed using the high-order in-house compressible multi-block structured flow solver, HiPSTAR [15]. The accuracy of the solver was previously validated on a variety of cases, including turbomachinery applications, e.g., Refs. [6,8,10,11]. For the cases considered here fourth-order accurate spatial and temporal discretization was used with implicit subgrid scale modeling. The fluid was modeled as a perfect gas with properties of cold air and the ratio of specific heats equal to 1.4. Inflow turbulence was generated using a compressible-flow extension of the digital filter approach of Klein et al. [16].
The computational domain around the compressor blade was meshed using an O-grid that was overlaid on a Cartesian background mesh with an overset method [17]. The majority of the cases were performed using an identical baseline mesh with approximately M points, shown in Fig. 2, with some cases incorporating a longer inlet and outlet to allow for more turbulence mixing. A spanwise extent of 30% of axial chord was used and 128 points were used to discretize the spanwise direction for all domains. Periodic boundary conditions were enforced in both spanwise and pitchwise directions. To validate the baseline mesh, an additional fine mesh was generated by doubling resolution in each direction resulting in M points. The wall resolution in viscous wall units for the validation mesh varied for the considered Reynolds numbers. For the lowest Reynolds number of k, the fine mesh resolution corresponded to a direct numerical simulation (DNS) level with maximum viscous wall units equal to: , , and and 95% of dissipation directly resolved as determined by the use of entropy budgets following the procedure of Przytarski and Wheeler [9]. For the highest Reynolds number of k, the fine mesh resolution corresponded to a wall-resolved LES with maximum viscous wall units equal to: , , and and 65% of dissipation directly resolved.
An instantaneous flow snapshot of one of the compressor cascade simulations is shown in Fig. 3 for which the spanwise vorticity contours were plotted. In addition, Fig. 7 shows a summary of turbulence decay along a freestream inviscid streamline for the on-design cases studied here. The figure demonstrates that a wide range of turbulent inflow conditions were tested. Later in the paper, this figure is also used to determine the amount of freestream turbulence decay between the inlet and the outlet plane of the control volume used in the analysis. To aid the analysis, the statistics were collected over at least 15 through flows for all the cases to ensure statistical convergence. This proved to be important to ensure enthalpy and entropy based quantities were fully converged, which is demonstrated in Fig. 4.

Instantaneous spanwise vorticity flowfield for an example case considered here: CDA at Re = 250 k and

Convergence of Gibbs equation (stagnation) budgets against the length of time-averaging period for an example case considered here: on-design CDA at Re = 250 k and
Gibbs Equation—Static
The Gibbs equation budget can then be constructed using time- and Favre-averaged quantities as follows:
—flux of entropy (entropy generation rate)
—flux of enthalpy
—flux of pressure
This is shown in Fig. 5 where we integrate the time- and spanwise-averaged Gibbs equation terms across the pitchwise extent of the computational domain and along the streamwise direction to obtain a line integral plot for the six on-design cases: three Reynolds numbers for two mesh resolutions each. These cases were setup with an identical turbulence inflow, but due to the mesh resolution, freestream turbulence evolution differed for these cases resulting in a slightly different turbulence intensity, , at the inlet reference plane located at 30% axial chord ahead of the blade leading edge.

Gibbs equation budget (static) for CDA cases at on-design conditions with at three Reynolds numbers: k (left), k (middle), and k (right) and two mesh resolutions: baseline 50 M and validation 400 M
For reference, this budget is reported in Supplemental Table 1 available in the Supplemental Materials on the ASME Digital Collection for the same six cases. It can be seen from the table that the budget is very well balanced for the cases shown here. This has also been verified to be true for the remaining 30 cases which were omitted from the paper for brevity reasons.
Gibbs Equation—Stagnation
Just as before we want to integrate the collected statistics within the volume to compute a full budget
These plots demonstrate that the entropy change does indeed only depend on stagnation pressure change. We will use this fact in the following section to define loss coefficients based on the Gibbs equation budget. It should be also pointed out that the change in entropy is almost identical for both versions of Gibbs equation, i.e., for the static and the stagnation states. For reference, we also report this budget in Supplemental Table 2 available in the Supplemental Materials which shows that, again, it is well balanced.

Gibbs equation budget (stagnation) for CDA at on-design conditions cases with at three Reynolds numbers: k (left), k (middle), and k (right) and two mesh resolutions: baseline 50 M and validation 400 M
Definition of Loss Coefficients
These two stagnation pressure loss coefficients were evaluated for the on-design LES cases considered here and compared in Fig. 8 with the left figure showing flux averaged stagnation pressure loss coefficient and the right one showing mass averaged stagnation pressure loss coefficient. To better elucidate how the loss coefficients vary for various cases we plot them against the freestream turbulence decay, i.e., the difference between the freestream turbulence intensity at the inlet plane and the outlet plane. This difference was determined from Fig. 7.

Comparison between flux averaged (left) and mass averaged (right) stagnation pressure loss coefficient for the on-design cases as a function of freestream turbulence intensity decay
Both plots show a clear trend of increased loss as the turbulence intensity decay increases, but also exhibit a considerably different sensitivity to turbulence decay with a slope of the sensitivity nearly three times higher for the flux averaged loss coefficient. It is interesting to note, however, that the loss trend for all the Reynolds numbers shows a linear behavior with relatively small spread despite the variation in turbulent lengthscale and mesh resolution for some cases. The slope also stayed constant regardless of the Reynolds number.
As it was asserted in the previous section, the flux averaged stagnation loss coefficient is the correct loss metric that matches the entropic loss. It is therefore of interest to understand the discrepancy between the mass averaged and flux averaged loss coefficients. This will be explored in the following section.
The Impact of the Choice of Averaging
—flux of stagnation pressure
—mass averaged flux of stagnation pressure
—pressure work
—convection of kinetic energy
—turbulent transport due to unsteadiness
—pressure diffusion
—pressure dilation
To determine which terms are responsible for the discrepancy in loss coefficients, we compute a budget of flux and mass averaged pressure. Table 2 shows this budget for the same six on-design cases (3 Re and 2 mesh resolution) that Gibbs equation budgets were reported for. For brevity, pressure dilation, diffusion, and unsteady pressure work terms were omitted from the table as they were found to be of negligible magnitude for all the cases considered here, which is a likely consequence of a low Ma number considered here. It can be seen that the budget is well balanced for all the considered cases. It is also clear from the table that the only non-zero term that is missing from the mass averaged stagnation pressure budget is the turbulence transport term .
Flux of stagnation pressure budget
Re | k | k | k | k | k | k |
Mesh | M | M | M | M | M | M |
7.845 | 6.823 | 6.006 | 8.157 | 7.462 | 6.894 | |
119.533 | 123.41 | 126.203 | 116.129 | 119.694 | 121.554 | |
124.091 | 126.649 | 128.502 | 120.476 | 122.932 | 124.057 | |
2.035 | 2.217 | 2.302 | 2.313 | 2.565 | 2.695 | |
1.332 | 1.466 | 1.525 | 1.559 | 1.760 | 1.823 | |
0.011 | 0.009 | 0.007 | 0.007 | 0.004 | 0.006 | |
LHS–RHS | 0.091 | 0.108 | 0.127 | 0.069 | 0.105 | 0.121 |
0.0345 | 0.0294 | 0.0254 | 0.0361 | 0.0323 | 0.0296 | |
0.0350 | 0.0298 | 0.0260 | 0.0364 | 0.0328 | 0.0300 | |
0.0290 | 0.0235 | 0.0195 | 0.0295 | 0.0251 | 0.0223 |
Re | k | k | k | k | k | k |
Mesh | M | M | M | M | M | M |
7.845 | 6.823 | 6.006 | 8.157 | 7.462 | 6.894 | |
119.533 | 123.41 | 126.203 | 116.129 | 119.694 | 121.554 | |
124.091 | 126.649 | 128.502 | 120.476 | 122.932 | 124.057 | |
2.035 | 2.217 | 2.302 | 2.313 | 2.565 | 2.695 | |
1.332 | 1.466 | 1.525 | 1.559 | 1.760 | 1.823 | |
0.011 | 0.009 | 0.007 | 0.007 | 0.004 | 0.006 | |
LHS–RHS | 0.091 | 0.108 | 0.127 | 0.069 | 0.105 | 0.121 |
0.0345 | 0.0294 | 0.0254 | 0.0361 | 0.0323 | 0.0296 | |
0.0350 | 0.0298 | 0.0260 | 0.0364 | 0.0328 | 0.0300 | |
0.0290 | 0.0235 | 0.0195 | 0.0295 | 0.0251 | 0.0223 |
At the bottom of Table 2, the stagnation pressure loss coefficient derived from the budget left-hand side () and the right-hand side () is compared and a good agreement is shown between these two loss coefficients. These loss coefficients are further compared with the ones computed globally, i.e., from the inlet and outlet reference planes and the summary of all the stagnation pressure loss coefficients is given in Table 3.
The summary of stagnation pressure loss coefficients
Re | k | k | k | k | k | k |
---|---|---|---|---|---|---|
Mesh | M | M | M | M | M | M |
0.0359 | 0.0306 | 0.0266 | 0.0378 | 0.0338 | 0.0313 | |
0.0345 | 0.0294 | 0.0254 | 0.0361 | 0.0323 | 0.0296 | |
0.0350 | 0.0298 | 0.0260 | 0.0364 | 0.0328 | 0.0300 | |
0.0298 | 0.0241 | 0.0199 | 0.0304 | 0.0258 | 0.0230 | |
0.0290 | 0.0235 | 0.0195 | 0.0295 | 0.0251 | 0.0223 |
Re | k | k | k | k | k | k |
---|---|---|---|---|---|---|
Mesh | M | M | M | M | M | M |
0.0359 | 0.0306 | 0.0266 | 0.0378 | 0.0338 | 0.0313 | |
0.0345 | 0.0294 | 0.0254 | 0.0361 | 0.0323 | 0.0296 | |
0.0350 | 0.0298 | 0.0260 | 0.0364 | 0.0328 | 0.0300 | |
0.0298 | 0.0241 | 0.0199 | 0.0304 | 0.0258 | 0.0230 | |
0.0290 | 0.0235 | 0.0195 | 0.0295 | 0.0251 | 0.0223 |
Turbulent Transport
Since the turbulent transport formula is a function of Reynolds stresses and mean strains, it is not surprising that for a given style of compressor and a limited amount of turning (19–) it is effectively linear with respect to the turbulent kinetic energy (TKE) decay. However, given the representative nature of the CDA cascade to the modern style compressor design and moderate levels of turning found in compressors in general, the authors expect this correlation to be applicable to a broad family of compressor geometries. As a result, it can be now used in conjunction with standard experimental practice of mass averaged stagnation pressure loss coefficient to correct for the terms neglected as part of the averaging procedure.

Difference between the flux and mass averaged stagnation pressure loss coefficient (left) and normalized turbulence transport, (right) for all the considered cases (both on- and off-design) as a function of freestream turbulence intensity decay
While the provided correlation can help correct for the highly turbulent steady inflow cascade experiments, it is not applicable for the flows featuring unsteady inflow disturbances such as wakes. Under these conditions turbulent transport by the turbulent flow term is non-zero as has been shown by Przytarski et al. [13], and can skew results accordingly depending on the wake strength and blade incidence.
While the turbulent transport has been show to play a significant role in determining stagnation pressure loss coefficient in compressors, the question remains as to whether it plays a similar role for the turbines. It is the authors’ belief that this term, while non-zero for highly turbulent steady inflow turbine flows, plays a relatively negligible role when compared to the overall turbine loss. This is a consequence of much larger momentum change across the turbine cascade.
For completeness, we also report the variation of flux averaged stagnation pressure budgets for all the on-design cases in Fig. 10. As was demonstrated in Eq. (20) and further supported by the integrated budgets in Table 2, the flux averaged stagnation pressure is primarily determined by the interplay of the mean pressure work , advection of mean kinetic energy , advection of turbulent kinetic energy , and turbulent transport (Reynolds stress work on boundaries) . The figure demonstrates that for a given amount of duty (turning), the mean pressure work is independent of turbulence decay and remains roughly constant for each Reynolds number. Advection of mean kinetic energy on the other hand varies with both turbulence decay and Reynolds number and tends to reduce loss as the turbulence decay increases or Reynolds number decreases. On the other hand, both the advection of turbulent kinetic energy and turbulent transport by the mean flow are Reynolds number independent and increase the loss as the turbulence decay increases (the more negative the value, the higher loss).

Variation of the flux averaged stagnation pressure budgets (dominant terms) with respect to the turbulence decay for all the cases
Conclusions
A series of high-fidelity simulations of a CDA compressor cascade with varying inflow conditions was carried out. Reynolds number, inflow turbulence intensity, turbulence lengthscale, and incidence were varied systematically to cover a range of conditions that are typically found in compressor cascade experiments. For each of the cases, Gibbs equation budget was computed. Subsequently, loss coefficients consistent with the Gibbs equation were derived in both transport equation form as well as in a global form by evaluating stagnation pressure flux on control volume boundaries. It was shown that while Gibbs equation derived flux averaged stagnation pressure loss coefficient was consistent with the entropy loss, the mass averaged stagnation pressure loss coefficient was found to be unreliable for cases considered here, under-predicting loss for one of the cases by as much as 30%. This was found to be due to the effect of the turbulent transport by mean flow term, also known as work of Reynolds stress on boundaries. This term, typically ignored as it is difficult to evaluate experimentally, was found to vary approximately linearly with freestream turbulence decay across the control volume and independent of Reynolds number and incidence. As a result, the paper provides a correlation for this term that is likely to be applicable to a range of compressor cascades and could be used to correct loss prediction from cascade experiments exposed to high levels of freestream turbulence. Lastly, the flux of stagnation pressure budget was shown for all the on-design cases revealing sensitivity of each term to inflow conditions. It was shown that both advection of TKE and turbulent transport by the mean flow terms are independent of Reynolds number and depend entirely on freestream turbulence decay which is important in the context of term modeling.
Acknowledgment
This project received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 101026928). We also acknowledge PRACE, which awarded access to the Fenix Infrastructure resources at CINECA, partially funded from the European Union’s Horizon 2020 research and innovation program through the ICEI project under the grant agreement No. 800858.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
This data are open access and available upon request.
Nomenclature
- =
enthalpy
- =
pressure
- =
entropy
- =
velocity
- =
quantity x
- =
temperature
- =
v integral of x
- =
mass flowrate
- =
time-average of x
- =
fluctuating component of x
- =
component of a vector quantity
- =
related to mean flowfield
- =
related to fluctuating flowfield
- =
related to the action of both mean and fluctuating flowfields
- =
related to the action of fluctuating flowfield on itself
- =
related to stagnation quantity
- =
related to enthalpy
- =
related to entropy
- =
related to pressure
- =
axial chord
- =
turbulence lengthscale
- =
entropy generation rate
- =
flux of stagnation pressure
- Ma =
Mach number
- Re =
Reynolds number based on inlet cond. and axial chord
- =
turbulence intensity
- LHS =
left-hand side
- RANS =
Reynolds-averaged Navier–Stokes
- RHS =
right-hand side
- TKE =
turbulent kinetic energy