## Abstract

In this article, we study the role of turbulence transport on loss prediction using high fidelity scale-resolving simulations. For this purpose, we use eight high-fidelity simulation datasets and compute flux of entropy and stagnation pressure transport equation budgets for all the cases. We find that under certain unsteady inflow conditions, the stagnation pressure coefficient is not a reliable loss metric even at low Mach number conditions. This is due to the turbulence transport terms. The impact of these terms, typically assumed to be negligible, made stagnation pressure loss coefficient underpredict loss from 0% to over 40% when compared to entropy loss coefficient for cases considered here. This effect was most pronounced for the cases with highly unsteady inflow conditions and at low Reynolds numbers.

## 1 Introduction

High-fidelity simulations provide the details of the entire flow field in the blade passage and therefore may be used to explore loss generation and loss generating mechanisms [1]. The most commonly used loss metric is the stagnation pressure loss coefficient. This is because it can be directly measured and is a workhorse for evaluating the performance of different blade designs experimentally. There are other available metrics, such as entropy [2] or mechanical work potential [3], which have an advantage of better representing the lost work due to their ability to account for heat transfer and thermal mixing effects. However, they rely on quantities, which are difficult to measure or have to be derived (e.g., entropy cannot be measured directly). As a result, they are difficult to implement experimentally and lead to high levels of measurement uncertainty. It is therefore of interest to investigate complete transport equations using high-fidelity simulations and compare loss drivers between different loss metrics.

Transport equations have been practically used since the $1980s$. Among the first, the work of Moore et al. [4] used mean part of the Reynolds decomposed transport equation of stagnation pressure (e.g., Ref. [5]) and applied it to the 2D-field measurements performed by means of hot-wire probes. They found that the integration of the production of the turbulence kinetic energy represented the generation of stagnation pressure loss well. Similarly, more recent articles, either experimental [6–8] or numerical [9,10], made strides to understand the mechanism by which turbulence extracts work from the mean flow via turbulence production. The interest in the production of turbulence kinetic energy is partially motivated by the relative ease of computing it from the experimental data with relatively low uncertainty. On the other hand, other terms appearing in the transport equations are difficult to obtain and are often neglected. For this reason, the transport equation of entropy that relates the entropy generation rate to the viscous dissipation and heat transfer terms has been computed in more recent high-fidelity simulations [11–16]. These authors showed that the full volume integration of the right- hand side of the entropy transport equation may provide a spatial breakdown of losses pointing out at their sources. In addition, Leggett et al. [15] provided a comparison of entropy and mechanical work potential terms appearing in the respective transport equations.

In the present article, we will compare different loss coefficients computed from several unsteady high-fidelity simulations of compressor blades, and we will further discuss the results computing the full transport equation of entropy and stagnation pressure flux. The article is structured as follow: (1) the global loss coefficients are introduced; (2) the numerical methods and simulation cases are summarized; (3) transport equations of entropy flux and stagnation pressure flux are analyzed to provide a rationale for a comparison of the metrics; and (4) the result section that discusses the different loss coefficients and loss budgets with a focus on the effect of turbulence on the different budgets. In particular, the article addresses the following questions:

What determines the stagnation pressure and entropy loss coefficients?

What drives the difference between the loss coefficients?

When can we expect the two coefficients to vary?

## 2 Computational Setup

### 2.1 Solver Details.

For this study, we use high-fidelity datasets of Przytarski and Wheeler [12,14]. These datasets were generated using the high-order compressible Navier–Stokes solver $3DNS$ [17]. In total, eight datasets were used which consisted of two $NACA65$ datasets at steady turbulent inflow conditions, three $NACA65$ datasets at unsteady inflow conditions akin to multistage environment, and three $CDA$ (Controlled Diffusion Airfoil) datasets unsteady inflow conditions akin to multistage environment. Table 1 summarizes the datasets and shows the running conditions of each case. For cases with unsteady inflow conditions reduced frequency $Fred$ was also reported, and the unsteadiness intensity was decomposed into periodic Pu and turbulent Tu components. The spanwise extent for $NACA65$ cases was 10% of the axial chord, while $CDA$ cases used spanwise extent equal to 15% of axial chord. Maximum viscous wall for these geometries are reported in Table 2 asserting the wall-resolved accuracy. Setup details and a more thorough description of each case, including a procedure used to mimic the multistage environment, can be found in Refs. [12,14]. An example of an instantaneous flowfield for steady and unsteady $NACA65$ profile as well as an unsteady $CDA$ profile is shown in Fig. 1 for which the spanwise vorticity contours were plotted.

Geometry | Case | Re | Ma | $Fred$ | Pu | Tu |
---|---|---|---|---|---|---|

NACA65 | Tu4 | 140k | $0.065$ | — | — | $4.0%$ |

NACA65 | Tu6 | 140k | $0.065$ | — | — | $6.0%$ |

NACA65 | Gap30 | 140k | $0.065$ | $1.71$ | $5.0%$ | $4.2%$ |

NACA65 | Gap40 | 140k | $0.065$ | $1.71$ | $3.6%$ | $3.3%$ |

NACA65 | Gap50 | 140k | $0.065$ | $1.71$ | $3.0%$ | $2.9%$ |

CDA | Gap30 | 250k | $0.20$ | $2.49$ | $4.9%$ | $3.1%$ |

CDA | Gap40 | 250k | $0.20$ | $2.49$ | $3.7%$ | $2.9%$ |

CDA | Gap50 | 250k | $0.20$ | $2.49$ | $3.2%$ | $3.0%$ |

Geometry | Case | Re | Ma | $Fred$ | Pu | Tu |
---|---|---|---|---|---|---|

NACA65 | Tu4 | 140k | $0.065$ | — | — | $4.0%$ |

NACA65 | Tu6 | 140k | $0.065$ | — | — | $6.0%$ |

NACA65 | Gap30 | 140k | $0.065$ | $1.71$ | $5.0%$ | $4.2%$ |

NACA65 | Gap40 | 140k | $0.065$ | $1.71$ | $3.6%$ | $3.3%$ |

NACA65 | Gap50 | 140k | $0.065$ | $1.71$ | $3.0%$ | $2.9%$ |

CDA | Gap30 | 250k | $0.20$ | $2.49$ | $4.9%$ | $3.1%$ |

CDA | Gap40 | 250k | $0.20$ | $2.49$ | $3.7%$ | $2.9%$ |

CDA | Gap50 | 250k | $0.20$ | $2.49$ | $3.2%$ | $3.0%$ |

## 3 Loss Metrics

As discussed by Denton [2], there are several loss coefficients that are used in turbomachinery. The most popular loss coefficients are entropy, enthalpy, and stagnation pressure loss coefficients. As asserted by Denton [2], entropy and enthalpy loss coefficients should report virtually identical results. However, the advantage of entropy loss coefficient is that it is more applicable in the engine setting. More in-depth discussion on the limitations of each loss coefficient can be found in the literature [18,19]. Despite the consensus that entropy loss coefficient is the most accurate for engine performance evaluation, the industrial practice often relies on stagnation pressure loss coefficient due to the ease of measuring it in an experimental setting. However, in our experience, it is often difficult to obtain consistent results from entropy and stagnation pressure loss coefficients. Given the availability of relevant high-fidelity datasets, it is desirable to study the predictive capability of these loss coefficients, what determines them and their limitations for the unsteady, turbulent turbomachinery flows.

### 3.1 Entropy Loss Coefficient.

### 3.2 Enthalpy Loss Coefficient.

### 3.3 Stagnation Pressure Loss Coefficient.

$\omega p$—for stagnation pressure including TKE

$\omega noTKEp$—for stagnation pressure without TKE

In this paper, we will explore the impact of turbulent kinetic energy on these predictions and the role turbulent transport plays in stagnation pressure loss coefficient in general.

## 4 Evaluation of Loss Coefficients

Table 3 gives a summary of above mentioned loss coefficients for all the cases considered here. For the steady cases, i.e., $NACA65$$Tu4$ and $Tu6,$ the reference planes that were used to compute the loss coefficients were set at $30%$ axial chord upstream of the leading edge and downstream of the trailing edge. For the unsteady cases, the reference planes were set at the domain inlet (20–40% upstream of the leading edge depending on the gap) and at the sampling plane, which was located at roughly $10%$ axial chord downstream of the trailing edge. As evident from the Table 3, both entropy and enthalpy loss coefficients result in closely matching estimates. On the other hand, the mass-averaged pressure loss coefficient ($\omega p$) results in values which are not only quantitatively different but also qualitatively misleading by reversing the loss trend for unsteady $NACA65$ cases. These values were highlighted in bold. The estimates are even worse when mass-averaged stagnation pressure loss coefficient without the inclusion of turbulent kinetic energy ($\omega noTKEp$) is considered (also highlighted in bold). This emphasizes the need of including turbulent kinetic energy for mass-averaged stagnation pressure estimates when highly unsteady flowfields are considered. To understand why the mass-averaged stagnation pressure coefficient ($\omega p$) is inaccurate, in the next section, we will examine entropy and stagnation pressure transport equations.

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Loss | Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 |

$\omega s$ | 0.0290 | 0.0383 | 0.0271 | 0.0259 | 0.0262 | 0.0229 | 0.0208 | 0.0208 |

$\omega s(T,p)$ | 0.0296 | 0.0390 | 0.0284 | 0.0267 | 0.0270 | 0.0234 | 0.0213 | 0.0212 |

$\omega h$ | 0.0296 | 0.0390 | 0.0284 | 0.0267 | 0.0270 | 0.0235 | 0.0213 | 0.0212 |

$\omega p$ | 0.0255 | 0.0287 | 0.0145 | 0.0186 | 0.0205 | 0.0207 | 0.0207 | 0.0206 |

$\omega noTKEp$ | 0.0225 | 0.0180 | 0.0067 | 0.0128 | 0.0161 | 0.0149 | 0.0151 | 0.0153 |

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Loss | Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 |

$\omega s$ | 0.0290 | 0.0383 | 0.0271 | 0.0259 | 0.0262 | 0.0229 | 0.0208 | 0.0208 |

$\omega s(T,p)$ | 0.0296 | 0.0390 | 0.0284 | 0.0267 | 0.0270 | 0.0234 | 0.0213 | 0.0212 |

$\omega h$ | 0.0296 | 0.0390 | 0.0284 | 0.0267 | 0.0270 | 0.0235 | 0.0213 | 0.0212 |

$\omega p$ | 0.0255 | 0.0287 | 0.0145 | 0.0186 | 0.0205 | 0.0207 | 0.0207 | 0.0206 |

$\omega noTKEp$ | 0.0225 | 0.0180 | 0.0067 | 0.0128 | 0.0161 | 0.0149 | 0.0151 | 0.0153 |

Note: Bold values correspond to typically computed mass-averaged stagnation pressure loss coefficients which for the cases considered here underestimate the loss when compared with similarly computed entropy and enthalpy loss coefficients.

## 5 Loss Transport Equations

### 5.1 Entropy Transport Equation.

*a*)

### 5.2 Stagnation Pressure Transport Equation.

*a*)

## 6 Loss Transport Equations Budgets and Loss Coefficients

*a*)

*b*)

*a*)

*b*)

The stagnation pressure loss coefficient arise as an interplay between dissipation due to mean strains $\Phi m$, turbulent dissipation $\Phi f+N$, turbulent transport due to mean and turbulent fields $Trmf$ and $Trff$, mean and turbulent viscous diffusion $VDm$ and $VDf$, as well as turbulent pressure work $PWf$, pressure diffusion $PDf,$ and pressure dilation $PLf$. It should be mentioned that since Moore et al. [4], the diffusive terms ($Trmf,ff$, $VDm,f$, $PWf$, $PDf$, $PLf$) are often neglected as it is assumed that their volume integral is negligible. Furthermore, there are practical difficulties estimating these terms experimentally. High-fidelity datasets are immune to such limitations and so are perfect test bed for verifying these assumptions under a variety of conditions. Table 4 shows the entropy transport equation budgets for all the cases. Below the budget, entropy loss coefficient derived from the budget is compared to the one computed globally at the reference frames. Very good agreement is shown between these two loss coefficients.

$TST\u2248\Phi m+\Phi f+N$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$TST$ | 7.564 | 9.954 | 6.980 | 6.668 | 6.726 | 2111.269 | 1916.130 | 1916.756 |

$\Phi m$ | 3.588 | 3.660 | 3.419 | 3.472 | 3.554 | 962.015 | 904.669 | 911.772 |

$\Phi f+N$ | 3.976 | 6.295 | 3.561 | 3.196 | 3.172 | 1149.254 | 1011.461 | 1004.984 |

$\omega s(TST)$ | 0.0295 | 0.0388 | 0.0272 | 0.0260 | 0.0263 | 0.0234 | 0.0213 | 0.0213 |

$\omega ms$ | 0.0290 | 0.0383 | 0.0271 | 0.0259 | 0.0262 | 0.0229 | 0.0208 | 0.0208 |

$TST\u2248\Phi m+\Phi f+N$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$TST$ | 7.564 | 9.954 | 6.980 | 6.668 | 6.726 | 2111.269 | 1916.130 | 1916.756 |

$\Phi m$ | 3.588 | 3.660 | 3.419 | 3.472 | 3.554 | 962.015 | 904.669 | 911.772 |

$\Phi f+N$ | 3.976 | 6.295 | 3.561 | 3.196 | 3.172 | 1149.254 | 1011.461 | 1004.984 |

$\omega s(TST)$ | 0.0295 | 0.0388 | 0.0272 | 0.0260 | 0.0263 | 0.0234 | 0.0213 | 0.0213 |

$\omega ms$ | 0.0290 | 0.0383 | 0.0271 | 0.0259 | 0.0262 | 0.0229 | 0.0208 | 0.0208 |

Similarly, Table 5 shows the stagnation pressure transport equation budgets for all the cases. At the bottom of the table, the stagnation pressure loss coefficient derived from the budget is compared to the one computed globally and, again, a good agreement is shown between these two loss coefficients for all the cases.

$\u2212FSP=\Phi m+\Phi f+N+Trmf+Trff\u2212VDm\u2212VDf\u2212PWf+PDf\u2212PLf$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$\u2212FSP=\u2212Advm\u2212Advf\u2212PWm$ | 6.691 | 7.576 | 3.811 | 4.801 | 5.308 | 1,816.369 | 1,891.532 | 1,929.971 |

$\Phi m$ | 3.588 | 3.660 | 3.419 | 3.472 | 3.554 | 962.015 | 904.669 | 911.772 |

$\Phi f+N$ | 3.976 | 6.295 | 3.561 | 3.196 | 3.172 | 1149.254 | 1011.461 | 1004.984 |

$Trmf$ | $\u22120.847$ | $\u22122.587$ | $\u22123.905$ | $\u22122.551$ | $\u22122.004$ | $\u2212387.640$ | $\u2212207.121$ | $\u2212208.588$ |

$Trff$ | — | — | 0.688 | 0.649 | 0.488 | 94.282 | 213.863 | 202.576 |

$\u2212VDm$ | $\u22120.012$ | $\u22120.009$ | $\u22120.014$ | $\u22120.015$ | $\u22120.015$ | $\u22122.451$ | $\u22122.229$ | $\u22122.240$ |

$\u2212VDf$ | $\u22120.008$ | $\u22120.017$ | $\u22120.005$ | $\u22120.005$ | $\u22120.005$ | $\u22121.881$ | $\u22121.577$ | $\u22121.544$ |

$\u2212PWf$ | — | — | — | — | — | — | — | — |

$PDf$ | — | - | 0.129 | 0.080 | 0.111 | 4.810 | 26.336 | 33.456 |

$\u2212PLf$ | — | — | — | — | — | — | — | — |

LHS–RHS | $\u22120.073$ | $\u22120.315$ | 0.054 | 0.016 | $\u22120.016$ | $\u221241.965$ | 13.668 | $\u221232.144$ |

$\omega p(FSP)$ | 0.0264 | 0.0298 | 0.0149 | 0.0187 | 0.0208 | 0.0204 | 0.0212 | 0.0217 |

$\omega mp$ | 0.0255 | 0.0287 | 0.0145 | 0.0186 | 0.0205 | 0.0207 | 0.0207 | 0.0206 |

$\u2212FSP=\Phi m+\Phi f+N+Trmf+Trff\u2212VDm\u2212VDf\u2212PWf+PDf\u2212PLf$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$\u2212FSP=\u2212Advm\u2212Advf\u2212PWm$ | 6.691 | 7.576 | 3.811 | 4.801 | 5.308 | 1,816.369 | 1,891.532 | 1,929.971 |

$\Phi m$ | 3.588 | 3.660 | 3.419 | 3.472 | 3.554 | 962.015 | 904.669 | 911.772 |

$\Phi f+N$ | 3.976 | 6.295 | 3.561 | 3.196 | 3.172 | 1149.254 | 1011.461 | 1004.984 |

$Trmf$ | $\u22120.847$ | $\u22122.587$ | $\u22123.905$ | $\u22122.551$ | $\u22122.004$ | $\u2212387.640$ | $\u2212207.121$ | $\u2212208.588$ |

$Trff$ | — | — | 0.688 | 0.649 | 0.488 | 94.282 | 213.863 | 202.576 |

$\u2212VDm$ | $\u22120.012$ | $\u22120.009$ | $\u22120.014$ | $\u22120.015$ | $\u22120.015$ | $\u22122.451$ | $\u22122.229$ | $\u22122.240$ |

$\u2212VDf$ | $\u22120.008$ | $\u22120.017$ | $\u22120.005$ | $\u22120.005$ | $\u22120.005$ | $\u22121.881$ | $\u22121.577$ | $\u22121.544$ |

$\u2212PWf$ | — | — | — | — | — | — | — | — |

$PDf$ | — | - | 0.129 | 0.080 | 0.111 | 4.810 | 26.336 | 33.456 |

$\u2212PLf$ | — | — | — | — | — | — | — | — |

LHS–RHS | $\u22120.073$ | $\u22120.315$ | 0.054 | 0.016 | $\u22120.016$ | $\u221241.965$ | 13.668 | $\u221232.144$ |

$\omega p(FSP)$ | 0.0264 | 0.0298 | 0.0149 | 0.0187 | 0.0208 | 0.0204 | 0.0212 | 0.0217 |

$\omega mp$ | 0.0255 | 0.0287 | 0.0145 | 0.0186 | 0.0205 | 0.0207 | 0.0207 | 0.0206 |

Some of the terms in the stagnation pressure transport equation budgets (mainly $PWf,PLf$) were not possible to compute and as a result were not considered. Despite omitting them in the budget, the difference between left-hand side and the right-hand side terms (LHS-RHS) is small, and it is therefore concluded that these terms can be considered negligible for the cases considered here.

In the next section, we will explore how the components of these loss budgets determine the overall loss coefficient.

## 7 Comparison of Loss Contributions

The first thing to note from Table 5 is that while most diffusive terms are indeed close to zero and negligible as typically assumed; however, the turbulent transport terms $Trmf$ and $Trff$ are consistently high and range from $15%$ of turbulence dissipation $\Phi f$ for the moderate turbulence $NACA65$$Tu4$ case, all the way to over $100%$ for the unsteady $NACA65$$Gap30$ case.

This is further demonstrated by integrating all the budgets along the streamwise direction to obtain a line integral plots for the steady $NACA65$$Tu4$ case, Fig. 2, unsteady $NACA65$$Gap40$ case, Fig. 3, and unsteady $CDA$$Gap40$ case, Fig. 4. For all the cases, the domain was normalized by the axial chord with $x=0$ coordinate corresponding to the leading edge and $x=1$ coordinate corresponding to the trailing edge.

Turbulent transport terms $Trmf$ and $Trff$ play a significant role in stagnation pressure transport equation for most of the considered cases. For $CDA$ cases, their impact appears to be limited as both terms are of similar magnitude and opposite sign and therefore lead to error cancellation.

It can be also noted that turbulent transport terms have a stronger impact on the cases that feature more unsteady/turbulent inflow. As a result, by the virtue of how stagnation pressure transport is computed and due to the terms’ negative contribution, they reduce the stagnation pressure loss coefficient, resulting in erroneous performance prediction.

To understand where the impact of turbulent transport terms is the strongest, we perform a domain decomposition and compute a loss budget for different regions. We determine the boundary layer and the wake edges with a vorticity criterion. Figure 5 shows the resulting region split. Figure 6 shows separate budgets for the combined regions of boundary layers and wake and for the freestream for the steady inflow $NACA65$$Tu4$ case. Similarly,Fig. 7 shows analogous budget for the unsteady inflow $NACA65$$Gap40$ case and Fig. 8 shows that for the unsteady inflow $CDA$$Gap40$ case. The overall loss comparison for the remaining cases is given in Table 6. It is clear from both figures that majority of turbulent transport $Trmf$ and $Trff$ happens in the freestream. As a result, the loss, as predicted by stagnation pressure, is well predicted for the boundary layers and wake regions; however, it reduces in the freestream, which skews the overall loss prediction. This was the case for most of the datasets considered here. For two $CDA$ cases with $Gap40$ and $Gap50,$ that effect was negligible as the two turbulent transport terms were of similar magnitude and opposite signs.

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$TST,tot$ | 7.564 | 9.954 | 6.980 | 6.668 | 6.726 | 2111.269 | 1916.130 | 1916.756 |

$\u2212FSPtot$ | 6.770 | 7.656 | 3.819 | 4.810 | 5.317 | 1860.353 | 1931.734 | 1972.560 |

$TST,blw$ | 6.289 | 6.616 | 5.836 | 5.688 | 5.789 | 1771.757 | 1610.546 | 1627.988 |

$\u2212FSPblw$ | 6.149 | 6.331 | 6.338 | 5.953 | 5.956 | 1449.368 | 1441.570 | 1424.509 |

$TST,free$ | 1.2752 | 3.3381 | 1.1441 | 0.9799 | 0.9377 | 339.5117 | 305.5845 | 288.7683 |

$\u2212FSPfree$ | 0.622 | 1.325 | $\u22122.519$ | $\u22121.143$ | $\u22120.639$ | 410.985 | 490.164 | 548.050 |

$\u2212FSPblw/TST,blw$ | 97.77% | 95.69% | 108.61% | 104.66% | 102.90% | 96.43% | 106.38% | 102.90% |

$\u2212FSPfree/TST,free$ | 48.76% | 39.70% | $\u2212220.16%$ | $\u2212116.63%$ | $\u221268.17%$ | 36.17% | 64.81% | 102.99% |

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$TST,tot$ | 7.564 | 9.954 | 6.980 | 6.668 | 6.726 | 2111.269 | 1916.130 | 1916.756 |

$\u2212FSPtot$ | 6.770 | 7.656 | 3.819 | 4.810 | 5.317 | 1860.353 | 1931.734 | 1972.560 |

$TST,blw$ | 6.289 | 6.616 | 5.836 | 5.688 | 5.789 | 1771.757 | 1610.546 | 1627.988 |

$\u2212FSPblw$ | 6.149 | 6.331 | 6.338 | 5.953 | 5.956 | 1449.368 | 1441.570 | 1424.509 |

$TST,free$ | 1.2752 | 3.3381 | 1.1441 | 0.9799 | 0.9377 | 339.5117 | 305.5845 | 288.7683 |

$\u2212FSPfree$ | 0.622 | 1.325 | $\u22122.519$ | $\u22121.143$ | $\u22120.639$ | 410.985 | 490.164 | 548.050 |

$\u2212FSPblw/TST,blw$ | 97.77% | 95.69% | 108.61% | 104.66% | 102.90% | 96.43% | 106.38% | 102.90% |

$\u2212FSPfree/TST,free$ | 48.76% | 39.70% | $\u2212220.16%$ | $\u2212116.63%$ | $\u221268.17%$ | 36.17% | 64.81% | 102.99% |

The results suggest that even for the cascades exposed to moderate levels of freestream turbulence, stagnation pressure loss coefficient may lead to incorrect predictions, especially when cases with varying or unsteady inflow conditions are considered.

We can also determine which components of the turbulent transport contribute the most. This is shown in Table 7, which demonstrates that for the cases considered here two terms associated with the $u\u2032u\u2032$ and $u\u2032v\u2032$ Reynolds stresses are responsible for the entire turbulent transport $Trmf$ (other terms were close to 0). This is encouraging as it suggests that at a mid-span section the turbulent transport term can be successfully estimated experimentally by considering only these two components alone and measuring Reynolds stress components associated with them. This has been demonstrated before by Perdichizzi et al. [23] or Jelly et al. [24]. The role of the turbulent transport term $Trmf$ was also previously recognized by Folk et al. [25,26] who used rapid distortion theory to estimate the magnitude of this term for turbine flow exposed to combustor turbulence. As far as $Trff$ term is concerned, it appears to be entirely determined by the streamwise fluctuations $u\u2032$ and turbulent kinetic energy $(1/2)ui\u2032ui\u2032$.

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$\u2202\u2202x[u\xaf(\rho u\u2032u\u2032\xaf)]/Trmf$ | 0.74 | 0.85 | 1.09 | 1.14 | 1.13 | 2.45 | 3.82 | 3.64 |

$\u2202\u2202x[v\xaf(\rho u\u2032v\u2032\xaf)]/Trmf$ | 0.26 | 0.15 | $\u22120.09$ | $\u22120.14$ | $\u22120.14$ | $\u22121.44$ | $\u22122.79$ | $\u22122.61$ |

$\u2202\u2202x[\rho u\u2032(12u\u2032u\u2032)\xaf]/Trff$ | — | — | 0.87 | 0.87 | 0.86 | 0.67 | 0.74 | 0.74 |

$\u2202\u2202x[\rho u\u2032(12v\u2032v\u2032)\xaf]/Trff$ | — | — | 0.07 | 0.07 | 0.07 | 0.25 | 0.22 | 0.21 |

$\u2202\u2202x[\rho u\u2032(12w\u2032w\u2032)\xaf]/Trff$ | — | — | 0.06 | 0.06 | 0.07 | 0.08 | 0.05 | 0.04 |

NACA65 | NACA65 | CDA | ||||||
---|---|---|---|---|---|---|---|---|

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$\u2202\u2202x[u\xaf(\rho u\u2032u\u2032\xaf)]/Trmf$ | 0.74 | 0.85 | 1.09 | 1.14 | 1.13 | 2.45 | 3.82 | 3.64 |

$\u2202\u2202x[v\xaf(\rho u\u2032v\u2032\xaf)]/Trmf$ | 0.26 | 0.15 | $\u22120.09$ | $\u22120.14$ | $\u22120.14$ | $\u22121.44$ | $\u22122.79$ | $\u22122.61$ |

$\u2202\u2202x[\rho u\u2032(12u\u2032u\u2032)\xaf]/Trff$ | — | — | 0.87 | 0.87 | 0.86 | 0.67 | 0.74 | 0.74 |

$\u2202\u2202x[\rho u\u2032(12v\u2032v\u2032)\xaf]/Trff$ | — | — | 0.07 | 0.07 | 0.07 | 0.25 | 0.22 | 0.21 |

$\u2202\u2202x[\rho u\u2032(12w\u2032w\u2032)\xaf]/Trff$ | — | — | 0.06 | 0.06 | 0.07 | 0.08 | 0.05 | 0.04 |

## 8 Gibbs Equation

*a*)

## 9 Conclusions

A series of high fidelity datasets of compressor cascades at varying inflow conditions was analyzed in this article. For each of the cases, a comparison between entropy, enthalpy, and stagnation pressure loss coefficient was carried out. To understand the difference between the entropy loss coefficient and stagnation pressure loss coefficient, a transport equation budget was performed for both terms. This resulted in the following list of conclusions. While entropy and enthalpy gave almost identical loss predictions, stagnation pressure loss coefficient was found to be unreliable loss metric for cases considered here underpredicting loss for one of the cases by as much as 40% when compared to the entropy loss coefficient ($NACA65$$Gap30$ case). This was the case despite all the simulations were run at adiabatic and low Mach number conditions. The use of stagnation pressure estimate discarding turbulent kinetic energy resulted in even more unreliable predictions highlighting the importance of including it when comparing experimental data with RANS for highly unsteady flowfields. To understand the discrepancy between loss coefficients, entropy and stagnation pressure transport equation budgets were performed. All the budgets were fully balanced and allowed to elucidate the origin of the discrepancy. This was traced to the turbulent transport terms that arise as part of the stagnation pressure transport equation. These terms were found to be primarily present in the freestream. A commonly used assumption of its negligible contribution to the overall budget was found to be wrong even when only moderate levels of inflow turbulence were present. The combined impact of these terms was found to be negative for a compressor, resulting in artificially lower loss estimates by the mass-averaged stagnation pressure loss coefficient. This was linked to flow deceleration in a compressor. For a turbine flow, loss estimates are likely to be artificially higher due to the flow acceleration. The use of stagnation pressure loss coefficient may still be valid when low levels of inflow unsteadiness are present and when turbulent transport is low. The impact also appears to be more pronounced for the cases at lower Reynolds number. In addition, it was shown that integrated turbulent transport terms were low within the boundary layers and wake. As a result, loss predicted by entropy and stagnation pressure in those regions was in good agreement. In experimental setting, however, such decomposition may be difficult to achieve, while estimating turbulent transport is beyond current standard experimental practice.

## Acknowledgment

The authors would also like to acknowledge the help of UK Turbulence Consortium funded by the EPSRC (Grant No. EP/L000261/1) and Cambridge Service for Data Driven Discovery system operated by the University of Cambridge Research Computing Service funded by EPSRC Tier-2 (Grant No. EP/P020259/1), whose HPC allocations have been used to obtain the results. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie (Grant No. [101026928]), and this support is also gratefully acknowledged. 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 (Grant No. 800858).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- $h$ =
enthalpy

- $p$ =
pressure

- $q$ =
heat transfer

- $s$ =
entropy

- $u$ =
velocity

- $x$ =
quantity x

- $m\u02d9$ =
mass flowrate

- $x\xaf$ =
time-average of x

- $T$ =
temperature

- $X$ =
volume integral of x

- $gax$ =
axial gap

- $tsT$ =
entropy generation rate

- $xf$ =
related to fluctuating flowfield

- $xff$ =
related to the action of fluctuating flowfield on itself

- $xi$ =
component of a vector quantity

- $xm$ =
related to mean flowfield

- $xmf$ =
related to the action of both mean and fluctuating flowfields

- $xt$ =
related to stagnation quantity

- $Cax$ =
axial chord

- $Fred$ =
reduced frequency

- $Xblw$ =
quantity integrated over the boundary layer and wake regions

- $Xfree$ =
quantity integrated over the freestream region

- $Xtot$ =
quantity integrated over the entire domain

- $x\u2032$ =
fluctuating component of x

- $xh$ =
related to enthalpy

- $xp$ =
related to pressure

- $xs$ =
related to entropy

- $f.sp.$ =
flux of stagnation pressure

- LHS =
left-hand side

- Ma =
Mach number

- Pu =
periodic intensity

- RANS =
Reynolds-averaged Navier–Stokes

- Re =
Reynolds number based on inlet cond. and axial chord

- RHS =
right-hand side

- Tr =
energy transfer between flowfields (turbulence production)

- Tu =
turbulence intensity

- TKE =
turbulent kinetic energy

### Greek Symbols

- $\Delta n+$ =
viscous wall units in wall-normal dir.

- $\Delta t+$ =
viscous wall units in streamwise dir.

- $\Delta z+$ =
viscous wall units in spanwise dir.

- $\epsilon $ =
artificial dissipation

- $\rho $ =
density

- $\tau $ =
shear stress

- $\varphi $ =
viscous dissipation

- $\Phi $ =
integrated viscous dissipation

- $\omega $ =
loss coefficient

- $\Omega $ =
volume of a domain

## Appendix: Double Decomposition of Kinetic Energy

*a*)

*b*)

and

adv: convection of kinetic energy, pw: pressure work, $t$: energy transfer between mean and fluctuating flowfield (turbulence production), tr: diffusion due to unsteadiness/turbulence, vd: viscous diffusion, $\varphi $: viscous dissipation, pd: pressure diffusion, pl: pressure dilation.

Table 8 shows that mean kinetic energy budgets is satisfied. For turbulent kinetic energy budget, the balance was achieved when artificial dissipation was added to the resolved dissipation, Table 9. This further validates the methodology taken in this study.

$Advm=\u2212PWm+Tmf\u2212Trmf+VDm\u2212\Phi m$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$Advm$ | 111.1445 | 110.9216 | 107.7270 | 109.5839 | 109.9521 | 46763.7137 | 46822.3687 | 46945.5559 |

$PWm$ | $\u2212105.2509$ | $\u2212106.1870$ | $\u2212105.9626$ | $\u2212106.2991$ | $\u2212105.8217$ | $\u221245480.7915$ | $\u221245432.2649$ | $\u221245490.2992$ |

$Tmf$ | 3.1712 | 3.3751 | 2.3372 | 2.4060 | 2.5776 | 695.0648 | 750.2811 | 780.8226 |

$Trmf$ | 0.8466 | 2.5871 | 3.9051 | 2.5509 | 2.0044 | 387.6402 | 207.1210 | 208.5881 |

$VDm$ | 0.0118 | 0.0090 | 0.0139 | 0.0145 | 0.0147 | 2.4507 | 2.2288 | 2.2402 |

$\Phi m$ | 3.5879 | 3.6597 | 3.4189 | 3.4715 | 3.5539 | 962.0147 | 904.6694 | 911.7720 |

LHS–RHS | $\u22120.0070$ | 0.2958 | $\u22120.0727$ | $\u22120.0273$ | 0.0179 | 15.9336 | $\u221255.4968$ | $\u221226.5095$ |

$Advm=\u2212PWm+Tmf\u2212Trmf+VDm\u2212\Phi m$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$Advm$ | 111.1445 | 110.9216 | 107.7270 | 109.5839 | 109.9521 | 46763.7137 | 46822.3687 | 46945.5559 |

$PWm$ | $\u2212105.2509$ | $\u2212106.1870$ | $\u2212105.9626$ | $\u2212106.2991$ | $\u2212105.8217$ | $\u221245480.7915$ | $\u221245432.2649$ | $\u221245490.2992$ |

$Tmf$ | 3.1712 | 3.3751 | 2.3372 | 2.4060 | 2.5776 | 695.0648 | 750.2811 | 780.8226 |

$Trmf$ | 0.8466 | 2.5871 | 3.9051 | 2.5509 | 2.0044 | 387.6402 | 207.1210 | 208.5881 |

$VDm$ | 0.0118 | 0.0090 | 0.0139 | 0.0145 | 0.0147 | 2.4507 | 2.2288 | 2.2402 |

$\Phi m$ | 3.5879 | 3.6597 | 3.4189 | 3.4715 | 3.5539 | 962.0147 | 904.6694 | 911.7720 |

LHS–RHS | $\u22120.0070$ | 0.2958 | $\u22120.0727$ | $\u22120.0273$ | 0.0179 | 15.9336 | $\u221255.4968$ | $\u221226.5095$ |

$Advf=\u2212PWf+Tmf\u2212Trff+VDf\u2212\Phi f\u2212PDf+PLf$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$Advf$ | 0.7976 | 2.8418 | 2.0468 | 1.5165 | 1.1772 | 533.4470 | 501.4282 | 474.7137 |

$PWf$ | — | — | — | — | — | — | — | — |

$Tmf$ | 3.1712 | 3.3751 | 2.3372 | 2.4060 | 2.5776 | 695.0648 | 750.2811 | 780.8226 |

$Trff$ | 0.0000 | 0.0000 | $\u22120.6882$ | $\u22120.6488$ | $\u22120.4883$ | $\u221294.2817$ | $\u2212213.8627$ | $\u2212202.5755$ |

$VDf$ | 0.0083 | 0.0167 | 0.0052 | 0.0050 | 0.0051 | 1.8809 | 1.5766 | 1.5437 |

$\Phi f+N$ | 3.9764 | 6.2947 | 3.5608 | 3.1964 | 3.1722 | 1149.2538 | 1011.4609 | 1004.9839 |

$PDf$ | — | — | $\u22120.1294$ | $\u22120.0801$ | $\u22120.1112$ | $\u22124.8095$ | $\u221226.3358$ | $\u221233.4561$ |

$PLf$ | — | — | — | — | — | — | — | — |

LHS–RHS | 0.0007 | -0.0610 | 0.0108 | 0.0021 | -0.0118 | -17.9523 | 1.6265 | 16.0645 |

$Advf=\u2212PWf+Tmf\u2212Trff+VDf\u2212\Phi f\u2212PDf+PLf$ | ||||||||
---|---|---|---|---|---|---|---|---|

NACA65 | NACA65 | CDA | ||||||

Tu4 | Tu6 | Gap30 | Gap40 | Gap50 | Gap30 | Gap40 | Gap50 | |

$Advf$ | 0.7976 | 2.8418 | 2.0468 | 1.5165 | 1.1772 | 533.4470 | 501.4282 | 474.7137 |

$PWf$ | — | — | — | — | — | — | — | — |

$Tmf$ | 3.1712 | 3.3751 | 2.3372 | 2.4060 | 2.5776 | 695.0648 | 750.2811 | 780.8226 |

$Trff$ | 0.0000 | 0.0000 | $\u22120.6882$ | $\u22120.6488$ | $\u22120.4883$ | $\u221294.2817$ | $\u2212213.8627$ | $\u2212202.5755$ |

$VDf$ | 0.0083 | 0.0167 | 0.0052 | 0.0050 | 0.0051 | 1.8809 | 1.5766 | 1.5437 |

$\Phi f+N$ | 3.9764 | 6.2947 | 3.5608 | 3.1964 | 3.1722 | 1149.2538 | 1011.4609 | 1004.9839 |

$PDf$ | — | — | $\u22120.1294$ | $\u22120.0801$ | $\u22120.1112$ | $\u22124.8095$ | $\u221226.3358$ | $\u221233.4561$ |

$PLf$ | — | — | — | — | — | — | — | — |

LHS–RHS | 0.0007 | -0.0610 | 0.0108 | 0.0021 | -0.0118 | -17.9523 | 1.6265 | 16.0645 |