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.

Graphical Abstract Figure
Graphical Abstract Figure
Close modal

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 [68] 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 [1116]. 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:

  1. What determines the stagnation pressure and entropy loss coefficients?

  2. What drives the difference between the loss coefficients?

  3. When can we expect the two coefficients to vary?

Finally, the article demonstrates why entropy loss coefficient does not match stagnation pressure loss coefficient despite adiabatic and subsonic flow conditions.

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.

Fig. 1
Instantaneous spanwise vorticity flowfield for three example cases considered here: NACA65−Tu4 (top), NACA65−Gap40 (center), and CDA−Gap40 (bottom)
Fig. 1
Instantaneous spanwise vorticity flowfield for three example cases considered here: NACA65−Tu4 (top), NACA65−Gap40 (center), and CDA−Gap40 (bottom)
Close modal
Table 1

Details of two steady and six unsteady inflow cases

GeometryCaseReMaFredPuTu
NACA65Tu4140k0.0654.0%
NACA65Tu6140k0.0656.0%
NACA65Gap30140k0.0651.715.0%4.2%
NACA65Gap40140k0.0651.713.6%3.3%
NACA65Gap50140k0.0651.713.0%2.9%
CDAGap30250k0.202.494.9%3.1%
CDAGap40250k0.202.493.7%2.9%
CDAGap50250k0.202.493.2%3.0%
GeometryCaseReMaFredPuTu
NACA65Tu4140k0.0654.0%
NACA65Tu6140k0.0656.0%
NACA65Gap30140k0.0651.715.0%4.2%
NACA65Gap40140k0.0651.713.6%3.3%
NACA65Gap50140k0.0651.713.0%2.9%
CDAGap30250k0.202.494.9%3.1%
CDAGap40250k0.202.493.7%2.9%
CDAGap50250k0.202.493.2%3.0%
Table 2

Maximum near-wall viscous units

CaseMeshΔn+Δt+Δz+
Steady NACA6573M0.977
Unsteady NACA65130M0.64.86.0
Unsteady CDA170M0.915.510.0
CaseMeshΔn+Δt+Δz+
Steady NACA6573M0.977
Unsteady NACA65130M0.64.86.0
Unsteady CDA170M0.915.510.0

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.

Entropy loss coefficient can be computed from high-fidelity data directly by taking time-averaged and flux-averaged entropy at the inlet and outlet of the integration domain:
ωs=T2(s2s1)ht,1h1
(1)
Alternatively, entropy can be approximated with time-averaged and mass-averaged pressure and temperature:
ssref=Cpln(TTref)Rln(ppref)
(2)
to compute the derived entropy loss coefficient:
ωs(T,p)=T2(s(T,p)2s(T,p)1)ht,1h1
(3)

3.2 Enthalpy Loss Coefficient.

Enthalpy loss coefficient (time averaged and mass averaged) can be computed according to:
ωh=h2h2sht,1h1
(4)
and is a popular choice for the design practice as it is independent of the Mach number.

3.3 Stagnation Pressure Loss Coefficient.

The stagnation pressure loss coefficient can be obtained experimentally by directly measuring total pressure, which can be time averaged and mass averaged:
ωp=pt,1pt,2pt,1p1
(5)
This is the most commonly used loss coefficient, and it is generally accepted that it is an accurate estimate, at least in the limit of low Mach numbers. When stagnation pressure is measured at sufficiently high frequency at incompressible limit, the stagnation pressure formula reads:
pt=p¯+12ui¯ui¯+12uiui¯
(6)
where (1/2)ui¯ui¯ represents mean kinetic energy and (1/2)uiui¯ represents turbulent kinetic energy. However, in practice, when unsteadiness is low enough, turbulent part of the kinetic energy is discarded resulting in:
pt,noTKE=p¯+12ui¯ui¯
(7)
Equations (6) and (7) are in practice used interchangeably, but at highly unsteady flowfields, they result in different stagnation pressure estimates and may have an impact in particular when comparing experimental results against Reynolds-averaged Navier–Stokes (RANS)-derived predictions. Definition in Eq. (6) arises from the total kinetic energy transport equation that will be shown later (Eq. (15)), while definition in Eq. (7) arises from the mean part of the kinetic energy transport equation, shown in Appendix A (Eq. (20a)). Both of these definitions can be used to obtain a mass-averaged stagnation pressure loss coefficient, which will be referred to as follows:
  • ωp—for stagnation pressure including TKE

  • ωnoTKEp—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., NACA65Tu4 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 (ωp) 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 (ωnoTKEp) 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 (ωp) is inaccurate, in the next section, we will examine entropy and stagnation pressure transport equations.

Table 3

The summary of loss coefficients for all the cases

NACA65NACA65CDA
LossTu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
ωs0.02900.03830.02710.02590.02620.02290.02080.0208
ωs(T,p)0.02960.03900.02840.02670.02700.02340.02130.0212
ωh0.02960.03900.02840.02670.02700.02350.02130.0212
ωp0.02550.02870.01450.01860.02050.02070.02070.0206
ωnoTKEp0.02250.01800.00670.01280.01610.01490.01510.0153
NACA65NACA65CDA
LossTu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
ωs0.02900.03830.02710.02590.02620.02290.02080.0208
ωs(T,p)0.02960.03900.02840.02670.02700.02340.02130.0212
ωh0.02960.03900.02840.02670.02700.02350.02130.0212
ωp0.02550.02870.01450.01860.02050.02070.02070.0206
ωnoTKEp0.02250.01800.00670.01280.01610.01490.01510.0153

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

The loss coefficients give us an estimate of losses in a global sense so that an overall performance of a system can be assessed. However, with all the information that high-fidelity datasets offer, these losses can be studied in more detail by examining their transport equations. Here, specifically, we will consider entropy transport equation and kinetic energy transport equation. To study these transport equations, all the quantities will be integrated in the volume, Ω, between the same reference planes as were used for the computations of loss coefficients. Throughout the article, the integrated quantities are capitalized, for instance, the integrated dissipation will be:
Φ=ΩϕdΩ
(8)
and since the simulation were compressible, all quantities used in the budgets are Favre-averaged:
f¯ρf¯ρ¯
(9)

5.1 Entropy Transport Equation.

First, we examine the entropy transport equation [20]:
ρDsDt=(1Tτijuixj)+κ((T)2T2)(qT)
(10)
To gain further insight, we may decompose the flowfield into mean and turbulent components by performing Reynolds decomposition:
f=f¯+f
(11)
For the purpose of comparing entropy with stagnation pressure loss metrics, we rearrange the equation so that the left-hand side is akin to TΔs term, while the right-hand side has contributions, which are due to the viscous dissipation and heat transfer terms. Following a similar procedure as the one outlined in Ref. [21], we obtain:
TρDsDt¯ts=τij¯ui¯xjϕm+τijuixj¯ϕf+κ1T¯(T¯xi)2irqmT¯xi(qi¯T¯)rqmT¯xi(qi¯T¯)rqf
(12a)
where ts is the entropy generation rate, ϕm is the viscous dissipation due to mean strains, ϕf is the turbulent viscous dissipation, irqm is the mean flow irreversible heat transport, rqm is the mean flow reversible heat flux, and rqf is the turbulent reversible heat flux
The first two terms on the right-hand side are the irreversible entropy changes due to viscous friction, while the terms three, four, and five are the irreversible and reversible changes in entropy due to heat transfer. For the cases considered in this article, integrated heat transfer terms are negligible (adiabatic, low Mach numbers). It is also important to point out that in a computational sense, another term comes about, εN, due to the combined effects of discretization errors and numerical filtering (artificial dissipation). As a result for cases considered here, the flux of entropy budgets can be computed as follows:
TρDsDt¯dΩTST(τij¯u¯ixj¯)dΩΦm+(τijuixj¯+εN)dΩΦf+N
(13)
Equation (13) suggests that for adiabatic flows at low Mach numbers, entropy comes about purely as a result of viscous dissipation. It also allows for the estimation of artificial dissipation term, εN, caused by the insufficient mesh resolution. This fact was previously used by Przytarski and Wheeler [14] to assess the simulation resolution and a more in-depth discussion of entropy budgets can be found there. We further assert that the artificial dissipation can be accounted to the turbulent viscous dissipation by performing Reynolds decomposition of kinetic energy transport equation and computing full budget for both mean and turbulent components. This is shown in Appendix A. Consequently, for the remainder of the article, we will refer to the combined resolved and unresolved turbulent viscous dissipation as Φf+N.

5.2 Stagnation Pressure Transport Equation.

To obtain the stagnation pressure transport equation (same as kinetic energy transport equation), momentum and continuity equations can be rearranged to obtain:
DptDtD(ρ(1/2)uiui)Dt+uipxj=uiτijxj
(14)
As mentioned earlier, we can gain more insight by decomposing the flowfield into mean and turbulent components by performing Reynolds decomposition and averaging with respect to time as shown in Ref. ([22], p. 71):
ui¯pt¯xifspui¯xj(ρ¯12ui¯ui¯)advm+ui¯xj(ρ¯12uiui¯)advf+ui¯p¯xipwm=τij¯ui¯xjϕmτijuixj¯ϕfxj[ui¯(ρuiuj¯)]trmfxj[ρuj(12uiui)¯]trff+τij¯ui¯xjvdm+τijui¯xjvdf+ui¯p¯xipwfpui¯xipdf+puixi¯plf
(15a)
where fsp is the mean flow of stagnation pressure, adv is the convection of kinetic energy, pw is the pressure work, tr is the turbulent transport due to unsteadiness, vd is the viscous diffusion, ϕ is the viscous dissipation, pd is the pressure diffusion, and pl is the pressure dilation.

6 Loss Transport Equations Budgets and Loss Coefficients

Entropy and stagnation pressure transport equations can be directly related to loss coefficients:
ωs(TST)=T2(s2s1)ht,1h11ht,1h1TSTm˙
(16a)
ωp(FSP)=pt,1pt,2pt,1p1ρrefpt,1p1FSPm˙
(16b)
as a result, we are now able to split each loss coefficient into its constituent components and comment directly on their relative importance:
TST=Φm+(Φf+N)
(17a)
FSP=Φm+(Φf+N)+Trmf+TrffVDmVDfPWf+PDfPLftypically assumed0
(17b)
As mentioned earlier, for cases considered here, the entropy loss coefficient is driven purely by the viscous dissipation, conveniently split into dissipation due to mean strains Φm and turbulent dissipation Φf+N (the sum of resolved and unresolved components).

The stagnation pressure loss coefficient arise as an interplay between dissipation due to mean strains Φm, turbulent dissipation Φf+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.

Table 4

Entropy transport equation budgets

TSTΦm+Φf+N
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
TST7.5649.9546.9806.6686.7262111.2691916.1301916.756
Φm3.5883.6603.4193.4723.554962.015904.669911.772
Φf+N3.9766.2953.5613.1963.1721149.2541011.4611004.984
ωs(TST)0.02950.03880.02720.02600.02630.02340.02130.0213
ωms0.02900.03830.02710.02590.02620.02290.02080.0208
TSTΦm+Φf+N
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
TST7.5649.9546.9806.6686.7262111.2691916.1301916.756
Φm3.5883.6603.4193.4723.554962.015904.669911.772
Φf+N3.9766.2953.5613.1963.1721149.2541011.4611004.984
ωs(TST)0.02950.03880.02720.02600.02630.02340.02130.0213
ωms0.02900.03830.02710.02590.02620.02290.02080.0208

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.

Table 5

Flux of stagnation pressure transport equation budgets

FSP=Φm+Φf+N+Trmf+TrffVDmVDfPWf+PDfPLf
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
FSP=AdvmAdvfPWm6.6917.5763.8114.8015.3081,816.3691,891.5321,929.971
Φm3.5883.6603.4193.4723.554962.015904.669911.772
Φf+N3.9766.2953.5613.1963.1721149.2541011.4611004.984
Trmf0.8472.5873.9052.5512.004387.640207.121208.588
Trff0.6880.6490.48894.282213.863202.576
VDm0.0120.0090.0140.0150.0152.4512.2292.240
VDf0.0080.0170.0050.0050.0051.8811.5771.544
PWf
PDf-0.1290.0800.1114.81026.33633.456
PLf
LHS–RHS0.0730.3150.0540.0160.01641.96513.66832.144
ωp(FSP)0.02640.02980.01490.01870.02080.02040.02120.0217
ωmp0.02550.02870.01450.01860.02050.02070.02070.0206
FSP=Φm+Φf+N+Trmf+TrffVDmVDfPWf+PDfPLf
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
FSP=AdvmAdvfPWm6.6917.5763.8114.8015.3081,816.3691,891.5321,929.971
Φm3.5883.6603.4193.4723.554962.015904.669911.772
Φf+N3.9766.2953.5613.1963.1721149.2541011.4611004.984
Trmf0.8472.5873.9052.5512.004387.640207.121208.588
Trff0.6880.6490.48894.282213.863202.576
VDm0.0120.0090.0140.0150.0152.4512.2292.240
VDf0.0080.0170.0050.0050.0051.8811.5771.544
PWf
PDf-0.1290.0800.1114.81026.33633.456
PLf
LHS–RHS0.0730.3150.0540.0160.01641.96513.66832.144
ωp(FSP)0.02640.02980.01490.01870.02080.02040.02120.0217
ωmp0.02550.02870.01450.01860.02050.02070.02070.0206

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 Φf for the moderate turbulence NACA65Tu4 case, all the way to over 100% for the unsteady NACA65Gap30 case.

This is further demonstrated by integrating all the budgets along the streamwise direction to obtain a line integral plots for the steady NACA65Tu4 case, Fig. 2, unsteady NACA65Gap40 case, Fig. 3, and unsteady CDAGap40 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.

Fig. 2
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the steady inflow NACA65Tu4 case
Fig. 2
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the steady inflow NACA65Tu4 case
Close modal
Fig. 3
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the unsteady inflow NACA65Gap40 case
Fig. 3
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the unsteady inflow NACA65Gap40 case
Close modal
Fig. 4
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the unsteady inflow CDA65Gap40 case
Fig. 4
Comparison of loss contributions to entropy (left) and stagnation pressure (right) loss metrics for the unsteady inflow CDA65Gap40 case
Close modal

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 NACA65Tu4 case. Similarly,Fig. 7 shows analogous budget for the unsteady inflow NACA65Gap40 case and Fig. 8 shows that for the unsteady inflow CDAGap40 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.

Fig. 5
Example domain decomposition for the NACA65Tu4 case
Fig. 5
Example domain decomposition for the NACA65Tu4 case
Close modal
Fig. 6
Comparison of loss contributions for the steady inflow NACA65Tu4 case for boundary layers and wake (top) and freestream (bottom)
Fig. 6
Comparison of loss contributions for the steady inflow NACA65Tu4 case for boundary layers and wake (top) and freestream (bottom)
Close modal
Fig. 7
Comparison of loss contributions for the unsteady inflow NACA65Gap40 case for boundary layers and wake (top) and freestream (bottom)
Fig. 7
Comparison of loss contributions for the unsteady inflow NACA65Gap40 case for boundary layers and wake (top) and freestream (bottom)
Close modal
Fig. 8
Comparison of loss contributions for the unsteady inflow CDAGap40 case for boundary layers and wake (top) and freestream (bottom)
Fig. 8
Comparison of loss contributions for the unsteady inflow CDAGap40 case for boundary layers and wake (top) and freestream (bottom)
Close modal
Table 6

Entropy and stagnation pressure loss prediction for the combined region of boundary layers and wake and for freestream

NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
TST,tot7.5649.9546.9806.6686.7262111.2691916.1301916.756
FSPtot6.7707.6563.8194.8105.3171860.3531931.7341972.560
TST,blw6.2896.6165.8365.6885.7891771.7571610.5461627.988
FSPblw6.1496.3316.3385.9535.9561449.3681441.5701424.509
TST,free1.27523.33811.14410.97990.9377339.5117305.5845288.7683
FSPfree0.6221.3252.5191.1430.639410.985490.164548.050
FSPblw/TST,blw97.77%95.69%108.61%104.66%102.90%96.43%106.38%102.90%
FSPfree/TST,free48.76%39.70%220.16%116.63%68.17%36.17%64.81%102.99%
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
TST,tot7.5649.9546.9806.6686.7262111.2691916.1301916.756
FSPtot6.7707.6563.8194.8105.3171860.3531931.7341972.560
TST,blw6.2896.6165.8365.6885.7891771.7571610.5461627.988
FSPblw6.1496.3316.3385.9535.9561449.3681441.5701424.509
TST,free1.27523.33811.14410.97990.9377339.5117305.5845288.7683
FSPfree0.6221.3252.5191.1430.639410.985490.164548.050
FSPblw/TST,blw97.77%95.69%108.61%104.66%102.90%96.43%106.38%102.90%
FSPfree/TST,free48.76%39.70%220.16%116.63%68.17%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.

To understand why turbulent transport by the mean flow Trmf is negative, we can inspect the formula behind the term:
trmf=xj[ui¯(ρuiuj¯)]
(18)
In the freestream, the Reynolds stress terms (ρuiuj¯) are expected to universally decay at the steady rate. For a compressor, this decay is combined with a flow deceleration (ui¯). As a result (also because of the sign of the term), the overall contribution to the stagnation pressure budget is negative, artificially lowering the predicted loss. We expect this to be the case when meaningful levels of turbulence/periodic unsteadiness(2–3% or above) is present. The erroneous stagnation pressure loss coefficient behavior happens irrespective of low Mach number conditions and is purely related to the freestream unsteadiness and turbulence transport term that arise due to it.
To understand why turbulent transport by the turbulent flow Trff is positive for the unsteady inflow cases considered here, we can inspect the formula behind the term:
trff=xj[ρuj(12uiui)¯]
(19)
The unsteady inflow cases consisted of periodic wakes, which have an appearance of u and v fluctuations. These fluctuations grow in size as the wake is initially accelerated and turned when entering the passage and subsequently diffused in the aft portion of it. Such behavior would result first in an increase of Trff term and then its slow reduction. Figures 3 and 4 suggest that the initial acceleration and turning dominate and determine the magnitude of this term, which then stays relatively constant for the remainder of the passage where it is diffused.

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 uu and uv 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 and turbulent kinetic energy (1/2)uiui.

Table 7

Contribution of different components to turbulent transport Trmf

NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
x[u¯(ρuu¯)]/Trmf0.740.851.091.141.132.453.823.64
x[v¯(ρuv¯)]/Trmf0.260.150.090.140.141.442.792.61
x[ρu(12uu)¯]/Trff0.870.870.860.670.740.74
x[ρu(12vv)¯]/Trff0.070.070.070.250.220.21
x[ρu(12ww)¯]/Trff0.060.060.070.080.050.04
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
x[u¯(ρuu¯)]/Trmf0.740.851.091.141.132.453.823.64
x[v¯(ρuv¯)]/Trmf0.260.150.090.140.141.442.792.61
x[ρu(12uu)¯]/Trff0.870.870.860.670.740.74
x[ρu(12vv)¯]/Trff0.070.070.070.250.220.21
x[ρu(12ww)¯]/Trff0.060.060.070.080.050.04

8 Gibbs Equation

The Gibbs equation accounts for the changes in fluid properties along the flow path and relates the change in enthalpy to the change in entropy and pressure. This relationship can be applied to stagnation properties of the flow as follows:
TtDsDt=DhtDt1ρtDptDt
(20)
When considering adiabatic flow through a stationary blade row, stagnation temperature is constant, consequently:
TtDsDt1ρtDptDt
(21)
This is demonstrated to be true in Fig. 9 for which a series of 500 instantaneous snapshots were used to compute Gibbs equation budgets in their compressible form. This assertion often leads to a conclusion that
ΔsRln(pt,2pt,1)
(22)
However, as shown earlier, this does not hold for the data presented. To explain why that is we perform Reynolds decomposition on incompressible stagnation pressure:
DptDtDDt(p+12ρuiui)
(23)
and find:
DptDt¯D¯Dt(ρ¯12ui¯ui¯)advm+D¯Dt(ρ¯12uiui¯)advf+ui¯p¯xipwmxj[ui¯(ρuiuj¯)]trmfxj[ρuj(12uiui)¯]trffui¯p¯xipwf+pui¯xipdfpuixi¯plf
(24a)
where
DDt=t+ujxj
(25)
and
DDt¯=uj¯xj
(26)
This decomposition allows us to distinguish between the mean flow of stagnation pressure (15) and the flux of stagnation pressure (24) budget. The first one is akin to the commonly used mass-averaged stagnation pressure loss coefficient, which is favored due to the relative ease of measuring it experimentally. The second one corresponds to the correct estimate of loss, but cannot be easily measured experimentally. The difference between the two, which has been highlighted earlier, is due to the presence of turbulent transport terms as well as pressure work, dilation, and diffusion terms. While the last three terms were found to be negligible, the turbulent transport terms, also known as the Reynolds stress work on boundaries, were found to play a significant role for the conditions considered here. These conditions are of relevance to many cascade experiments that feature high levels of unsteadiness at the inflow.
Fig. 9
Gibbs equation budget for the unsteady inflow CDA65Gap40 case
Fig. 9
Gibbs equation budget for the unsteady inflow CDA65Gap40 case
Close modal

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 (NACA65Gap30 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˙ =

mass flowrate

x¯ =

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 =

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

Δn+ =

viscous wall units in wall-normal dir.

Δt+ =

viscous wall units in streamwise dir.

Δz+ =

viscous wall units in spanwise dir.

ε =

artificial dissipation

ρ =

density

τ =

shear stress

ϕ =

viscous dissipation

Φ =

integrated viscous dissipation

ω =

loss coefficient

Ω =

volume of a domain

Appendix: Double Decomposition of Kinetic Energy

Considerable insight can be drawn from the analysis of kinetic energy budget. Using Reynolds decomposition, kinetic energy can be decomposed into mean (m) and fluctuating (f) components (full derivation available, e.g., in Ref. [27]):
D¯Dt(ρ¯12ui¯ui¯)advm=ui¯p¯xipwm(ρuiuj¯ui¯xj)tmfxj[ui¯(ρuiuj¯)]trmf+τij¯ui¯xjvdmτij¯ui¯xjϕm
(A1a)
D¯Dt(ρ¯12uiui¯)advf=+(ρuiuj¯ui¯xj)tmfxj[ρuj(12uiui)¯]trff+τijui¯xjvdfτijuixj¯ϕfui¯p¯xipwfpui¯xipdf+puixi¯plf
(A1b)
where
D¯Dtt+uj¯xj
(A2)

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, ϕ: 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.

Table 8

Double decomposition—mean kinetic energy budget

Advm=PWm+TmfTrmf+VDmΦm
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
Advm111.1445110.9216107.7270109.5839109.952146763.713746822.368746945.5559
PWm105.2509106.1870105.9626106.2991105.821745480.791545432.264945490.2992
Tmf3.17123.37512.33722.40602.5776695.0648750.2811780.8226
Trmf0.84662.58713.90512.55092.0044387.6402207.1210208.5881
VDm0.01180.00900.01390.01450.01472.45072.22882.2402
Φm3.58793.65973.41893.47153.5539962.0147904.6694911.7720
LHS–RHS0.00700.29580.07270.02730.017915.933655.496826.5095
Advm=PWm+TmfTrmf+VDmΦm
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
Advm111.1445110.9216107.7270109.5839109.952146763.713746822.368746945.5559
PWm105.2509106.1870105.9626106.2991105.821745480.791545432.264945490.2992
Tmf3.17123.37512.33722.40602.5776695.0648750.2811780.8226
Trmf0.84662.58713.90512.55092.0044387.6402207.1210208.5881
VDm0.01180.00900.01390.01450.01472.45072.22882.2402
Φm3.58793.65973.41893.47153.5539962.0147904.6694911.7720
LHS–RHS0.00700.29580.07270.02730.017915.933655.496826.5095
Table 9

Double decomposition—turbulent kinetic energy budget

Advf=PWf+TmfTrff+VDfΦfPDf+PLf
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
Advf0.79762.84182.04681.51651.1772533.4470501.4282474.7137
PWf
Tmf3.17123.37512.33722.40602.5776695.0648750.2811780.8226
Trff0.00000.00000.68820.64880.488394.2817213.8627202.5755
VDf0.00830.01670.00520.00500.00511.88091.57661.5437
Φf+N3.97646.29473.56083.19643.17221149.25381011.46091004.9839
PDf0.12940.08010.11124.809526.335833.4561
PLf
LHS–RHS0.0007-0.06100.01080.0021-0.0118-17.95231.626516.0645
Advf=PWf+TmfTrff+VDfΦfPDf+PLf
NACA65NACA65CDA
Tu4Tu6Gap30Gap40Gap50Gap30Gap40Gap50
Advf0.79762.84182.04681.51651.1772533.4470501.4282474.7137
PWf
Tmf3.17123.37512.33722.40602.5776695.0648750.2811780.8226
Trff0.00000.00000.68820.64880.488394.2817213.8627202.5755
VDf0.00830.01670.00520.00500.00511.88091.57661.5437
Φf+N3.97646.29473.56083.19643.17221149.25381011.46091004.9839
PDf0.12940.08010.11124.809526.335833.4561
PLf
LHS–RHS0.0007-0.06100.01080.0021-0.0118-17.95231.626516.0645

References

1.
Sandberg
,
R. D.
, and
Michelassi
,
V.
,
2022
, “
Fluid Dynamics of Axial Turbomachinery: Blade-and Stage-Level Simulations and Models
,”
Annu. Rev. Fluid. Mech.
,
54
, pp.
255
285
.
2.
Denton
,
J. D.
,
1993
, “
Loss Mechanisms in Turbomachines
,”
ASME J. Turbomach.
,
115
(
4
), pp.
621
656
.
3.
Miller
,
R. J.
,
2013
, “
Mechanical Work Potential
,” Turbo Expo: Power for Land, Sea, and Air, ASME Paper No. GT2013-95488, American Society of Mechanical Engineers.
4.
Moore
,
J.
,
Shaffer
,
D.
, and
Moore
,
J.
,
1987
, “
Reynolds Stresses and Dissipation Mechanisms Downstream of a Turbine Cascade
,”
ASME J. Turbomach.
,
109
(
2
), pp.
258
267
.
5.
Tennekes
,
H.
, and
Lumley
,
J.
,
1972
,
A First Course in Turbulence
,
MIT Press
,
Cambridge, MA
.
6.
Stieger
,
R. D.
, and
Hodson
,
H. P.
,
2005
, “
The Unsteady Development of a Turbulent Wake Through a Downstream Low-Pressure Turbine Blade Passage
,”
ASME J. Turbomach.
,
127
(
2
), pp.
388
394
.
7.
MacIsaac
,
G.
,
Sjolander
,
S.
, and
Praisner
,
T.
,
2012
, “
Measurements of Losses and Reynolds Stresses in the Secondary Flow Downstream of a Low-Speed Linear Turbine Cascade
,”
ASME J. Turbomach.
,
134
(
6
), p.
061015
.
8.
Lengani
,
D.
,
Simoni
,
D.
,
Ubaldi
,
M.
,
Zunino
,
P.
,
Bertini
,
F.
, and
Michelassi
,
V.
,
2017
, “
Accurate Estimation of Profile Losses and Analysis of Loss Generation Mechanisms in a Turbine Cascade
,”
ASME J. Turbomach.
,
139
(
12
), p.
121007
.
9.
Wu
,
X.
, and
Durbin
,
P. A.
,
2001
, “
Evidence of Longitudinal Vortices Evolved From Distorted Wakes in a Turbine Passage
,”
J. Fluid Mech.
,
446
, pp.
199
228
.
10.
Michelassi
,
V.
, and
Wissink
,
J. G.
,
2015
, “
Turbulent Kinetic Energy Production in the Vane of a Low-Pressure Linear Cascade With Incoming Wakes
,”
Int. J. Rot. Mach.
,
2015
, p.
650783
.
11.
Wheeler
,
A. P.
,
Sandberg
,
R. D.
,
Sandham
,
N. D.
,
Pichler
,
R.
,
Michelassi
,
V.
, and
Laskowski
,
G.
,
2016
, “
Direct Numerical Simulations of a High-Pressure Turbine Vane
,”
ASME J. Turbomach.
,
138
(
7
), p.
071003
.
12.
Przytarski
,
P. J.
, and
Wheeler
,
A. P.
,
2020
, “
The Effect of Gapping on Compressor Performance
,”
ASME J. Turbomach.
,
142
(
12
), p.
121006
.
13.
Leggett
,
J.
,
Richardson
,
E.
,
Priebe
,
S.
,
Shabbir
,
A.
,
Michelassi
,
V.
, and
Sandberg
,
R.
,
2020
, “
Loss Analysis of Unsteady Turbomachinery Flows Based on the Mechanical Work Potential
,”
ASME J. Turbomach.
,
142
(
11
), p.
111009
.
14.
Przytarski
,
P. J.
, and
Wheeler
,
A. P.
,
2021
, “
Accurate Prediction of Loss Using High Fidelity Methods
,”
ASME J. Turbomach.
,
143
(
3
), p.
031008
.
15.
Leggett
,
J.
,
Zhao
,
Y.
,
Richardson
,
E. S.
, and
Sandberg
,
R. D.
,
2021
, “
Turbomachinery Loss Analysis: The Relationship Between Mechanical Work Potential and Entropy Analyses
,” Turbo Expo: Power for Land, Sea, and Air, ASME Paper No. GT2021-59436,
American Society of Mechanical Engineers
.
16.
Miki
,
K.
, and
Ameri
,
A.
,
2024
, “
Numerical Investigation of the Effect of Trailing Edge Thickness of Simulated CMC Blades on Loss Profiles
,”
ASME J. Turbomach.
,
146
(
9
), p.
091009
.
17.
Wheeler
,
A. P. S.
,
Dickens
,
A. M. J.
, and
Miller
,
R. J.
,
2018
, “
The Effect of Non-equilibrium Boundary Layers on Compressor Performance
,”
ASME J. Turbomach.
,
140
(
10
), p.
101003
.
18.
Brown
,
L. E.
,
1972
, “
Axial Flow Compressor and Turbine Loss Coefficients: A Comparison of Several Parameters
,” ASME Conference Proceedings, ASME Paper No. 72-GT-18.
19.
Jardine
,
L.
,
2019
,
The Effect of Heat Transfer on Turbine Performance
,
University of Cambridge
,
Cambridge, UK
.
20.
Cantwell
,
B.
,
2013
,
AA200: Applied Aerodynamics
,
Stanford University
,
Stanford, CA
.
21.
Zhao
,
Y.
, and
Sandberg
,
R. D.
,
2020
, “
Using a New Entropy Loss Analysis to Assess the Accuracy of RANS Predictions of an High-Pressure Turbine Vane
,”
ASME J. Turbomach.
,
142
(
8
), p.
081008
.
22.
Hinze
,
J. O.
,
1975
,
Turbulence
,
McGraw Hill
,
New York
.
23.
Perdichizzi
,
A.
,
Ubaldi
,
M.
, and
Zunino
,
P.
,
1990
, “
A Hot Wire Measuring Technique for Mean Velocity and Reynolds Stress Components in Compressible Flow
,” Measuring Techniques in Turbomachinery, Paper No. MTT1090,
American Society of Mechanical Engineers
.
24.
Jelly
,
T. O.
,
Day
,
I. J.
, and
di Mare
,
L.
,
2017
, “
Phase-averaged Flow Statistics in Compressors Using a Rotated Hot-Wire Technique
,”
Exp. Fluids
,
58
(
48
), pp.
1
7
.
25.
Folk
,
M.
,
2019
,
The Impact of Combustor Turbulence on Loss Mechanisms in the High Pressure Turbine
,
University of Cambridge
,
Cambridge, UK
.
26.
Folk
,
M.
,
Miller
,
R. J.
, and
Coull
,
J. D.
,
2020
, “
The Impact of Combustor Turbulence on Turbine Loss Mechanisms
,”
ASME J. Turbomach.
,
142
(
9
), p.
091009
.
27.
Aupoix
,
B.
,
Blaisdell
,
G. A.
,
Reynolds
,
C. W.
, and
Zeman
,
O.
,
1990
, “
Modelling the Turbulent Kinetic Energy Equations for Compressible, Homogeneous Turbulence
,” Proceedings of the Summer Program 1990,
Center for Turbulence Research
, Paper No. N92-30653.