Nomenclature

$A$

flow area

$b$

bias

$B$

bias uncertainty

$\boldsymbol{B}$

mxn Broyden matrix

${C_p}$

specific heat at constant pressure

${C_v}$

specific heat at constant volume

${C_V}$

velocity coefficient

$\boldsymbol{E}$

m-vector of mass and energy imbalances

${F_g}$

gross thrust

${F_n}$

net thrust

$h$

specific enthalpy

$\boldsymbol{J}$

mxn Jacobian matrix

$\dot m$

mass flow

${{MN}}$

Mach number

$n$

sample size

$N$

rotational speed

$NL$

rotational speed for the LP spool

$NH$

rotational speed for the HP spool

$P$

pressure

$PR$

pressure ratio

$Rline$

auxiliary coordinate in compressor maps

$s$

specific entropy

${s_p}$

precision index

$SFC$

specific fuel consumption

${t_{95}}$

inverse of student’s t distribution (95% confidence)

$T$

temperature

$U$

uncertainty

$WAR$

water-to-air ratio

$\dot W$

shaft power

$\boldsymbol{x}$

n-vector of independent parameters

${\boldsymbol{x}^*}$

solution n-vector

$Y$

generic cycle parameter

$\bar z$

sample average

Greek letters

$\phi $

random errors

${\rm{\Phi }}$

random errors uncertainty

$\beta $

bypass ratio

$\theta $

nondimensional temperature

$\eta $

efficiency

${\rm{\Delta }}$

difference

$\tau $

convergence tolerance

$\gamma $

specific heat ratio

$\varepsilon $

error(s)

Subscripts

$A$

accuracy

${\rm{conf}}$

corresponding to engine configuration/architecture

${\rm{corr}}$

corrected

${\rm{fuel}}$

parameter associated with the fuel entering the combustor

${\rm{ICAO}} - {\rm{SA}}$

ICAO standard atmosphere

${\rm{input}}$

corresponding to model inputs

${\rm{map}}$

corresponding to turbomachinery map

${\rm{num}}$

corresponding to numerical methods

$p$

precision

${\rm{pri}}$

engine primary stream

${\rm{phy}}$

corresponding to physics modeling

${\rm{phy}}\& num$

corresponding to physics and numeric modeling

${\rm{thermo}}$

corresponding to thermodynamic properties

${\rm{sec}}$

engine secondary (or bypass) stream

${\rm{std}}$

standard day condition

0

total (or stagnation) thermodynamic property (e.g. h ₀ , T ₀ , P ₀ )

Abbreviations

AGCM

Aerothermodynamic Generic Cycle Model

DP

design point

GTE

gas turbine engine

HPC

high-pressure compressor

HPT

high-pressure turbine

LARCASE

Laboratory of Applied Research in Active Control, Avionics and AeroServoElasticity

LPC

low-pressure compressor

LPT

low-pressure turbine

MSC

model subject of comparison

NPSS

numerical propulsion system simulation

OD

off-design

RM

reference model

2.0 Model description

Both the MSC and the RM are intended to represent a generic two-spool turbofan engine with separate exhausts (see Fig. 1) and their details are treated in Ref. [Reference Gurrola-Arrieta and Botez9]. In Fig. 1, the gas path thermodynamic properties (i.e. T, P, $\dot m$ , etc.) are identified using numerals across the different stations throughout the engine. For example, the total pressure at the high-pressure compressor (HPC) exit and the maximum cycle total temperature are referred to as P _0,030 and T _0,040 , respectively.

Figure 1.

Turbofan generic model schematic.

The MSC and the RM are both composed of two sub-models, one for DP and the other for OD performance analyses. These sub-models encompass the exact thermodynamic modeling, e.g. mass and energy balances. For the purpose of this work, the MSC-OD and RM-OD were considered and their necessary inputs, e.g. aerothermodynamic DP, turbomachinery maps and scaling factors, were taken from Ref. [Reference Gurrola-Arrieta and Botez9].

The aerothermodynamic DP encompasses assumptions to define the engine power ( $\beta$ , $PR$ s, ${T_{0,040}}$ ), the engine component efficiencies ( $\eta$ s, ${C_V}$ s, etc.), and the off-takes (HPC bleed and shaft(s) power extractions) of a middle-class thrust turbofan engine (about 12,700 lb_f/56,492 N of take-off thrust). These assumptions permit the computation of the nozzle exhaust areas (i.e. ${A_{080}}$ and ${A_{180}}$ ) and the turbomachinery component maps scaling factors. A map encompasses a series of correlations that relate the overall thermodynamic characteristics of the turbomachinery (e.g. ${\dot m_{corr}}$ , $PR$ , $\eta$ ) to its rotational speed ( ${N_{corr}}$ ). For this work, the same maps were used by both the MSC and the RM. The maps are modified by their scaling factors to represent the intended middle-class thrust engine.

One essential capability of any cycle model is to determine the values of its relevant thermodynamic properties (T, P, h, γ, etc.) across the engine gas path. Cycle models typically use a set of equations or tabulated values to obtain the desired thermodynamic properties for air and combustion products. Herein, the names given to these equations or tabulated values are thermodynamic packages. The RM (i.e. NPSS) has preloaded four thermodynamic packages to choose from: allFuel, GasTbl, CEA and JANAF; their descriptions are presented in Ref. [31]. For this work, only allFuel and GasTbl were considered, as they are typically used in GTE analyses for propulsion applications, and their setup is seamless and straightforward. The MSC has two thermodynamic packages, thermo_package1 and thermo_package2. These packages were put together based on the thermodynamic properties’ tabulated values presented in Refs. [Reference Gordon32, Reference Pelton and Hannah33], respectively. For this research, only thermo_package1 is treated. Moreover, a subset of the NPSS allFuel and GasTbl were obtained by reverse engineering. These tables were implemented in the MSC and labeled as AGCM_allFuel and AGCM_GasTbl, respectively, to avoid any confusion from their original source in the NPSS. The errors in the calculated thermodynamic properties between AGCM_allFuel vs. allFuel and AGCM_GasTbl vs. GasTbl were about 1x10^–10; thus the former tables were deemed equivalent to the latter. Finally, for the scope of this work only dry air and combustion products were considered (WAR = 0.0).

3.0 Uncertainty, bias and random errors

Throughout this discussion, the definitions of uncertainty, bias and random errors presented in Ref. [Reference Abernethy and Thompson34] are taken as reference. These definitions were presented for experimental measurements in GTEs, however, they were adapted for this work, as explained next.

The total uncertainty ( $U$ ) is the maximum error expected between the MSC and the RM, and is a measure of the precision of the former, as depicted in Fig. 2. According to Ref. [Reference Abernethy and Thompson34], the $U$ provides an estimate of the largest error that might reasonably be expected. In our case, the $U$ represents an estimate of the interval where the largest error is expected to fall for a given performance parameter (e.g. ${F_n}$ , ${\rm{SFC}}$ , etc.).

Figure 2.

Graphical interpretation of precision and accuracy.

The $U$ in Equation (1) is expressed as the sum of the uncertainty due to biases ( $B$ ) and the corresponding to precision random errors ( ${\rm{\Phi }}$ ). The expression in Equation (1) was presented in Ref. [Reference Abernethy and Thompson34] and it is adopted in this work due to its simplicity. The biases ( $b$ ) are systematic or repeated errors that appear when comparing two models, hence they produced an uncertainty ( $B$ ). In the case of thermodynamic cycle models, the $B$ might be a function of power setting, e.g. engine rotational corrected speed ( $N{L_{corr}}$ ), $OPR$ , etc., and they may not be symmetric, i.e. their magnitude and sign may vary with engine power. The first term on the right side of Equation (1) must be interpreted as the net sum of the uncertainty biases, positive ( ${B^ + }$ ) and negative ( ${B^ - }$ ). As stated in Ref. [Reference Abernethy and Thompson34], a bias cannot be determined unless it is compared with the true value to be characterised. In this paper, the true value is defined as the true relative value, obtained from the RM (i.e. for checking precision). The term true value is reserved in this paper to the performance obtained from engine testing (i.e. for checking accuracy), as depicted in Fig. 2.

(1) \begin{equation}U = {\rm{\;}}B \pm {\rm{\Phi }}\end{equation}

The ${\rm{\Phi }}$ in Equation (2), is the uncertainty associated with random errors ( $\phi $ ) in any process. The ${\rm{\Phi }}$ typically follow a bell shape probability distribution around the true relative value being characterised (as depicted in Fig. 2). In the absence of any $B$ , a measurement process is only influenced by the ${\rm{\Phi }}$ , which establish a measurement of the error prediction’s closeness (i.e. precision). Random errors might appear due to different sources; thus, the total ${\rm{\Phi }}$ in Equation (2) is determined by the root sum squared of j = 1, …, m sources. The precision index ( ${s_p}$ ), Equation (3), is the sample standard deviation of the errors ( $\varepsilon$ ), in which $\bar z$ is the average sample error. Given that the datapoints (n) available for the models’ comparison might be limited (n ≤ 30), the ${\rm{\Phi }}$ _j need to be corrected due to a reduced sample; thus, the statistic called inverse student t-test ( ${t_{95}}$ ) is used. For large samples, ${t_{95}}$ tends to converge asymptotically to that of a normal distribution, ${t_{95}}\;$ = 2.0. The $B$ and ${\rm{\Phi }}$ can be estimated based on the $\varepsilon$ established between the MSC and the RM, as shown in Equation (4); where $Y$ represents any cycle parameter of interest (e.g. ${F_n},$ ${\rm{SFC}}$ , etc.) computed in both the MSC and the RM.

(2) \begin{equation}{\rm{\Phi }} = {\rm{\;}}\sqrt {\mathop \sum \limits_{j = 1}^m \left( {{t_{95}}{s_p}} \right)_j^2} \end{equation}

(3) \begin{equation}{s_p} = \sqrt {\frac{{\mathop \sum \nolimits_i^n \!{{\left( {{\varepsilon _i} - {\rm{\;}}\bar z} \right)}^2}}}{{n - 1}}} \end{equation}

\begin{equation}\varepsilon {\rm{\;}} = {\rm{\;}}\left( {\frac{{{Y_{MSC}}{\rm{\;}} - {\rm{\;}}{Y_{RM}}}}{{{Y_{RM}}}}{\rm{\;}}} \right) \times 100{\rm{\% \;\;}}\left( {for{\rm{\;}}all{\rm{\;}}parameters{\rm{\;}}but{\rm{\;}}T} \right)\end{equation}

(4) \begin{equation}\varepsilon {\rm{\;}} = {\rm{\;}}{Y_{MSC}}{\rm{\;}} - {\rm{\;}}{Y_{RM}}{\rm{\;\;}}\left( {for{\rm{\;}}T} \right)\end{equation}

The cycle model accuracy is measured relative to its true value, e.g. relative to a specific engine test or engines’ sample (see Fig. 2). Original engine manufacturers (OEMs) might use a test from a development campaign, i.e. specific test(s); or data from acceptance test production engines (engine sample) to establish the true value. This process is typically called engine matching and is outside the scope of this work. For more information about engine matching accuracy, the reader is referred to Roth et al. [Reference Roth, Doel, Mavris and Beeson35].

Someone facing an engine matching task should first validate the cycle model precision, provided it has yet to be validated, then deal with the engine matching itself. Due to the scope of this work, the accuracy uncertainties due to bias ( ${B_A}$ ) and random errors ( ${{\rm{\Phi }}_A}$ ), presented in Fig. 2, are considered to be zero.

Next, the proposed definitions for the $B$ and ${\rm{\Phi }}$ are discussed. These definitions are accompanied by examples observed in the literature from the so-called groups 2 and 3 discussed in Section 1.

Configuration bias uncertainty ( ${B_{conf}}$ ). This uncertainty is associated with differences in the engine configuration/architecture being simulated, i.e. the number of components (compressors, turbines, ducts, etc.), spools (one, two, three), exhaust type (e.g. mixed or separate), off-takes (bleed and power extractions) and cooling type (cooled vs. uncooled turbines, chargeable vs. non-chargeable cooling).

A ${B_{conf}}{\rm{\;\;}}$ may occur, among other scenarios, when assumptions such as HPC bleed extraction and shaft power extraction used to satisfy parasitic (i.e. engine) and aircraft demands during actual engine operation are ignored. In the case of parasitic needs, the HPC bleed extraction is used to cool down the parts in the hot section, and the shaft power extraction is needed to drive accessories systems such as engine oil and fuel pumps. Concerning the aircraft, HPC bleed extraction is used to meet the demands of its environmental control system (ECS). Ignoring, deliberately or by omission, the assumptions concerning HPC bleed and shaft power extractions (for engine and aircraft) would impose non-negligible errors (i.e. bias) in the comparisons between the two models.

In various studies reported in the literature, ${B_{conf}}$ are embedded in the results; however, they are not always explicitly acknowledged. One example in which a ${B_{conf}}$ might occur is when comparing two significantly different configurations. For example, in Ref. [Reference Bardela and Botez36], a turbofan engine model with separate exhaust was designed to match a mixed stream turbofan engine data, such as the RR AE3007C. An optimisation technique was used to minimise the error between the model and the experimental engine data. Although the optimisation technique solved the problem, the thermodynamic processes of single vs. separate exhaust turbofan engines differ. For a single exhaust engine, the primary (hot) and secondary (cold) streams are mixed before expanding in a single exhaust nozzle. The thermodynamic mixing process is not accounted for within the boundaries of a separate exhaust turbofan, such as the one depicted in Fig. 1. The result is the induction of a ${B_{conf}}$ , however, this is not explicitly acknowledged by Bardela and Botez [Reference Bardela and Botez36]. It is believed that a ${B_{conf}}$ influenced the values of the independent parameters used to minimise the errors between the model and the engine data.

Input bias uncertainty ( ${B_{input}}$ ). This uncertainty is caused by differences in input parameters to the cycle model, e.g. in DP analyses: total engine flow ( ${\dot m_0})$ , $\beta$ , $\eta$ s, $PR$ s, ${T_{0,040}}$ . In OD analyses, ${B_{input}}$ could arise due to differences in the nozzle exhaust areas ( ${A_{080}}$ , ${A_{180}}$ ), and in turbomachinery component maps or their scaling factors, as well as due to differences in power setting ( $N{L_{corr}},$ $OPR$ , etc.).

An example of a ${B_{input}}$ effect is found in Ref. [Reference Gaudet27] for the steady-state compressor operating line validation, where a consistent discrepancy was found in the operating lines obtained from the MSC and RM (GasTurb 10). According to Gaudet [Reference Gaudet27], the discrepancy’s root cause was the differences in the input demanded shaft power, variable in the MSC, and constant in RM. Once the input discrepancy was amended (considering constant input power in both models), the operating lines from both models seemed to overlap.

${B_{conf}}\;$ and ${B_{input}}$ can lead to large, systematic errors of possibly unknown magnitudes that are difficult to reconcile and which must be avoided. As discussed in the different examples in this paper, using a known RM is an excellent way to detect ${b_{conf}}\;$ and ${b_{input}}$ . The researcher in charge of the model comparison must ensure that all due diligence has been exercised to prevent these type of biases.

Thermodynamic bias uncertainty ( ${B_{thermo}}$ ). This uncertainty is due to differences in the assumptions used to define thermodynamic properties, such as h, C _p , γ, etc. Differences in assumptions, such as air and fuel composition (number of moles and species included in the chemical reaction), gas dissociation, the coefficients of the equations (e.g. high-order polynomials) used to set the C _p (T) and C _v (T), ideal vs. non-ideal gas behaviour; all these differences could cause a ${B_{thermo}}$ .

The potential influence of the thermodynamic package (i.e. ${B_{thermo}}$ ) in model comparisons has been acknowledged in the literature [Reference Gurrola-Arrieta and Botez9, Reference Alexiou and Mathioudakis24, Reference Gaudet27, Reference Gazzetta Junior, Bringhenti, Barbosa and Tomita30]; however, the fact that ${B_{thermo}}$ can significantly impact their cycle model comparisons has only been partially addressed. For example, in Ref. [Reference Gaudet27], the ${B_{thermo}}$ influence is acknowledged after significant errors in fuel flow (e.g. 3.19%) and delivered shaft power (e.g. 1.18%) were found between the MSC (dynamic model) and the RM (GasTurb 10). These errors were attributed to gas dissociation in combustion products not accounted for in the former. In Ref. [Reference Gurrola-Arrieta and Botez9], the differences in thermodynamic packages between the MSC (AGCM) and the RM (NPSS) are discussed. Gurrola-Arrieta and Botez [Reference Gurrola-Arrieta and Botez9] observed that while the absolute values of the thermodynamic properties, e.g. specific enthalpy (h), between packages are not the same, their derivatives (e.g. $\frac{{\partial h}}{{\partial T}}$ ) are overall consistent; thus, a small impact due to thermodynamic package differences was expected.

Physics modeling random errors uncertainty ( ${{\rm{\Phi }}_{phy}}$ ). This uncertainty concerns the thermodynamic modeling differences between the MSC and the RM (i.e. mass, energy, entropy and momentum balances) that occur either at the component or engine level. Provided that no misconception of the underlying physics exists, and assuming no other $B$ or ${\rm{\Phi }}$ are present, two different thermodynamic models should give close enough results.

The ${{\rm{\Phi }}_{phy}}$ is a good measure to identify shortcomings in MSC thermodynamic modeling; however, it is difficult to measure/quantify alone. High-fidelity cycle model calculations use numerical methods to solve the unknowns posed by either DP or OD simulations. For example, in the case of OD simulations, for finding the appropriate operating point in the turbomachinery engine components (e.g. compressors, turbines, etc.) that vanish the mass and energy imbalances within the engine. If the aim is to neatly quantify ${{\rm{\Phi }}_{phy}}$ , the influence of the numerical method must be forestalled, which brings significant complexity. From the literature review, only in Ref. [Reference Chapman, Lavelle, Litt and Guo26], a methodology was proposed to preclude the numerical method influence between the model comparisons (T-MATS vs. NPSS). However, given that Chapman et al. [Reference Chapman, Lavelle, Litt and Guo26] only used one data point in their comparisons, no conclusions can be drawn about the ${{\rm{\Phi }}_{phy}}$ .

Numeric random errors uncertainty ( ${{\rm{\Phi }}_{num}}$ ). This uncertainty is associated with the mathematical formulation to solve the imbalances in the model (e.g. mass, energy, etc.). Both the formulation and the method to solve the problem might differ between the MSC and the RM; however, it is expected that both models solve the mass and energy imbalances.

In the case of the MSC used in the present work, the problem to be solved is formulated in Equation (5), in which the m-vector E ( x ) represents the mass and energy imbalance errors, ${\left\| \cdot \right\|_2}$ represents the Euclidian norm, and τ represents a numerical tolerance. The n-vector x of independent parameters, Equation (6), represents those parameters that are varied by the numerical method to find the solution to Equation (5). These parameters include the total engine flow ( ${\dot m_0}$ ), engine bypass ratio ( $\beta$ ), fuel flow ( ${\dot m_{fuel}}$ ) and the turbomachinery maps parameters such as $Rline$ s, $N{H_{corr}}$ , $PR$ s.

(5) \begin{equation}||\boldsymbol{E}{( \boldsymbol{x} )||_2} \le {\rm{\tau }}\end{equation}

(6) \begin{equation}{\boldsymbol{x}^T} = \left[ {{{\dot m}_0},\beta ,Rlin{e_{fan}},Rlin{e_{LPC}},Rlin{e_{HPC}},N{H_{map,corr}},{{\dot m}_{fuel}},P{R_{map,HPT}},P{R_{map,LPT}}} \right]\end{equation}

The numerical method programmed and implemented in the MSC is a quasi-Newton type-class method, which is intended to find a solution vector ( x ^*) that satisfies Equation (5). As discussed in Ref. [Reference Gurrola-Arrieta and Botez8], this method presents significant advantages in terms of execution speed compared to other gradient-based methods.

Regarding the RM, some solver characteristics can be inferred, though still uncertain, based on the information extracted from some model’s files and the information presented in Ref. [37]. It is worth mentioning that most of the time, it is unlikely that the details of the numerical methods used in the MSC or the RM are known, as they may be proprietary and thus not readily available. Instead, it is likely that two different models will use different formulations and or numerical methods to solve the engine imbalances, thereby creating the possibility of ${{\rm{\Phi }}_{num}}$ , or even a ${B_{num}}$ to exist, as discussed later in this paper.

4.0 Cycle model precision uncertainty methodology

This section defines the methodology to estimate the figures of the biases and random errors uncertainties discussed in Section 3. When developing this methodology, it was assumed that the RM is a well-known model/platform, meaning that the researcher has confidence in its results. Moreover, it is assumed that the RM can be manipulated to some extent, which may not be the case with a limited privileges RM (e.g. trial licenses with restricted functionalities). For this research, the available NPSS license allowed manipulation of the model to add or remove engine components, customise its solver, and select different thermodynamic packages, which was sufficient for the scope of this work.

The total uncertainty ( $U$ ) in a given performance parameter is proposed to be established from Equation (1) along with the biases and random errors uncertainties discussed in the previous section; thus, the $U$ can be computed by the sum of the individual uncertainties, $B$ and ${\rm{\Phi }}$ , as expressed in Equation (7).

(7) \begin{equation}U = {\rm{\;}}B_{conf}^{} + B_{input}^{} + {\rm{\;}}B_{thermo}^{} \pm \left( {{\rm{\Phi }}_{phy}^{} + {\rm{\Phi }}_{num}^{}} \right)\end{equation}

As noted in Section 3, ${b_{conf}}\;$ and ${b_{input}}$ could induce large systematic errors of unknown magnitude and should be avoided. In this work, both ${B_{conf}}\;$ and ${B_{input}}$ have been implicitly forestalled from the model comparisons. The MSC and the RM were intended to represent the turbofan engine depicted in Fig. 1 (hence, ${B_{conf}}$ = 0.0). Both models were given the same inputs: aerothermodynamic DP, turbomachinery component maps and their scaling factors; henceforth, ${B_{input}}$ = 0.0. Moreover, it is assumed that ${B_{conf}}$ and ${B_{input}}$ = 0.0 hold throughout this discussion, given that the engine configuration and the main engine assumptions discussed in Section 2 remain invariant.

To establish the remaining uncertainties, i.e. ${B_{thermo}}$ , ${{\rm{\Phi }}_{phy}}$ , and ${{\rm{\Phi }}_{num}}$ , it was necessary to define the methodology that allows to compute one uncertainty while supressing the effects of the remaining ones. For example, to compute ${{\rm{\Phi }}_{phy}}$ , both ${B_{thermo}}$ and ${{\rm{\Phi }}_{num}}$ had to be forestalled (i.e. ${B_{thermo}}$ and ${{\rm{\Phi }}_{num}}$ = 0.0). A similar rationale can be devised when computing the effects of the other two uncertainties.

To calculate ${B_{thermo}}$ , ${{\rm{\Phi }}_{phy}}$ and ${{\rm{\Phi }}_{num}}$ , a sample of data points was proposed (see Table 1). This sample encompasses several flight conditions (altitude, MN and $\Delta {T_{ICAO - SA}}$ ) and power setting scenarios tested in both the MSC and the RM. The power setting was selected as the LP spool corrected speed ( $N{L_{corr}}$ ), defined as the ratio of the rotational speed of the LP spool ( $NL$ ) to the square root of the nondimensional temperature ( $\theta $ ) at the fan inlet (station 120 in Fig. 1). The nondimensional temperature ( ${\theta _{120}}$ ) is the ratio of ${T_{0,120}}$ and ${T_{std}}$ = 518.67 R (288.15 K). The $N{L_{corr}}$ ranged from 52.5-100.0% with $\Delta N{L_{corr}}$ = 2.5% for Ground; the lower limit of the $N{L_{corr}}$ range had to be adjusted for Flight 1 and Flight 2 altitudes to avoid convergence troubles.

Table 1.

Flight conditions for model comparison

4.1 Random errors uncertainty ( ${{\bf{\Phi }}_{\boldsymbol{phy}}}$ and ${{\bf{\Phi }}_{\boldsymbol{num}}}$ )

To compute ${{\rm{\Phi }}_{phy}}$ , the effects of ${{\rm{\Phi }}_{num}}$ and ${B_{thermo}}$ first need to be supressed. A specific running mode was used in both the MSC and the RM to preclude ${{\rm{\Phi }}_{num}}$ . A running mode is defined, informally herein, as a way of modifying the cycle model execution to meet the desired intent. This mode will be called one-pass, the name given in the RM, which was also adopted in this paper.

The thermodynamic calculations execution on each component depicted in Fig. 1 requires several passes on each iteration. These calculations are performed left-to-right in Fig. 1 during each pass, first on the secondary stream and then on the primary. The numerical method might call for several passes, in which the vector x is perturbed by small amounts ( $\Delta \boldsymbol{x}$ ), to construct either the Jacobian ( J ) or Broyden ( B ) matrix. The J (or B ) is then used to compute an updated x . The one-pass mode, on the other hand, allows to execute the model calculations, for a given x , only once, thus avoiding the call to the numerical method (henceforth ${{\rm{\Phi }}_{num}}$ = 0.0). In this work, the solution vector ( x^* ) was provided deliberately to both models. The x^* on the flight conditions considered in Table 1 were obtained from an independent run of the RM. Additionally, to preclude ${B_{thermo}}$ (i.e. ${B_{thermo}}$ = 0.0), both the MSC and the RM considered equivalent thermodynamic packages, i.e. AGCM_allFuel and allFuel, respectively (discussed at the end of Section 2).

To isolate the ${{\rm{\Phi }}_{num}}$ effect, it is required to forestall all other terms in Equation (7). Assuming equivalent thermodynamic package are used during the comparison, then ${B_{thermo}}$ = 0.0. Moreover, to make ${{\rm{\Phi }}_{phy}}$ = 0.0, the same model (either MSC or RM) must be considered during the comparisons. However, to determine the influence of the numerical method (i.e. ${{\rm{\Phi }}_{num}}$ ), it is needed that the chosen model run with its own numerical method and the others.

Isolating ${{\rm{\Phi }}_{num}}$ was deemed cumbersome, impractical and, most likely, impossible to achieve. The programming code of the numerical method used by the RM cannot be exported to be used in the MSC. On the other hand, the RM does not allow for the addition of another solver than its own. To overcome the shortcomings in determining ${{\rm{\Phi }}_{num}}$ its uncertainty was combined with ${{\rm{\Phi }}_{phy}}$ , and the lumped uncertainty was designated as ${{\rm{\Phi }}_{phy\& num}}$ . The ${{\rm{\Phi }}_{phy\& num}}$ was obtained by comparing the runs from both the MSC and the RM using their corresponding numerical methods and only precluding ${B_{thermo}}$ , i.e. considering AGCM_allFuel and allFuel, respectively.

It is worth noting that when allowing the numerical methods to find x^* , the uncertainty in ${{\rm{\Phi }}_{num}}$ is combined with ${{\rm{\Phi }}_{phy}}$ , and cannot be separated. In other words, ${{\rm{\Phi }}_{phy}}$ and ${{\rm{\Phi }}_{num}}$ are not linearly independent, i.e. ${{\rm{\Phi }}_{phy\& num}} \ne {{\rm{\Phi }}_{phy}} + {{\rm{\Phi }}_{num}}$ . Based on the previous discussion, Equation (7) had to be rewritten to account for ${{\rm{\Phi }}_{phy\& num}}$ , as shown in Equation (8).

(8) \begin{equation}U = {\rm{\;}}B_{conf}^{} + B_{input}^{} + {\rm{\;}}B_{thermo}^{} \pm {\rm{\Phi }}_{phy\& num}^{}\end{equation}

4.2 Thermodynamic bias uncertainty $(\;{\boldsymbol{B}_\boldsymbol{thermo}}$ )

To establish ${B_{thermo}}$ alone, the influence of the last term in Equation (8) must be eliminated, i.e. deliberately making ${{\rm{\Phi }}_{phy\& num}}$ = 0.0. The ${B_{thermo}}$ was assessed by using solely the MSC, thus precluding ${{\rm{\Phi }}_{phy\& num}}$ . The physics modeling and mathematical formulation to define the engine mass and energy imbalances and the numerical method to find x ^* remain invariant when using the same model. To compute ${B_{thermo}}$ , the AGCM_allFuel package was taken as reference when comparing the other two thermodynamic packages of interest, namely, AGCM_GasTbl and thermo_package1. A similar approach could be followed to determine the ${B_{thermo}}$ for any other thermodynamic package of interest. For thermo_package1, the thermodynamic properties were normalised based on the corrections provided by Gurrola and Botez [Reference Gurrola-Arrieta and Botez9]. These corrections allowed to reduce the ${B_{thermo}}$ between thermo_package1 and AGCM_allFuel. Finally, a summary of the overall proposed methodology is presented in Fig. 3.

Figure 3.

Uncertainty methodology summary.