Digital twin

The term “Digital Twin” is widely used and has a range of accepted definitions. The term generally refers to an integration of data between physical and virtual components (Fuller et al., 2020). In the current context of the manufacturing process of gas turbine blades, a Digital Twin incorporates structured-light scans and aerodynamic performance (primary functional requirement) predictions. The scan data traces the unique geometry of each component and the aerodynamic predictions can be used to sentence components and compensate for manufacturing variability (Lee et al., 2019a).

Aerodynamic simulation hierarchy

The aerodynamic environment of a turbine blade is complex and can only be captured using physics-based simulations. Computational Fluid Dynamics (CFD) tools that encompass a range of fidelities have been used by various published studies and they will be discussed here in the order of increasing fidelity.

Considering the high volume of manufactured blades, rapid CFD tools like the Multiple blade Interacting Streamtube Euler Solver (MISES) (Drela, 1985) are attractive. MISES is a streamtube Euler solver coupled to an integral boundary layer code that is limited to quasi-three-dimensional (Q3D) blade sections. It has been used to assess compressor blades (Garzon and Darmofal, 2003; Dow and Wang, 2015; Goodhand et al., 2015) and turbine blades (Duffner, 2008).

Reynolds-Averaged Navier-Stokes (RANS) CFD solves the time-averaged Navier-Stokes equation. By incorporating source terms, turbulence models, and mixing planes, full three-dimensional (3D) geometries of multi-passage and multi-blade-row turbomachinery can be studied. However, RANS CFD is not able to resolve time-dependent phenomena and is vulnerable to turbulence and transition modelling. With today’s computational resources, RANS CFD is the favoured tool for predicting the aerodynamics performance of manufactured gas turbine blades (Lange et al., 2012; Högner et al., 2016).

Unsteady RANS (URANS) CFD allows time-accurate fluctuations in the flow field to be resolved. URANS can capture large-scale movements in the fluid but does not resolve the vortical structures in the boundary layer. In addition, the computational cost of URANS models often make them impractical for the purposes of design, let alone for high-volume assessments. If resources permit, URANS CFD could provide useful insights for dynamic blade loading (Clark et al., 2018).

The highest degrees of fidelity are offered by Large Eddy Simulations (LES) and Direct Numerical Simulation (DNS). These tools can resolve flow features at the length scales of surface roughness (Kapsis and He, 2018). However, given their immense computational cost, these are not currently practical options for Digital Twins, and are therefore not considered in this paper.

Increasing the fidelity of the CFD methods improves the realism and accuracy of the predictions but it is also associated with higher computational cost and complexity. Given the high volume of parts that need to be assessed, a balance must be struck between computational expense and accuracy. This paper therefore examines the question: what is the trade-off between cost and accuracy of different aerodynamic simulation fidelities?

Paper aims & outline

This paper compares the accuracy and computational cost of a range of aerodynamic simulation methods. Using the geometric data from a set manufactured turbine blades, the predictions of key aerodynamic performance parameters are compared across the range of simulation fidelity. The following sections detail:

The paper concludes with recommendations on the suitable aerodynamic simulations to support Digital Twins.

Euler solver: MISES

Version 2.56 of MISES was used and the models were run to the prescribed pressure ratio. The models were run for 10 inviscid iterations before introducing viscous effects for a further 15 iterations. This made the model more robust by allowing the flow field to stabilise before introducing the boundary layers (Drela and Youngren, 2008).

RANS solver: Hydra

Hydra’s spatial discretisation is based on an upwind edge-based finite volume scheme and is second-order accurate. Hydra has been experimentally validated for a similar geometry (Zamboni and Adami, 2016) and across a range of turbine designs (Coull, 2017). Rolls-Royce’s Parametric Design & Rapid Meshing (PADRAM) (Shahpar and Lapworth, 2003) was used to generate multi-block, structured meshes. Unless otherwise specified, the computational domain is a 1-stage simulation of the High Pressure Turbine (HPT) stage, as illustrated in Figure 2. Sealing flows and film cooling are modelled using secondary inlets and source terms on the blade respectively.

Figure 2.

Meridional view of the 1-stage CFD model.

The k−ω SST turbulence model (Menter, 1992) with wall functions was used favouring its robustness across the range of freestream and highly viscous flow regimes typically found in a HPT. The PADRAM mesh has wall y+ values of approximately 30 or higher on the blades and endwalls, in line with the minimum y+ requirements of the wall functions. Mesh sensitivity studies on a similar case have been previously published (Lee et al., 2019a) which showed residual mesh sensitivity of ∼0.01% for the HPT stage inlet capacity, Q and ∼0.5 in scaled HPT stage efficiency, η. These residuals are a small fraction of the observed range of performance variations (see Figure 10).

Figure 10.

Plots of the averaged single-passage (ASP) values against multi-passage (MP) values from the multi-passage CFD models for the: (a) mass-averaged βexit, (b) Q, (c) ψ, and (d) scaled η. The mean and standard deviation are marked on the appropriate axes with a circle and an error bar respectively.

This approach contains approximations, including: the inability of RANS models to resolve time-dependent phenomena (e.g. TE shedding), film cooling being represented by source terms, and the turbulence model. Broadly speaking, these uncertainties behave systemically and assuming a uniformly distributed impact on the results, the effect of these approximations on the predicted performance changes from the design-intent is significantly reduced.

Key performance parameters

For a turbine stage, the performance quantities of interest are: the exit flow angle, βexit, the stage inlet capacity, Q, the stage efficiency, η, and the stage loading coefficient, ψ. βexit is defined in Equation (1) and computed from the mass-averaged tangential velocity, Vϑ, and the mass-averaged axial velocity, Vx, at the rotor exit plane. The βexit of the rotor is particularly important and it controls the produced work, the passage mass flow, and the stage reaction.

Q is defined in Equation (2) as a quasi-non-dimensional where m˙inlet is the mass flow at the stage inlet, and T0,inlet and P0,inlet are the mass-averaged stagnation temperature and stagnation pressure at the stage inlet.

For the purposes of this study, η is defined in Equation (3) and obtained by calculating the rotor torque by numerically integrating forces over all the specified rotating wall faces (surface, S) and summing the pressure, p, and viscous shear, τ, terms (Rolls-Royce plc, 2009).

ψ is defined in Equation (4) as the ratio of the change in total enthalpy, h0, across the stage divided by the square of rotational speed of the rotor blade, U. The value of U is area-averaged at the rotor exit plane.

Part I: Q3D Euler vs. single-passage 3D RANS

The ability to accurately predict the βexit is vital as the bulk flow direction is one of the primary means by which the turbine’s functional requirements on work, capacity, reaction, and efficiency are met. Figure 5 compares the trends of mass-averaged βexit that were obtained from the following models:

Figure 5.

Predictions of mass-averaged βexit from the MISES and single-passage Hydra models.

The MISES results were mass-averaged across the span and the Hydra results were mass-averaged at the HPT rotor exit plane. Figure 5 shows that the trend from MISES does not match those of the 3D Hydra models completely and with notable discrepancies for blade IDs 18, 24, and 25. This is primarily due to MISES treating each Q3D blade section in isolation and not accounting for the effects of the hub and tip vortices. In addition, the trends from the unshrouded and shrouded models show that changes in the βexit is well-estimated even while omitting the shroud.

Part II: multi-row and multi-stage effects

For civil aero engines, the aerodynamic changes on one blade row will have a “knock-on” effect on all downstream blade rows. To study this, 3D RANS CFD models of an extended computational domain were created, incorporating the Intermediate Pressure (IP) NGV, Intermediate Pressure Turbine (IPT) rotor blade, and the first Low Pressure (LP) NGV. Figure 6 shows the radial profiles of the inlet and exit flow angles, referenced to the mass-averaged values across the respective inlet and exit planes, and gives a visual appreciation of the how small geometry changes in the HPT rotor blade row affects the flow angles of the entire IPT stage (IP NGV and IPT rotor blade).

Figure 6.

Radial profiles of flow angles from the HPT rotor exit and the IPT stage. “Abs.” denotes “absolute” and “Rel.” denotes “relative”. Refer to Figure 8 for the corresponding efficiency plots.

Although the geometry changes were only imposed on the HPT rotor blade, Figure 6 shows that the downstream effects only diminish by the IPT rotor blade exit. Immediately downstream of the HPT rotor, the flow angle changes caused by the geometric variability imposed on the HPT rotor blade lowers the inlet flow angle to the IP NGV (in the absolute frame) by up to ∼6∘ around the mid-span. Although the IP NGV helps reduce the variability in the flow angle by the time the flow arrives at IP NGV exit plane, the IPT rotor’s inlet flow angles are still being affected by up to ∼3∘ in the mid-span (under-turning) and tip (over-turning) regions.

Examining the radial profile of βexit from the HPT rotor in Figure 6, three of the 25 blades show a distinct shift in βexit between 70%<r<95% (coloured magenta in Figure 6) and this is due to a change in the behaviour of the horseshoe vortex (HSV) (Lee et al., 2019b). The change in the HSV flow structure is illustrated in Figure 7. In “Type A” flow (design-intent) the HSV is largely held within the upstream cavity by the blades’ pressure field. The legs of adjacent HSVs emerge together close to the suction-surface of the blade; thus the sealing flow remains close to the blade suction-surface.

Figure 7.

Illustration of the flow structures due to the blades’ proximity to the cavity, showing (Left) the HSV partly in the cavity and (Right) the HSV outside the cavity.

If the influence of the blades’ pressure field is weakened, either by a change in blade loading and/or a change in the position of the blade, the pressure side leg of the HSV emerges from the cavity and wraps around the blade leading edge. This alternative mode for the HSV is shown in Figure 7 as “Type B” flow. A more detailed discussion surrounding the mechanism behind this change in the flow field has been previously published (Lee et al., 2019b).

The changes in the flow angles in the IPT stage affects the efficiency of the stage and Figure 8 plots the changes in the scaled η for the HPT stage (2.5-stage and 1-stage simulations) and the IPT stage. As shown, the IPT stage also responds to changes in the geometry of the HPT rotor blade. In Figure 8, two types of behaviour are highlighted and the corresponding profiles of βexit have also been highlighted in Figure 6:

Figure 8.

Scaled stage efficiencies obtained from the 1-stage and 2.5-stage Hydra RANS models. Refer to Figure 6 for the corresponding radial profiles of the flow angles.

For blade ID 12, the HPT rotor exit flow angle was significantly reduced between 30%<r<70%. This under-turning results in a lower work output and consequently, a lower HPT stage efficiency. This under-turning cascades downstream to the IP NGV and redistributes the inlet flow angle profile to the IPT rotor blade. This shifts the work across the span towards the IPT rotor blade tip where losses are generally higher, reducing the IPT stage efficiency.

For blade IDs 13, 20 and 23, the change in HSV behaviour causes the tip region of the HPT rotor blade to under-turn the flow, producing less work and reducing HPT stage efficiency. However, unlike for blade ID 12, the IP NGV is effective is removing most of the flow angle discrepancy at the IP NGV inlet and the exit flow angle from the IP NGV is similar to the design-intent, minimising the impact of the change in HSV behaviour on the IPT stage efficiency.

The small differences in the trends between the 2.5-stage and 1-stage predictions of scaled η of the HPT stage is due to a small change in the operating point of the turbine. Although the absolute numbers are slightly different, the trend of the change in HPT stage scaled η is shared between the two types of simulations, as expected.

Part III: multi-passage effects

In an engine, adjacent blades will have slightly different manufacturing deviations. This interaction with neighbouring blades is not captured by single-passage calculations which assume periodicity over one blade pitch. Multi-passage simulations were therefore used to explore these interactions: 26 1-stage models each of 3-passage and 5-passage shrouded HPT rotor blades were created and solved using Hydra. The 26 cases include one case (case index 1) assembled using only blades that exhibit Type B behaviour, and a further 25 cases constructed using Latin Hypercube Sampling where geometric data from the 25 STL meshes were used to populate each blade position once.

As with the single-passage calculations, multi-passage cases also exhibited Type A and Type B flow. However, the flow structure for each multi-passage calculation does not simply depend on the presence of blade(s) that exhibited Type B behaviour in a single-passage calculation. Figure 9 compares:

Figure 9.

Flow structures observed in the multi-passage (MP) CFD models and the “average” flow structure from the single-passage (ASP) CFD models of the corresponding constituent blades.

Examining the trend for the 5-passage cases, Type B flow is observed in multi-passage cases 1, 3, 5, 6, 7, 9, 12, 15, 17, 18, 23, and 24 but cases 6, 9, 12, and 24 do not contain blade(s) that gave Type B behaviour in a single-passage calculation. In addition, multi-passage cases 10, 14, 16, 21, 25, and 26 all contain a blade that gives Type B behaviour but all demonstrate Type A flow. Similar observations can be made for the 3-passage cases.

The above comparison shows that the detailed flow behaviour in the multi-passage environment is somewhat different to the single-passage environment. Nonetheless, if the overall performance can be estimated from an average of single-passage calculations, then the cost and complexity of simulations will be reduced enormously. Figure 10 shows this assessment for the four key performance parameters mentioned earlier:

Figure 10a and 10b show good agreement between the ASP and MP results for the mass-averaged βexit and Q as the data points lie along the one-to-one line. The mean mass-averaged βexit and mean Q from the 3-passages and 5-passages cases are consistent and do not show a noticeable shift when the number of blade passages is increased. The standard deviation, however, reduces with an increasing number of blade passages. The insensitivity of the mass-averaged βexit and Q to the flow structure changes (multi-passage model that exhibit Type B flow are denoted by the subscript “B” and filled-in markers in Figure 10) supports the validity of averaging single-passage results to predict multi-passage behaviour.

In contrast, Figure 10c shows significant discrepancy between the predicted stage loading ψ from the ASP and MP approaches. For the cases that exhibit Type B flow, the MP values of ψ are typically higher than the ASP values. The exception is the case in the top-right corner of Figure 10c, case index 1 in Figure 9, which was assembled from blades that only exhibited Type B flow. The dependency on the flow structure is also shown by how the cases that exhibit Type B behaviour form a distinct cluster which causes the mean of the MP models to shift (compared to the mean of the ASP values) and increases the standard deviation in the MP results. For efficiency η, Figure 10d shows a weak correlation between the predictions of η from the ASP and MP approaches. This shows a significant inter-dependence between the blades and averaging single-passage results is only capable of giving a rough estimate of the multi-passage stage efficiency.

To summarise, the HSV flow structure near the vicinity of the HPT rotor blade tip is bimodal: Type A and Type B flow are observed as two distinct states with no clear intermediate state between the two. The behaviour is sensitive to interaction between adjacent blades and their pressure fields. Considering overall performance parameters, flow angle and capacity are well approximated by averaging the individual single-passage predictions of each blade (R2>0.86). This approach is significantly less accurate for the stage loading coefficient ψ and efficiency η (R2<0.54).

Part IV: accuracy and computational costs across the hierarchy

Having examined some of the differences between the predictions obtained from aerodynamic simulation methods of various degrees of realism, attention now returns to the central question motivating this paper, namely what is the trade-off between cost and accuracy of different aerodynamic simulation fidelities?

Figure 11 plots the predicted changes in capacity Q and scaled HPT stage efficiency η for two of the 25 blades: blade ID 2 that exhibits Type A flow and blade ID 20 that exhibits Type B flow. The performance changes are plotted on a log-scale for the number of CPU hours that were used to run each simulation. To ensure a valid comparison, the RANS and URANS cases were calculated using only the HPT stage as the computational domain and the same blade (either blade ID 2 or blade ID 20) was used to populate every blade position in the cases where the HPT rotor has more than one blade passage.

Figure 11.

Comparison of the computational costs (CPU core hours) and the predicted performance changes by the various aerodynamic simulations across the aerodynamic simulation hierarchy.

On a log-scale, the Euler, RANS, and URANS CFD models occupy three distinct regions. Compared to the predictions from RANS and URANS, Euler models are very quick and take only ∼15 CPU core seconds per blade section. However, the Euler models fall short in the ability to capture the performance changes observed in the higher fidelity methods. The RANS models are all of similar order of magnitude in terms of simulation cost (50–200 CPU core hours) and the URANS models require ∼5,500 core hours. As the URANS models required the same size sector for the NGV and rotor blade rows, the URANS models had 6 HP NGVs and 11 HPT rotor blades. In addition to these raw computational costs, the complexity in running the models in Figure 11 generally increases with fidelity. Rising computational cost is therefore accompanied by an increase in the time and expertise required to build and manipulate these models.

As a general principle, the required aerodynamic modelling fidelity is driven by the need to capture the aerodynamic phenomena that drive the performance quantities of interest. While it is difficult to draw more specific universal rules for simulation, the following recommendations can be made for Digital Twins of gas turbine blades (in particular, a HPT rotor blade):