Design of Aerodynamically Balanced Transonic Compressor Rotors

It is well known that the aerodynamic design of multistage transonic compressor rotors is incredibly difficult. The reason is that as the blade relative inlet Mach number approaches unity, the incidence range over which a blade row can operate efficiently reduces and becomes restrictively small [1]. At the same time, transonic flow is highly three-dimensional, and as such transonic rotor blades exhibit significant 3D radial flow variations [2]. This spanwise 3D flow within the blade passage becomes increasingly important in setting the shock structure at transonic inlet relative Mach numbers, particularly approaching the sonic condition [3]. An added problem is that the three-dimensional flow structure within the blade passage also varies significantly with the compressor's operating condition.

In industry, this aerodynamic design process is usually handled by engineers with years of experience who iteratively redesign the compressor through a manual process [1]. This process is often laborious, and the quality of the final design is dependent on the experience of the designer. Moreover, to tackle the complexity of the design process several multi-objective inverse, adjoint, and machine learning optimization schemes have been recently proposed [4–6], but these are not simple and lack physical intuition. This paper addresses the transonic aerodynamic design problem by proposing a novel physics-based method that acts to greatly simplify the design process. This method enables the design of multistage transonic compressor rotors which have both a wider operating range and an improved design efficiency.

The reason that the design process of transonic compressors and fan rotors is so complex is that they have a shock which extends over much of the span. This shock structure controls both the blade’s operating range and design performance. The strength and location of the shock depend on several geometric parameters, including radial variations in the distribution of suction surface angle, thickness-to-chord, and pitch-to-chord. An added difficulty is that the flow is often highly three-dimensional and this three-dimensionality changes significantly as the blade row’s geometry is changed, the blade row is throttled, or the blade speed is changed. It is the sheer number of geometric design parameters and the complexity of the flow field which makes the design process so difficult.

Lefas and Miller [3] showed that for a fixed blade inlet relative Mach number, the shock structure across each annular streamtube is fixed primarily by the throat to inlet flow area of the streamtube, A_throat/A_inlet, which for simplicity will be referred to as A_t/A₁ in the remainder of this paper. This allows a compressor or fan rotor to be plotted on a relative inlet Mach number area ratio plot, like that shown in Fig. 1. The black lines on the left-hand side of the figure, Fig. 1(a), show the area ratio at which a streamtube chokes or the suction surface of the blade separates by a strong shock. This shows that as the inlet relative Mach number increases, the range of area ratios over which a blade section can operate efficiently reduces.

The red solid lines with circles in Fig. 1(a) show the design of a typical transonic compressor rotor. Each red line in Fig. 1(a) shows the variation of streamtube area ratio from hub to tip at a particular throttle setting. As the rotor is throttled from choke to stall, the red line can be seen to shift vertically. This is because as the inlet incidence into the blade row increases the inlet area A₁ drops; pushing A_t/A₁ to higher values. For the most closed red throttle setting, near the tip of the rotor, the area ratio can be seen to be higher than the shock-boundary layer separation limit. In Fig. 1(b), the spanwise shock structure location and suction surface streamlines at this closed throttle setting are shown. The shock can be seen to be strong enough to separate the suction surface boundary layer toward the tip.

In this paper, Lefas and Miller’s previous work is used to develop a new design method. The method involves extracting the real streamtube A_t/A₁ area ratio from the three-dimensional computational fluid dynamics (CFD) using streamtube tracking. This area ratio A_t/A₁ is shown graphically using Fig. 2 and can be changed in two ways. First, by flow turning, in the blade-to-blade sense, and second by radial mass flow redistribution in a 3D spanwise sense. In this paper, it will be shown that during design both these contributions are of equal first-order importance. Hence, extracting A_t/A₁ from 3D CFD ensures that the true three-dimensional structure of the streamtubes is accurately captured. The correct spanwise variation in area ratio that controls the shock structure can then be plotted, as just shown in Fig. 1(a), for the baseline compressor rotor in red.

As already discussed, the baseline rotor’s area ratio toward the tip was shown to be too high resulting in a shock strong enough to separate the suction surface boundary layer. Therefore, to improve the blade’s performance, the design intent would be to close down the throat area in the tip region and therefore reduce A_t/A₁, moving the design away from the shock-boundary layer separation limit.

After a small number of iterations, the new rotor design shown in blue dashed lines with squares is achieved, where for the same closed throttle setting as the baseline, the A_t/A₁ value near the tip lies below the shock-boundary layer separation line. As a result, the shock strength is reduced and moves further forward whilst avoiding separation of the suction surface boundary layer.

It will be shown that this design method produces rotors with an increased operating range and an increased design efficiency. By keeping the area ratio within the ideal range over the widest range of operating conditions as the compressor is throttled, as shown by the blue dashed lines in Fig. 1(a), the new design method is said to produce blades which are “aerodynamically balanced.”

After explaining the new method, the paper will demonstrate its application using two examples. The first example is a transonic rotor from the front stage of a multistage civil compressor. Because the rotor has variable stator vanes (VSVs) upstream of it, the design aim is to maximize the design efficiency along the working line and operating range across a wide range of shaft speeds by optimizing both the rotor’s geometry and the VSV settings. The second example is an embedded rotor within a multistage military fan with an inlet distortion. The fan has no upstream VSVs. In this rotor’s design, the working line mass flow point is set by a safe stall margin criterion. To achieve this, the working line point is forced to operate away from peak efficiency. The design challenge, in this case, is to increase the operating range of the blade row toward stall so that the working line of the machine can be operated closer to the rotor’s peak efficiency for the same safe stall margin.

In both examples shown in the paper, the new design method allows significant design changes, which can sometimes be counter-intuitive and would normally take an expert designer a significant time, to be undertaken quickly and in an automated manner. The final design in both cases is shown to have a higher performance than could be achieved with the current design methods. The new design method can also be seen to provide a physically intuitive way of understanding how the design was improved.

The two rotor examples presented in this paper are Rolls-Royce proprietary geometry and by intention have been chosen because they have widely different design specifications to demonstrate the universality of the method. For example, relative to the civil front-stage rotor, the embedded stage military-style rotor has a 30% smaller hub-to-tip ratio: r_hub/r_tip, operates at an 18% lower flow coefficient: V_m/U_mid, at an 8% higher non-dimensional blade speed: U_mid/√T_o1, and delivers a 25% higher design stagnation pressure ratio: P_o2/P_o1.

The multistage CFD was run in HYDRA, Rolls-Royce’s proprietary solver [7]. The new design method was then applied to the rotor of interest isolated using the CFD solver TURBOSTREAM [8], with boundary conditions extracted from the multistage HYDRA solutions. The final rotor was then rerun in multistage HYDRA to validate the performance benefits.

TURBOSTREAM is a structured multi-block Reynolds-averaged Navier–Stokes (RANS) solver based upon TBLOCK and implemented for parallel GPU operation. The turbulence model used was the Spalart-Allmaras [9]. The meshing process was automated using AUTOGRID. A y⁺ lower than one was ensured at every blade surface and a mesh sensitivity study was performed for every case considered. HYDRA is an unstructured RANS solver. The turbulence model used was again the Spalart-Allmaras. The meshing process for the multistage was done automatically with PADRAM [10]. For both HYDRA and TURBOSTREAM transition was not modeled, and the boundary layers are turbulent.

The new design method is applied to isolated rotor CFD cases run with the inlet and outlet boundaries placed 1.5 axial chords upstream and downstream, respectively. The real hub and casing walls are used and extracted from the multistage civil and military compressors of interest. To ensure the prescribed profile did not change from the inlet plane of the isolated rotor case to the equivalent mixing plane in the multistage CFD, the walls were enforced to be slip walls between those two planes. The profiles were also physically checked to ensure that they were indeed the same.

All the 3D blades shown in this paper are schematics, and this is to protect Rolls-Royce proprietary information. Finally, the shock locations relative to the leading edge, surface streamlines, multiple blade speed characteristics, isentropic Mach number plots, and radial spanwise distributions shown in this paper have been derived from TURBOSTREAM and are not schematics.

This section provides an overview of the new design method. The practical implementation of the method in the 3D CFD is given in the Sec. 4. The first example, the front-stage rotor of a multistage civil compressor, is used to illustrate how the new design method works.

Lefas and Miller [3] showed that the strength of the shock on each streamtube is primarily set by the area ratio A_t/A₁, already shown graphically in Fig. 2. This can be understood by considering a simpler two-dimensional sectional view, where the radial streamtube contraction is fixed, as shown in Fig. 3. In this case, the inlet relative Mach number to the blade row is 0.95 with a shock forming ahead of the passage throat location. In Fig. 3(a), the flow field within the blade passage is shown and is rotated to be horizontal for clarity. Three streamlines going through the passage are plotted with the static pressure along these streamlines plotted in Fig. 3(b).

The green streamline in Fig. 3 connects the inlet and the throat planes via an isentropic streamline. It should be noted that the streamline just passes through the sonic patch of the blade but undergoes both an isentropic expansion and compression. An isentropic streamline exists in every transonic blade row as the shock never extends all the way across the passage. However, even for relatively small supersonic Mach numbers below 1.20, where the shock does extend across the passage, a streamline exists which is close to isentropic and so the same behavior is observed.

At the throat plane, marked by a cross in the lower part of the figure, the static pressure is the same across all three streamlines showing that the pressure is virtually uniform normal to the throat streamlines. This is because, for transonic blade rows of representative loading coefficients, the streamlines are effectively straight from the throat to the outlet, as can be seen in Fig. 3(a).

Because the pressures upstream of the blade and at the blade throat are uniform and connected by an isentropic streamline, the pressure ratio between the throat and inlet plane is fixed by the area ratio A_t/A₁.

The universality of these assumptions is demonstrated for a wide range of transonic blade designs using a web interactive demo: Transonic compressor dataset (whittle.digital²) powered by dbslice [11,12]. In this demo, a dataset of more than 400 transonic blade designs operating at an inlet relative Mach number of 0.95 is presented. It can be seen from the histogram within the demo that these designs cover a wide range of A_t/A₁.

These designs represent a wide range of blade styles as they were achieved not only by changing the camber distribution but also by varying both key geometric and aerodynamic design parameters such as thickness-to-chord, pitch-to-chord, blade loading, inlet relative flow angle, and radial contraction to the throat and outlet. Going through each design in turn shows that each A_t/A₁ design corresponds to a particular pressure rise across the shock and that this primarily sets the pre-shock Mach number irrespective of the detailed blade geometry.

The pressure ratio and efficiency characteristics of the front-stage rotor within the multistage civil compressor, before (solid) and after (dashed) redesign, are shown in the top half of Fig. 4, where significant benefits in working line efficiency and operating range can be seen. The bottom half of the figure shows the two designs plotted on the same A_t/A₁ against inlet relative Mach number graph shown in Fig. 4, but with the real A_t/A₁ lines obtained from 3D CFD. For both cases, two blade speeds of operation are considered: part-speed and design speed. The baseline design will be discussed in this section and the process of redesigning the rotor to be “aerodynamically balanced” will be discussed in Sec. 3.3.

In Fig. 4(a), three operating points on each blade speed’s characteristic are shown by the cross, square, and circle symbols. For each of these operating points, the baseline rotor design’s spanwise distributions of A_t/A₁ have been extracted from the 3D CFD and are plotted in Fig. 4(b). Examining Fig. 4(b), for both design and part-speed, three A_t/A₁ lines can be observed. The solid line with crosses corresponds to the spanwise distribution of A_t/A₁ at the peak efficiency working line point. The dot-dashed line with squares corresponds to the point on the characteristic close to the maximum pressure-rise limit and the dashed line with circles to the point close to the choking limit.

Figure 4(b) also plots two black “limit” lines. The top line is the one at which shock-boundary layer separation occurs and the bottom line at which choking occurs. These “limit” lines were derived in the authors' previous work [3], using the well-validated quasi-3D CFD solver MISES [13].

Transonic blade rows can operate outside these limit lines, but inefficiently. If a blade row has an A_t/A₁ value above the shock-boundary layer separation black line, the shock is strong enough to separate the boundary layer. Once shock-boundary layer separation occurs there is a significant drop in efficiency until at even lower mass flowrates, the blockage from the separation increases, and the blade row eventually stalls. If a blade row has an A_t/A₁ value below the choking black line, indicating that the blade has less than a 1% choke margin, the losses within the blade row rise rapidly causing the blade to choke. As a result, keeping the rotor’s A_t/A₁ spanwise distribution within these limits over the maximum possible range of compressor, throttle positions will act to increase the overall operating range of the compressor.

Figure 4(b) clearly shows that the baseline rotor design is not optimal because at the working line condition, the spanwise A_t/A₁ distribution, shown by the red cross solid line in Fig. 4(b), is too high toward the shock-boundary layer separation limit near the tip and too close to the choking limit near the hub region.

Hence, when the compressor is throttled up its characteristic, the baseline rotor design prematurely crosses the shock-boundary layer separation limit line toward the tip. This occurs at both part-speed and design speed. This indicates that a more low mass flow operating range could be achieved toward stall. In addition, it can be seen that at design speed as the compressor is throttled toward choke, close to the hub, the design crosses the choking limit line. This also indicates that more higher mass flow operating range toward the choke could be achieved at the design speed.

The purpose of the redesign is to change the rotor geometry up the span such as to move the spanwise distribution of A_t/A₁ within the efficient operating region, set by the two limit lines, as shown in Fig. 4(c). This was achieved by changing the spanwise distribution of the throat area across the span. In this case, the geometric throat area of the redesigned rotor blade was closed down toward the tip and opened up toward the hub relative to the baseline blade, such as to make the gradient of the spanwise distribution of A_t/A₁ more horizontal. The amount by which this was done was determined whilst accounting for the radial flow changes occurring by the throat plane from 3D CFD; as will be discussed in the next section.

Because the spanwise mass flow distribution varies, especially when the blade is operating close to the limit lines, the 3D CFD was then rerun and the new spanwise distribution of A_t/A₁ was extracted. Usually, only three to five iterations are required using this design process before a rotor design is achieved which doesn’t cross the limit lines at any speed for the same compressor throttle positions as the baseline.

The benefits in working line design efficiency and operating range of this redesigned (dashed) rotor compared to the baseline (solid) are shown in Fig. 4(a). The redesigned blade shows a larger operating range at low mass flow toward stall and high mass flow toward choke at both speeds, as well as improvements in the working line efficiency of 0.25% and 0.20% at design and part-speed respectively.

A design of a blade which lies within the two limit lines over the widest possible range of compressor throttle conditions is said to be spanwise “aerodynamically balanced” and results in significant performance improvements compared to an unbalanced design.

The practical implementation of the method will now be discussed. Key to the method is first, the accurate extraction of the area ratio A_t/A₁ from 3D CFD solutions and second, splitting A_t/A₁ into its blade-to-blade component, due to flow turning, and its component due to spanwise streamtube contraction. Splitting it in this way is necessary to determine how to improve the 3D blade design.

The process of calculating A_t/A₁ can be split into two steps that will now be summarized with the help of Fig. 5. The first step is to determine the geometric throat plane location, as shown in Fig. 5(a) by the green line. In this paper, the geometric throat plane is defined by taking normals to the 3D blade suction surface at each section including the displacement from the boundary layer.

In this case, when area-averaging the throat plane, the meridional rather than relative velocity must be used and the blockage δ_t introduced by the blade thickness must also be included plus the displacement from the boundary layer thickness at the point where the throat is intersecting with the blade, also shown in Fig. 5(a). It should be noted that errors are small if the boundary layer is not included in this analysis, as the blockage term δ_t is mostly governed by the blade thickness.

The redesign process involves changing the blade-to-blade geometry, altering o/scosα_1rel, to achieve the required change in A_t/A₁. But this does not include the effect of the radial 3D flow redistribution. The 3D CFD is then rerun and the real change in A_t/A₁ is extracted. The change in the 3D spanwise flow, AVDR_t is then calculated independently from Eq. (3) and can be used to make a more informed decision on the next o/scosα_1rel change needed to achieve the target A_t/A₁. In practice, the design target spanwise distribution of A_t/A₁ can typically be achieved within three to five iterations.

The first example is a transonic rotor from the front stage of a multistage civil compressor. Because the rotor has variable stator vanes (VSVs) upstream of it, the design aim is to maximize the design efficiency and operating range across a wide range of shaft speeds by optimizing both the rotor’s geometry and the VSV settings. In this section, the redesign of the rotor’s geometry at a fixed blade speed, the design speed, will be described. Varying blade speeds will be considered in the next section. The baseline rotor and the redesigned rotor blade were already shown using Fig. 4. For clarity, the baseline design and the redesign are reproduced together in Fig. 7 for the design speed only.

The geometric design changes made in this paper are those that would typically be allowed during the latter stages of design when optimizing a transonic rotor for efficiency and range without compromising its mechanical rigidity. These geometric changes are shown schematically in Fig. 8 as (a) the blade metal inlet angle (χ_LE), (b) the proportion of front loading up to the throat, χ_LE–χ_t, and finally (c) the blade metal trailing edge angle (χ_TE).

The spanwise thickness distribution, true chord distribution, blade number, 3D radial stacking and hub and casing lines have been kept fixed. Since these geometric parameters are fixed, the allowed changes in blade metal suction surface angle χ have been achieved by changing the blade’s spanwise camber distribution.

Here, the difference in the geometric and 3D flow component of A_t/A₁ between the baseline and redesigned blade will be presented. Particular attention will be paid to the complex 3D flow changes occurring within the blade passage because of the redesign. The actual geometric spanwise changes made will be discussed in the next section.

Figure 9 plots the spanwise distribution of (a) o/scosα_1rel and (b) AVDR_t for the final redesign and baseline front-stage compressor rotor. The target aerodynamically balanced redesign was achieved within three design iterations and is shown by the dashed line in Fig. 9. The baseline design is shown by the solid line. Using Eq. (2), these contributions can be combined to give the resulting A_t/A₁ spanwise distributions for the redesigned and baseline blade at the design speed peak efficiency point shown by the solid line with crosses earlier in Fig. 7.

Looking back at Fig. 7, to achieve the design intent A_t/A₁ distribution of the redesigned blade, A_t/A₁ must be reduced at the tip and increased at the hub. For this to happen, the geometric throat area, o, has to be reduced at the tip and increased at the hub. Hence, in Fig. 9(a), the geometric component of o/scosα_1rel from midspan to casing has decreased and below midspan increased. The shaded region shows the difference between the two designs.

However, changes in the geometry of transonic blade rows also result in a significant change in 3D radial flow at the throat plane: AVDR_t. The difference between the two designs is now shaded in blue in Fig. 9(b) and can be seen to be comparable to the o/scosα_1rel change shaded in red in Fig. 9(a). Importantly, Fig. 9 shows that the change in AVDR_t counteracts the effect of changes in o/scosα_1rel and significantly contributes to A_t/A₁.

In other words, if the designer changes the blade geometry, with the aim of changing A_t/A₁, the change in AVDR_t will naturally act to cancel it. This mechanism was also observed in the authors’ previous work and was identified as critical in ensuring that at off-design conditions, the blade remained between the choking and shock-boundary layer separation limits [3].

Examining Fig. 9(b) in more detail, it can be seen that AVDR_t lies below unity everywhere along the span and is on average equal to 0.925 for both redesigned and baseline rotors. This is because the casing contracts across this rotor, as is typical for multistage compressors.

In addition, the streamtubes are contracting less within the 75% span-casing region, where the redesigned blade has a lower value of A_t/A₁ compared to the baseline blade in Fig. 7. A lower value of A_t/A₁ means that the pressure rise across the shock has decreased. As a result, so has the density rise from the inlet to the throat for the redesigned blade. This density drop when compared to the axial velocity increase is greater in terms of magnitude and is dominating. This means that by continuity in the top 25% region of the blade, the streamtubes of the redesigned blade must contract less to the throat compared to the baseline blade to fit the same amount of mass flow.

The opposite effect can be seen in the region from midspan to 75% span, where the redesigned blade streamtubes can be seen to contract more relative to the baseline design. The overall effect is that the average radial contraction across the span has remained the same for both the redesigned and baseline rotors. This is expected since the overall radial contraction across the rotor at the throat plane is still set by the hub and casing lines across the rotor.

It should be noted that throughout this redesign process, the spanwise distribution of AVDR_t within the blade passage changes, although there is no change in the spanwise distribution of AVDR between the blade inlet and exit. This is because the design intent exit relative flow angle (i.e., loading distribution) is fixed during the design process.

Figure 9 shows that the product of the o/scosα_1rel and AVDR_t distributions is A_t/A₁, which primarily sets the shock strength. Because of the counteracting effect between o/scosα_1rel and AVDR_t, small changes in the baseline design are not sufficient to change A_t/A₁ even by the modest amount of 0.02 shown in Fig. 9(c). For example, as will be discussed in more detail in Sec. 5.3, to redesign A_t/A₁ at the tip, 6 deg of camber had to be removed by the throat.

The importance of one-dimensional area ratios, and particularly A_t/A₁, in establishing the design flow in transonic compressors has been mentioned by Wright and Miller [15] and Smith [16]. In Wright and Miller’s paper, correlations based only on the geometric throat area o/scosα_1rel are presented, but, as shown in this paper, this considers only half the design problem. Neglecting the AVDR_t changes between different designs could explain the significant scatter present in Wright & Miller’s paper.

In Smith’s paper, spanwise A_t/A₁ derived using GE’s circumferential average flow determination (CAFD) radial equilibrium solver, with internal axial and radial stations, i.e., a throughflow, is used to start the design process and compare designs with past experience, but not as part of detailed design. In this case, A_t/A₁ is determined by multiplying the geometric throat area by AVDR_t as calculated by the throughflow.

However, based on the new information presented in this paper that A_t/A₁ controls the shock structure and that the permissible A_t/A₁ between choke and shock-boundary layer separation narrows restrictively close to the sonic condition, it can be concluded that a precise calculation of A_t/A₁ is required to inform detailed transonic design. For example, from Fig. 1 it can be seen that at the sonic condition, just a 5% change in A_t/A₁ can be the difference between choking and a strong shock separating the boundary layer.

The complex 3D nature of transonic flow within internal blade passages with strong shocks upstream, as shown in Fig. 9(b), means that a higher fidelity method than a throughflow with several internal axial and radial stations is necessary. In this case, to get an accurate representation of the internal flow field at the throat plane, which is important in setting the shock structure, the novel method presented here of extracting A_t/A₁ from the 3D CFD would need to be employed.

The extraction of A_t/A₁ from 3D CFD presented is both simple and accurate and thus can be easily included in the detailed design phase. In fact, as shown in this design example what is unique about the new A_t/A₁ method, is that it accurately captures the 3D flow redistribution that counteracts the geometric changes made. Hence, it allows for significant redesign changes, discussed in detail in Sec. 5.3, to be made simply and quickly. This makes it a practical design tool which goes beyond a starting point for transonic rotor design as proposed by Smith in his paper.

There are two ways to change the geometry up to the throat, o, within the design restrictions enforced in this paper. First, is changing χ_t, shown schematically in Fig. 8(b), and the second is χ_LE, shown schematically in Fig. 8(a). Reducing the amount of camber up to the throat, i.e., a decrease in χ_t will act to decrease the blade’s throat area, o. Opening the blade, i.e., a reduction in χ_LE will act to increase the throat area.

Using the iterative design process explained earlier in Sec. 4, χ_t is adjusted up the span first, with a fixed spanwise distribution of χ_LE. If the iterative design process fails to deliver the desired target aerodynamically balanced blade, then the spanwise distribution of χ_LE is changed, and the process of iterating on spanwise χ_t repeated until the target aerodynamically balanced blade is achieved. χ_TE is also changed to maintain the exit relative flow angle at the outlet to the rotor such as to maintain the rotor’s pressure ratio and preserve compressor matching.

In Fig. 10, the geometric differences between the final redesigned (dashed) and baseline (solid) front-stage compressor rotor designs in terms of spanwise χ_LE–χ_t, χ_LE, and χ_TE are presented. It is worth noting that large changes in the geometry are required to achieve even the relatively modest differences in the redesigned and baseline A_t/A₁ spanwise design profiles presented earlier in Fig. 7. This is because as already mentioned the variation in AVDR_t within the passage acts to negate any changes in the 3D geometry.

Figure 10(a) shows that for the redesigned blade χ_LE–χ_t is smaller from 35% span to casing and greater from hub to 35% span. This means χ_t is smaller for the redesigned blade from 35% span to casing and greater from hub to 35% span. In other words, camber has been removed from the 35% span to the tip and added from the hub to 35% span, with nearly 6 deg removed at the tip and an increase of about 2 deg made toward the hub. This acts to reduce o/s toward the casing and increase o/s toward the hub, as already seen in Fig. 9(a).

Just making changes in χ_t using the baseline blade’s χ_LE distribution did not converge on the target aerodynamically balanced spanwise A_t/A₁ distribution. This is because the blade choked prematurely within the 60%-85% span region of the blade. The solution was to open up the redesigned blade locally, i.e., decrease χ_LE by around 1–2 deg within this region relative to the baseline blade; as can be seen in Fig. 10(b). The design process of iterating on spanwise χ_t was then restarted and the distribution of χ_LE–χ_t shown in Fig. 10(a) was converged upon.

Finally, the trailing edge angle χ_TE, shown in Fig. 10(c), was at the same time adjusted to maintain the design intent exit relative flow angle. This ensured that the design spanwise pressure ratio was maintained. During this process, the trailing edge metal angle was reduced in the tip region. This resulted in a slightly higher subsonic diffusion downstream of the throat. It should be noted that this increased diffusion was not close to the subsonic rear diffusion limit, and the blade’s operation was still limited by shock-boundary layer interaction.

To assess and understand the performance benefit of the redesign against the baseline rotor design, it is necessary to examine the isentropic Mach numbers at pertinent blade sections along the span at the design speed peak efficiency/working line condition. For this purpose, three sections of the redesigned and baseline blade will be considered here, at 30%, 60%, and 85% span. When comparing the isentropic Mach number distributions between the two designs at these sections particular attention will be paid to (1) the peak Mach number, (2) the Mach number at the point where the throat plane intersects the blade suction surface, and (3) the shock location, as these are linked physically to A_t/A₁.

In regions of the blade where the redesigned blade has a greater value of A_t/A₁ compared to the baseline blade, the shock will be stronger. Hence, the peak Mach number on the isentropic Mach number plot should be greater. In addition, because A_t/A₁ primarily controls the pressure along the throat plane, the greater the A_t/A₁ the larger the expected pressure at the throat. Hence, on the isentropic Mach number plots where the throat plane intersects the suction surface, the isentropic Mach number should be lower. The opposite applies to regions of the redesigned blade with an A_t/A₁ value smaller than the baseline blade.

To test whether these predictions from using A_t/A₁ hold, Fig. 11 plots the isentropic Mach numbers along the 30%, 60%, and 85% span section from the leading edge to the trailing edge. These sections have been picked purposely because, referring back to Fig. 7, at 30% span the relative inlet Mach number is 0.85, and the redesigned blade has a greater value of A_t/A₁ relative to the baseline blade. At 60% span, where the inlet relative Mach number is around 0.90, the redesigned blade has a slightly smaller but comparable value of A_t/A₁, and finally, at 85% span, the A_t/A₁ of the redesigned blade is much smaller compared to the baseline blade and the inlet relative Mach number is about 0.97. In Fig. 11, the location of the throat is indicated by the dot-dashed line, with the Mach number on the suction surface at the throat plane also indicated by the red symbols.

From Fig. 11(a), it can be seen that as expected from A_t/A₁ at the 30% section the redesigned blade’s shock strength has increased and the throat Mach number has decreased relative to the baseline blade. This increase is manageable because the 30% section operates with a lower relative inlet Mach number and a much wider range of A_t/A₁ is possible than near the tip. If we examine the same plot, but now at 60% span, the shock has moved further forward but is of similar strength. A slight increase in the Mach number at the throat can be seen, but this is not significant. Again, this is as expected from A_t/A₁. Finally, looking at the 85% section, the shock has also moved further forward and the pre-shock Mach number has decreased from 1.25 to 1.18, with the throat Mach number on the suction surface also larger relative to the baseline blade. This reduction in tip shock Mach number results in significantly less loss at the tip and is the primary reason why the redesigned blade performs more efficiently at the design working line condition compared to the baseline blade. Hence, at all sections, the expected physical behavior from A_t/A₁ is observed.

To understand why the shock has moved further forward, particularly at 60% and 85% span, each section’s corresponding blade suction surface angle χ distributions relative to the inlet flow angle α_1rel is also plotted in Fig. 12. It can be seen that for the 60% (Fig. 12(b)) and 85% span section (Fig. 12(c)), the redesigned blade has an increased global incidence α_1rel–χ_LE because the blade has been opened, reducing χ_LE in this region as already shown in Fig. 10(b). Yet this is small and compensated by removing camber up to the throat. As a result, the overall amount of flow turning α_1rel–χ at the shock foot has decreased, and the shock strength is reduced. The shock has moved further toward the leading edge for the redesigned blade because a larger proportion of the overall flow turning up to the shock is achieved around the blade leading edge rather than along the blade suction surface for the redesigned rotor compared to the baseline rotor.

In summary, the redesigned “aerodynamically balanced” blade has a shock moved further toward the leading edge because of the increased leading edge blade angle. Conventional thinking would have rejected such a design, as a rotor having a 3D shock structure closer to the leading edge at peak efficiency would be assumed to have a small positive incidence range. This is because a shock closer to the leading edge is conventionally interpreted as operating with a larger positive incidence closer to the stalling condition [1]. However, as it has been shown, designing a blade which is aerodynamically balanced, though acting to move the shock forward in the tip region, also acts to reduce the strength of the shock, resulting in an increase in the blade’s positive incidence range.

The first example, the transonic rotor from the front stage of a multistage civil compressor, will now be investigated over a range of blade speeds. Crucial for the performance of front-stage rotors is extending their range off-design, both in terms of stall margin and choking capacity, whilst also maintaining if not improving on efficiency. The objective here is to optimize the VSV settings at different blade speeds and assess a given candidate blade’s performance with optimum VSV settings quickly at multiple blade speeds.

As the compressor changes its blade speed, the inlet relative Mach number and relative flow angle into the rotor swing significantly. Figure 13 shows this variation in inlet relative Mach at design, part-speed, and maximum climb for a representative transonic rotor within a modern multistage civil compressor. Similar changes in the relative flow angle into the rotor also occur.

In modern civil multistage compressors, these are corrected in the front stages by variable stator guide vanes (VSVs) that must be staggered appropriately to ensure optimal operation at each blade speed. This section will consider the three representative VSV settings plotted in Fig. 14. These three VSV settings will be used to show how to identify the best VSV setting at part-speed for a given fixed blade design.

In Fig. 14, the solid line indicates the VSV setting at the design blade speed. At part-speed, this will be shown to result in a blade operating with a too-high positive incidence and thus lies very close to its maximum pressure-rise limit. As a result, in practice at part-speed, the VSV is closed relative to the design speed setting, as shown by the dashed and dot-dashed line VSV settings. As this happens, two things change.

First, the absolute flow angle at the exit to the VSV changes (Fig. 14(a)), changing the inlet area A₁ into the rotor. Second, the inlet relative Mach number into the rotor changes as shown in Fig. 14(b). However, it should be noted that most of the change in inlet relative Mach number across the rotor at part-speed is because the compressor blade speed has been reduced as shown in Fig. 13. The change from closing the VSV is additional and of secondary importance.

Both the change in the inlet area and relative Mach number can be accounted for when using the design methodology based on A_t/A₁ proposed in this paper. To illustrate this, Fig. 15 plots for the redesigned blade and each of the VSV settings are shown in Fig. 14: (a) the spanwise distributions of A_t/A₁ in blue and (b) the corresponding pressure-rise mass flow characteristics at part-speed operation.

Examining the A_t/A₁ spanwise distributions using Fig. 15(a), two things can be observed. First, the spanwise distributions of A_t/A₁ now cover a lower range of inlet relative Mach numbers from hub to tip when compared to the design speed shown previously in Fig. 7. This is because it operates at a reduced compressor blade speed. In addition, as the VSV is closed, the mean relative inlet Mach number also decreases slightly. This is because of the secondary effect that closing the VSV has on the inlet relative Mach number across the rotor, observed in Fig. 14(b). Second, as the VSV is closed at the same time, the spanwise distribution of A_t/A₁ moves more toward the choking limit because of the change in A₁. As a result, taking these two effects of changing the inlet relative Mach number and inlet area into account, it can be concluded that the intermediate dashed VSV setting is the optimal one for range as it lies between the two limits of choking and shock-boundary layer separation.

The characteristics shown in Fig. 15(b) agree with the analysis based on A_t/A₁ above. As predicted, when comparing the same mass flow point (cross), the fixed design VSV setting is very close to the maximum pressure-rise limit, whilst that of the dot-dashed VSV setting is closer to the choking condition. Hence, the intermediate dashed VSV setting is optimal as the key for front-stage rotors is to ensure adequate margin toward stall and choke.

The final step even after selecting the best VSV setting, for each blade speed, is to optimize the 3D blade design such that it operates effectively at multiple blade speeds. For this purpose, plotting the spanwise distribution of A_t/A₁ over the blade speeds of interest whilst the compressor is being throttled provides a physically intuitive way of understanding how the design can be improved. This design process has already been demonstrated at part and design speed using Fig. 4. In this section, the maximum climb blade speed condition is considered, where the rotor in Fig. 13 becomes supersonic from 70% span and above.

Figure 16 plots the spanwise distribution of A_t/A₁ for the baseline blade (red) and the redesigned blade (blue) at the maximum climb blade speed as the compressor is being throttled. From Fig. 16(a), it can be seen that because the blade is operating across the sonic condition, the limits of efficient operation between shock-boundary layer separation and choking are extremely narrow. In addition, as the compressor is being throttled, significant complex variations in the spanwise 3D flow along the passage can be observed.

It can be seen that at the throttling condition close to stall, the baseline blade is completely separated in the tip region. In fact, for the baseline blade even at the working line peak efficiency condition, the tip region is just about separated. In addition, close to choking the baseline blade has already choked in the near hub region. Hence, as for part and design speed, the throat area of the baseline blade needs to be reduced at the tip and increased at the hub. The redesigned rotor blade, shown in Fig. 16(b), has this desired area ratio distribution and can be seen to be aerodynamically balanced at the maximum climb speed as well.

The characteristics of efficiency and pressure ratio against mass flowrate at part-speed, design speed, and maximum climb speed are shown using Fig. 17 for the redesigned and baseline rotor design. In terms of operating range, the maximization of which is the main design objective for front-stage rotors, the redesigned blade has more mass flow capacity and stalls at a lower mass flow over all blade speeds considered; including the maximum climb blade speed just discussed. This is as predicted just by looking at the spanwise distributions of A_t/A₁ in Fig. 4 and Fig. 16.

The efficiency at the working line design point is also higher at every blade speed (Fig. 17(a)). This is primarily because of the reduction in tip shock Mach number (Sec. 5.4—Fig. 11), because close to the casing A_t/A₁ is lower for the redesign compared to the baseline design. This results in significantly less loss at the tip; particularly at the maximum climb condition, where the efficiency has improved by 0.7%.

Finally, along the working line, the mass flow and pressure rise between the two designs have successfully been kept the same at all speeds considered. This is important because it means that this redesigned front-stage rotor can be inserted into the multistage civil compressor without causing matching problems.

The second design example is an embedded rotor within a multistage military fan operating with an inlet distortion. The fan has no upstream VSVs. The inlet distortion is caused by the inlet ducting upstream of the fan creating a stagnation pressure deficit that persists through the three-stage fan. This case is important because it is characteristic of the problem faced by designers of supersonic fans or compressors with large inlet distortions. In these cases, the spanwise area distribution A_t/A₁ in the distorted region operates well outside the optimum range of A_t/A₁ and an aerodynamically balanced design is not possible. In fact, in this specific case, a different application of the design method will be demonstrated where further off-design improvements are investigated whilst keeping A_t/A₁ fixed.

Fig. 18 shows the A_t/A₁ spanwise distributions from hub to tip of this fan rotor as it is throttled up its characteristic. In the distortion region, the low total pressure causes the incidence into the blade row to increase, and this causes A_t/A₁ to be extremely high. This coupled with the fact that the inlet relative Mach number in this region is supersonic, between 1.14 and 1.23, results in a strong shock on the suction surface that is high enough to cause the boundary layer to separate. The consequence is that the total-to-total efficiency in the tip region, shaded in red, drops significantly to well below 90%, as shown later in Fig. 24.

The efficiency and pressure rise of the baseline and the redesigned military-style rotor are shown in Fig. 19 at design and part-speed. Because of a pre-set safe stall margin criterion, the dark gray solid working line in Fig. 19(b) is forced to intersect the choke side of the design speed characteristic, as shown by the circle. As can be seen from Fig. 19(a), this is far from this fan rotor’s peak efficiency but is a necessary compromise.

The redesign will be shown to increase both the operating range and peak efficiency of the fan. At the design speed, this can be seen to potentially increase the design efficiency by 1.2%. Around a 0.3% increase in efficiency comes from a reduction in the strength of the shock at a fixed mass flow. And another 0.9% increase in efficiency comes from trading in the increased operating range of the redesigned rotor. For example, for the same pre-set safe stall margin, this increased operating range could allow the fan’s working line to be raised, as shown by the dashed gray solid line. By doing this the new design point of the fan, shown by the cross can be moved to a lower mass flow, closer to this rotor’s peak efficiency.

The first choice of a designer faced with this challenge should be to increase A_t/A₁ within the hub to midspan region and drop the A_t/A₁ in the casing region. Because of the severity of the distortion, it would not be possible to lower A_t/A₁ enough to stop the suction surface boundary layer of the blade separating, but doing so would act to improve the fan rotor’s operating range, as was already shown using the previous example of a front-stage civil compressor rotor. However, with this example, a different application of the design method will be demonstrated.

In redesigning this specific rotor, in addition to the geometric limitations described in Sec. 5.1, changes were also restricted to the top 25% of the blade span. This is to ensure the mechanical rigidity of the blade. This meant that since the rotor's maximum mass flow capacity cannot be compromised and the hub to midspan sections cannot be opened to allow for more mass flow, decreasing A_t/A₁ any further in the tip distorted region is not an option. In such cases, the method described in this paper can be flipped to investigate further off-design improvements. Specifically, A_t/A₁ will actively be kept fixed whilst adjusting the geometry.

For a given A_t/A₁ distribution, different 3D rotor geometries can be explored, whose detailed geometry could become more important toward the stall. This is because there are two ways by which A_t/A₁ can be changed. First, by changing the spanwise geometry o/scosα_1rel and second by varying the spanwise 3D radial flow distribution. In this design example, these two components of A_t/A₁ will be traded off with one another such that the spanwise distribution of A_t/A₁ is kept fixed.

In Fig. 18 the spanwise hub-to-casing variation of A_t/A₁ between the baseline in blue and redesigned blade in red can be seen to have been kept fixed. As discussed, this is by design intent and is for two reasons. First, to maintain first-order design performance, as A_t/A₁ sets the pressure rise across the shock. Second, so that it does not compromise the blade’s choking capacity.

The design objective here is to adjust the spanwise o/scosα_1rel such as to minimize the total amount of blade metal angle turning up to the shock foot, which could become increasingly important toward the extreme off-design stalling condition, whilst keeping A_t/A₁ fixed to preserve design performance. This can be achieved in two ways. The first way is by going into the detailed profile design, which might bring about marginal improvements. The second way, and the one shown in this paper, is by finding the optimum combination of the blade leading edge angle, χ_LE and camber up to the throat, χ_t up the span within the top 25% of the blade span. The resulting redesigned blade is the one shown in red in Figs. 18 and 19 and is presented in more detail next.

Figure 20 plots the spanwise distribution of (a) o/scosα_1rel and (b) AVDR_t for the final redesign (red dashed) and baseline (blue solid) embedded stage military-style fan rotor at the design speed close to peak efficiency. Using Eq. (2), these contributions can be combined to give the resulting spanwise distribution of A_t/A₁ already shown by the solid line with crosses in Fig. 18 to have been kept the same for the redesigned (red) and baseline (blue) designs. It can be seen from Fig. 20 that the throat area has been reduced from 75% to 95% span by removing camber up to the throat and that this has become possible, whilst keeping A_t/A₁ fixed because the streamtubes now contract less within that region.

To understand how this desired change in AVDR_t for the final redesign was achieved, a schematic of the blade geometry changes made is presented using Fig. 21. Just removing camber up to the throat results in a decrease in geometric A_t: o/scosα_1rel. So to increase A_t back to the datum value, a change in the 3D spanwise component of A_t: AVDR_t, needs to occur. This can be achieved by finding the right combination of χ_LE. As shown in the schematic of Fig. 21, the combination of χ_LE that worked in this case was opening the tip region locally by about 3 deg, whilst closing the region of the span between 75% and 90% span by about 1 deg. The result of this is a radial streamtube expansion, as shown by the vertical arrows in the schematic, that increases A_t by the exact amount needed to counteract the effect of removing camber. As a result, keeping A_t/A₁ fixed.

In Fig. 22, the detailed spanwise variation in blade leading edge angle χ_LE and its resulting effect on the blade’s leading edge and 3D shock axial location are plotted for the baseline (blue) and redesigned (red) blades. Examining the effect of the local χ_LE changes on the blade’s leading edge axial location in Fig. 22(b), it can be observed that the redesigned blade has some effective sweep. This forward-swept leading edge has not been achieved by changing the radial stacking line, but by twisting the blade locally at its leading edge. Because of this sweep, a spanwise 3D mass flow redistribution occurs by the throat plane, as already described. Despite this mass flow redistribution, as seen from Fig. 22(b), because A_t/A₁ has been maintained, the blade’s 3D shock axial location remains unchanged between designs. Finally, as for the front-stage civil compressor rotor, the trailing edge angle χ_TE was also adjusted to match the spanwise design intent exit relative flow angles and pressure ratio.

The overall effect of these geometry changes is, as desired, to reduce the amount of turning up to the shock foot, α_1rel–χ_sh, within the distorted supersonic region of the blade from 75% span to the casing. This is demonstrated using Fig. 23, which plots the α_1rel–χ chordwise distribution from the leading edge to the trailing edge of three sections at (a) 75%, (b) 85%, and (c) 95% span, for the red dashed redesigned and blue solid baseline design. The cross shows the location of the shock for the three sections of each design at the working line condition, which has not moved significantly. At this shock location, all three redesigned red spanwise sections have a reduced level of α_1rel–χ_sh.

The 75% and 85% leading edge sections had been closed, i.e., α_1rel–χ_LE has decreased, whilst the 95% section had been opened, i.e., α_1rel–χ_LE has increased. This can be seen to be the case in Fig. 23. For the 95% section, Fig. 23(c), more camber has been removed by the shock foot than the blade opened which means that α_1rel–χ_sh is still reduced relative to the baseline design.

Reducing the amount of turning α_1rel–χ_sh by the supersonic shock foot within the top 25% of the span of this supersonic rotor, results in a less strong shock and a more efficient interaction with the suction surface boundary layer. To prove this Fig. 24 plots the spanwise profiles of total–total efficiency across the rotor at (a) the peak efficiency point and (b) a point near stall at the design speed and part-speed.

These profiles have been obtained from running the baseline and new rotor in the whole multistage fan system environment using HYDRA [7] where the inlet profiles are allowed to change as the blade goes up its characteristic; as happens for this type of fan system in reality. A clear drop in total–total efficiency is observed near the tip region, where there is an inlet distortion. As also observed from the spanwise A_t/A₁ profiles already shown in Fig. 18 this is unavoidable.

It can be concluded from Fig. 24(a), that the geometric changes made do indeed result in a rise in efficiency within the expected top 25% distortion region at the peak efficiency point for both design and part-speed. Since no geometric blade changes have been made from the hub to 75% span, in this region the baseline and redesigned blade operate almost identically. Figure 24(b) shows that toward stall these efficiency benefits become more pronounced and even extend down to midspan for both speeds considered.

This second design case example is significant for two reasons. First, it demonstrates how the design method can be used even in extreme cases where the inlet flow is distorted, and as a result A_t/A₁ is well outside the limits of efficient blade operation. Second, it shows how the design method can be flipped to control A_t/A₁ whilst redesigning only part of the blade’s span. Through this approach, it is possible to study the influence of key 3D geometric levers on the off-design performance of a given rotor.

This paper describes a simple and efficient physics-based method for designing optimal transonic multistage compressors and fan rotors. Up until now, the design of transonic blades has been a laborious process requiring experienced designers or complex optimization schemes, due to the large number of geometric parameters that can affect the shock and the complex three-dimensionality of the flow. However, the new method simplifies this process and can be used to achieve highly satisfactory results, whilst providing a physically intuitive way of understanding how the design was improved.

The key to this novel method is that it is based on the physical parameter A_throat/A_inlet, which sets the shock strength along each 3D streamtube and that this one-dimensional parameter can be easily and accurately extracted along a blade’s span from 3D CFD.

The paper also introduces the concept of an “aerodynamically balanced” blade design; one which operates at every section with an A_throat/A_inlet value which is between its choking and shock-boundary layer separation limit over the widest range of operating conditions. This is significant because it allows a designer to simply understand when a rotor is close enough to its optimal performance. Potential further off-design improvements toward stall are possible and can be investigated by changing the 3D blade geometry whilst keeping this area ratio fixed.

Finally, this paper has demonstrated, using two very different rotor examples of industry-relevant multistage machines, how physically based insights based on A_throat/A_inlet can be used to improve the flow and result in designs with increased efficiency and operating range over multiple blade speeds. The authors believe that this finding is general and that the performance of nearly all multistage transonic compressors and fans could be improved using the new method.

The authors would like to thank Rolls-Royce plc for their permission to publish this work and the EPSRC for their financial support through an IAA Research Fellowship. A special thanks are extended to Nick Cumpsty for his direct involvement and in instigating many of the ideas presented in this paper; as well as from Rolls-Royce: Robert Corin, Chris Hall, John Bolger, Mark Wilson, Paul Hield, Harry Simpson, and Phil Athayde for their invaluable advice, discussions, and feedback on the results. We would also like to thank Harry Simpson and Phil Athayde for providing us with the multistage HYDRA calculations. Finally, we would like to thank James Taylor, Aljaz Kotnik, and Graham Pullan from the Whittle Laboratory for their help with setting up the dbslice demo and the drafting of the final manuscript.

There are no conflicts of interest.

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

c =

chord

o =

2D section throat passage length

r =

radius

s =

pitch

A =

3D area

M =

Mach number

P =

pressure

U =

blade speed

V =

velocity

y⁺ =

dimensionless wall distance

ps/ss =

pressure/suction surface

AVDR =

axial velocity density ratio: ρ₁V_m1/ρ₂V_m2

CFD =

computational fluid dynamics

LE/TE =

leading/trailing edge

PR =

pressure ratio

VSV =

variable stator vanes

α =

flow angle measured from meridional

δ =

blade blockage including the boundary layer

η =

total–total efficiency across the rotor

ρ =

density

χ =

blade suction surface angle

3D =

three-dimensional

 des =

at design condition

is =

isentropic

m =

meridional

mid =

midspan

rel =

relative frame

sh =

shock foot

x =

axial component

t =

evaluated at the throat plane

1 =

evaluated at the inlet plane