## Abstract

Reaction is the fundamental parameter by which the asymmetry of the velocity triangle of a stage is set. Little is understood about the effect that a reaction has on either the efficiency or the operating range of a compressor. A particular difficulty in understanding the effect of the reaction is that the rotor and stator have a natural asymmetry caused by the centrifugal effects in the rotor boundary layer being much larger than that in the stator boundary layer. In this paper, a novel approach has been taken: McKenzie’s “linear repeating stage” concept is used to remove the centrifugal effects. The centrifugal effects are then reintroduced as a body force. This allows the velocity triangle effect and centrifugal force effect to be decoupled. The paper shows the surprising result that, depending on how the solidity is set, a 50% reaction stage can either result in the maximum, or the minimum, profile loss. When the centrifugal effects are removed, 50% reaction is shown to minimize endwall loss, maximize stage efficiency, and maximize operating range. When the centrifugal effects are reintroduced, the compressor with the maximum design efficiency is found to rise in the reaction by 5% (from 50% reaction to 55% reaction) and the compressor with the maximum operating range is found to rise in the reaction by 15% (from 50% reaction to 65% reaction).

## 1 Introduction

In the central stages of a multistage compressor, it is typically argued that symmetrical rotor and stator velocity triangles maximize the stage efficiency. Horlock [1] and Cumpsty [2] say this is because the static-pressure rise is split equally between the rotor and stator and so the adverse pressure gradient is balanced.

There is also an argument for symmetrical velocity triangles maximizing stage efficiency based on balancing the relative inlet velocities into the rotor and stator. Denton [3] says that this is because the relative inlet velocities into the rotor and stator are equal. At any other value of the reaction, the relative inlet velocities into the rotor and stator are not equal. As the increase in specific entropy due to the surface boundary layers on each blade is proportional to the cube of the surface velocity, the blade with the increased relative inlet velocity will have a greater increase in specific entropy than the reduction in the other blade. There is, therefore, a reduction in stage efficiency for asymmetric velocity triangles.

These views were the views of the authors until they received a personal communication from Dr. L. H. Smith (LHS) on Oct. 15, 2015:

“I have found that 50% reaction does not always give the highest efficiency. For a given flow coefficient and work coefficient, 75% reaction gives higher efficiency than 50% reaction. This happens because the lower solidities can be used with high reaction blading, holding $Deq*$ constant.”

LHS went on to reference a discussion he wrote for Lieblein’s paper [4] where he used Lieblein’s effective diffusion ratio $Deq*$ to set the solidity of a stage and this showed that 50% reaction did not produce a compressor with the highest design efficiency. There is further evidence for this by Casey [5] who used a preliminary design system to study the effect of the reaction.

In real compressors, the reaction is not an independent design variable. This is because at the inlet and exit of a multistage compressor, there is zero absolute swirl and this results in high reaction. This corresponds to designs with typically high reactions in the range 70–90% [6]. In the central stages of the compressor, it is possible to reduce the reaction to 50% by raising the interstage swirl through the first few stages and dropping it through the last few stages.

However the reaction is chosen, it is important to understand how it effects the overall efficiency and operating range of the compressor. Figure 1 compares the stage velocity triangles for a 50% (symmetric) and 70% reaction design, both with equal flow and work coefficient. It shows why reaction is the fundamental parameter which sets the asymmetry between the rotor and stator velocity triangles.

Fig. 1
Fig. 1
Close modal

There is an added natural asymmetry between the rotor and stator caused by rotational forces. This asymmetry is due to two effects of rotation, shown in Fig. 2. First, a bulk passage effect, where the centrifugal forces in the freestream are balanced by a radial pressure gradient. This causes the bulk passage flow in the rotor to move radially outward and in the stator to move radially inward. Second, a differential boundary layer effect where the differential effect of centrifugal and Coriolis forces causes the boundary layers in the rotor to be differentially accelerated toward the casing [7]. This effect is critical to this paper as it acts as a natural asymmetry between the way in which the rotor and stator boundary layers develop.

Fig. 2
Fig. 2
Close modal

There is some experimental evidence in the literature [1,8] to show that high reaction designs are advantageous due to the presence of rotational forces in the rotor. However, in these studies, it is difficult to decouple the effects of the reaction on the velocity triangle and on the rotational force in the boundary layer because they are linked. To overcome this problem, a new rotation model has been developed which uses McKenzie’s “linear repeating stage” concept to first remove the effects of rotation. The differential boundary layer effects of rotation are then reintroduced as a body force, in a controlled way.

In real compressors, the choice of the reaction also introduces asymmetry into the stage Mach number triangles. For a constant axial velocity ratio, it is thought that 50% reaction maximizes the stage efficiency based on balancing the relative inlet Mach numbers into the rotor and stator. This becomes more important as the Mach numbers increase due to the increased peak Mach numbers on the rotor and stator and increased shock losses. To decouple the effects of the reaction from the effects of the Mach number in this study, a blade speed Mach number of 0.3 was chosen so that the flow can be considered incompressible.

This paper is split into three parts. First, the effect of the reaction on profile loss is investigated. Second, the effect of the reaction on the endwall with rotation switched off is investigated. Finally, the effect of the reaction on the endwall with rotation switched on is investigated.

## 2 Methodology

The performance of a compressor stage is a function of many non-dimensional parameters. In this paper, the number of parameters is reduced for simplicity so that the effect of the reaction and rotation can be studied in isolation. The design is limited to the typical design choices available to a compressor designer trying to design the central stages of a multistage compressor. Equation (1) describes the typical choices available:
$(ψ′,η)=f(Φd,Ψd,Λd,Mu,σ,AR,t/c,ε/c,Rec)$
(1)
In this paper, a design flow coefficient Φd of 0.597 and a work coefficient Ψd of 0.436 are chosen, and these values are the same as used by To and Miller [9]. The blade speed Mach number Mu is set to 0.3. The aspect ratio AR is set as 2.0. The values of t/c and ɛ/c are fixed as 0.05 and 0.01, respectively. The stage length is constant, which fixes the rotor and stator Reynolds numbers $Rec$. At 50% reaction, the rotor and stator $Rec$ are equal to 106. The remaining design choices are described by Eq. (2):
$(ψ′,η)=f(Λd,σ)$
(2)
where Λd is the design reaction and σ is the solidity of the stage. It will be shown that the optimal choice of the reaction depends on the way in which the solidity is set. In this paper, three different methodologies of setting solidity will be used: (1) by fixing the solidity, (2) by fixing the equivalent diffusion ratio $Deq*$ equal to 1.78 [4], or (3) by fixing the shape factor of the suction-surface boundary layer at the blade trailing edge Hte. These three methodologies have common values at 50% reaction, where the level of solidity is set to achieve a diffusion factor of approximately 0.45. Appendices  A and  B explain how the solidity is set using fixed $Deq*$ and Hte.

### 2.1 Computational Fluid Dynamics Setup.

The two-dimensional (2D) loss calculations in this paper are computed using the program MISES, a coupled Euler boundary layer solver [10]. The boundary layers are considered to be fully turbulent.

The three-dimensional (3D) loss calculations are computed using the program TBlock, a multi-block structured grid computational fluid dynamics (CFD) program developed by Denton [11]. TBlock is a fully 3D, Reynolds-averaged Navier–Stokes (RANS) finite volume program. Steady mixing planes are used, and the boundary layers are considered to be fully turbulent.

The airfoil geometries are of a controlled diffusion airfoil type, designed using MISES so that the stagnation streamline always bifurcates on the nose of the airfoil. The airfoils are designed with a “linear shape factor philosophy” where the suction-surface shape factor increases from the peak suction point to the trailing edge linearly. The compressor designs are cantilevered with a plane annulus. The rotor tip and stator hub clearances are equal and set to 1% of the airfoil chord length, hence 50% reaction rotor and stator geometries are identical.

### 2.2 Linear Repeating Stage Model.

Smith [12] showed that in a multistage compressor, the spanwise stage inlet conditions repeat after three to four stages in a well-matched compressor. McKenzie [6] developed this into a “linear repeating stage” concept, and it has been implemented computationally by Auchoybur and Miller [13] and To and Miller [9]. There are three elements to its implementation in this paper.

First, the bulk passage effect of rotation is removed by choosing a compressor geometry which is at a span-to-radius ratio of 0, i.e., the rotor and stator are rectilinear cascades of blades. This allows coupled-influence between the rotor and stator and removes the variation in velocity triangles up the span.

Second, a 1.5-stage compressor model (rotor-stator-rotor) is calculated using 3D CFD. The stator exit conditions are copied to the rotor inlet, and this is repeated at the beginning of each time-step. This means that in each converged calculation, the stator exit conditions are identical to the rotor inlet conditions.

Third, a blade speed Mach number of 0.3 was chosen so that the flow can be considered incompressible.

### 2.3 Rotation Model.

A central part of this paper is the ability to switch on and off rotational forces in a controlled way. The effects of rotation are first removed by choosing a compressor geometry which is at a span-to-radius ratio of 0, i.e., the rotor and stator are rectilinear cascades of blades. The rotational forces are then reintroduced as body forces.

As described above, the rotational forces have two effects. First, the bulk passage effects cause the flow in the rotor to move radially outward and the flow in the stator to move radially inward. Second, a differential boundary layer effect caused by the centrifugal and Coriolis forces in the boundary layer. This causes the boundary layers in the rotor to be differentially accelerated toward the casing. This effect is critical to this study as it acts as an asymmetry between the way in which the rotor and stator boundary layers develop.

It was decided that the rotation model should only model the differential boundary layer effect. This is because the bulk passage effects cause small incidence variations across the span of the blade. In a real design, the blade profile would be varied along its span to compensate for this effect. In this controlled study, detailed redesign of the blade across the span must be avoided and so it was decided that the bulk passage effects of rotation would not be modeled.

To model only the differential boundary layer effect, the rotation model adds a body force per unit volume into the CFD calculation of the form
$ρVθ2r−ρVθ2r¯|(x,r)$
(3)
This new term models the perturbation centrifugal forces. The second term in Eq. (3) is defined as the pitchwise volume-averaged value of $ρVθ2/r$ at the same meridional position, i.e., the same axial and radial coordinates and r is an effective radius. The new model was introduced into TBlock as a source term
$ρDVDt−ρ(Vθ2r−Vθ2r¯|(x,r))=−∂p∂r+viscous+Fr$
(4)

The model allows an effective radius to be set. By setting a relatively high value, the blades act as if part of a rectilinear cascade. By setting the effective radius equal to the radius of a real compressor, the perturbation centrifugal forces are equal to those in a real compressor. The benefit of using the model, rather than changing the real radius of the compressor, is that the geometry of the stage remains unchanged as the magnitude of the body force is varied.

## 3 Profile Loss

The lost efficiency of a stage due to profile loss alone can be written as
$TΔsΔh0=(TΔs)rotor+(TΔs)statorΔh0$
(5)
The total entropy generation in the attached boundary layer on each blade, either rotor or stator, can be calculated by integrating the entropy production in the boundary layer over the blade surface
$S˙=∑c[∫01CdρV03Tdxc]$
(6)
where the summation is across both blade surfaces and V0 is the velocity at the boundary layer edge. Writing Eq. (6) in the form of the lost efficiency of a row of blades gives
(7)
where we follow Denton [3] and write the enthalpy change in terms of an isentropic stage reference velocity C0
$C0=2Δh0$
(8)
and Vin,ref is the relative inlet velocity into the blade row.

The lost efficiency given by Eq. (7) is made up of three terms. The first term, the solidity, represents the effect of changing the number of blades in a row. The second term represents the effect of changing the velocity triangle on loss. This term is high for a blade row with a high relative inlet velocity. This term shows that one of the key aims for a designer is to minimize the ratio of the cube of the relative inlet velocity into the blade rows relative to the enthalpy rise of the stage. The third term is the single blade loss coefficient. This term is high if a blade has a high surface velocity, relative to the blade inlet velocity, or a large wetted area.

In Secs. 3.13.3, the three terms in Eq. (7) will be used as a framework through which we can understand the effect of the reaction on the lost efficiency of a stage. It will be shown that changing reaction changes all three terms. Only by controlling how these terms change with the reaction, the effect of the reaction on the lost efficiency of a stage can be understood.

### 3.1 Constant Solidity.

It is commonly believed that compressors of 50% reaction have the highest stage efficiency. This way of thinking is based on the idea that the blade solidity is held constant. The lost efficiency of each blade row for the case of constant solidity is plotted in Fig. 3. The shape of each line is mainly determined by changes in term 2 in Eq. (7), the velocity triangle term. Moving from 50% to 70% reaction, term 2 changes by approximately +60% and term 3 changes by −7%. The shape of the lines is caused by the way in which the velocity triangle controls the relative inlet velocity into each blade row and the fact that loss scales with the cube of the relative inlet velocity into the blade row. The black line in Fig. 4 shows the lost efficiency of this stage. It is clear that 50% reaction must be the most efficient stage because it minimizes the sum of the $Vin,ref3$ into both blade rows. The variation in lost efficiency is symmetrical about 50% reaction. Increasing reaction from 50% to 70% reduces the stage efficiency by 0.39%.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

### 3.2 Effect of Solidity.

The view developed in Sec. 3.1 was the view of the authors until they received the personal communication from LHS quoted previously. In this section, the solidity is controlled in two ways. First, as LHS proposed, the equivalent diffusion ratio $Deq*$ of each blade row is held constant. Second, the shape factor of the suction-surface boundary layer at the trailing edge Hte of each blade row is held constant.

The effect of holding the equivalent diffusion ratio $Deq*$ constant and equal to 1.78 is shown as the red line in Fig. 4. The change in the solidity is shown in Fig. 5. The line shows that, as LHS said, 50% reaction is now the most inefficient compressor. In fact, at 50% reaction, the compressor has an efficiency which is 0.22% lower than an equivalent compressor at 70% reaction. Figure 5 shows that this is caused by dropping the solidity in both the rotor and stator by approximately 60%. It seems surprising that the solidity in both blade rows drops simultaneously. This effect will be explained later in the section.

Fig. 5
Fig. 5
Close modal

The effect of holding the shape factor of the suction-surface boundary layer at the trailing edge Hte constant is shown as the blue line in Fig. 4. The line shows that the change in lost efficiency is almost independent of the reaction. Increasing reaction from 50% to 70% reduces the stage efficiency by only 0.13%. Figure 5 shows that increasing reaction from 50% to 70% reduces the solidity in both blade rows by approximately 30%.

It is clear that as LHS said, solidity plays an important role in determining the impact of the reaction on compressor efficiency. However, to understand this effect, a choice must be made about how the solidity is varied as the design of the blade is changed.

To understand why there is a reduction in rotor and stator solidity either side of 50% reaction, first consider the diffusion factor (DF) equation [14] in its simplest form
$DF=1−DH+ΔVθ2Vinσ$
(9)
which rearranges to Eq. (10):
$σ=ΔVθ/2VinDF−(1−DH)$
(10)
We can then define the top and bottom of Eq. (10) as two terms given by Eqs. (11) and (12):
$term1=ΔVθ/2Vin$
(11)
and
$term2=DF−(1−DH)$
(12)

Term 1 represents a loading term relative to the relative inlet velocity into the blade row. Term 2, for a fixed diffusion factor DF equal to 0.45, is proportional to the blade row de Haller number DH. Figures 6 and 7 show the variation of term 1 and term 2, defined by Eqs. (11) and (12), relative to their values at 50% reaction for the rotor and stator, respectively. The maximum solidity occurs when the two lines meet at a tangent. This occurs at approximately 50% reaction.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

Term 1, in Fig. 6, varies almost linearly with the reaction. The reason for this is that the work coefficient and blade speed are constant so that $ΔVθ$ is constant. However, as the reaction rises, the relative inlet velocity into the rotor Vin rises. This steady rise in Vin causes the approximately linear rise in term 1.

Term 2, in Fig. 6, varies almost parabolically with the reaction. This variation is driven by the variation in de Haller number DH. One might expect that as the reaction is increased, the rotor DH would continually drop, but it does not. Above a reaction of approximately 60%, it starts to rise again. This is because as the reaction rises, the rotor static-pressure rise continually increases; however, Vin also increases. The two effects combine to set DH. As the reaction increases beyond approximately 60%, the increase in static-pressure rise across the rotor is weak relative to the increase in Vin. This results in the de Haller number rising. The inflection point in term 2 is at a reaction of approximately 60%; however, the gradient of term 1 results in the two lines meeting at a tangent at approximately 50% reaction. Looking once again at Fig. 5, we can see that for the case of constant $Deq*$ and constant Hte, the maximum solidity occurs close to, but not quite at, 50% reaction.

Finally, Fig. 8 shows how the shape factor of the suction-surface boundary layer at the trailing edge Hte changes with the reaction, for the case of constant solidity and constant equivalent diffusion ratio $Deq*$. It is clear that setting $Deq*$ constant is undesirable because as the reaction moves away from 50%, the boundary layers are driven toward separation. Equally, it is clear that holding solidity constant is undesirable because as the reaction moves away from 50%, the stage becomes overbladed.

Fig. 8
Fig. 8
Close modal

### 3.3 Effect of Work and Flow Coefficient.

It is clear from Sec. 3.2 that whether 50% reaction is the most, or least, efficient compressor, depending on a trade between the solidity effect (term 1 in Eq. (7)) and the velocity triangle and blade loss coefficient effects (term 2 and term 3 in Eq. (7)). This trade depends on the particular work and flow coefficient at which the compressor is designed.

The Smith charts in Figs. 9 and 10 show the effect of the work and flow coefficient on whether 50% reaction is the most, or least, efficient compressor. Figure 9 shows the case of constant solidity, and Fig. 10 shows the case of constant $Deq*$. The contours show the difference in the efficiency between a compressor of 70% reaction and a compressor of 50% reaction. Blue means that 50% reaction is most efficient, and red means that 50% reaction is the least efficient. The variation of lost efficiency with the reaction at the three points A, B, and C in both Figs. 9 and 10 is shown in Fig. 11. Appendix  C explains how the lost efficiency is calculated for these cases. The black dotted lines are lines of constant de Haller number at 50% reaction.

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

From Figs. 9 and 10, a number of points can be made. First, as the work and flow coefficient are raised, point C, in both Smith charts, is a blue region. This implies that in this region, the 50% reaction compressor is more efficient than the 70% reaction compressor by approximately 0.31% for in case of constant solidities and 1.08% in the case of constant $Deq*$.

Second, the region close to the work and flow coefficient explored earlier in this paper, point B, the effect of the reaction is very sensitive to how the designer selects solidity. By changing the way the solidity is set, 50% reaction can switch between the most, and the least, efficient compressors.

Finally, as the work and flow coefficient are dropped, point A, both Smith charts once again show a blue region. This shows that the 50% reaction compressor is more efficient than the 70% reaction compressor. In fact, for the case of constant solidity, a 50% reaction compressor is shown to be approximately 3.11% more efficient than a 70% reaction compressor.

## 4 Endwall With Rotation Model Switched Off

In this section, the effects of the reaction on the endwall flow, with the effects of rotation switched off, are presented. The solidity of both the rotor and the stator has been set by fixing the boundary layer shape factor at the trailing edge of the suction-surface Hte. The endwall loss is defined as the total loss minus the profile loss.

### 4.1 Design Loss.

The effect of the reaction on the hub endwall loss is shown in Fig. 12. The black line shows the case with zero clearance, and the red line shows the case with a 1% stator hub clearance. For clarity, the casing endwall loss has not been plotted. It is identical to the hub endwall loss except that the x-axis is one minus reaction, 1 − Λ. The hub endwall loss can be seen to rise as the reaction rises.

Fig. 12
Fig. 12
Close modal
The cause of the rise can be understood by considering the loss which would occur in a turbulent boundary layer over the hub endwall. The boundary layer edge velocity is considered to vary axially, and to be equal to the circumferentially mass-averaged blade midspan relative velocity W(x). The entropy generation rate in such a boundary layer, per unit pitch, is given by Eq. (13):
$S˙=∫CdρW3(x)Tdx$
(13)
where the value of Cd is set as 0.002 [3]. Writing Eq. (13) in the form of the hub lost efficiency gives the blue line in Fig. 12. The exact form of the equation plotted is derived in Appendix  D. A comparison of the blue and black line shows that the rise in loss, as the reaction is raised, is simply caused by the rise in the freestream velocity relative to the hub endwall.

The cause of the rise in the freestream velocity relative to the hub endwall can be understood from the velocity triangles in Fig. 1. As the reaction is raised, the relative velocity into both the rotor and the stator, W1 and W2, rise. This can be understood more intuitively from Fig. 13. The figure shows a schematic of the time-averaged mid-height streamline in the relative frame. As the reaction is raised, the rotor stagger rises. Because the rotor is the blade which is connected to the hub endwall, it sets the mean flow angle relative to that endwall. Raising the rotor stagger, for a constant axial velocity, therefore raises the endwall relative velocity Win into both the rotor and stator.

Fig. 13
Fig. 13
Close modal

To a first order, a designer can, therefore, estimate whether the loss on an endwall is either high or low, simply by looking at the stagger of the blade row which is connected to that endwall.

A secondary effect of the reaction on endwall loss is shown in Fig. 12. Comparing the black line, the case with no stator hub clearance, and the red line, the case with a 1% stator hub clearance, it can be seen that the effect of the hub gap on endwall loss drops as the reaction is raised. On the hub endwall, the clearance gap is on the stator hub and so this shows that as the stagger of the stator is reduced, the hub leakage loss drops. An identical behavior was observed on the casing. When the stagger of a blade was reduced, the leakage loss was found to drop.

Figure 14 shows the effect of the reaction on the total lost efficiency of the stage. For both the cases without and with rotor and stator clearances, the 50% reaction compressor is the most efficient. Table 1 summarizes the results in Fig. 14 by comparing the difference in lost efficiency between the 70% and 50% reaction stage. As expected from the findings earlier in this paper, fixing solidity by setting a constant Hte results in the profile loss of the stage becoming relatively independent of the reaction. Table 1 shows that changing reaction from 70% to 50% reduces the endwall lost efficiency, causing an increase in stage efficiency of 0.49%, for the case with clearances, and 0.58%, for the case without clearances.

Fig. 14
Fig. 14
Close modal
Table 1

Summary of the effect of the reaction on design efficiency (3D CFD)

$ηΛ=70%−ηΛ=50%(%)$0% clearances1% clearances
Endwall−0.47−0.38
Profile−0.11−0.11
Total−0.58−0.49
$ηΛ=70%−ηΛ=50%(%)$0% clearances1% clearances
Endwall−0.47−0.38
Profile−0.11−0.11
Total−0.58−0.49

### 4.2 Operating Range.

The effects of the reaction on the operating range of the compressor, for the cases without clearances and with clearances, are shown in Figs. 15 and 16. On each plot, the dotted line shows the maximum pressure rise throttle characteristic at the point at where the CFD solution started to diverge.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
To compare the operating range between compressors, the maximum pressure rise throttle coefficient, k, is used. This is a measure of the exit area at maximum pressure rise and is defined in Eq. (14):
$k=ψ′Φ2$
(14)

Figure 17 gives a summary of how the maximum pressure rise throttle coefficient k varies with the reaction. For both cases, the 50% reaction compressor has the largest operating range. It can be seen that the addition of clearances reduces the operating range at all reactions, by a similar amount.

Fig. 17
Fig. 17
Close modal

The cause of the reduction in the maximum pressure rise throttle coefficient is shown in Fig. 18. The figure shows the limiting surface streamlines at close to maximum pressure rise for the stator of the 50% and 70% reaction compressor stages with zero clearance. This shows the surprising result that the reduction in maximum pressure rise, as the reaction is raised, is the result of an increase in the size of the stator hub separation. This is unexpected because as the reaction rises, the pressure rise across the rotor increases and the pressure rise across the stator decreases.

Fig. 18
Fig. 18
Close modal

Exactly the same, but opposite, occurs as the reaction is dropped from 50% to 30%. In this case, the reduction in maximum pressure rise occurs because the size of the separation in the rotor casing increases. This case is not shown for brevity.

The cause of the increase in the size of the stator hub corner separation at high reaction can be understood by looking at the spanwise distribution of the local static-pressure rise coefficient across the stator, shown in Fig. 19. Here, we follow Auchoybur and Miller [13] and define the local static-pressure rise coefficient Cp as
$Cp=Δp12ρVlocal2$
(15)
where Vlocal is the relative inlet velocity to the stator. The bottom of Eq. (15) is the inlet dynamic pressure into the stator at each span fraction. Figure 19 shows that as the reaction is raised, the local static-pressure rise coefficient in the stator hub rises. This rise in the local static-pressure rise coefficient causes the rise in the size of the stator hub separation.
Fig. 19
Fig. 19
Close modal

This raises the question of why the local static-pressure rise coefficient in the stator hub rises as the reaction rises. The rise is caused by a drop in stator inlet velocity, Vlocal in Eq. (15), in the hub endwall region. The cause of this drop can be understood by looking at the stator inlet velocity triangle. This shows the re-energising effect, caused by the change in frame of reference, described by Koch [15] and Auchoybur and Miller [13].

Figure 20 shows the freestream and hub endwall stator inlet velocity triangles for 30% and 70% reaction. The freestream and hub endwall velocities have been extracted from the CFD by mass-averaging the velocities over 25–75% of the mass flux and 0–25% of the mass flux, respectively. The figure shows that as the reaction rises from 30% to 70% the relative difference between $Vfs2$ and $Vhub2$ rises. This is caused by two effects. First, as the reaction is raised, the axial velocity in the hub region drops. Second, as the reaction rises, the magnitude of freestream stator inlet velocity Vfs drops and therefore any drop in Vhub causes a larger fractional change in the dynamic pressure entering the stator in the hub endwall region.

Fig. 20
Fig. 20
Close modal
Finally, it is necessary to explain why the axial velocity in the hub endwall region drops as the reaction is raised. Consider once again the hub streamtube (0–25% of the mass flux) used to create Fig. 20. Now the mass-averaged change in stagnation enthalpy, stagnation pressure, and the entropy across the stage is extracted from the CFD. As the flow is incompressible, these three are related by the fundamental thermodynamic relation
$Δh0U2=Δp0ρU2+T0ΔsU2$
(16)

In the endwall region, there are two restrictions on Eq. (16). First, the second term, the stagnation pressure rise coefficient, must be constant as the reaction is changed, shown in Fig. 21(a). This is because in a repeating stage, the stagnation pressure rise coefficient is constant across the span and all stages have been designed to achieve the same stagnation pressure rise coefficient. Second, the first term in Eq. (16), the work coefficient, must always collapse onto the same characteristic, shown in Fig. 21(b). This is because in the endwall region, the deviation is found to be small, approximately 1 deg, and is found to be relatively independent of the reaction. This means that the work coefficient of the endwall region must collapse onto the same characteristic set by the blade metal angles. It should be noted that as the reaction is changed, the gradient of the work coefficient verses flow coefficient characteristic does not change.

Fig. 21
Fig. 21
Close modal

These restrictions on the work and stagnation pressure rise coefficient, caused by Eq. (16), are shown graphically in Fig. 21. They result in the magnitude of the endwall loss, the third term in Eq. (16), fixing the endwall flow coefficient. This effect was also observed by Auchoybur and Miller [13]. As discussed in Sec. 4.1, as the reaction rises, the hub endwall loss rises and this causes the mass flow in the endwall to drop.

To summarize, the reduction in the maximum pressure rise of a stage, as the reaction rises, is caused by an increase in the size of the stator hub separation. This is caused by a rise in the local static-pressure rise coefficient of the stator hub. In turn, this is caused by a drop in the axial velocity in the stator endwall region as the stator endwall loss rises.

This behavior is fundamental to all cantilever compressors and shows that as the reaction is raised, the stator hub will limit the maximum pressure rise of the stage, and as the reaction is dropped, the rotor casing will limit the maximum pressure rise of the stage.

## 5 Endwall With Rotation Model Switched On

This section investigates the effect of centrifugal forces on the efficiency and operating range of a compressor. The centrifugal forces equivalent to a real compressor of hub-to-tip ratio equal to 0.8 have been introduced.

### 5.1 Design Loss.

The effect of rotation on the total lost efficiency of the stage is shown in Fig. 22. It can be seen that rotation has a relatively small effect on design loss. However, the reaction which achieves the optimal design efficiency increases by around 5% reaction (from 50% reaction to 55% reaction). It is also important to note that the range of reactions over which the efficiency only varies by 0.01% is relatively wide, between 50% reaction and 60% reaction.

Fig. 22
Fig. 22
Close modal

### 5.2 Operating Range.

The effects of rotation on the operating range of a compressor are much larger than the effect on design loss. Figures 23 and 24 show the effect of switching on rotation on compressors with and without clearance. The effect of rotation on the maximum pressure rise throttle coefficient is shown in Figs. 25 and 26. The effect of rotation is to increase the reaction which achieves the maximum pressure rise, by around 15% (from 50% reaction to 65% reaction).

Fig. 23
Fig. 23
Close modal
Fig. 24
Fig. 24
Close modal
Fig. 25
Fig. 25
Close modal
Fig. 26
Fig. 26
Close modal

The effect of switching on rotation on the surface limiting streamlines is shown in Figs. 27 and 28. The figures show the compressor at a flow coefficient which is just before maximum pressure rise (Φ = 0.449). The figures show that the effects of rotation are much larger in the rotor than in the stator.

Fig. 27
Fig. 27
Close modal
Fig. 28
Fig. 28
Close modal

To understand why the effects of rotation are much larger in the rotor than in the stator, it is necessary to understand how, and where, rotation introduces perturbation centrifugal forces into the rotor and stator boundary layer. To do this, a dimensionless parameter is defined, the dimensionless perturbation centrifugal force Fc, which is a measure of relative magnitude of the perturbation centrifugal forces in the boundary layer.

The dimensionless perturbation centrifugal force Fc is defined as the difference between the centrifugal force per unit volume on the blade surface and in the freestream, $ρΔVθ2/r$, non-dimensionalised by the blade speed squared U2, density ρ, and the blade span Δr. This gives the dimensionless perturbation centrifugal force Fc as
$Fc=ρ(Vθ2r)surface−ρ(Vθ2r)fsρU2/Δr=Δrr(ΔVθ2U2)$
(17)

The first term on the right-hand side of Eq. (17) controls the overall magnitude of the perturbation centrifugal forces in the stage. This shows that if a stage has a low span-to-radius ratio, Δr/r → 0, then the perturbation centrifugal forces in the boundary layer approach zero. This term can also be rewritten as the hub-to-tip ratio of the compressor.

The second term on the right-hand side of Eq. (17), $ΔVθ2/U2$, varies across the blade surfaces and is a measure of the relative local magnitude of the perturbation centrifugal forces in the boundary layer.

For the rotor, this second term can be written as
$ΔVθ2U2=U2−(Vθ2)fsU2$
(18)
because the fluid on the rotor blade surface moves at the blade velocity. For the stator, it can be written as
$ΔVθ2U2=0−(Vθ2)fsU2$
(19)
because the fluid on the stator blade surface is stationary.

Each term in Eqs. (18) and (19) is plotted in the upper half of Fig. 29. The red dashed lines show the square of the blade speed of the rotor U2 and the stator, 0. The lower half of Fig. 29 shows the overall magnitude of the terms in Eqs. (18) and (19). The lower half of Fig. 29 shows that, as expected, the perturbation centrifugal forces in the rotor and stator have opposite signs. They are radially outward in the rotor and radially inward in the stator. However, the important point to note is that perturbation centrifugal forces in the rotor are between two and four times larger than in the stator. The cause of this difference is the difference in the tangential velocity squared, shown in Eqs. (18) and (19). Finally, Fig. 30 shows how raising reaction changes the magnitude of the differential boundary layer effects caused by rotation.

Fig. 29
Fig. 29
Close modal
Fig. 30
Fig. 30
Close modal

We can now explain the cause of the large increase in maximum pressure rise coefficient at high reaction. Figure 31 shows the effect of rotation on the stator limiting streamlines. Rotational effects increase the maximum pressure rise by reducing the size of the stator separation. The cause of this reduction is shown in Fig. 31. The rotational effects in the rotor cause a radial migration of high loss fluid in the hub toward the midspan. This in turn increases the endwall dynamic pressure entering the stator hub. This causes the local static-pressure rise coefficient in the stator hub, shown in Fig. 32, to rise.

Fig. 31
Fig. 31
Close modal
Fig. 32
Fig. 32
Close modal

### 5.3 Effect of Varying Level of Rotation.

The effect of varying the level of rotation on the maximum pressure rise of the compressor is shown in Fig. 33. The figure shows the effect of changing the effective radius of the compressor from 0.03% of a real compressor (a rectilinear cascade) to 250% of a real compressor. The figure shows that the increase in the maximum pressure rise occurs when the effective radius changes between approximately 25% and 125% of a real compressor.

Fig. 33
Fig. 33
Close modal

## 6 Application to Multistage Compressors

We are now in a position to understand how the choice of the reaction affects the overall lost efficiency of a multistage core compressor. This is an important industry question because the requirement for axial flow at the inlet and exit of a multistage compressor naturally results in high reaction. The designer, therefore, must decide whether to tolerate this high reaction through the compressor or to aim for a more optimal reaction in the central stages of the compressor.

To answer this question, the lost efficiency of a multistage machine can be written as
$(TΔsΔh0)compressor=∑1n(TΔs)stageΔh0$
(20)
where the summation of loss is across all n stages and Δh0 is the isentropic work input to the machine. We will consider a hypothetical n = 10 stage compressor, with axial flow at the inlet and exit. For conventional levels of work (Ψd = 0.436) and flow coefficient (Φd = 0.597), axial flow at the inlet and exit of the compressor corresponds to a reaction of approximately 75%. The lost efficiencies found in this paper can be used in Eq. (20) to estimate the overall lost efficiency of a multistage compressor.

To understand how the choice of the reaction affects the overall lost efficiency, we will consider three cases. The stagewise distribution of the reaction in these three cases is shown in Fig. 34.

Fig. 34
Fig. 34
Close modal

Case A represents a historic design philosophy of having 50% reaction in all the stages. To achieve this, an inlet guide vane (IGV) and outlet guide vane (OGV) are required. This adds additional loss. We will consider the loss coefficient of the IGV and OGV to be 0.04.

Case B represents a second historic design philosophy where the reaction is maintained at 75% through all stages. This benefits from having no IGV or OGV; however, it suffers from having a reaction which has a higher design loss. This case was found to have a design efficiency which is 0.07% higher than case A. It should be noted that this design philosophy would have a better operability than case A, due to the increased maximum pressure rise of its stages. This explains why many historic compressors, with high reactions, had a relatively good design efficiency and operating range.

Case C represents the compressor with the maximum design efficiency. This has central stages which have a reaction of 55%, which is shown in Sec. 5.1. This was found to result in a design efficiency which was 0.65% higher than case A.

It is, therefore, clear that, if maximizing efficiency is the aim of the designer, then the reaction of the central stages should be 55%. However, it is important to note that if the central stages have a reaction between 50% and 60%, the change in efficiency is relatively small. This provides the designer with a useful degree-of-freedom.

## 7 Conclusions

There is considerable debate over the effect of the reaction on compressor design efficiency and operating range. This study shows that the confusion is due in part to the inability to decouple the effects of the centrifugal force and the effects of changing the velocity triangle, in a controllable way.

A unique approach has been taken in which the centrifugal forces have been removed by using McKenzie’s concept of a “linear repeating stage” [6]. The perturbation centrifugal force has then been reintroduced using a body force. This has allowed the two asymmetries, centrifugal force and velocity triangle, to be decoupled and has allowed their effect on compressor performance to be studied independently for the first time.

The effect of the reaction on profile loss has been shown to be highly dependent on the methodology by which the solidity is set. When the solidity is set by the shape factor of the suction-surface boundary layer at the blade trailing edge, and conventional levels of work and flow coefficient are used (Ψ = 0.44 and Φ = 0.60), the profile loss has been shown to be independent of the reaction.

Reaction is shown to have a major effect on endwall loss. This is because it controls the freestream velocity of the flow relative to the endwall, at the edge of the endwall boundary layer. When the centrifugal effects are removed, this results in 50% reaction compressors having the lowest endwall loss and thus the highest design efficiency.

A surprising conclusion of the study is that the maximum pressure rise capability of high reaction compressors is limited not by the rotor, but by the stator. This is counterintuitive because at high reaction, the pressure rise in the rotor is greater than in the stator. The cause of this is due to the way reaction changes endwall loss, and the way reaction changes the re-energizing effect provided by the change in the reference frame, described by Koch [15] and Auchoybur and Miller [13].

When the centrifugal forces are reintroduced, the compressor with the maximum design efficiency is found to rise in the reaction by 5% (from 50% reaction to 55% reaction) and the compressor with the maximum operating range is found to rise in the reaction by 15% (from 50% reaction to 65% reaction). If a designer aims to maximize the design efficiency of a compressor, the reaction of its central stages should, therefore, be 55%. However, it is important to note that the maximum efficiency is a weak function of the reaction between reactions of 50% and 60%.

It is worth considering the impact of the study on design. Currently, many preliminary design systems do not differentiate between rotors and stators in terms of loss and operating range. This means they are unlikely to correctly predict the true optimal reaction. Second, the asymmetry between centrifugal forces in the rotor and stator boundary layers implies that three-dimensional blade design philosophies of rotating and stationary blades should differ.

The study shows that LHS’s statement that reactions higher than 50% were optimal was correct. However, it has been shown that his reasoning was incorrect and that the primary cause is the asymmetry in the magnitude of the perturbation centrifugal forces in the rotor and stator boundary layers.

It is interesting to note that historically high reaction compressors operated with relatively high design efficiency and operating range. The study explains why this is the case. It is due to the higher centrifugal forces in the rotor acting as a form of boundary layer control.

It is also interesting to note that high reaction stages have lower optimal blade solidities. This significantly reduces the number of blades in the compressor. This implies that in the future, high reaction compressors could be optimal in applications where cost and weight are the primary drivers.

## Acknowledgment

The authors would like to thank John Denton, Chris Hall, Nick Cumpsty, Simon Gallimore, Tom Hynes, Tony Dickens, James Taylor, Ho-On To, John Adamczyk, and Dr. L. H. Smith (LHS) for their support throughout this project. The authors would also like to thank Rolls-Royce plc. and the Engineering and Physical Sciences Research Council (EPSRC) for funding this work.

## Conflict of Interest

There are no conflict of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

## Nomenclature

### Symbols

• c =

chord

•
• k =

throttle coefficient

•
• p =

pressure

•
• r =

•
• t =

•
• T =

temperature

•
• U =

•
• V =

absolute velocity

•
• W =

relative velocity

•
• $S˙$ =

entropy creation

•
• C0 =

isentropic stage reference velocity

•
• Cd =

dissipation coefficient

•
• Cp =

local static-pressure rise coefficient

•
• $Deq*$ =

equivalent diffusion ratio

•
• Fc =

perturbation centrifugal force

•
• Fr =

•
• Hte =

trailing edge boundary layer shape factor

•
• Mu =

Mach number based on blade speed

•
• AR =

aspect ratio

•
• DF =

diffusion factor

•
• DH =

de Haller number

•
• Rec =

Reynolds number based on chord

•
• Δh0 =

change in specific stagnation enthalpy

•
• Δr =

•
• Δs =

change in specific entropy

### Subscripts

• 0 =

stagnation

•
• 1 =

rotor inlet

•
• 2 =

stator inlet

•
• bl =

boundary layer

•
• d =

design

•
• fs =

freestream

•
• in =

inlet

•
• x =

axial

•
• θ =

circumferential

### Greek Symbols

• ɛ =

clearance

•
• η =

isentropic efficiency

•
• Λ =

reaction

•
• σ =

solidity

•
• Φ =

flow coefficient

•
• Ψ =

work coefficient

•
• ψ′ =

static-pressure rise coefficient

### Appendix A

Holding constant the equivalent diffusion ratio $Deq*$ [4] in Eq. (A1) allows the blade row solidity σ to be set. This requires knowledge of the stage velocity triangles only
$Deq*=1DH[1.12+0.61cos(βin)ΔVθVin,refσ]$
(A1)
where βin is the relative tangential inlet flow angle and Vin,ref is the relative velocity into the blade row.

### Appendix B

Alternatively, the blade row solidity can be set by holding constant the suction-surface boundary layer shape factor at the trailing edge Hte. A program was written by To [16] to find the optimum airfoil profile that yields the lowest profile loss.

### Appendix C

To calculate the lost efficiencies in Sec. 3.3 only, a low order loss model is used. Equation (8) in Ref. [4] correlates the equivalent diffusion ratio $Deq*$ against the wake momentum thickness (θ/c)2. Using this value of (θ/c)2 and knowledge of the stage velocity triangles and solidities, Equation (11) in Ref. [4] can be used to estimate the blade row total-pressure loss coefficients $ω~$. The stage lost efficiency can then be calculated from
$TΔsΔh0=2(Win,rotor2ω~rotor+Vin,stator2ω~stator)Δh0$
(C1)

### Appendix D

The total entropy generation, per unit pitch, in either the hub or casing endwall, can be calculated by
$S˙=L∫01CdρV03Td(xL)$
(D1)
where L is the stage length and V0 is the boundary layer edge velocity. Writing Eq. (D1) in the form of lost efficiency gives
$(TΔsΔh0)endwall=2C0VxLΔr∫01Cd(V0,refC0)3dxL$
(D2)
where L is the stage length and V0,ref is the boundary layer edge velocity relative to the endwall.

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