Profile loss

Approximating the blade as a flat plate and following Denton's approach (Denton 1993), we can express the profile loss due to friction on the blade surfaces as

With the dissipation coefficient cd defined as

the profile loss coefficient can be derived as

The parameters ρ, V, h and m˙ in Equation (5) are kept constant. The same holds true for the term nBc, as blade number increases proportionally, when chord length is reduced. Thus the wetted area remains constant and a change in profile loss due to a change in aspect ratio solely depends on the average dissipation coefficient

This only holds true, if the non-dimensional cascade geometry remains constant. Hence, this approach does not account for a change of relative leading- and trailing edge thickness. Denton (1993) states for laminar flow, that

According to Schlichting (2006), the momentum thickness θ in laminar flow depends on axial length as

Combining Equations (7) and (8), we can derive a relation between the average dissipation coefficient and the blade chord Reynolds number as

From Equation (5) we can finally derive the dependency

for laminar flow. For turbulent flow no such analytical solution exists. According to To and Miller the loss coefficient for turbulent flow over a flat plates varies with

This means that for an increase in aspect ratio, profile loss increases accordingly. Considering blade geometry and loading, the exponent of Rec in Equation (11) can vary significantly. For the considered stator vane profile, a Reynolds number dependency

with n=0.3 has been determined (see part II). As this is a 2D problem, the value was obtained from a Reynolds number variation using the 2D flow solver MISES (Drela, 1986). The result is in good agreement with 3D CFD calculations.

Clearance loss

According to Storer and Cumpsty (1994) clearance loss in incompressible flow can be approximated by

When aspect ratio is altered, all parameters occurring in this equation such as clearance t, solidity σ, leakage angle ε, blade height h and stagger angle βSt remain unchanged. Experimental investigations in cascades and rotating compressor test rigs - e.g. Inoue et al. (1986) - also show that clearance loss depends mainly on clearance-to-blade height ratio. As changing aspect ratio only influences clearance-to-blade chord ratio, clearance loss is expected to be unaffected. This argument is supported by Peters (2019) and To and Miller (2015).

Secondary loss

For a simplified case corresponding to an incompressible linear cascade, Mellor and Wood (1971) derive the following integral boundary layer equation:

This corresponds to the von Kármán momentum integral with the additional occurrence of blade forces. The so called “blade force deficit” is defined as

It represents a measure for the difference in blade force (represented through the pressure difference between pressure and suction side Δp and the blade tangential stress τb,x) between free stream and boundary layer.

In axial compressors the axial velocity component often varies only slightly between row inlet and outlet. The change in axial velocity is thus neglected for the following analysis and Equation (14) can be reduced to

The endwall shear stress usually shows as a weak dependency on axial length or rather Reynolds number as an approximation of Schlichting (2006) for turbulent pipe flow

reveals. The shear stress is therefore approximated as constant between row inlet and outlet. Regarding the blade force deficit, the parameters Δp and τb,x are assumed to be similar for blades with different aspect ratios. The effect on τb,x due to a change in blade Reynolds number is neglected in this case. Considering these simplifications, the integrand in Equation (15) is exchanged by an average value

which is independent from aspect ratio. This gives

In Equation (19) an approximately constant shape factor H is assumed between row inlet and outlet, as supported by the numerical results in this study and flat plate theory according to Schlichting (2006) respectively. Further, the dependency between momentum and displacement thickness is approximated by the 1/7 power law according to the Blasius-solution (Blasius, 1908) for turbulent pipe flow. In the next step, Equation (16) is integrated in axial direction from row inlet i to outlet o:

Herein the average momentum thickness for the integral term on the right-hand side was approximated by a linear average, assuming an approximately linear increase of boundary layer thickness between inlet and outlet.

If we consider the definition of stagger angle βSt (see Figure 1) and solidity σ and rearrange the terms, we can derive the following equation for the growth of the momentum thickness:

This suggests a linear relation between inlet and outlet momentum- and thus boundary layer thickness. Endwall boundary layer thickness and secondary loss are usually closely coupled – especially, when the blade loading remains constant. For this reason, secondary loss is expected to vary proportionally to momentum or rather boundary layer thickness. This theory of boundary layer growth incorporates many simplifications, thus a numerical verification for the cascade geometry is presented in part II.

Loss mechanisms with minor impact

The loss increase due to a larger relative leading edge thickness can be explained by a larger portion of the chord being subjected to the higher velocities of the leading edge flow region. At the trailing edge higher mixing loss occurs due to a larger trailing edge thickness compared to the blade spacing. To investigate the effect, aspect ratio (and thus chord length) was changed, while leading and trailing edge thickness was kept constant.

The resulting change in profile loss (compared to the base configuration) as a function of aspect ratio is depicted in Figure 2. For the considered geometry, the effect of increased relative leading and trailing edge thickness on profile loss is up to 40% compared to the effect of the change in Reynolds number. As this effect strongly depends on the individual profile geometry, a general correlation cannot be derived.

Figure 2.

Effect of leading and trailing edge thickness on profile loss.

A numerical investigation of the effect of increased endwall slope on secondary loss can be found in Peters (2019). A detailed description of the results cannot be given in this paper as the change in secondary loss strongly depends on endwall design. However, a general conclusion from the investigation is that a stronger endwall slope leads to a weaker reduction of secondary loss when changing the blade aspect ratio. The effect of endwall slope on secondary loss is found to be about 20% to 40% of the effect of a thinner endwall boundary layer that will be discussed later in this paragraph. For a high accuracy computation of secondary loss the effect of endwall slope is not negligible. However, in order to quantitatively determine its impact, 3D CFD simulations of the exact geometry seem inevitable.

According to Meyer et al. (2012) the effect of an increased relative fillet radius (ratio of fillet radius to chord length) has a beneficial effect for small fillet radii (due to a suppressed corner stall) and a detrimental effect for large fillet radii (due to higher local trailing edge loss). To investigate this effect for the problem at hand, the aspect ratio was doubled from 2.0 to 4.0 for four base configurations with different fillet radii from 0 to 10% chord length. When aspect ratio was increased, the fillet radius remained constant, leading to an increased relative fillet radius rfillet/c.

The results can be observed in Figure 3. Up to a fillet radius of 5% chord length, the decrease in secondary loss due to an increase in aspect ratio is slightly higher compared to a configuration without fillet radius. For 10% chord length however, the change in secondary loss due to an increase in aspect ratio drops from 15% to 12%. As fillet radii for HPC front stages are usually smaller than 10% chord length, this shows, that the effect of fillet radius can be neglected, if one is interested in the effects of first order.

Figure 3.

Effect of fillet radius on secondary loss.

For all three above mentioned loss mechanisms a detailed description of methodology and results can be found in Peters (2019). In this paper however only the most dominant and at the same time generally relevant effects will be examined in detail: the effect of Reynolds number on profile loss and the effect of a thinner endwall boundary layer on secondary loss.

Reynolds number effect

The change in profile loss was found to be almost independent from endwall boundary layer thickness. This can be explained by the fact, that the midspan region is approximately independent from the endwall regions as long as aspect ratio does not fall below a value of ∼1.0 - see Schmidt et al. (2017) and To and Miller (2015). Thus only the results for δ∗/c=0.018 are presented in Figure 4. In addition to the numerical CFD results, the MISES midspan solution as well as the analytical theory based on Equation (12) are shown. All results are normalised with respect to the base configuration value.

Figure 4.

Profile loss for different aspect ratios.

We can observe good agreement between MISES, 3D CFD and analytical theory, if the coefficient n=0.3 is chosen. As the MISES calculations feature a transition model, while 3D CFD is modelled fully turbulent, the slope of the MISES curve is slightly smaller than that of the analytical and 3D CFD results. If transition is considered, the portion of the laminar boundary layer grows with decreasing Reynolds number and thus increasing aspect ratio. This leads to lower profile loss antagonising the above described Reynolds number effect. In combination, an increase in aspect ratio considering transition leads to an approximately 10% smaller profile loss variation compared to a case without transition. This relatively small influence of transition is supported by compressor cascade wind tunnel experiments of Walraevens and Cumpsty (1995).

Considering the airfoil as a flat plate, the increased profile loss for a shorter chord length can be explained as follows. For the low and high aspect ratio plate, the wetted area is the same (as plate length decreases but plate number increases accordingly due to constant solidity). The skin friction coefficient (scaling with Rec−n) decreases with increasing plate length. Consequently the skin friction at the trailing edge and thus its average coefficient is larger for the short plate compared to the long plate, causing larger frictional loss for the short plate and hence the high aspect ratio airfoil.

Effect of thinner endwall boundary layer

The last herein discussed loss component is the variation in secondary loss due to a thinner endwall boundary layer. As the chord length is the characteristic length for the endwall boundary layer, it is evident, that its thickness and thus secondary loss will decrease, when chord length is reduced. However, when deriving a quantitative correlation for the change in secondary loss, the question how to treat the incoming boundary layer thickness as an input variable arises. In contrast to the before mentioned parameters, two ways of handling the incoming boundary layer seem technically relevant.

The first approach would be to keep the ratio of inlet boundary layer thickness-to-chord length constant when increasing aspect ratio. This corresponds to a repeating stage concept as e.g. applied in To and Miller (2015). In a repeating stage compressor, it is assumed that after about the 4^th compressor stage, the endwall boundary layer shows no more significant growth for the downstream stages. As (for sufficiently large aspect ratios) the chord length is the most relevant length scale for this problem, it is reasonable to assume, that this “equilibrium” boundary layer thickness scales with the average chord length of the compressor. So the approach of a constant relative boundary layer thickness approximates a multistage compressor with a uniform change in aspect ratio for all stages. The works of Watzlawik (1991) and To and Miller (2015) are based on this approach and show that secondary loss varies inversely with aspect ratio.

However the uniform change of aspect ratio in a multistage compressor does not seem to be the only technical relevant case. In many cases a change of only one blade row or a non-uniform change of aspect ratio seems more beneficial due to the different aerodynamic properties of the different compressor stages. Another case to investigate could thus be a single blade row. If this blade row is the only row to change, the absolute thickness of the inlet boundary layer would necessarily remain constant (as all upstream geometries also remain unchanged). In the following investigation, this approach is called the single row approach. In contrast to the repeating stage approach the ratio of boundary layer-to-blade height instead of the ratio of boundary layer-to-blade chord is kept constant. To the author's best knowledge, a similar approach is not yet available in the literature. In the following paragraph, both approaches are investigated numerically.

For the repeating stage approach, the cascade incoming relative displacement thickness is fixed at δ∗/c=0.018. This is realized by varying the inlet duct length accordingly, when aspect ratio is altered. For the single row approach, the displacement thickness is fixed at δ∗/h=0.009. For the base configuration this corresponds to δ∗/c=0.018. A constant δ∗/h and thus a constant δ∗ can simply be realized by keeping a constant inlet duct length. The integral results for both approaches are displayed in Figure 5.

Figure 5.

Effect of selected inlet boundary layer approach on secondary loss.

The curve for the repeating stage approach perfectly fits the correlation

which is well known from the work of Watzlawik (1991) and To and Miller (2015). For the single row approach however, a significantly weaker dependency of secondary loss on aspect ratio is visible.

Figure 6 shows the radial loss distribution for both cases for the maximum aspect ratio compared to the base configuration. Both curves feature the same increase in profile loss due to an increase in Reynolds number. However the secondary loss region for the repeating stage approach is smaller compared to the single row approach, while the base configuration clearly has the largest secondary loss region.

Figure 6.

Radial loss distribution for different inlet boundary layer approaches.

As the size of the secondary loss region scales with boundary layer thickness, these patterns can be explained. For the single row approach the same inlet boundary layer thickness is applied as for the base configuration. However the shorter blade chord leads to a smaller growth in boundary layer and thus the outlet and average thickness is smaller than that of the base configuration. Compared to the single row approach, the repeating stage approach additionally features a thinner inlet boundary layer, which once again reduces the average boundary layer thickness and thus secondary loss.

In order to quantitatively describe this effect, two relations need to be determined: (a) the relation between inlet and outlet boundary layer thickness and (b) the dependency of secondary loss on boundary layer thickness. Considering the first problem, a simultaneous variation of normalized inlet displacement thickness δ∗/c and aspect ratio for the investigated compressor cascade has been conducted. The resulting outlet displacement thickness is depicted in Figure 7.

Figure 7.

Relation between inlet and outlet displacement thickness.

The relation between inlet and outlet boundary layer thickness is independent from aspect ratio, which seems realistic, as boundary layer growth should be independent from blade height for sufficiently large aspect ratios. In addition, the relation is linear, which is not immediately evident. However this linear dependency has already been derived in Equation (21). For a fixed non-dimensional geometry and the assumptions already made, all parameters in this equation are independent from aspect ratio and can thus be handled as constants except for the inlet and outlet momentum thickness θx,i and θx,o as well as the chord length c. It was already assumed, that momentum, displacement and boundary layer thickness can be approximated as proportional. Thus from Equation (21) we can derive, that

with the constants A and B. The dependency between boundary layer thickness and secondary loss is evaluated in Figure 8. For this assessment the same calculations as in Figure 7 were considered. The figure shows, that for the considered cascade simulations, secondary loss seems to be almost proportional to displacement thickness to blade height ratio at row outlet. The slight offset between both curves in Figure 8 can be explained by the effect of Reynolds number on secondary loss. This effect is neglected as its impact on secondary loss is one order of magnitude smaller than the effect of boundary layer thickness.

Figure 8.

Relation between secondary loss and boundary layer thickness.

The relation also seems to be independent from inlet boundary layer thickness yielding

Together with Equation (23) the relation

can be derived. Experimental investigations on turbine cascades (Shammaa, 1978) support this linear dependency between inlet boundary layer thickness and secondary loss.

For the special case of the repeating stage approach δi/c is constant and thus Equation (25) turns into Equation (22). Hence, this approach is independent from inlet boundary layer thickness. The same does not apply for the single row approach, where δi/h is constant instead. Only for the case of a vanishing inlet boundary layer thickness both approaches deliver the same results. For the single stage approach it is useful to rewrite Equation (25) as

For the geometry in this investigation, the constants A and B can be determined via Figure 7. Then Equation (26) can be evaluated for arbitrary values of boundary layer thickness and aspect ratio. The results are shown in Figure 9.

Figure 9.

Evaluation of secondary loss for single row approach based on Equation 26.

A strong dependency of secondary loss on incoming boundary layer thickness can be observed. For an incoming boundary layer thickness of zero, an increase in aspect ratio of 100% leads to a reduction in secondary loss of 50% (corresponding to the repeating stage approach). The same increase in aspect ratio reduces secondary loss by only 12% if δi∗/h is increased to 0.02 (a value that is easily reached in a multistage compressor).

Considering the significant difference between both approaches, the question arises which approach to use best in what situation. The repeating stage approach neglects the fact, that the first compressor stages experience an inlet boundary layer that does not scale with the chord length, while it gives a good approximation for the rear stages. In reality, the first stages will have higher secondary loss, which leads to an underestimation of the total secondary loss. The single row approach gives a good approximation for the row considered, but it neglects downstream effects, such as the reduction of the next row's inlet boundary layer leading to an overestimation of secondary loss.

So, on the one hand a conservative estimation can be made using both approaches to obtain a minimum and maximum value for the possible change in secondary loss. On the other hand, depending on the situation, the better fitting approach can be selected. For a multistage compressor with a similar change in aspect ratio for all blades and vanes, the repeating stage approach would be suitable. If a single stage compressor is considered or a change in aspect ratio of a single blade row or stage in a multistage compressor, the single row approach should be preferred.