#### 4.1. Impact of Volute Ascpect Ratio on MFP

Initial analysis of volute aspect ratio was completed at constant volute A/r. The volute A/r determines the rotor inlet flow angle and the swallowing capacity of the stage. Therefore, maintaining A/r is expected to maintain Mass Flow Parameter (MFP) and therefore ensure aerodynamic similarity.

Presented in

Figure 9 are the hysteresis loops for all three volute aspect ratios at 20, 40 and 60 Hz pulse frequencies. It is clear that maintaining volute

A/r did not ensure a constant MFP and that the volute aspect ratio has a significant impact on stage performance. At all three tested frequencies the MFP of the volute reduces with increasing aspect ratio. The maximum variation in cycle averaged MFP between the three designs at constant

A/r was 4.3% under the 60 Hz pulse frequency. This impact of volute aspect ratio is important as volute cross sectional shape is often compromised due to packaging requirements of the turbocharger. If the volute cross section shape is altered for such a design, ensuring constant

A/r alone will therefore not ensure a constant MFP. A similar effect was observed by Yang et al. [

9]. The variation observed in that study was approximately 2%. Despite the discrepancy in MFP of the two volute designs the authors still compared them in terms of performance.

Figure 10 presents the variation in LE incidence throughout the pulse for each of the volute aspect ratios at 60 Hz pulse frequency. It can be seen that increasing the volute aspect ratio increases the mean incidence throughout the pulse. The variation between the three designs is approximately 6°. This observation, along with that of varying MFP with aspect ratio, indicates that reducing volute aspect ratio gives the same effect as increasing

A/r and vice versa. More specifically, it is believed that increasing the centroid radius reduces the radial flow component of velocity due to less flow guidance from the volute outer wall.

#### 4.2. Aspect Ratio Effect with Constant MFP

To ensure aerodynamic similarity between the designs, thereby giving a fair comparison of design performance, stage MFP must be matched. To achieve this the volute

A/r’s were modified.

Figure 11 shows

A/r as a function of the azimuth angle for each volute design. The

A/r ratio of the

AR = 1 and

AR = 2 volutes was increased slightly to enable constant MFP.

The resulting MFP hysteresis loops are presented in

Figure 12 for the three designs at the three tested pulse frequencies. The resulting cycle averaged MFP varied by less than 0.5% between all three housing designs. The designs were judged to be aerodynamically similar for the purpose of analyzing turbine performance.

The rotor efficiencies achieved over the pulse cycle for each configuration at the three tested pulse frequencies are presented in

Figure 13. A clear reduction in the range of rotor operating velocity ratios with increasing frequency can be observed. This range reduction almost exclusively happens at the high

$U/{c}_{s}$ end of the spectrum. The maximum

$U/{c}_{s}$achieved at 60 Hz was only 0.851 as apposed to 0.994 at 20 Hz. The range of rotor efficiency measured also reduced with increasing frequency.

The variation in rotor efficiency between the tested designs is small, but at all frequencies the volute AR = 0.5 design was less efficient across the range of operation. However, the AR = 0.5 design gave a greater maximum rotor $U/{c}_{s}$ than the other two designs at all frequencies. This can be attributed to a greater volute loss occurring in the smallest AR design resulting in a lower pressure ratio acting over the rotor. The efficiency of the AR = 1 and AR = 2 are similar over the range of rotor operation at all tested frequencies.

Table 1 presents the cycle averaged efficiencies for each design under the three tested pulse frequencies. The cycle averaged rotor performance of the largest aspect ratio design was consistently the greatest. The maximum improvement of 1.04% over the

AR = 0.5 design occurred at 20 Hz pulse frequency. Increasing volute

AR also resulted in an increase in cycle averaged stage efficiency with the greatest improvement over the

AR = 0.5 design being 1.47%, again at 20 Hz pulse frequency. The larger improvement in stage efficiency between designs was the result of the volute total pressure loss being significantly greater in the

AR = 0.5 design. The greatest difference in normalized volute loss coefficient between designs was 13.83%. In all cases the performance of the

AR = 1 design showed performance levels similar to that of the

AR = 2 design. The efficiency of the rotor at the minimum and maximum

$U/{c}_{s}$ running points is also presented. Consistently, the variation in efficiency between the three aspect ratio designs is greatest at the maximum velocity ratio point. The largest variation at this running point was 7.59% at 20 Hz pulse frequency. At the minimum

$U/{c}_{s}$ running point the maximum variation in performance was only 1.17%. As the maximum amount of pulse energy is available at the minimum

$U/{c}_{s}$ running point, it is the performance here that has the largest contribution towards cycle averaged performance.

To understand the loss generation within the rotor, the passage can be broken down into regions of interest and the entropy generated in each region calculated. While the definition of the regions is somewhat arbitrary, the entropy generated within each region can be attributed to a dominant loss mechanism. The method used is similar to that implemented in both [

24,

25].

Figure 14 shows the control volumes created in the rotor passage. The entropy generation in each region is calculated from net entropy flux on the surfaces and summed over the control volumes.

The resulting entropy generation is plotted in

Figure 15 and

Figure 16 at the minimum and maximum running points respectively.

Comparing the losses at the minimum incidence running point shows that the maximum losses occur in the pressure surface (PS) shroud and passage regions. This can be attributed to the high level of PS separation occurring at the highly negative incidence. At this running point the level of PS loss is much greater in the

AR = 0.5 volute design. This result correlates with that presented in

Table 1 which shows a large efficiency difference between the

AR = 0.5 design and the other designs at this pulse point. The

AR = 2 design shows a less significant improvement than the

AR = 1 design.

At the maximum incidence running point, the major entropy generation occurs in the suction surface (SS) passage and shroud regions due to the LE losses associated with positive incidence. The variation between the three designs is smaller in this case. The most significant difference occurs in the SS shroud region where there are greater losses for the AR = 0.5 design. The main loss contribution in this region was shroud tip leakage.

#### 4.3. Secondary Flow Structures

Understanding the impact of volute aspect ratio on turbine performance requires an understanding of the secondary flows within the volute passage.

Figure 17 shows contours of radial velocity and streamlines at seven planes around the volute for each design, at the minimum incidence running point for the 40 Hz pulse. The angles of the planes are defined from the tongue in the direction of flow, as indicated in the figure. The planes are selected to sit directly between the blades to reduce the impact of blade proximity. The exception is the 72° plane which is included to show the peak secondary flow activity.

A turbine volute is an example of a complex pipe as both the radius of curvature and volute cross sectional area reduce with the azimuth angle. Also in the case of a turbine volute there is an added pressure force acting out of the volute and in to the rotor region which will resist flow reversal in the volute.

The development of the secondary flows within the volute can be attributed to the Dean effect. Dean vortices were first observed by Dean [

26,

27]. These flow structures occur when the flow encounters a bend, with the low inertia flow at the walls turning readily into the radial direction within the pipe. In contrast, the higher inertia flow in the bulk of the passage does not turn as readily resulting in the development of counter rotation vortices with flow moving radially downwards at the walls and upwards in the passage center. The Dean number,

$De$, is defined as the balance of forces responsible for the development of the Dean vortices in pipe bends given by:

where:

$r$ is the pipe radius,

${R}_{C}$ is the radius of curvature of the pipe bend and

$R{e}_{D}$ is the Reynolds number based on the pipe hydraulic diameter and is given by:

where

$\mu $ is dynamic viscosity,

$\rho $ is density, u is bulk velocity and

${D}_{H}$ is the hydraulic diameter. The Reynolds number expresses the ratio of inertial to viscous forces. The remaining term in Equation (4),

$\sqrt{\frac{r}{{R}_{C}}}$ expresses the centripetal force acting on the fluid. Therefore the Dean number represents the impact of the inertia, viscous and centripetal forces acting on the fluid around the pipe bend [

28].

Figure 17 shows that the

AR = 0.5 volute results in a significant increase in the size of the counter rotating vortices present. The vortices can be seen high in the volute passage at 18° and increase in size and move down the plane up to the 72° position. Between the 72° and 126° positions the vortices continue to move radially inwards but decrease in size ahead of the volute outlet. Beyond this point the vortices no longer exist, even though clear variation in radial velocity across the plane still exists. In the

AR = 1 design vortices are only visible at the 72° position and in the

AR = 2 design no vortex development occurs. Despite the lack of coherent vortex structures, the streamlines in both the

AR = 1 and

AR = 2 volutes show deviations from ideal flow through the volute. The variation in radial velocity over the cross section is also evident in all cases with the velocity around the volute walls noticeably greater due to the Dean effect. In the

AR = 0.5 case, the reduced volute width results in a greater shear between the wall and bulk passage flow and hence strong vortices develop. It should also be noted that increasing the volute

AR also increases the wall curvature. Therefore, the flow close to the walls in the

AR = 0.5 case is predominantly radial, whereas the wall flow in the

AR = 2 case is only radial at the volute center, beyond this the flow turns axially with the passage walls. The change in wall velocity components can be expected to affect secondary flow development as well as vortex shape and orientation. The effect of this can be seen in the relative orientation of the vortices in the

AR = 1 and

AR = 0.5 cases. In the former, vortices are positioned at a tangent to the local wall curvature (approximately 40° from the vertical). In the latter, the vortices are orientated radially due to the reduced wall curvature.

The development of the vortex core present in the housing of aspect ratio 0.5 is presented in

Figure 18 using the lambda 2 criterion. This parameter shows the development of a vortex core and was used by Hellstrom and Fuchs [

20] to show the development of vortices in the exhaust manifold ahead of a radial turbine. In the

AR = 0.5 volute, particularly at the maximum velocity ratio running point, a clear vortex core can be seen to develop in

Figure 18a. The planes depicting streamlines are labeled with their angle from the volute tongue. The vortex region starts at around the 54° position where the streamlines show strong secondary flows. With increasing azimuth angle, the vortex cores move radially inwards towards the rotor LE and in towards the passage center. Just beyond 126° the vortex core no longer exists. At the maximum incidence running point shown in

Figure 18b, the vortex core is much less distinct. Although vorticity exists in the same area of the volute, no clear core is established. The streamlines also show a large reduction in vortex size.

Figure 19 shows the lack of clear vortex development in the

AR = 1 and

AR = 2 volutes at minimum incidence. In both cases, high levels of vorticity only exist close to the volute walls.

The spanwise distribution of absolute flow angle at the volute exit is presented in

Figure 20 for each of the volute aspect ratios at the 54°, 90°, 126° and 162° positions at both minimum and maximum incidence running points. The distance across the span of the volute exit has been normalized with the wheel hub being defined as the start of the span. The first observation to be made is that the spanwise variation in absolute flow angle at the exit of the volute is significant at all azimuth angles presented in all designs. Secondly, the spanwise variation is greater at the minimum incidence running points, hence at the peak energy running points the variation in exit flow absolute angle over the span reduces. The vortex development at maximum incidence is also reduced as shown in

Figure 18b for the

AR = 0.5 design.

Figure 20 also shows significant circumferential variation, with the absolute flow angle range peaking between the 90° and 126° positions reaching angles up to 85°. This position coincides with the angle at which the vortex core is close to the volute exit as shown in

Figure 18. At these two angular positions, variation in the three designs can be observed, which is greatest at the 90° plane at minimum incidence. The variation at this point reaches approximately 8.5°. Beyond this position the variation reduces with all the housing designs resulting in similar distributions at the 162° plane. This very similar spanwise distribution for the three designs continues around the remainder of the volute.

The impact of volute exit variation on circumferentially averaged rotor LE incidence is presented in

Figure 21,

Figure 22 and

Figure 23 at both the minimum and maximum incidence points of the pulse. The incidence is calculated from the circumferentially mass averaged velocities at the LE. This is therefore the variation in incidence averaged around the rotor inlet and not at specific positions as shown in

Figure 20.

At minimum incidence, the lowest flow angles are achieved at the rotor hub, increasing to a maximum between 15% and 20% span. Incidence reaches a local minima at the span center in the 60 Hz case, 60% span in the 40 Hz case and 70% in the 20 Hz case, before increasing again to a maximum at the shroud side. With increasing pulse frequency, a greater variation between the volute aspect ratio designs can be observed. In the 60 Hz case a variation of approximately 4° in incidence was observed at the passage center between the

AR = 0.5 and the other two larger aspect ratio designs which show very similar incidence distributions at all frequencies. The notable variation in incidence between the frequencies can be attributed to the considerably different maximum rotor velocity ratio achieved under each tested pulse as shown in

Figure 13.

At the maximum incidence point, a distribution with a parabolic shape is formed at all pulse frequencies with the maximum incidence achieved in the center of the LE. Between the tested frequencies, the distribution shows little variation; only under the 60 Hz pulse does the peak incidence achieved show a slight reduction. It was observed that the AR = 0.5 volute design consistently results in an increase in maximum incidence measured at the LE center of approximately 2°. Away from the center of the LE, the AR = 0.5 design achieved lower incidence angles by up to 3° when compared to the AR = 1 and AR = 2 designs which show very similar distribution at maximum incidence under all pulse frequencies. This variation in LE incidence results in a greater range of incidence in the AR = 0.5 case. However, the averaged incidence measure varies by less than 0.9° at all frequencies.

It should be noted that the change in LE axial flow component between the aspect ratio designs was negligible. Therefore, the change in incidence angle observed was not a result of changing blade angle but the result of relative flow angle variation.