Stabilisation number

A basis for evaluating the pressure recovery of a diffuser design are empirical diffuser charts, e.g., Sovran and Klomp (1967) or ESDU (1990). However, these charts rely on simplified inflow conditions and require additional, semi-empirical corrections to account for non-uniform inflow conditions. For example, Vassiliev et al. (2011) showed that the diffuser performance depends on the inlet Mach number, total pressure distribution, flow angle, and turbulence characteristics.

In addition to the static inlet conditions Kluß et al. (2009) numerically investigated the effects of unsteady wakes and secondary flows shed from rotating cylindrical spokes upstream of the diffuser on the pressure recovery. In contrast to predictions made using diffuser design charts assuming static inlet conditions, the unsteady inflow caused a re-attachment of the boundary layer. This results in an increase in pressure recovery exceeding static predictions. The findings were experimentally validated by Sieker and Seume (2008). Kuschel and Seume (2011) conducted additional experiments using a NACA profile instead of cylindrical spokes. A detailed analysis of these experimental results can be found in Kuschel et al. (2015) and Drechsel et al. (2015). They conclude that the increase in pressure recovery is caused by the interaction of tip vortices of the rotor blades with the diffuser boundary layer. The origin of theses vortices in the rotor tip region is further investigated by Drechsel et al. (2016). All these findings show that a combined design methodology for the last turbine stage and the diffuser is beneficial in achieving more efficient designs.

The stabilising properties of tip leakage vortices generated in the last rotor row and their effect on the boundary layer characteristics have been examined in Mimic et al. (2018). A correlation between the pressure recovery of the diffuser and integral rotor parameters of the last stage has been established, based on analytical considerations, numerical simulations, and experimental data. Parts of the experimental data have previously been published in Kuschel (2014). These rotor parameters are loading coefficient Ψ, flow coefficient Φ, and reduced frequency f_red (see Equations A1 to A3, as detailed in Appendix A).

Both experimental data and scale-resolving simulations, carried out with the SST-SAS method, showed excellent agreement with the correlation. The three parameters have been further condensed into the stabilisation number Σ which is defined as

The number represents a measure for the prediction of the diffuser effectiveness ϵ. With the definition of the stabilisation number Σ presented in Mimic et al. (2018), the correlation then gives

as depicted in Figure 1. The correlation shows that operating points with higher values for the stabilisation number Σ—i.e., essentially higher blade loading, more circumferential trajectories of the blade tip vortices, and more vortex passings per unit time—exhibit an increased diffuser effectivity. A change in Σ can be attributed to deliberate blade design choices regarding the last turbine stage as well as part-load operation for a given engine design.

Figure 1.

Increase in diffuser effectiveness from reference, Δϵ, against stabilisation number Σ (Reproduced from Mimic et al., 2018).

Total pressure loss coefficient

The total pressure loss in diffusers can be attributed to several sources. From a design point of view the losses caused by struts within the flow path, especially with high incidence flow during part-load, and the losses in the Carnot diffuser between the annular and conical diffuser are of great interest. The great amount of research conducted makes this evident, see for example Hirschmann et al. (2012), Vassiliev et al. (2014), Schäfer et al. (2014) or Seume and Drechsel (2015). In this paper, however, we consider just the annular part of the diffuser without struts.

The scope of this paper comprises the total pressure losses produced in the shroud boundary layer of the annular diffuser downstream of the rotor. Babu et al. (2011) were able to demonstrate that rotor tip leakage, achieved by means of flow injection in the shroud region, allow to decrease total pressure loss in comparison to an idealised uniform inflow. In the investigation presented here, the tip jet is achieved by a rotor instead. Therefore, a trade-off can be expected between the losses introduced by the tip vortex and the loss reduction by the homogenisation of the boundary layer and its delayed separation. As flow separation can generally be linked to a decrease in pressure recovery and an increase in total pressure loss, we propose the following hypothesis:

The total pressure loss coefficient ζ decreases for increasing values of the stabilisation number Σ.

Note that because this investigation takes place immediately downstream of the rotor, additional mixing losses, that are introduced by the wakes of the rotor entering the diffuser domain, must be taken into account.

Boundary layer losses

For an ideal gas, the energy equation of a flow can be given in the following form:

where s_ij is defined as the traceless strain rate tensor, i.e.,

The last term De on the right-hand side of Equation (9) describes the dissipation of kinetic energy as heat. Therefore, a reduction of the wall-normal velocity gradients reduces the dissipation losses in the boundary layer. As can be seen in Figure 2, this is exactly what happens to the boundary layer velocity profile of the diffuser, when the rotor outflow is characterised by a high stabilisation number Σ.

Figure 2.

Mass-flow-weighted, circumferentially averaged radial profile of non-dimensional axial velocity at 50% of diffuser length (Reproduced from Mimic et al., 2018).

Secondary flow losses

While the increased diffuser effectiveness and reduced total pressure losses in the boundary layer are certainly favourable, the effect of additional secondary flow structures in the diffuser would benefit from further examination. From the dot product of the vorticity ω_i and the vorticity equation, i.e.,

follows the enstrophy equation, namely

with the enstrophy E being defined as

and giving a measure for the rotational kinetic energy of the flow field. Here, the last term on the right-hand side of Equation (12), DE, represents enstrophy dissipation and is strictly non-negative. Thus, in the presence of high vorticity gradients—as they are typically found in the perturbations caused by tip leakage flow—enstrophy is dissipated as heat. This entails increased vortex-induced losses for operating points with high values of Σ.

Note, however, that dissipation of energy and dissipation of enstrophy should not be understood as two summands of some kind of total, or combined, dissipation. The same as enstrophy describes a filtered feature of the velocity field, namely its rotation, the dissipation of enstrophy gives a measure for the portion of energy dissipation that is caused by said rotation.

Because the interactions between the boundary layer in the diffuser and the blade tip vortices are complex and not necessarily linear—meaning that they cannot simply be superimposed onto each other—a mere analytical prediction would be inaccurate, at best. Therefore, we present experimental data and numerical simulations to evaluate the extent of the individual influence factors and to assess their effect on overall total pressure loss production.

Computational domain

The numerical domain represents one pitch of the rotor. The numerical simulations feature—different than in the experiments—blade counts of 25, 30 and 40. The blade has a symmetric NACA0020 profile and is unloaded at the aerodynamic design point. The rotor domain is followed by an annular diffuser with a half-opening angle of 15° leading to a numerical outlet section that is made up of a divergent and a straight duct. The entire numerical domain, as shown in Figure 4, is in a rotating frame of reference. The numerical reference planes for the evaluation of the overall diffuser total pressure loss coefficient ζ match the experiment. They are located 15 mm downstream of the diffuser inlet and at diffuser outlet.

Figure 4.

Computational Domain (coarse mesh for display): The entire domain is simulated inside the rotating system (Reproduced from Mimic et al., 2018).

The mesh consists of 1.7–2.4 million overall cells, depending on the blade count of the respective mesh. The mesh is refined in and around the tip gap as well as in the diffuser shroud region. Here, unsteady effects are to be expected due to vortex generation and massive boundary layer separation. A numerical outlet section—downstream of the nominal diffuser outlet—coarsened in axial direction leads to locally elevated numerical dissipation. This way, eventual disturbances interacting with the outlet boundary condition are damped. Numerical convergence and stability are enhanced, as a result. Because the whole numerical domain is part of the rotating relative system, no interface is needed between rotor and diffuser. Static pressure values are given as outlet boundary conditions. The outlet pressure is adjusted to match the required mass flow rate for each individual operating point. An extensive grid convergence study with a similar grid and the SST model has been carried out by Drechsel et al. (2015).

Rectified total pressure loss

The absolute rotor outflow angle α depends on the operating point. Thus, the fluid travels a longer distance in the diffuser with regards to a swirl-free rotor outflow. Trigonometry shows that the length of a streamline is inversely proportional to cos α. Because total pressure losses are expected to increase with the distance covered, the total pressure loss coefficients for the respective operating points are rectified to an equivalent, swirl-free pressure loss coefficient.

Because the definition of the stabilisation number Σ is based on the behaviour of the tip leakage vortices, it cannot account for the effect of wakes generated by the rotor. A simple numerical analysis reveals that the wakes become more accentuated for higher flow coefficients Φ—this equals a higher incidence of the rotor inflow—which manifests in greater wake velocity deficits as shown in Figure 5. As can be seen, some oscillations are present in the free-stream region. Even though they bear no significance on the overall result, this is a typical issue that arises when RANS computations are performed using central differencing schemes. It is caused by inaccuracies in the performance of the blending function from Strelets (2001).

Figure 5.

Time-averaged circumferential distribution of non-dimensional wake velocity deficit at Euler radius at 8% of diffuser length.

Additionally it is assumed that the pressure loss coefficients associated with wake mixing are roughly proportional to the square of the absolute rotor outflow velocity cII2 for the parameter range discussed in this paper. For the above reasons, we propose a rectified total pressure loss coefficient ϖ which is defined as

where Υrel, Φ_rel and cII,rel2 are defined as

Note that the relative rotor outflow velocity cII,rel is still in the absolute frame of reference and must not be confused with the rotor outflow velocity in a relative frame of reference.

Of course, the assessment of wake mixing losses can be done in greater detail, which is, however, beyond the scope of this paper. Examples for such calculations are given by Denton (1993) as well as Rose and Harvey (1999).

The correlation

Figure 6 shows for the numerical simulations performed that the rectified diffuser pressure loss coefficient ϖ decreases in a linear way for increasing values of the stabilisation number Σ. The following correlation can be given from the numerical samples:

Figure 6.

Rectified diffuser total pressure loss ϖ against stabilisation number Σ.

It is, however, physically impossible for ϖ to attain negative values. Anyhow, for excessively increasing values of Σ, one may expect drastically increasing secondary flow losses.

The absolute difference in rectified total pressure loss coefficient to the respective reference variants for simulations and experiments,

facilitates the comparison between record sets. The result is shown in Figure 7. Again, a linear correlation for both, experiment and numerical simulations, between Δϖ and Σ becomes apparent—here with a slightly steeper descent than just for the numerical results. The correlation gives

Figure 7.

Absolute difference in rectified diffuser total pressure loss from reference, Δϖ, against stabilization number Σ.

or, for a regression that goes through the point of origin

Since ϖref equals ζ_ref, the correlation can be simplified to

Here, we introduce the loss rectification factor Λ to quantify the effect of wakes and other non-vortex-induced phenomena. For the rotor investigated, it is defined as

We suspect that the exact definition of Λ is dependent on the exact geometry of the rotor and may require additional rectification factors. Regardless, this requires further studies. While the correlation proposed in Equation (20) indicates a clear relation between total pressure losses generated in the diffuser and the degree of boundary layer stabilisation its physical underpinnings shall be further discussed.

Dissipation

As initially discussed, the dissipation terms of the energy equation and the enstrophy equation, De and DE, respectively, indicate loss production. While the former gives a measure for the entirety of losses produced, the latter allows to detect the losses caused by vortical structures. An analysis of the respective distributions of both terms in the numerical solutions allows to better understand where irreversible flow processes lead to total pressure drops.

Unlike the term given for De in Equation (9), the turbulent energy dissipation,

shall be used in the following. In this formulation, the viscosity μ has been replaced by the sum of molecular and turbulent viscosity μ + μ_t to account for the influence of turbulence. Figure 8 shows the circumferentially averaged relative magnitude of De,t in the shroud near region for different operating points.

Figure 8.

Circumferentially averaged distribution of non-dimensional energy dissipation De,t/De,t,ref,avg for different values of Σ.

Apparently, the dissipation of energy decreases in intensity and expanse for increasing diffuser stability. This is consistent with the homogenisation of the velocity profiles seen in Figure 2. The five variants with the lowest stabilisation numbers (from bottom to top in Figure 8)—which are also located close to each other in Figure 6—are fairly similar, whereas a considerable weakening of the dissipation can be observed from Σ = 0.0613 to Σ = 0.1678.

Conversely, Figure 9 depicts the turbulent dissipation of enstrophy for different, representative operating points, that is,

Figure 9.

Meridional plane showing the time averaged distribution of non-dimensional enstrophy dissipation DE,t/DE,t,ref,avg for different values of Σ.

The contour plots are taken for meridional planes, which are oriented so that the trailing edge of the blade lies directly behind the plane. The slanted, white bar at the left of the individual figure frames is caused by the wakes and indicates the position of the trailing edge. One can see that the highly vortical tip leakage flow causes considerable enstrophy dissipation. An effect that becomes more accentuated for increasing values of Σ, as the tip leakage vortex becomes more powerful.

While the magnitudes of De,t and DE,t cannot be adequately compared to each other, the vastly different spatial dimensions make it clear that the effect of vortex-induced losses is more locally limited and almost vanishes in comparison to the dominating boundary layer and separation losses for the diffuser rig discussed.

Symbols

Subscripts