Exhaust diffusers significantly enhance the available power output and efficiency of gas and steam turbines by allowing lower turbine exit pressures. The residual dynamic pressure of the turbine outflow is converted into static pressure, which is referred to as pressure recovery. Since total pressure losses and construction costs increase drastically with diffuser length, it is strongly preferred to design shorter diffusers with steeper opening angles. However, these designs are more susceptible to boundary layer separation. In this paper, the stabilizing properties of tip leakage vortices generated in the last rotor row and their effect on the boundary layer characteristics are examined. Based on analytical considerations, for the first time, a correlation between the pressure recovery of the diffuser and the integral rotor parameters of the last stage, namely, the loading coefficient, flow coefficient, and reduced frequency, is established. Experimental data and scale-resolving simulations, carried out with the shear stress transport scale-adaptive simulation (SST-SAS) method, both show excellent agreement with the correlation. Blade tip vortex strength predominantly depends on the amount of work exchanged between fluid and rotor, which in turn is described by the nondimensional loading coefficient. The flow coefficient influences mainly the orientation of the vortex, which affects the interaction between vortex and boundary layer. The induced velocity field accelerates the boundary layer, essentially reducing the thickness of the separated layer or even preventing separation locally.

## Introduction

Diffusers downstream of gas and steam turbines significantly influence the available power output of the turbine. As shown by Farokhi [1], a linear dependency exists between the power output as well as the efficiency of the turbine and the pressure recovery of the diffuser—the latter characterized by the nondimensional pressure recovery coefficient
$cp=pout−pinptot,in−pin$
(1)
For an inviscid ideal diffuser under uniform inflow and outflow conditions, cp solely depends on the ratio of the cross-sectional areas at the inlet and the outlet of the diffuser AR = A2/A1, the swirl angle $α=tan−1(cϑ/cm)$ at the inlet, and the change in Euler radius across the diffuser
$cp,ideal=1−1AR2+tan2α(r1r2)21+tan2α$
(2)
The ideal pressure recovery coefficient cp,ideal represents the theoretical limit for the pressure recovery of a given diffuser geometry. Based on the ideal pressure recovery coefficient, the effectiveness of the diffuser is defined as
$ϵ=cpcp,ideal$
(3)

Considering viscous effects, the available pressure recovery decreases with the displacement thickness δ1 of the boundary layer at diffuser outlet as it reduces the effective cross-sectional area. The displacement thickness in turn depends on pressure and frictional losses generated in the boundary layers. For a given area ratio AR, friction losses are most relevant for relatively long diffusers with small opening angles. In short diffusers with high opening angles and resulting high positive pressure gradients, pressure losses due to boundary layer separations can significantly increase the displacement thickness. Nevertheless, although high pressure gradients increase the risk of a fully detached diffuser flow with no pressure recovery, highly loaded diffusers with boundary layers close to separation or even with small intermittent separations are technically interesting as they provide the highest cp for a given AR [2]. Against this background, empirical diffuser charts, e.g., of Sovran and Klomp [3] and the ESDU [4], provide a valuable and reliable tool for the design of efficient annular or conical diffusers facing steady homogeneous axial inflow.

However, these inflow boundary conditions are rather idealized for a diffuser downstream of a turbine as its outflow is characterized by swirl for most of the operating points and inhomogeneities due to wakes and secondary flows. Consequently, several researchers have emphasized the need to design the final turbine stage and the diffuser together [5].

### Past Research About Turbine–Diffuser Interaction.

The effect of swirl on the performance of, e.g., annular diffusers has been studied comprehensively in Refs. [68] and it was shown that moderate swirl angles increase the pressure recovery by reducing or preventing boundary layer separation—i.e., stabilizing the boundary layer—at the diffuser shroud. However, for higher swirl angles, the destabilization of the hub boundary layer can, in turn, lead to a reduction of the recovery coefficient. The effect of tip jets, which can be observed in the case of unshrouded turbine blades, was modeled in several experimental setups by injecting fluid under realistic conditions (viz., angle and momentum) into the shroud boundary layer [7,9,10]. In general, a highly positive effect on the pressure recovery in the case of no swirl was observed with a decreasing impact on cp for increasing swirl angles. Similarly, Babu et al. identified a generally positive effect of tip injections on diffuser performance and a strong sensitivity toward inflow conditions [11]. Likewise, hub injections have been shown to lead to improvement in pressure recovery [12]. Further positive effects on the pressure recovery were observed by altering the shape of the (circumferentially homogeneous) radial total pressure inlet profile [13] and by increasing turbulence intensity at the diffuser inlet [14]. While flow control devices such as vortex generators may come to mind, it must be ensured that they require a minimum of maintenance even under prolonged exposure to hot exhaust fumes. Thus, a combined approach toward the design of turbines and diffusers seems promising.

Although all those parameters have an empirically proven and (analytically) plausible effect on diffuser stability and pressure recovery, respectively, their idealized application is not able to fully describe the pressure recovery and losses in a diffuser downstream of a turbomachinery rotor. The investigations of, e.g., Sieker and Seume [15,16], Kluß et al. [17], and Kuschel et al. [18,19] for unshrouded wake generators with varying bladings upstream of different highly loaded diffusers have shown that there is no distinct correlation for cp with, e.g., swirl number, tip jets, and turbulence intensity. The comprehensive measurements and the analysis of Kuschel et al. revealed, however, the existence of a rotor-independent correlation between cp and the magnitude of $τ̃$ close to the shroud at diffuser inlet (see Fig. 1), where $τ̃$ is characterized by the sum of the Reynolds shear stresses with a radial component: $cx′cr′¯min$ and $cr′cϑ′¯min$. Correlations were obtained for annular diffusers with half-opening angles of 15 deg and 20 deg. Drechsel et al. [20] could reproduce this correlation with simulations applying the scale-adaptive simulation (SAS) turbulence treatment [21]. Again, by means of scale-resolving simulations, Drechsel et al. [22] pointed out that the relevant Reynolds shear stresses are produced by coherent secondary flow vortices formed at the rotor tip gap, e.g., the tip leakage vortex. Keeping the background of previous tip jet investigations in mind, it is worth mentioning that this production of Reynolds shear stresses and their correlation with the pressure recovery also applies in the case of no distinct tip jets [20].

Fig. 1
Fig. 1
Close modal

These latest investigations have identified a strong correlation between the pressure recovery coefficient of the diffuser and the magnitude of the Reynolds shear stresses with a radial component. Turbulent kinetic energy, however, does not bear remarkable effects on pressure recovery [19]. A better understanding in which way this behavior is linked to the design parameters of the final turbine stage could allow to isolate and reproduce the exact causes for the increase in pressure recovery. This would considerably facilitate the combined design of the final turbine stage and the diffuser, effectively leading to shorter diffusers, which raises two crucial questions:

1. (1)

What are the mechanisms that explicitly cause the correlation between anisotropic turbulence with a dominant radial component and reduced boundary layer separation?

2. (2)

What are the main parameters, in terms of the turbine, that influence these mechanisms?

## Analytical Considerations

To answer the aforementioned questions, hereinafter a series of analytical considerations is presented with the intent to elucidate the link between unsteady phenomena that take place in the boundary layer of the diffuser, and the blade configuration and operating point of the last stage of the low-pressure turbine. Coming from the observations made in Refs. [19], [20], and [23], the stabilizing effect of certain Reynolds stress components on the boundary layer is explained and linked back to the presence of highly unsteady vortical structures. These structures are subsequently related to secondary flow phenomena in the tip gap region, ultimately establishing a connection to integral parameters of the turbine stage.

### Generation of Turbulence.

The first of the two questions posed above requires a thorough examination at the origin of anisotropic turbulent structures, such as those that were measured by Kuschel.

Turbulent structures are called eddies and can vaguely be described as regions of highly unsteady, rotational flow. The rotation of a flow is quantified by the vorticity vector$Ω$ which is defined as the curl of the velocity field $Ω=∇×c$, or using index notation
$Ωi=εijk∂ck∂xj$
(4)
with εijk being the three-dimensional Levi-Civita symbol. Using the identity
$−∂∂x2(c1′c2′¯)=c2′Ω3′¯−c3′Ω2′¯$
(5)
the nexus between the physical quantities of the Reynolds shear stress $ci′cj′¯$ and that of the vorticity fluctuations become abundantly clear.
From the vorticity equation
$∂ Ωi∂t︸①+cj∂ Ωi∂xj︷②=Ωj∂ci∂xj︸③+ν∂2Ωi∂xj∂xj︷④$
(6)
which can be obtained by taking the curl of the Navier–Stokes equations, it follows that the local rate of change ① of the vorticity vector is determined by the convective transport ②, the vorticity diffusion rate ④, and term ③ that describes the deformation of a vortex line under the influence of a velocity gradient.
The decomposition of this term ③ yields
$Ωj∂ci∂xj=Ωj∂ci∂xj|i=j+Ωj∂ci∂xj|i≠j$
(7)
with the first term of the right-hand side being the axial stretching or compression of a vortex line parallel to the local velocity field and its gradient, whereas the second term denotes the case that the vortex line is orthogonal to the velocity vector yet parallel to its gradient and the vortex line is “bent” or “tilted,” i.e., a (smaller) vorticity component is generated in the same direction as the local velocity field. Both cases, vortex stretching and vortex bending, are illustrated in Fig. 2. This means that every interaction between vortex lines and velocity gradients creates smaller vortical structures in a multitude of new directions, eventually leading to isotropic turbulence. Davidson explains this concept in much greater detail in Ref. [24].
Fig. 2
Fig. 2
Close modal

There is yet another insight that can be deduced from Eq. (6): since the generation of vorticity described by term ③ depends on the local magnitude of already existing flow field rotation, the term does not quite constitute a vorticity production term but rather a vorticity amplification. With ④ being a diffusion term, merely redistributing vorticity, this in turn leads to the conclusion that the vorticity in a control volume has to be brought in by means of convection from an external source if there was no vorticity to begin with. In the case of the diffuser, one may argue that the boundary layer is a major source of vorticity. While this is certainly true, the orientation of the vorticity it generates is nearly exclusively circumferential and there is no significant source of axial vorticity in the diffuser. The stabilizing vortices, in turn, run predominantly in the axial direction. Furthermore, the authors infer that the rotational energy of the turbulent structures, and consequently, the measured Reynolds stresses are a result—or rather a symptom—of vortical inflow into the diffuser. This is plausible because the measured Reynolds stresses not only feature statistical velocity fluctuations but also rather deterministic fluctuations caused by coherent vortex structures. While the term “Reynolds stresses” is used throughout the literature (and, for the greatest part, in this paper, too) for the sake of simplicity, a broader term such as “unsteady apparent stresses” would be technically more correct.

However, the anisotropy of the measured Reynolds stresses, namely, their dominant components $c1′c2′¯$ and $c2′c3′¯$, implies wall-normal (radial in the diffuser) velocity fluctuations $c2′$. It follows that the orientation of the vortex axes is perpendicular to this, i.e., a primarily wall-parallel (axial-circumferential in the diffuser) orientation of the vortex axes, whereas the wall-normal component of the vortex axes is minimal.

### Stabilization of the Boundary Layer.

Knowing that the Reynolds stresses measured by Kuschel and Seume [18] in the shroud near region indicated the presence of vortices, one may ask, in which way a vortex delays, or even prevents, the separation of a boundary layer under the influence of an adverse pressure gradient. In order to answer this, a closer look at the flow physics of vortices and boundary layers is necessary.

#### Vortices.

Vortices, in the broadest sense, may be described as circular flow structures. They may roughly be divided into a core region, where flow is rotational and determined by viscous effects, and an outer, irrotational, circular potential flow. This so-called Rankine vortex has an azimuthal velocity cazv of
$cazv(r)={Γr/2πrcore2if r≤rcoreΓ/2πrif r>rcore$
(8)

For two-dimensional considerations, this azimuthal velocity may be superimposed onto the local velocity field as induced velocity cind = cazv.

#### Boundary Layer Thickness.

As follows from the boundary layer equations for steady, incompressible flow, the displacement thickness δ1 of a turbulent boundary layer on a flat plate grows in the flow direction in the absence of a pressure gradient. So do the boundary layer thickness δ and the momentum thickness δ2, since they are proportional to each other. However, the thickness decreases upon acceleration of the boundary layer. Clearly, if a velocity field induced by a vortex is superimposed, δ1 diminishes for positive cind. In addition to the decreased thickness, the velocity induced by the vortex reduces the tendency of the boundary layer to separate; the separation point is defined as the point where the wall normal gradient of the velocity becomes zero at the wall, i.e.,
$∂c1∂x2|x2=0=0$
(9)
Equation (8) shows that outside the border of the vortex core, cind is strictly decreasing or, from the walls perspective, strictly increasing, i.e.,
$∂cind∂x2|x2=0>0$
(10)
which means that a superposition of cind moves the boundary layer state away from flow separation as can be seen in Fig. 3.
Fig. 3
Fig. 3
Close modal

The following can be concluded: an influx of Reynolds stresses with a radial component may indicate the presence of vortices with wall-parallel axes. These vortices, under the right circumstances, reduce the boundary layer thickness and delay or even prevent flow separation.

In order to give an answer to the second question, it is necessary to reiterate the mechanisms that ultimately lead to the generation of these stabilizing vortices. The major source of vorticity entering the diffuser can be attributed to secondary flows generated in the final turbine stage. Drechsel et al. demonstrated in Ref. [20] that the tip leakage vortex plays the greatest role in stabilizing the diffuser shroud boundary layer.

The tip leakage vortex is caused by fluid flowing over the blade tip from the pressure side to the suction side as a result of the pressure difference between the two sides. Another explanation that lends itself to further considerations can be found in the potential theory, where the flow around an airfoil is obtained from the superposition of a parallel flow with a relative velocity w and a potential vortex with a circulation Γ just strong enough to keep the rear stagnation point at the trailing edge of the airfoil. According to the Kutta–Joukowski theorem, the blade force is then computed as
$F=−ρ∞hw∞Γ$
(11)
with ρ as the free stream density and h as the blade height. Since Helmholtz's second theorem states for inviscid flows that a vortex line—in this case the airfoil-bound vortex—cannot end in a fluid but only on the boundary of a fluid, at infinite distance, or must form a closed path, it follows that a vortex filament is shed from the blade tip in the main flow direction. In real fluids, this is of course accompanied by viscous effects. Circulation around the boundary line $∂S$ of a closed surface $S$ is defined as the line integral of the velocity, and by Stokes' theorem, can be reformulated as the surface integral of the vorticity as follows:
$Γ=∮∂Swi dsi=∬SΩi dSi$
(12)

Therefore, the vorticity, or strength of the blade tip vortex, directly depends on the blade force. Likewise, the rotational kinetic energy contained in the vortex correlates with the work performed in the rotor, i.e., more work performed in the rotor leads to stronger stabilizing tip leakage vortices.

The work performed in the rotor is expressed per Euler's simplified turbine and pump equation as
$arotor=u(cϑ,I−cϑ,II)$
(13)
$Ψ≡au2=cϑ,I−cϑ,IIu$
(14)
It is a function of the flow deflection angles in the row (since $cϑ,I−cϑ,II=wϑ,I−wϑ,II$) [25]. Magnitude and sign of the tip leakage vorticity can be attributed to Ψ, as they both depend on the deflection angles.

### Flow Coefficient.

The eventual benefit or detriment of the tip leakage vortex does obviously not only depend on its magnitude but also its trajectory and consequently the orientation of its induced velocity field relative to the boundary layer flow direction. Since the vortex convects downstream with cx, yet at the same time the origin of the vortex—the blade tip—moves with utipu, as shown in Fig. 4, the trajectory of the vortex core is proportional to the flow coefficient
$Φ≡cxu$
(15)
Fig. 4
Fig. 4
Close modal

This implies that, for higher Φ, the vortex displays an increasingly axial trajectory, whereas for smaller Φ it follows a more circumferential path, which is beneficial for boundary layer stabilization.

Additionally, the orientation of the induced velocity field depends on the swirl angle of the vortex. Following the definition given by Hall [26], the vortex swirl angle ϕsw is
$ϕsw≡arctan(wazvwaxv)$
(16)

Since the greater part of the tip leakage vortex consists of fluid from the shroud boundary layer, waxv can be estimated as being roughly proportional to cx. The relative azimuthal velocity component of the vortex wazv is determined by the pressure difference between suction side and pressure side of the moving blade. For a given blade geometry, this pressure difference depends on the incidence angle. If cx is kept constant, the incidence angle—and by extension wazv—is essentially a function of u, as depicted in Fig. 4. It follows that ϕsw is in the order of magnitude of 1/Φ. This means that smaller Φ leads to a more advantageous streamline orientation inside the vortex. Consequently, the stabilizing effect of the tip leakage vortex is concluded to be proportional to 1/Φ2.

### Reduced Frequency.

The last factor that has to be taken into account is the reduced blade passing frequency fred. It is defined analogous to the Strouhal number
$fred≡fbpℓccx=nNℓc160cx$
(17)

where n is the blade count, N is the rotor speed in rpm, and c is the blade chord length. The reduced blade passing frequency fred indicates the number of vortices that pass in a certain time frame relative to the main convective velocity. Averaged over time, it can be interpreted as the “density” of vortices interacting with the boundary layer. Hence, the boundary layer stabilization is expected to increase with fred.

### Stabilization Number.

For the reasons mentioned earlier, the authors propose a novel stabilization number Σ, which is defined as
$Σ≡ΨfredΦ2$
(18)

It represents a measure for the prediction of the diffuser effectiveness ϵ. It must be noted, and cannot be stressed enough, that actual values of Ψ, Φ, and fred have to be taken into account—including incidence and deviation—not the ones at the design point.

### Other Influence Factors.

Of course, there are other influence factors that are not covered by this simplified approach yet are expected to occur in a real turbine. These encompass, but are not limited to, phenomena such as transition, flow separation on the blades, secondary flows from the previous rows, or turbulent entry conditions. They may affect the generation and decay of the tip leakage vortex. Similarly, it may strongly vary in strength with different tip gap heights or tip designs, and completely vanishes when shrouded rotors are used. However, this in turn could lead to other circulation-induced phenomena.

Based on the previous contemplations, it can be assumed that the generation of tip leakage vortices is a result of blade circulation. Besides vortex strength, trajectory, swirl angle, and frequency are important factors and depend on the turbine parameters: loading coefficient, work coefficient, and reduced frequency.

## Test Facility

The experimental investigations are conducted by means of the low-speed axial diffuser test rig at the Institute of Turbomachinery and Fluid Dynamics. It represents a 1:10 scaled heavy-duty exhaust diffuser. The diffuser consists of an annular and a conical part, as depicted in Fig. 5. The half-opening angle of the annular diffuser is 15 deg. Thus, according to the diffuser charts of Sovran and Klomp [3], the boundary layer of the annular diffuser is susceptible to separation for steady homogeneous inflow conditions. The turbine outflow is modeled by a rotating wake generator which is able to reduce separations and increase the pressure recovery of the diffuser as the previous investigations have shown [19]. Two interchangeable rotating wake generators consist of 30 and 15 symmetric NACA0020 blades, respectively, that are unloaded at the aerodynamic design point. Additional test rig parameters are given in Table 1.

Fig. 5
Fig. 5
Close modal
Table 1

Geometric properties of the test rig

Rotor propertiesDiffuser properties
Hub/tip stagger angle43 deg/58 degrCD,hub35 mm
Rotor propertiesDiffuser properties
Hub/tip stagger angle43 deg/58 degrCD,hub35 mm

## Numerical Method

All presented simulations were carried out using the noncommercial solver TRACE 8.2 (Turbomachinery Research Aerodynamics Computational Environment) developed by the Institute of Propulsion Technology at the German Aerospace Center (DLR). The SAS method by Menter and Egorov [27] is used in conjunction with the kω-shear stress transport (SST) turbulence model by Menter [28]—hereafter referred to as SST-SAS—in order to resolve unsteady vortical structures. The wall boundary layers are treated as fully turbulent. A stagnation point anomaly fix as per Kato and Launder [29] is employed. Detailed examples for scale-resolving simulations can be found in Refs. [30] and [31].

According to Fröhlich and Terzi [32], the SAS method is classified as a second generation unsteady Reynolds-averaged Navier–Stokes (2G-URANS) model, which can be regarded as an intermediate step between conventional URANS models and large eddy simulation (LES) models. While the former suppress the formation of unsteady flow features triggered by instabilities through a local increase in turbulent viscosity, the latter are inherently grid-dependent. 2G-URANS models represent a way to avoid explicit grid-dependence while still resolving a greater part of unsteady effects.

In order to achieve this, the SST-SAS approach provides for an additional production term in the dissipation equation of the kω-SST model. Consequently, the local turbulent viscosity decreases. This allows for unsteady, turbulent structures to develop that otherwise would be subjected to damping. The production term is mainly driven by the ratio
$Lt,kωLvK$
(19)

where Lt, is the modeled turbulent length scale and LvK is the von Kármán length scale. The von Kármán length scale is proportional to the ratio of the first to the second derivative with regard to space of the velocity vector ci and gives a measure for the unsteadiness of flows; a decrease in LvK signalizes an increase in instantaneous velocity fluctuations. An extensive derivation of the SAS production term and the von Kármán length scale is given in Ref. [24]. A more in-depth dissection of the SAS method can be found in Ref. [32]. It was shown by Drechsel et al. [20] that SAS, in contrast to URANS, is able to predict the flow of the diffuser to a greater accuracy.

### Discretization.

The solver provides a second-order accuracy implicit finite volume spatial discretization and second-order accuracy Euler Backward time discretization. Numerical robustness is further increased by using a third-order differencing scheme with a van Albada square limiter for the computation of the state variables. As for the solution method, incomplete lower upper (ILU) factorization showed far greater stability with regard to symmetric successive over-relaxation (SSOR).

Upwind schemes cause considerable damping of rotational flows. Scale-resolving regions are discretized using a central scheme in order to keep numerical dissipation low, thus allowing vortical structures to develop.

In smooth flow regions, however, the numerical damping of upwind schemes may be beneficial, inasmuch as it prevents the emergence of unphysical, high-frequency oscillations in the solution. This greatly reduces the stability of the computation. Hence, in smoother, URANS-dominated regions, a second-order upwind scheme is used to reduce these oscillations and to maintain high stability.

#### Blending Function.

The implementation of a blending function provides a smooth transition between the central and upwind scheme. The inviscid fluxes are calculated as a weighted sum of both schemes
$Finv=(1−σ)Fcentral+σFupwind$
(20)
with σ as the blending function; a detailed explanation of the utilized blending method is given by Strelets [33]. In brief, the function is defined between zero in SAS-regions, where it leads to a central scheme, and unity, where an upwind scheme is used. The blending function is mainly influenced by the ratio of the local grid size to the turbulent length scale Lt,σ, i.e., coarse grids and small values for Lt,σ promote an upwind scheme. The length scale Lt,σ in turn increases as the sum of molecular and turbulent viscosity goes up and as the strain rate and vorticity tensors go down. This increase in Lt,σ, however, is limited by a “characteristic convective time” τ [33]. For the presented simulations, τ is estimated from the chord length of the blade and the mean inlet velocity, even though in the present cases, variations of τ did not show any mentionable impact on the solution.

### Computational Domain.

As mentioned earlier, the numerical domain contains one pitch of the rotating wake generator consisting of a symmetric NACA0020 blade that is unloaded at the aerodynamic design point, an annular diffuser with a half-opening angle of 15 deg, and a numerical outlet section consisting of a divergent and a straight part, as shown in Fig. 6. Pressure data for the computation of the cp values have been extracted 15 mm downstream of the diffuser inlet and at the diffuser outlet to match the experimental measuring planes.

Fig. 6
Fig. 6
Close modal

Depending on the simulated blade count, the mesh has between 1.7 × 106 and 2.4 × 106 overall cells. It is refined in the tip gap region and adjacent to the diffuser shroud, where unsteady effects due to vortex generation and massive boundary layer separation are to be expected. The numerical outlet section is coarsened in the axial direction to promote higher numerical dissipation in order to reduce the magnitude of disturbances interacting with the outlet boundary condition. The entire numerical domain is inside the rotating relative system in order to facilitate the analysis of time-resolved Reynolds stresses and to obtain a one-on-one connectivity between the blade passage and the diffuser. A static pressure outlet boundary condition is used. The outlet pressure is adjusted to match the required mass flow rate. An extensive grid convergence study with a similar grid and the SST model has been carried out by Drechsel et al. [20].

## Analysis

In order to validate the postulated relation between the stabilization number Σ and the diffuser effectiveness ϵ, variations in Ψ, Φ, and fred are presented. This was achieved through different combinations of the parameters blade count n, rotor speed N, and mass flow rate $m˙$. This was done for a selection of both, numerical (NUM) and experimental (EXP) as well as supplementary experimental (SUP) test cases, which are listed in Table 2. The EXP test cases have been presented in Refs. [19] and [23]. Identical operating points of SUP that were measured multiple times are shown as averaged values. It must be noted that EXP and SUP were measured years apart and with different probes; thus, they are presented separately. Additionally, between the measurement of EXP and SUP as well as in between measurement runs of SUP, the test rig was dis- and reassembled multiple times and different rotors were used. This inevitably leads to small changes in the operation of the test rig. However, the correlation, which is presented in the following, does not exhibit strong sensitivity to the variations. This implies a high repeatability and supports the validity and reliability of the correlation.

Table 2

Test cases

$N$o.nN in rpm$m˙$ in kg/sΨΦfredΣ
1EXP3015004.80.411.100.45−0.0081
23015005.30.521.210.410.0893
33025005.3−0.010.730.670.1434
43025006.30.120.880.560.1521
5SUP1515004.80.191.100.220.0355
61515005.30.271.210.200.0370
71525005.3−0.010.730.34−0.0079
81525006.30.070.860.280.0277
93015004.80.331.080.460.1300
103025005.30.000.740.66−0.0032
113015005.30.451.210.410.1252
123025006.30.090.850.580.0681
13NUM2518755.50.201.100.370.0613
142525005.50.030.820.500.0187
153025005.10.000.780.630.0014
163025005.20.010.790.620.0118
173025005.30.020.810.610.0208
183025005.50.040.830.590.0357
194018755.50.381.140.580.1678
204025005.50.100.850.770.1080
$N$o.nN in rpm$m˙$ in kg/sΨΦfredΣ
1EXP3015004.80.411.100.45−0.0081
23015005.30.521.210.410.0893
33025005.3−0.010.730.670.1434
43025006.30.120.880.560.1521
5SUP1515004.80.191.100.220.0355
61515005.30.271.210.200.0370
71525005.3−0.010.730.34−0.0079
81525006.30.070.860.280.0277
93015004.80.331.080.460.1300
103025005.30.000.740.66−0.0032
113015005.30.451.210.410.1252
123025006.30.090.850.580.0681
13NUM2518755.50.201.100.370.0613
142525005.50.030.820.500.0187
153025005.10.000.780.630.0014
163025005.20.010.790.620.0118
173025005.30.020.810.610.0208
183025005.50.040.830.590.0357
194018755.50.381.140.580.1678
204025005.50.100.850.770.1080

The flow coefficient is equal to or higher than the design flow coefficient for all cases. Most samples display a positive work coefficient, as this would be expected for a turbine. The magnitude of the respective flow coefficients and loading coefficients is rather small with regard to real turbine applications yet provides a reasonable starting point for future investigations. Samples 3, 7, 10, and 15 are basically free from incidence and flow deflection. The negative values for the experimental samples 3 and 7 are most likely due to inaccuracy of measurement; their deviation is negligible. Variants 3 and 15 are thus used as the respective references for experiments and simulations; samples 7 and 10 serve as references for the supplementary data.

### Influence of the Stabilization Number.

In Fig. 7, the diffuser effectiveness ϵ (see Eq. (3)) is plotted against the stabilization number Σ (see Eq. (18)). Linear correlations can be inferred for both, experimental and numerical data. It follows as a correlation for the experimental samples
$ϵcorr, EXP (Σ)=2.5421Σ+0.3107 with R2=0.968$
(21)
Fig. 7
Fig. 7
Close modal
The coefficient of determination R2 indicates that 96.8% of the results are predictable from the proposed correlation [34]. Based on the simulations, linear regression gives a correlation function of
$ϵcorr,NUM(Σ)=2.4354Σ+0.1259 with R2=0.9611$
(22)

Compared to the experiments, the diffuser effectiveness is systematically underpredicted in the numerical simulations. This is not surprising: in Refs. [20] and [22], Drechsel et al. showed that SAS gives distinctly lower turbulent kinetic energy as well as anisotropic Reynolds stresses—and ergo the tip leakage vortex—with regard to the experiments. However, the slopes of ϵcorr,EXP and ϵcorr,NUM show astonishingly good agreement.

In order to derive a more general correlation, the respective reference variants—i.e., 3 and 15—are subtracted from the samples. The result gives the absolute increase in diffuser effectiveness compared to the deflection-free zero-work variant, in the following called Δϵ, as shown in Fig. 8. If all samples, also the supplementary experimental data, are included, the correlation gives
$Δϵcorr(Σ)=2.364Σ+0.0104 with R2=0.9267$
(23)
or, if the regression line crosses the point of origin, approximately
$Δϵcorr(Σ)≈2.45Σ with R2≈0.92$
(24)
Fig. 8
Fig. 8
Close modal

### Utilization.

The correlation shown in Eq. (23) gives the increase in diffuser effectiveness Δϵ, rather than the absolute value, in order to facilitate the combined design of turbine and diffuser—independently from the specific rotor geometry. For a conservative approach, the usage of this correlation would be as follows:

1. (1)

Estimate the pressure recovery coefficient from empirical diffuser charts, e.g., Sovran and Klomp [3].

2. (2)

Divide by the swirl-based ideal pressure recovery coefficient cp,ideal to obtain the effectiveness (see Eqs. (2) and (3)).

3. (3)

Add the increase in diffuser effectiveness Δϵ from the presented correlation Eq. (23) for the given stabilization number Σ of the last stage.

4. (4)

Multiply with cp,ideal (see Eq. (3)). The result is the pressure recovery coefficient cp for the specific operating point.

The procedure is outlined in Fig. 9. Clearly, there are certain limitations to this correlation. For one, it is obvious that ϵ cannot be greater than unity. Moreover, it is based on turbine-specific operating points, i.e., Ψ ≥ 0. For Ψ values that are clearly negative in turn, which would suggest compressor-specific behavior of the rotor, the underlying flow physics might differ. The current correlation is given for a diffuser with a half-opening angle of 15 deg; different half-opening angles will most likely yield different slopes for the correlation. Also, further influence factors, such as tip gap heights and different tip designs, will lead to more complex and more precise correlations.

Fig. 9
Fig. 9
Close modal

### Bridging the Gap to the Boundary Layer.

For a more profound understanding of the matter at hand, the initially postulated nexus between the generated tip leakage vorticity and the unsteady boundary layer characteristics yet remains to be numerically dissected.

Figure 10 confirms the notion that greater values for the stabilization number Σ do, in fact, stabilize the boundary layer. Shown are the mass-flow-weighted, circumferentially averaged radial profiles of the nondimensional axial velocity at 50% of the diffuser length. The thickness of the shroud boundary layer is visibly reduced for higher Σ. For the variant Σ = 0.1678, the extent of the flow separation is also significantly smaller. As a result, the free stream velocity decreases when the stabilization number increases. This, of course, is more pronounced in the upper half of the passage but is still detectable in the hub near region.

Fig. 10
Fig. 10
Close modal
The reason for this becomes clear, if the vortex strength at the rotor outflow is considered: a suitable measure for this is the enstrophy
$E=ΩiΩi$
(25)
as it takes into account the positive and the negative direction of rotation. The mass-flow-weighted, circumferentially integrated radial enstrophy profile immediately at the diffuser inlet is shown in Fig. 11 for the top 20% of the channel height. The values above 98% of the channel height are omitted, because the evaluation of the velocity gradients directly at the wall is not reliable and produces unphysically high values. As it is to be expected, the more stable variants with high values for Σ exhibit far greater enstrophy near the shroud wall. The profile for Σ = 0.0613 may appear as an exception, between 92.5% and 96.4%, where it attains higher enstrophy values than Σ = 0.1080.
Fig. 11
Fig. 11
Close modal

The nondimensional blade pressure distributions shown in Fig. 12 reveal why: in fact, the generated blade force depends on the loading coefficient Ψ. Variant Σ = 0.0613, which is listed as sample 13 in Table 2, has a distinctly stronger acceleration on the suction side than Σ = 0.1080 or sample 20. Due to the incidence in the case of sample 13, the deflection is increased. This implies a higher loading coefficient. Yet, at the same time, sample 13 displays a fairly low reduced frequency fred because of the low blade count and rotor speed. Compared to sample 20, less stabilizing vortices are produced per time unit or axial distance (see Fig. 13). Sample 13 also has a rather high flow coefficient that leads to a more axial vortex trajectory (see Fig. 13) and far greater vortex swirl angles ϕsw, as can be seen in Fig. 14. As a result, one can see on the left of Fig. 13 how the flow vectors are deflected slightly upward by the tip leakage vortex. As a consequence, the axial component of the vortex flow is reduced and the stabilizing interaction with the boundary layer is reduced.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

### Comparison to the Experiment.

Even though numerical simulations and experiments show the same increase in diffuser effectiveness Δϵ for given stabilization numbers Σ (see Fig. 8), there is still an offset between both, considering the absolute diffuser effectiveness ϵ (see Fig. 7). Figure 15 shows the axial velocities in the diffuser inlet plane—viewed from the rotating frame of reference—for the experimental design point and the corresponding simulation, both for a mass flow rate of 5.3 kg/s and a rotor speed of 2500 rpm. The velocity deficit in the wake of the numerical solution (on the right) is much more prominent. However, the vectors that indicate secondary flows have a stronger radial component in the experiment (on the left).

Fig. 15
Fig. 15
Close modal

Figure 16 shows the flow field at 8% of the diffuser length—again, viewed from the rotating frame of reference. Generally, in contrast to the plane at diffuser inlet (see Fig. 15), the numerical solution (on the right) experiences more wake mixing and a stronger decay of the secondary flow than its experimental counterpart (on the left). The experiment, in turn, still exhibits distinct secondary flow structures. Thus, it may be concluded that the higher pressure recovery in the experimental setup, i.e., the offset seen in Fig. 7, is a result of more pronounced vortical structures, whereas the numerical simulations generate a more homogeneous flow field in the diffuser. Therefore, the boundary layer separation is delayed further in the experiment. It may be assumed that the diffusivity of the chosen numerical method is too high, which would lead to a rather quick degradation of the vortices; similar results were obtained by Drechsel et al. [22], where the SAS method underpredicted the intensity of the Reynolds stresses, albeit showing better results than simple URANS simulations. However, this does not pose a problem, as the differences in vortex strength and diffuser effectiveness between the numerical variants are adequately rendered by the chosen numerical method.

Fig. 16
Fig. 16
Close modal

## Conclusions

Anisotropic turbulence with dominant wall-normal Reynolds shear stress components has been shown to correlate with boundary layer stabilization in highly loaded axial diffusers. Analytical considerations that identify the measured apparent shear stresses as blade tip vortices are presented. In this paper, their stabilizing properties are linked back to vortex strength, orientation, trajectory, and frequency.

A wide range of experimental and numerical data points confirm the qualitative predictions derived from the analysis: namely, that an increase in the values of the loading coefficient Ψ and of the reduced frequency fred, and, likewise, a decrease in the value of the flow coefficient Φ positively affect pressure recovery in the subsequent axial diffuser. This is mainly the result of a rise in blade tip vortex generation and changes in vortex orientation and trajectory. The developing vortices induce a velocity field with an axial component, resulting in an acceleration of the boundary layer. It is demonstrated that this leads to a reduction in the strength and size of the flow separation, even up to the point that separation is inhibited locally. The turbine design parameters Ψ, Φ, and fred are combined into a nondimensional stabilization number Σ that serves as an indicator for the increase in pressure recovery of the diffuser. For the first time, based on the experimental and numerical results, a correlation between the stabilization number Σ and the increase in diffuser effectiveness Δϵ is formulated.

Subsequent investigations shall focus on expanding the presented correlation to different diffuser opening angles and consider the impact of vortex-induced boundary layer stabilization on the total pressure losses. With further generalization as the main objective, future research should include a wide range of turbine-specific blade geometries and tip clearances, with the influence of transition and flow separation on the blades being taken into account. The use of scale-resolving simulations such as delayed detached eddy simulation (DDES) or large eddy simulation (LES), experimental measurements of blade pressure distributions, and wall shear stress measurements in the diffuser seems promising.

## Acknowledgment

The authors would like to acknowledge the substantial contribution of the DLR Institute of Propulsion Technology and MTU Aero Engines AG for providing TRACE. Specifically, the authors appreciate the support from Martin Franke (DLR) for implementing the SST-SAS turbulence model into TRACE and giving valuable remarks to this work. The authors thank Christoph Jätz and Lars Wein for their valuable thoughts and especially for posing the right questions.

## Nomenclature

• a =

specific work

•
• A =

area

•
• AR =

area ratio of the diffuser

•
• ci, wi =

flow velocities in absolute and relative frame

•
• cp =

pressure recovery coefficient

•
• $f, F$ =

function, discrete function

•
• F =

•
• fbp, fred =

•
• h =

•
• k =

turbulent kinetic energy

•
• l, , c =

length, characteristic length, chord length

•
• Lt =

turbulent length scale

•
• Lt, =

turbulent length scale (kω-SST model)

•
• Lt,σ =

turbulent length scale (Strelets [33])

•
• LvK =

von Kármán length scale

•
• $m˙$ =

mass flow rate

•
• n =

•
• N =

rotational speed

•
• p =

pressure (default: static)

•
• $r,r$ =

•
• R2 =

coefficient of determination

•
• $S, ∂S$ =

closed surface and its boundary

•
• dS, ds =

volume form of $S$, length parametrization of $∂S$

•
• t =

time

•
• u =

rotational velocity

•
• x, xi =

axial coordinate, generalized spatial coordinate

•
• α =

swirl angle in absolute frame of reference

•
• Γ =

circulation

•
• δ, δ1, δ2 =

boundary layer/displacement/momentum thickness

•
• ε =

Levi-Civita symbol

•
• ϵ =

diffuser effectiveness

•
• ϑ, θ =

circumferential coordinate, single pitch

•
• ν =

kinematic viscosity

•
• ξ =

relative chord length

•
• ρ =

density

•
• σ =

Strelets blending function [33]

•
• Σ =

stabilization number

•
• τ =

characteristic convective time [33]

•
• $τ̃$ =

Reynolds shear stress sum [19]

•
• ϕsw =

vortex swirl angle

•
• Φ,Ψ =

•
• ω =

specific turbulent dissipation rate

•
• $Ω, Ωi, ΩSW$ =

vorticity, streamwise vorticity

•
• $E$ =

enstrophy All quantities are time-averaged, unless noted otherwise

### Subscripts

Subscripts

annular diffuser, conical diffuser

•
• ave =

average

•
• axv, azv =

axial/azimuthal with respect to vortex axis

•
• core =

vortex core quantity

•
• corr =

correlated

•
• dyn =

dynamic quantity

•
• i, j, k =

generic indices

•
• in, out =

diffuser inlet/outlet

•
• ind =

induced

•
• tot =

total quantity

•
• I, II =

rotor inlet plane, rotor exit plane

•
• ∞ =

free stream quantity

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