total pressure loss reduction in annular diffusers · 2019. 5. 13. · total pressure loss...

Total Pressure Loss Reduction in Annular Diffusers

Dajan Mimic1,2,3, Christoph Jätz2, Philipp Sauer2 and Florian Herbst1,2

1 Junior Research Group Multiphysics of Turbulent Flows

2 Institute of Turbomachinery and Fluid Dynamics

Leibniz Universität Hannover Appelstrasse 9, 30167 Hanover, GERMANY

3 E-mail: [email protected]

ABSTRACT Power output and efficiency of gas turbines depend strongly

upon the achievable pressure rise in the subsequent diffuser. In combination with the requirement to keep diffuser length to a minimum, ever steeper opening angles are sought, while avoiding diffuser stall.

In terms of diffuser pressure rise, the boundaries of what is achievable can be pushed further if the tip leakage vortices from the last stage are used to re-accelerate the diffuser boundary layer, thus delaying separation onset. Such measures have been shown to decrease total pressure losses as well.

In this paper, we show that the benefit of total pressure loss re-duction in vortex-stabilised diffusers becomes more pronounced for steeper opening angles by means of a numerically and experimen-tally validated approach. In extension, we provide evidence that the loss production in highly loaded vortex-stabilised diffusers, which would stall otherwise, can be brought down to the level of non-stalling diffusers. Furthermore, we present a detailed analysis of the different loss mechanisms and their response to vor-tex-stabilisation of the diffuser.

NOMENCLATURE Symbols � cross-sectional area of the diffuser AR area ratio of the diffuser �, � flow velocity �� pressure recovery coefficient �� specific isobaric heat capacity �r reduced frequency ℎ enthalpy (default: static) ℓ chord length

meridional coordinate

number of blades � rotational speed in revolutions per minute � pressure (default: static) Euler radius � specific gas constant � temperature (default: static) � rotational velocity � generalised spatial coordinate � axial coordinate � flow angle, whirl angle

curve

diffuser half-opening angle Δ difference

diffuser effectiveness

total pressure loss coefficient circumferential coordinate

Lamé constant Λ loss rectification number dynamic viscosity � kinetic energy coefficient � rectified total pressure loss coefficient � density � generalised spatial vector Σ stabilisation number , Ψ flow coefficient, loading coefficient

Subscripts I, II rotor inlet/outlet plane eff effective corr correlated in, out diffuser inlet/outlet ref reference rel relative t turbulent quantity tot total quantity

enthalpy-induced

dilatational shearing-induced

thermodynamic � vorticity-induced

INTRODUCTION Without the use of exhaust diffusers, the expansion of hot gas

achievable in turbines is bounded by the ambient pressure. Only the subsequent conversion of kinetic energy into static pressure, real-ised by an increase in cross-sectional area in the diffuser, allows for considerably higher expansion ratios in the turbine. As a conse-quence, the power output and—assuming constant heat in-put—efficiency increase.

The resulting main aerodynamic design goal of exhaust gas dif-fusers is to convert as much kinetic energy as possible into static pressure, i.e., maximise the ratio of the static pressure rise over the diffuser to the kinetic energy at diffuser inlet. Diffuser designers tend to call this ratio pressure recovery and express it in terms of the non-dimensional pressure recovery coefficient

�� = � − �i � ,i − �i . (1)

International Journal of Gas Turbine, Propulsion and Power Systems April 2019, Volume 10, Number 2

Manuscript Received on October 2, 2018 Review Completed on April 10, 2019

Copyright © 2019 Gas Turbine Society of Japan

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If an incompressible, inviscid and irrotational flow is assumed, it can be shown that the pressure recovery in an annular diffuser is determined by

��,i al = − AR + tan � i + tan � . (2)

This ideal pressure recovery coefficient is a function of the area ratio of inlet to outlet of the diffuser AR = � �i⁄ , whirl angle � = tan− � /�� and ratio of Euler radii i / . It represents the absolute upper limit of pressure recovery realisable by a certain diffuser geometry for a given whirl angle.

To compare between diffusers across different geometries, it is advisable to compare how close their pressure recovery coefficients come to the respective upper limits. This is called the diffuser effectiveness and is defined as

= ��,i al. (3)

Since a full conversion of kinetic energy into total pressure would require an infinitely long diffuser, the exit flow of a finite diffuser always contains an amount of residual kinetic energy, whose coef-ficient is given by

� = � , − �� ,i − �i . (4)

Real diffuser flows are, in contrast to ideal ones, viscous and rota-tional and, therefore, produce entropy. Entropy characterises the devaluation of energy—or the loss of exergy, i.e. usable energy. For an adiabatic flow, it can be equated to a total pressure loss via

Δ = −� ln � , � ,i⁄ . (5)

Again, in its non-dimensional form, the total pressure loss coeffi-cient is defined as

= � ,i − � ,� ,i − �i . (6)

From Bernoulli’s equation, it can be derived that those three coef-ficients always fulfil

�� + � + = . (7)

It follows that a change of this pressure budget, e.g., a reduction of the total pressure loss, affects all its constituents. This interde-pendence bears several implications, of which two shall be ad-dressed briefly.

1. Because longer diffusers yield increased frictional total pres-sure losses and higher investment costs, diffuser designer seek to keep diffuser length minimal. Since pressure recovery depends primarily on the area ratio, shorter diffusers neces-sitate more aggressive aerodynamic loading by means of steeper opening angles. These, in turn, render the flow more susceptible to boundary layer separation or even stall. In case of flow separation, total pressure losses would increase and pressure recovery decrease drastically.

2. Residual kinetic energy at the diffuser outlet is often regarded as waste energy when the mere gas turbine is considered. If, however, the system boundaries are expanded to include the entire power plant, the residual kinetic energy is necessary to overcome the flow resistance imposed by the subsequent exhaust stack and, more specifically, heat recovery steam generator in the case of a combined gas-and-steam turbine cycle. Here, a sufficient kinetic energy of the flow is crucial for the combined-cycle efficiency.

Evidently, future increases in overall cycle power and efficiency will require combined design approaches that include the diffuser and its neighbouring components.

Diffuser research A first step towards more elaborate design approaches for dif-

fusers lies in the development of empirical design charts that ac-count for the effect that boundary layer thickening and separation have on diffuser performance. Examples for such design charts for steady, homogeneous inflow conditions are [1,2] or, with a multi-tude of additional corrections for different flow conditions, [3].

With the advancement of computational fluid dynamics, the in-vestigation of more complex, highly rotational flows, e.g., due to vortex generators, and their impact on diffuser performance be-came of interest (e.g. [4]).

The ongoing increase in affordable computing power, made the analysis of unsteady, heterogeneous inflow conditions viable. Kluß et al. showed by means of numerical simulations that unsteady wakes and secondary flow can lead to a rise of the pressure recov-ery of an annular diffuser above the values predicted by empirical design charts [5]. These findings were confirmed by simulations and measurements [6–9].

Vassiliev et al. ascertained that diffuser performance is influ-enced by inlet Mach number, total pressure distribution, flow angle, and turbulence characteristics of the inflow [10].

These aggregated observations necessitate the conclusion that the outflow conditions of the upstream stage have to be taken into account for the design of efficient diffusers. Mihailowitsch et al. demonstrated in a numerical study that allowing some tip leakage flow in the last turbine stage can improve the combined efficiency of turbine and diffuser by enhancing diffuser performance [11].

Stabilisation number Mimic et al. showed that the stabilising mechanism is the ve-

locity field induced by tip leakage vortices generated in the turbine rotor [12]. Elevated turbulence levels prevailing in the tip region may be casually taken for the underlying nexus, as they correlate with the vortex strength; they are, however, merely a symptom and may even act to the detriment of the diffuser performance, as will be shown in this paper.

A reduced-order model, that predicts changes in diffuser effec-tiveness Δ due to a variation of the stage-operating point, was devised in [12]. For that purpose, a novel stabilisation number Σ was defined based on the dimensionless stage parameters loading coefficient Ψ, flow coefficient Φ, and reduced frequency �r (Eqs A1 to A3, as detailed in Appendix A):

Σ ≡ Ψ�r . (8)

The aerodynamic intuition behind this number is that it captures essential characteristics of the blade tip vortices that depend on the stage operating point: vortex strength due to blade loading, cir-cumferential vortex trajectory and vortex streamline orientation due to deflection, and spatial as well as temporal vortex packing due to reduced frequency.

It was shown that the increase in effectiveness from reference operating point of the rotor Δ correlates linearly with the stabili-sation number Σ. This was done for an annular diffuser with a half-opening angle of 15°, which yields

∆ rr Σ ≈ . Σ (9)

or, using the pressure coefficient ��, rr,

��, rr ≈ . Σ + r ��,i al (10)

with r being the diffuser effectiveness at reference operating point. The underlying Σ model was subsequently tested and ex-panded to correlations for various diffuser opening angles in [13].

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Fig. 1 Diffuser effectiveness against stabilisation number Σ for various diffuser half-opening angles

(adapted from [13])

Figure 1 shows some examples of correlations obtained. For better clarity, only numerical samples are shown and the absolute effec-tiveness is used.

Total pressure losses

The use of the diffuser effectiveness instead of the pressure recovery coefficient �� enables us to compare diffusers across a wide parameter range, independently from whirl angle, opening angle and diffuser length, by relating the pressure recovery coeffi-cient to the pressure recovery coefficient of an ideal diffuser under the same flow conditions.

A similar premise has to be defined for the total pressure loss coefficient . The ideal diffuser, however, is loss-free by definition and cannot be used as a benchmark for the total pressure losses. Instead, a quantifier is needed, which selects for losses that stem from the phenomena studied, i.e., blade tip vortices and flow sepa-ration, yet ignores the effect of other factors, such as diffuser whirl and wake mixing.

For that reason, Mimic et al. introduced the rectified total pres-sure loss coefficient [14]:

� = Λ , (11)

where Λ represents the loss rectification factor. For the rotor inves-tigated, it is defined as

Λ ≡ �II,r l r lr l (12)

with

r l ≡ cos �cos �r , r l = r and �II,r l = �II�II,r . (13)

Here, r l accounts for the change in flow path length due to dif-ferent diffuser whirl angles. The ratio r l incorporates a simple

Fig. 2 Absolute difference in rectified diffuser total pressure loss

from reference Δ� against stabilisation number Σ (adapted from [14])

estimate for the velocity deficit. The influence of the dynamic pressure on wake mixing is characterised by �II,r l. Details on the definition of Λ can be found in [14]. It was demonstrated that the rectified total pressure loss coefficient � correlates well with Σ for numerical and experimental data, as seen in Fig. 2, with the corre-sponding equation

∆� rr Σ ≈ − . Σ. (14)

METHOD

Test facility Since the experimental validation of the Σ model has already

been presented in [12–14], the test rig shall be addressed just briefly in this paper.

The diffuser is divided into an annular and a conical section, of which the former is the object of study in this paper. The conical section has a half-opening angle of 15° and an area ratio of 1.78 between diffuser inlet and outlet. Further details are given in Tab. 1 in Appendix B. Following empirical diffuser design charts, like those provided in [2], it can be seen that the diffuser is very sus-ceptible to stall under steady, homogeneous inflow conditions.

To investigate the behaviour under unsteady, heterogeneous in-flow, the test rig is equipped with a NACA0020-bladed rotor up-stream of the diffuser. The symmetric, twisted blades are aerody-namically unloaded at design operating point and, when operated at underspeed, provide blade tip vortices similar to those found in turbines. The rotor can be equipped with either 15 or 30 blades.

Under design conditions, the inlet Mach number is approxi-mately 0.1 and the rotor spins at 2500 RPM. Further details con-cerning the test rig and instrumentation are found in [8].

Simulation We conducted the numerical simulations used for this study with

the non-commercial solver TRACE 8.2 (Turbomachinery Research Aerodynamics Computational Environment) from the Institute of

0

0.1

0.2

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

ϵ

Σ

Simulations 5° Simulations 10°


Linear (Simulations 5°) Linear (Simulations 10°)


-0.1

-0.08

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-0.05 0 0.05 0.1 0.15 0.2

Δϖ Σ

Experiment 15° Simulation 15° Linear (All)

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Propulsion Technology at the German Aerospace Center (DLR). In order to capture secondary flow and separation to the greatest extent possible, yet to maintain a reasonable level of computational effort, we used the Shear Stress Transport Scale-Adaptive Simula-tion (SST-SAS) modelling approach proposed by Menter and Egorov in [15] with the �-�-SST turbulence model [16] in the version published in [17], fully turbulent wall treatment and a stagnation point anomaly fix [18]. To further encourage the for-mation of rotational flow structures, inlet turbulence intensity was kept below 0.1 %. The reasons for choosing the SST-SAS approach for this diffuser test case have been discussed in [19]. A more detailed description of the numerical setup is presented in [12]. Detailed analyses of scale-resolving simulations in TRACE have been performed by Franke and Morsbach [20].

The numerical domain emulates a single pitch of the rotor and the annular section of the diffuser test rig. Geometry and aerody-namic design point are identical to the test rig. In addition, the domain is extended for a coarsened outlet section. Oscillations that would otherwise cause instability of the solution are thus prevented from reaching the outlet. For the sake of illustration, three rotor pitches are shown in Fig. 3. The individual meshes comprise be-tween 1.7106 and 2.4106 cells. All meshes are refined signifi-cantly in the region where the tip leakage vortex forms and where flow separation occurs or high velocity gradients are expected. A verbose grid convergence study of a similar mesh and the SST model is presented in [9].

To cover a wide range of values for the stabilisation number Σ, we chose to vary rotor speed, mass-flow rate and solidity, i.e., blade count of the rotor. With regard to the diffuser part, we performed simulations with half-opening angles of 5°, 10°, 12° and 15° and a constant diffuser length, thus resulting in different area ratios. All diffusers, except 5°, exhibit flow separation when no vor-tex-induced stabilisation occurs.

Because we adjusted the outlet pressure for the target mass flow and due to slow convergence rates, coupled with the highly non-linear stall behaviour of the diffuser, it proved difficult to reach evenly distributed values of Σ.

Validation The predictive power of the Σ model has been discussed thor-

oughly in previous publications: it was shown in [12] that the model is able to predict the increase in pressure recovery of a 15° diffuser due to vortical inflow conditions. The predictions match experi-mental as well as numerical data. The underlying assumptions of the Σ model were then tested against experimental results of a 20° diffuser and numerical results of diffusers with half-opening angles of 5°, 10° and 12° [13]. As a result, the model proved successful in rendering accurate predictions of the increase in pressure recovery and additional correlations for these test cases were derived.

Fig. 3 Computational domain (coarse mesh and multiple rotor

pitches for display, flow direction from right to left) [13]

Regarding the total pressure losses in the diffuser, it was demonstrated in [14] that the model is likewise able to predict the reduction in total pressure losses for a 15° diffuser, as seen in ex-periment and simulations. For the time being, no total pressure loss data for the 20° diffuser is available to us.

Because the diffuser variants 15° and 20° are characterised by a violently separated flow in the vicinity of the aerodynamic design point of the rotor, we consider these the most critical and difficult test cases. This becomes also apparent in the stability issues that we encountered when simulating those diffuser variants.

However, considering that even those critical diffuser test cases could be predicted accurately using the Σ model and the 15° dif-fuser could also be rendered numerically, it is safe to assume that correlations for shallower opening angles with significantly smaller or even no separation can be derived from purely numerical data. Hence, without further mention, only numerical data will be used in the following.

RESULTS

Correlations Negative linear correlations between the stabilisation number Σ

and the rectified total pressure loss coefficient � can be formed for all opening angles investigated in the same way as presented in [13]. Figure 4 shows that an increase in Σ, i.e., reduced flow sepa-ration, leads to a reduction of the rectified total pressure losses in the diffuser for all diffuser opening angles investigated, with the exception of the 5° diffuser. Since this diffuser exhibits fully at-tached flow for all operating points, its rectified total pressure loss coefficient remains virtually unchanged across different values of Σ, bar a negligible increase due the incoming vortices.

The �-intercept, that is the rectified total pressure loss coeffi-cient of a non-stabilised diffuser, grows increasingly with the dif-fuser half-opening angle . Similarly, the slopes of the graphs vary in approximately the same manner with the half-opening angle. For the greatest values of the stabilisation number Σ investigated, the rectified total pressure losses of the most highly loaded diffuser ( =15°) can be reduced almost to the level exhibited by shallower diffusers with significantly smaller or even no flow separation ( =5°). Although not shown here, this is also valid for the non-rectified total pressure losses. The correlations corresponding to Fig. 4 are given in Eqs (C1) to (C4) in Appendix C.

Fig. 4 Rectified total pressure losses � against stabilisation

number Σ for various diffuser half-opening angles

0

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

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Entropy production Total pressure loss quantifies the loss of exergy, i.e., usable en-

ergy in a flow and, as such, satisfies the definition of an increase in entropy. An analysis of the entropy gradient not only allows us to locate exactly where exergy losses occur, but also enables us to determine by which mechanism they are caused.

In order to do this, we make use of Crocco’s theorem, which can be derived analytically from the momentum equation of a fluid and the second law of thermodynamics. For a steady flow, this gives

∇ = � ∇ℎ + � ∇ × � × � + �� ∇ ⋅ ∇⨂� T+ +�� ∇ ∇ ⋅ � , (15)

where is the so called Lamé constant and characterises the influ-ence of the bulk viscosity. We invoke Stokes’ hypothesis and ignore the bulk viscosity, so that = − ⁄ . This yields the following simplified form of Eq. (15):

∇ = � ∇ℎ + � ∇ × � × � + �� ∇ ⋅ ∇⨂� T+ �� ∇ ∇ ⋅ � . (16)

To account for the effect of turbulence we substitute with the effective viscosity , which is the sum of molecular and turbulent viscosity, i.e.,

= + . (17)

Equation (16) can now be rewritten as

∇ = � ∇ℎ + � ∇ × � × � + �� ∇ ⋅ ∇⨂� T+ �� ∇ ∇ ⋅ � . (18)

For better clarity, we can write Eq. (18) in a more compact form as

∇ = ∇ + ∇ � + ∇ + ∇ . (19)

Here, ∇ is the entropy gradient induced by the gradient of total enthalpy ℎ . Unlike the original formulation of the theorem by Crocco in [21], we include the entropy gradient caused by mo-mentum diffusion ∇ and volume dilatation ∇ . The second term, ∇ �, arises from the decomposition of the advective term in the momentum equation, using the Lagrangian vector identity:

� ⋅ ∇⨂� T = ∇ × � × � + ∇ � ⋅ � . (20)

The second term of Eq. (20), i.e., the gradient of kinetic energy, is absorbed, together with the enthalpy gradient, into the gradient of total enthalpy, wherefore it does not appear in Eq. (15). The first term represents the Lamb vector. Its formulation resembles that of the Coriolis acceleration, which is not surprising, as it acts in quite the same way. This means that, where the velocity vector and vor-ticity vector are not collinear, a fluid parcel receives an acceleration perpendicular to its original path. Because such a process implies a redistribution of momentum across an increasing number of de-grees of freedom, it manifests itself in the development of an en-tropy gradient. This entropy gradient, which is expressed in the term ∇ �, occurs in every rotational flow, even if it is inviscid. The notable exception to this is the Beltrami flow, where the velocity vector and vorticity vector are aligned throughout the entire do-main. The case of a Beltrami flow, however, must not be confused with that of a longitudinal vortex. This insight is significant and so is the hypothesis that we can deduce from it:

If we assume that a rotational flow experiences a rise in entropy, even in case of an inviscid fluid, introducing reverse rota-tion into said flow, reduces its entropy compared to the initial, non-stabilised flow.

To test this hypothesis, we computed the entropy sources as represented by the individual terms in Crocco’s theorem and will discuss them for a total of four exemplary variants: 10° and 12° half-opening angle, each one for low and high values of the stabi-lisation number Σ.

Computation of entropy. The entropy distribution itself can be obtained by utilising a fundamental property of gradient vector fields: since entropy is a scalar quantity, its gradient field is irrota-tional, scil. conservative. Therefore, any line integral is solely dependent on its end points; otherwise it is path-independent.

Let be any curve from � to �. Then the entropy at point � can be evaluated using

� = ∫ ∇ � ⋅ d� + ��[�0,�] . (21)

For the problem at hand, we performed the integration numerically using the post-processing and flow-visualisation software Tecplot and pragmatically, beginning from diffuser inlet along lengthwise grid lines. To yield the boundary value � , we computed the entropy distribution at the diffuser inlet from the thermodynamic definition of the entropy, namely � �i :

� ∶= � �i = �� ln (� �i�̅i ) − � ln (� �i�̅i ), (22)

where �̅i and �̅i represent the spatially averaged static tempera-ture and pressure at the diffuser inlet, �i denotes a location at the diffuser inlet and �� stands for the specific isobaric heat capacity. The flow fields represent temporal averages over one blade passing. Figure 5 shows the integrated entropy distribution in a representa-tive meridional plane of the diffuser. Iso-lines of � are shown for comparison. It can be seen that the results of both, the thermody-namic definition of entropy and the integration the entropy gradient obtained by Crocco’s theorem, show very good agreement. Some smearing of the result, visible as longitudinal stripes, occurs in the case of the integration, which can be attributed to the limited accu-racy of the software used. This issue will be addressed in future works, as the approach seems very promising for the detailed analysis of loss mechanisms.

Because scalar fields and their gradient vector fields allow su-perposition, the constituents of the overall entropy gradient, i.e., ∇ , ∇ �, ∇ and ∇ , can be integrated in exactly the same way to obtain the respective entropy distributions. Since for the entropy components, we aimed to compute only the increase in the diffuser domain, we chose zero as boundary condition for the integration and denote the resulting scalars as differences:

Δ � = ∫ ∇ � ⋅ d� +�[�0,�] . (23)

As a consequence, the following equation is fulfilled:

� = Δ + Δ � + Δ + Δ + � �i . (24)

Because all entropy terms calculated refer to the reference state at diffuser inlet or zero as a reference, negative values are possible.

Overall entropy. The left column of Fig. 5 shows that in the absence of stabilising blade tip vortices, as indicated by the low Σ value, the shroud boundary layer regions of the 10° diffuser, and even more so of the 12° diffuser, exhibit a considerable amount of

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Fig. 5 Contour map: entropy , as per Eq. (21) Iso-lines: thermodynamic entropy �, as per Eq. (22)

Fig. 6 Contour map: enthalpy-induced entropy Δ , as per Eq. (23) Iso-lines: thermodynamic entropy �, as per Eq. (22)

entropy. The presence of vortices, as seen on the right for high values of Σ, leads to a slight increase of entropy in their vicinity. However, boundary layer entropy decreases remarkably. For the 12° diffuser, this leads to a strong reduction of overall entropy, while in the case of the 10° diffuser a significant redistribution takes place. This outcome agrees with the total pressure losses and their respective sensitivities obtained.

For the purpose of further analysis, we will address the indi-vidual terms from Eq. (24). As expected, the entropy Δ , which is caused by volume dilatation, vanishes compared to the other com-ponents due to the low Mach numbers and will, therefore, be omit-ted from this analysis.

Total enthalpy. Both diffuser variants, 10° and 12°, exhibit ele-vated entropy levels in nearly the entire main flow and especially near the outer edge of the shroud boundary layer, as shown in Fig. 6. Stabilisation of the boundary layer, and the resulting reduc-

Fig. 7 Contour map: vorticity-induced entropy Δ �, as per Eq. (23) Iso-lines: thermodynamic entropy �, as per Eq. (22)

Fig. 8 Contour map: shear-induced entropy Δ , as per Eq. (23) Iso-lines: thermodynamic entropy �, as per Eq. (22)

tion in flow separation, leads to a reduction of sharp gradients of the stagnation enthalpy and, therefore, to an overall reduction in en-tropy in comparison to the non-stabilised diffuser. The exception to this correlates with the location of the vortex trajectories, where entropy levels are still slightly elevated, yet lower than in the non-stabilised case.

Lamb vector. While the 12° diffuser experiences a strong reduc-tion of vorticity-induced entropy Δ � in the boundary layer region upon stabilisation due to reduced vorticity, Δ � in the non-stabilised 10° diffuser is already very small, to begin with, as shown in Fig. 7. Here, stabilisation grants merely a residual benefit.

In contrast, both diffuser variants exhibit a distinct increase in Δ � in the vicinity of the vortex trajectories. In the case of the 10° diffuser this increase seems even to outweigh the reduction achieved in the boundary layer. Therefore, the hypothesis stated earlier can be confirmed only partly. Even though the blade tip

Iso-lines: � in J/ � ⋅ kg)

= 10° Σ = 0.0144

in J/ � ⋅ kg): -1 0 1


= 10° Σ = 0.1328

in J/ � ⋅ kg): -1 0 1

= 12° Σ = 0.0123


in J/ � ⋅ kg): -1 0 1 -1 0 1 in J/ � ⋅ kg):


= 12° Σ = 0.1323


= 10° Σ = 0.0144

Δ in J/ � ⋅ kg): -0.8 0 0.8


= 10° Σ = 0.1328

Δ in J/ � ⋅ kg): -0.8 0 0.8

= 12° Σ = 0.0123


Δ in J/ � ⋅ kg): -0.8 0 0.8 -0.8 0 0.8 Δ in J/ � ⋅ kg):


= 12° Σ = 0.1323


= 10° Σ = 0.0144

Δ � in J/ � ⋅ kg):

-0.6 0 0.6


= 10° Σ = 0.1328

Δ � in J/ � ⋅ kg): -0.6 0 0.6

= 12° Σ = 0.0123


Δ � in J/ � ⋅ kg):

-0.6 0 0.6 -0.6 0 0.6 Δ � in J/ � ⋅ kg): Iso-lines: � in J/ � ⋅ kg)

= 12° Σ = 0.1323


= 10° Σ = 0.0144

Δ in J/ � ⋅ kg):

-0.2 0 0.2


= 10° Σ = 0.1328

Δ in J/ � ⋅ kg): -0.2 0 0.2

= 12° Σ = 0.0123


Δ in J/ � ⋅ kg): -0.2 0 0.2 -0.2 0 0.2 Δ in J/ � ⋅ kg):


= 12° Σ = 0.1323

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vortices do reduce entropy locally, they can also increase entropy locally, if placed in an irrotational velocity field, as it is the case outside of the boundary layer region. Whether the entropy increases or decreases globally, is determined by the rotation of the initial, non-stabilised velocity field.

Momentum diffusion. The entropy caused by momentum diffu-sion is relatively small in comparison to the other components. Nevertheless, a significant reduction is observed in Fig. 8 for both opening angles, whereas no increase due to the blade tip vortices occurs.

In our formulation, the shear-induced entropy depends not only on the molecular, but also on the turbulent viscosity. It follows that an increased turbulence level exacerbates the entropy caused by momentum diffusion. We presume that—all other parameters re-maining constant—stronger blade tip vortices would be required to negate these additional losses. If vortex strength increases, the intensity of the Lamb vector increases correspondingly, along with the associated entropy. Based on this, we conjecture the following design objective for boundary-layer–stabilisation devices:

To minimise loss production, boundary-layer–stabilisation devices must generate coherent, well-oriented vortices, while keeping turbulence production minimal.

CONCLUSIONS In this paper, we demonstrated that the Σ model for loss predic-

tion is applicable to a wide range of diffuser opening angles and we devised corresponding correlations. The sensitivity of the rectified total pressure loss coefficient � towards Σ grows disproportion-ately with the diffuser opening angle. As a result, the losses gener-ated in the steepest diffuser could be reduced to the level of the shallowest, non-stalling diffuser.

This leads to the important realisation that steeper vor-tex-stabilised diffusers can achieve higher pressure recovery than shallower diffusers, while maintaining a comparable level of recti-fied total pressure losses. Therefore, the notion of the optimal diffuser needs to be reconsidered with regards to stabilising vortical inflow.

Based on Crocco’s theorem, we introduced an approach to evaluate the entropy of a flow field that distinguishes between different sources of entropy. The approach proved to be promising for future application.

Vorticity-induced losses, as characterised by the Lamb vector, can be reduced locally by means of reverse flow field rotation relative to the prevalent direction of rotation. However, this reverse rotation may lead to a redistribution of entropy or even a global increase. In contrast, entropy caused by momentum diffusion and by gradient of total enthalpy was reduced globally.

Future works will focus on a refinement of the analysis approach presented, as well as application to more complex flow problems.

ACKNOWLEDGEMENTS We would like to acknowledge the valuable contribution of the

DLR Institute of Propulsion Technology and MTU Aero Engines AG for providing TRACE. We appreciate the support from Martin Franke (DLR) for implementing the SST-SAS turbulence model into TRACE.

APPENDIX A: ROTOR DESIGN PARAMETERS The rotor design parameters are defined as follows (see [22]):

Ψ ≡ � ,I − � ,II� . (A1)

≡ �� . (A2)

�r ≡ � ℓ �� . (A3)

APPENDIX B: TEST RIG GEOMETRY Table 1 Geometric properties of the test rig

Rotor Annular diffuser Aerofoil NACA0020 Area ratio 1.78

Number of blades 15/30 Half-opening angle 15° Hub diameter 280 mm Length 235 mm

Blade height 97 mm Hub radios 140 mm

Tip gap 1 mm Shroud radius ( inlet) 237 mm

Hub/tip stagger angle 43°/58° Shroud radius (outlet) 300 mm

APPENDIX C: CORRELATIONS

As shown in Fig. 4:

∆� rr Σ, δ = ° = . Σ + . . (C1)

∆� rr Σ, δ = ° = − . Σ + . . (C2)

∆� rr Σ, δ = ° = − . Σ + . . (C3)

∆� rr Σ, δ = ° = − . Σ + . . (C4)

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