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Research Papers

# Aerodynamic Similarity Principles and Scaling Laws for Windmilling FansPUBLIC ACCESS

[+] Author and Article Information
Dilip Prasad

Pratt and Whitney,
East Hartford, CT 06108
e-mail: dilip.prasad@pw.utc.com

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received August 11, 2018; final manuscript received August 28, 2018; published online October 15, 2018. Editor: Kenneth Hall.

J. Turbomach 140(12), 121004 (Oct 15, 2018) (10 pages) Paper No: TURBO-18-1197; doi: 10.1115/1.4041375 History: Received August 11, 2018; Revised August 28, 2018

## Abstract

Windmilling requirements for aircraft engines often define propulsion and airframe design parameters. The present study is focused on two key quantities of interest during windmill operation: fan rotational speed and stage losses. A model for the rotor exit flow is developed that serves to bring out a similarity parameter for the fan rotational speed. Furthermore, the model shows that the spanwise flow profiles are independent of the throughflow, being determined solely by the configuration geometry. Interrogation of previous numerical simulations verifies the self-similar nature of the flow. The analysis also demonstrates that the vane inlet dynamic pressure is the appropriate scale for the stagnation pressure loss across the rotor and splitter. Examination of the simulation results for the stator reveals that the flow blockage resulting from the severely negative incidence that occurs at windmill remains constant across a wide range of mass flow rates. For a given throughflow rate, the velocity scale is then shown to be that associated with the unblocked vane exit area, leading naturally to the definition of a dynamic pressure scale for the stator stagnation pressure loss. The proposed scaling procedures for the component losses are applied to the flow configuration of Prasad and Lord (2010). Comparison of simulation results for the rotor-splitter and stator losses determined using these procedures indicates very good agreement. Analogous to the loss scaling, a procedure based on the fan speed similarity parameter is developed to determine the windmill rotational speed and is also found to be in good agreement with engine data. Thus, despite their simplicity, the methods developed here possess sufficient fidelity to be employed in design prediction models for aircraft propulsion systems.

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## Introduction

When an aircraft engine ceases to be operational under flight conditions, the fan rotor undergoes “windmill” behavior, wherein it continues to rotate in the same direction as the powered state, but without the addition of work to the flow. Although this situation is an exceptional one, it defines propulsion- and aircraft-system parameters [1]. For example, the vertical stabilizer is sized to allow for engine-out at takeoff, while engine-out drag at cruise sets the nacelle lines. Additionally, the maximum continuous thrust requirement for a twin-engine aircraft with one engine inoperational can influence the choice of core size. Thus, the necessity to meet windmill requirements affects both weight and fuel burn.

The reduction in fan pressure ratio and the accompanying bypass ratio increase in recent years have accentuated the influence of windmill-related design parameters. It is, therefore, not surprising that the study of windmilling has received greater attention [28]. Traditional approaches to the study of windmilling were based on empirical scaling and cycle matching of component maps [9,10]. While these methods can provide useful estimates of design parameters, they often lack physical insight and their extensibility to new designs may be limited. This motivated the study of Prasad and Lord [2], wherein engine test data were analyzed to demonstrate that the fundamental parameter governing windmill rotational speed is the throughflow velocity rather than the flight Mach number, as had been previously assumed [11]. In addition, it was found that the stagnation pressure loss across the fan stage increases sharply with throughflow. Numerical simulations were carried out to elucidate the test results. It was found that the losses across the rotor and core splitter are relatively small, with the bulk of the loss occurring across the stator. At this unpowered condition, the only work performed by the rotor is that required to overcome the relatively small bearing friction and windage. Consequently, the work coefficient at windmill is two orders of magnitude less than at idle. With no swirl in the absolute flow entering the rotor, this state of near-zero work implies that the absolute flow leaving the rotor is also swirl-free in a mean sense. The stator, therefore, operates at a strong negative incidence and a gross separation ensues, producing significant loss. More recent work [5,8] has verified these observations and further clarified the loss mechanisms.

The windmill parameters of interest to the designer are the fan rotational speed and stage loss. The challenge in determining these quantities arises from the fact that the flows encountered are severely off-design, making their prediction using steady numerical methods difficult. The prediction of windmill performance has, therefore, relied entirely on unsteady simulations [2,7,8]. When these simulations must be carried out over a range of operating conditions, considerable computational expense is required, making them impractical for use in design optimization. This deficiency of the state of the art has also been pointed out by Dufour and Thollet [7], who proposed the use of a body-force model in conjunction with Euler simulations to reduce the computational cost. One aspect of the problem that has seen progress with respect to model development is the determination of the fan rotational speed. Specifically, a reduced-order model was developed in Ref. [2], making use of the requirement of zero rotor work, and the same principle was incorporated into the pseudo-time stepping computational fluid dynamics-based procedure of Gunn and Hall [8].

## Scope of Paper

The objective of the present study is to develop a predictive scheme to more expediently determine the fan stage flow characteristics and losses under windmill conditions. Specifically, procedures that avoid the need for multiple, computationally intensive simulations are sought. The approach adopted consists of first determining the fundamental scaling laws and parameters of the flow, and then using these to deduce the losses and fan rotational speed. To facilitate the scaling, the intrastage flow between the rotor and the stator and that through the vane are studied separately.

Commencing with the former, an asymptotic model for the rotor exit radial profiles is developed. This model serves to bring out the self-similar nature of the intrastage flow, an aspect that has escaped attention in the previous studies. In particular, a similarity parameter for the rotational speed is identified, and the loss associated with the rotor and splitter is shown to scale with the dynamic pressure.

An analogous scaling-based approach is employed for the stator flow. Although this flow is highly unsteady owing to the previously mentioned separation, considerable insight can be gained by examining the time-averaged properties. Thus, it is found that the blockage induced by the separated flow in the vane passage varies little over a broad range of throughflow rates. The flow through the unblocked area defines a characteristic velocity, use of which collapses the spanwise profiles of the vane exit velocity. By analogy with the approach used for the rotor, this suggests that the natural scale for the vane stagnation pressure loss is the dynamic pressure associated with the unblocked vane exit flow.

Based on the similarity principles investigated in the present study, scaling procedures for the component losses and fan rotational speed are developed. Application of these procedures to the flow configuration of Ref. [2] reveals good agreement with the simulation results for the losses and with engine data for the rotational speed. In particular, the stator loss is shown to feature a choking-type behavior, exhibiting sharp growth at elevated throughflow rates, reminiscent of previous observations [2]. These results demonstrate the feasibility applying the present methods to estimate the windmill parameters in design procedures for aircraft propulsion systems.

## Intrastage Flow

The configuration to be studied is illustrated in meridional view in Fig. 1, which has been adapted from Ref. [2]. Cylindrical-polar coordinates are used with x, θ, and r denoting the axial, tangential, and radial directions, respectively. The axial and radial coordinates are normalized by Rt, the fan leading edge tip radius. The system comprises the inlet, fan rotor, core- and bypass-ducts, and the fan stator. Engine stations are also indicated in Fig. 1, with 1, 2A, and 14 indicating the free stream, fan inlet, and fan stage exit, respectively. As in Ref. [2], the intermediate stations RE and VI, located at the rotor exit and vane inlet, respectively, are also employed.

###### Rotor Exit Flow.

The study commences with an examination of the flow behavior between the fan rotor and stator in the region away from the core splitter. Focusing attention on the region immediately downstream of the rotor, the flow is taken to be inviscid, axisymmetric, and steady with ideal-gas properties. The use of the relationship between radial profiles of flow quantities and gross parameters such as rotor work is well known at design conditions, and a similar approach is taken here. Thus, circumferential averages of the flow quantities are considered, which are dependent on the radial coordinate, r. Since the bypass ratio during windmill is an order of magnitude larger than at the design point [2], the flow that enters the core can be neglected. The region of interest is the outer, bypass portion of the annulus, Rs < r < Rb, where Rs and Rb represent the duct radii corresponding to the splitter and the fan tip trailing edge, respectively.

Based on earlier simulation results [2], it is assumed that the radial velocity is small relative to its axial counterpart. Furthermore, noting that the flow in the region of interest is nearly uniform, the absolute velocity components (cx, cθ) and stagnation temperature TT may be written as Display Formula

(1a)$cx(r)=c¯x[1+εF(r)]$
Display Formula
(1b)$cθ(r)=εc¯xG(r)$
Display Formula
(1c)$TT(r)=T¯T,2A[1+εQ(r)]$

where $T¯T$ is the mass-averaged stagnation temperature, ε ≪ 1, and F, G, and Q are all O(1). It should be noted that no work is performed upstream of the rotor, so that $T¯T,2A=TT,1$. The quantity $c¯x$ is taken to be the area-averaged axial velocity, which requires that Display Formula

(2)$∫RsRbrF(r)dr=0$

The fan angular speed is taken to be Ω and the rotor exit relative flow angle, denoted by β = β(r), is assumed to be known from the blade geometry [2]. The flow quantities in Eqs. (1a)(1c) are to be determined subject to the prescription of the bypass mass flow rate, $m˙b$ and the constraint that no work is done by the rotor, expressed, respectively, as Display Formula

(3a)$2π∫RsRbρcxrdr=m˙b$
Display Formula
(3b)$∫RsRb(TT−T¯T,2A)ρcxrdr=0$

where ρ = ρ(r) is the density.

In the Appendix, an asymptotic model for determining the perturbation profiles in Eq. (1) is developed, based on the assumption of quasi-steady isentropic flow behind the rotor. The model results show that

$εF(r)=kr(kr−tan β)1+krεG(r)=−kr−tan β1+krεQ(r)=c¯x2CpT¯T,2Akr(kr−tan β)1+kr$

where Cp is the specific heat at constant pressure and $k=Ω/c¯x$ is determined by numerical solution of Eq. (A13). This latter equation features only blade and duct geometric parameters so that k depends on these quantities but not on the throughflow. Thus, for a given fan design, it follows that $Ω∝c¯x$, consistent with the findings of Ref. [2]. Defining the fan speed parameter, λ, according to Display Formula

(4a)$λ=ΩRt/c¯x$
it is evident that, in view of the preceding discussion, λ is constant for any given flow configuration.

Returning to Eqs. (1a) and (1b), the axial and tangential velocity profiles are found to be given by Display Formula

(4b)$cx/c¯x=1+kr(kr−tan β)/(1+kr)$
Display Formula
(4c)$cθ/c¯x=−(kr−tan β)/(1+kr)$

The above relations show that, for a given geometric configuration, the radial profiles of axial and tangential velocity are independent of the throughflow, when scaled by the mean throughflow velocity. This is an indication of self-similar behavior and is a feature that has not thus far been noted in the previous studies.

Turning next to the expression in Eq. (1c) for the stagnation temperature profile, it is found that Display Formula

(4d)$Cp(TT−T¯T,2A)/c¯x2=kr(kr−tan β)/(1+kr)$

Equation (4d) reveals that the stagnation enthalpy change across the rotor is also self-similar in nature under a scaling based on the square of the mean throughflow velocity. The similarity laws described by Eqs. (4a)4(d) provide a fundamental insight into the nature of the rotor flow under windmilling conditions. In what follows, the previous simulation results will be examined to assess the extent to which these laws apply to realistic flows.

The windmill rotational speed is first examined in Fig. 2. The operating point is characterized by the flow parameter, Φ, a variant of the corrected flow per unit area, defined as Display Formula

(5)$Φ=m˙2A(RT¯T,2A)1/2p¯T,2AA2A$

where $R$ is the gas constant and A represents the duct area. (The definition of Φ in Eq. (5) differs from that of Ref. [2] in using the stagnation pressure at station 2A rather than that at station 1; however, since the inlet loss is negligible, the difference in small.) The fan speed parameter, λ, is expected be independent of Φ, based on the scaling derived earlier. The veracity of this hypothesis is investigated in Fig. 2 in which values of λ derived from the engine data reported in Ref. [2] are plotted as a function of Φ. Despite some minor scatter, it is observed that λ remains essentially constant over the entire range of Φ. The plot in Fig. 2 also depicts values of λ obtained using the present asymptotic theory. In addition, the value of λ using the numerical model of Ref. [2] is shown and is seen to be essentially constant as well. This latter estimate of λ is in very good agreement with the data, with an error of 2.4%. The error in the asymptotic model has a somewhat higher error (8.5%) but nevertheless provides a good first approximation for the rotational speed.

Next, in order to verify self-similarity, the simulations of Ref. [2] are interrogated for different values of the flow parameter, Φ. Radial profiles of the circumferentially averaged, normalized axial velocity and stagnation temperature at rotor exit (station RE) are depicted in Figs. 3(a) and 3(b). The fan inlet stagnation sonic speed ($a¯T,2A$) and temperature ($T¯T,2A$) are used, respectively, as normalization factors. Both sets of profiles are observed to exhibit significant variation with Φ. However, when the scaling based on Eqs. (4a) and (4c) is adopted, a collapse of the axial velocity and stagnation temperature profiles is observed, as illustrated in Figs. 4(a) and 4(b) demonstrating their self-similar nature. The profile collapse occurs over the entire span, even in the outboard separated region. Figure 4(a) also depicts the scaled tangential velocity profiles, which are seen to exhibit self-similar behavior as well. Finally, the theoretical profiles for the velocities and stagnation temperature given by Eqs. (4a)(4c) are shown in Figs. 4(a) and 4(b) along with the corresponding profiles determined using the model of Ref. [2]. The model of Ref. [2] is found to collapse to a common profile that is in good agreement with the simulations away from the splitter region and the separated region, where radial redistribution, not captured by the model, can be expected to occur. It is also apparent from Fig. 4(a) that the present asymptotic theory provides a good approximation to the model of Ref. [2] and to the simulations.

###### Stator Inlet Flow.

The focus of the study now shifts further downstream to the bypass duct, and the question of whether flow self-similarity remains valid for the flow entering the fan exit guide vane is addressed. This is carried out using the velocity magnitude, c and absolute angle, α rather than the axial and tangential velocities. Since the magnitude of the mean tangential velocity downstream of the rotor is zero, we have $c¯x=c¯$, i.e., the mean velocity and the mean axial velocity are identical. It then follows that $c/c¯$ and α should exhibit self-similar behavior. This is confirmed in Figs. 5(a) and 5(b) in which profiles of the scaled velocity and absolute angle obtained from the simulations of Ref. [2] are plotted at vane inlet (station VI) for the same Φ-values as those in Fig. 3. The presence of a separation on the splitter is obvious and is seen to occupy nearly 25% of the span at the inner wall of the duct. Nevertheless, the scaled profiles all collapse to a common profile even in this region.

Turning next to the stagnation temperature and making use of the scaling in Eq. (4d), the similarity parameter for this quantity is proposed as Display Formula

(6)$Θ(r)=Cp(TT−T¯T,2A)/c¯2$

The spanwise behavior of $Θ$, evaluated at vane inlet (station VI), is shown in Fig. 5(c). Like the velocity and flow angle, it too is found to behave in a self-similar manner, with the profiles at different values of Φ collapsing to a common shape. Departures from this profile are observed in the neighborhood of the splitter separation, but these are small. It may, therefore, be concluded that $Θ$ is the appropriate similarity parameter for the stagnation temperature.

The final quantity worthy of investigation is the similarity parameter corresponding to the stagnation pressure. The form of this parameter can be motivated from a consideration of isentropic flow across the rotor and splitter, consistent with the model assumptions. At vane inlet (station VI), this approximation yields

$pTp¯T,2A=(TTT¯T,2A)γγ−1=[1+c¯2CpT¯T,2AΘ(r)]γγ−1$

where the definition of $Θ(r)$ in Eq. (6) has been used. From Fig. 5(c), it is observed that $Θ(r)$ is small so that, upon carrying out a Taylor series expansion in $Θ$ and retaining only the first term of this expansion, we obtain at vane inlet (station VI)

$pT=p¯T,2A+ρ¯T,2Ac¯2Θ(r)$

The Mach number of the flow through the rotor and downstream of it is generally small, permitting use of the incompressible approximation, $ρ¯T,2A≈ρ¯$. Hence, for isentropic flow, the stagnation pressure difference scales with the dynamic pressure, $ρ¯c¯2$. However, losses are known to occur in the rotor and splitter flows [2,6,8], and one may reasonably expect these losses to also scale with the dynamic pressure. In view of the foregoing considerations, the stagnation pressure similarity parameter is defined as Display Formula

(7)$Π(r)=pT−p¯T,2Aρ¯c¯2$

The spanwise variation of Π(r) obtained from the previous numerical simulations [2] at vane inlet (station VI) is illustrated in Fig. 5(d). Over the range of Φ considered, the profiles are observed to display self-similar behavior across the entire span, justifying the definition of the similarity parameter, Π, in Eq. (7).

###### Rotor-Splitter. Loss

In line with the earlier reasoning concerning the losses accumulated in the rotor and splitter flows, it is observed from Fig. 5(d) that the mean value of Π over the duct span is negative. The overall rotor-splitter loss may be quantified in terms of the change in the mass-averaged stagnation pressure between fan inlet (station 2A) and vane inlet (station VI). It should be noted that, for the relatively low-speed flow in this part of the flow domain, the mass-averaged stagnation pressure loss is equivalent to the more rigorous entropy-averaged value [12]. Like the stagnation pressure similarity parameter, the mass-averaged stagnation pressure loss across the rotor and splitter, denoted by ΔpT,rs, scales with the dynamic pressure owing to the similarity of the flow profiles at vane inlet (station VI). Hence, it follows that Display Formula

(8)$ΔpT,rs=12ρ¯VIc¯VI2Zrs$

where Zrs is a loss coefficient that is independent of the operating condition. The operating condition is characterized by the mass flow rate, expressed in nondimensional form as the flow parameter Φ, defined by Eq. (5). The value of Zrs determined from the numerical simulations for different values of Φ is shown in Table 1. The loss coefficient varies by about 2% and can, therefore, be taken as constant. The implication is that, upon making use of the scaling in Eq. (8), the results of a simulation performed for any one windmilling point may be used to determine Zrs and to thus deduce the loss for any other windmilling condition. A procedure to accomplish this is described later in the paper.

## Stator Flow Scaling

The flow through the stator is studied next, with the objective of developing a scaling procedure for the loss across this component. Owing to the strong negative incidence on the stator, the flow is highly separated and, therefore, unsteady. However, it proves convenient to consider temporal averages of the flow quantities in order to develop insight into the physical phenomena at play.

Making use of the results of Ref. [2], it is possible to construct a schematic description of the time-averaged flow configuration in the vane-to-vane direction. This is depicted in Fig. 6 at an outer-span location, where the incidence on the vane is highly negative. The incident flow stagnates on the convex surface of the airfoil at the point S and separation occurs near the leading edge, B. The flow over the convex surface of the airfoil remains attached until it reaches the vicinity of the trailing edge, T, where separation again occurs. The flow in the separated region that lies between the free streamlines emanating from B and T is essentially stagnant. This behavior remains qualitatively similar at lower spans than the one shown in Fig. 6. At these spans, the incidence is less severe and the width of the wake is smaller. Near the hub, the incidence is still smaller and the solidity is higher so that, although the flow separates close to the leading edge as in Fig. 6, it reattaches near midchord, and the wake is almost nonexistent.

With this qualitative understanding, it is instructive to begin study of the vane flow by examining the wake behind the stator at vane exit (station 14). This is done in Fig. 7 for flow parameter values of 0.194, 0.246, and 0.313. Since these flow parameter values correspond to significantly different mean velocities, the wake is visualized using the quantity $|v|/max|v|$, where v represents the absolute velocity, and the maximum is taken over the axial plane at vane exit. The size and the shape of the separated wake are observed from Fig. 7 to be very similar over a broad range of the flow parameter, Φ. In all of the cases examined, the blocked area occupies a large portion of the annulus in the outer span; this extent diminishes in the lower span and completely disappears below span fractions of approximately 25%.

The wake behavior along the span can be characterized using a blockage parameter, which is defined as the fraction of the flow area in which the flow is essentially stagnant. Thus, referring to Fig. 6, where the airfoil spacing is denoted by d, the width of the wake is μd, and that of the unobstructed flow is (1 − μ)d. While the blockage, μ, can be quantitatively estimated in different ways with varying degrees of complexity, we adopt here the relatively simple definition that μ at any given spanwise location is the fraction of the annulus for which

$|v|max|v|<12$

The variation of blockage with span, estimated using this criterion, is shown in Fig. 8 for the values of Φ simulated previously [2]. Consistent with the behavior in Fig. 7, we observe that the largest blockage occurs in the vicinity of 80%-span, where the wake occupies nearly half the passage width. Away from this location, the blockage decreases almost monotonically toward the inner diameter. There is no significant wake below 25–35% span. Figure 8 also illustrates that the variation of μ at a given spanwise location is relatively insensitive to the throughflow rate, quantitatively confirming the wake visualization in Fig. 7.

The separated flow through the stator results in the formation of a virtual throat in the vane passage near the trailing edge. This was pointed out in Ref. [2] and is also apparent in the schematic illustration of Fig. 6. As a consequence, the largest velocities occur in the vicinity of the vane exit. Furthermore, the blockage is of sufficiently large magnitude that compressibility effects will be significant at high throughflow. The velocity scale for the vane flow is, therefore, likely to be the speed through the unblocked area at vane exit (station 14). For convenience, the quantities associated with this unblocked area are denoted by the subscript $14′$. On a one-dimensional basis, it then follows that $A14′=(1−μ¯)A14$, where the mean blockage, $μ¯$, is determined as

$μ¯=∫14μ(r)r dr∫14r dr$

The values of $μ¯$ for the flow coefficients considered in Ref. [2] are shown in Table 2 and are observed to vary by less than 2%, consistent with the wake blockage shown in Fig. 8.

Next, the swirl angle of the flow through the virtual throat at the vane exit is examined. The spanwise profiles of the unblocked vane exit area-averaged absolute flow angle, $α14′$, obtained from the simulation results are shown in Fig. 9. It is observed that they are clustered around a common profile, especially for span fractions greater than 75%. The profiles exhibit differences in the range of 20–40% span, and this is not surprising in view of the variations in the wake width for different values of Φ, as was observed in Fig. 8. Over the entire span of the vane, the magnitudes of $α14′$ are small, with values of less than 5 deg below 30%-span and values between 5 deg and 8 deg above 30%-span. It is, therefore, reasonable to approximate the vane exit flow as swirl-free so that, upon taking $ρ¯14′$ to be the unblocked area-averaged density, the characteristic velocity, $c¯14′$, is given by

$c¯14′=m˙bρ¯14′(1−μ¯)A14$

Having defined $c¯14′$, the next step is to validate the conjecture that this is the appropriate velocity scale. To this end, the circumferential average of the velocity over the unblocked portion of the annulus, $c14′(r)$ is determined as a function of span and its behavior under different normalizations is examined. Figure 10 illustrates spanwise profiles of $c14′$ normalized by the area-averaged vane inlet velocity, $c¯VI$, and by the unblocked vane exit area-averaged velocity, $c¯14′$. These profiles are shown for different values of the flow parameter, Φ, and it is clear that with the $c¯VI$-normalization, there is significant variation of the profiles with Φ. The $c¯14′$-normalization, on the other hand, collapses the profiles, illustrating their self-similar nature under this normalization.

In view of the fact that the relevant velocity scale is $c¯14′$, one would expect the overall stagnation pressure loss across the vane to scale with the exit dynamic pressure of the unblocked flow, $12ρ¯14′c¯14′2$. This can be expressed as Display Formula

(9)$ΔpT,v=12ρ¯14′c¯14′2Zv$

where Zv is a loss coefficient that, like its rotor-splitter counterpart, Zrs, is expected to be independent of flow conditions. The values of Zv determined from the numerical simulations are shown in Table 2 for different values of Φ and are found to be essentially constant, with a variation of only about 2%. This validates the scaling proposed in Eq. (9).

The overall vane loss, ΔpT,v, is the quantity of primary interest. However, with a view to developing a scaling procedure for determining the loss at an arbitrary operating condition, we also examine the loss incurred between vane inlet (station VI) and unblocked vane exit (station $14′$). Like the overall loss, this quantity is expected to also scale with the dynamic pressure, so that we may write Display Formula

(10)$p¯T,VI−p¯T,14′=12ρ¯14′c¯14′2ζ$

where ζ is a loss coefficient between the two stations. Table 2 shows the values of ζ determined from the numerical simulations, and we see that it, too, is essentially independent of Φ, with a variance of about 3%. The magnitude of ζ is roughly half that of Zv, showing that the loss incurred by mixing downstream of the vane is of the same order of magnitude as that which occurs within the vane passage, consistent with the results of Ref. [2].

## Scaling Methodology

The discussion has thus far centered on the similarity quantities for the windmilling fan stage. In particular, the windmill rotational speed was shown to scale with the throughflow velocity and the stagnation pressure losses with the dynamic pressure. While these scalings are instructive at a fundamental level, for practical use, it is necessary to relate the losses to a more global quantity; following [2], this quantity is taken to be the flow parameter, Φ.

###### Stagnation Pressure Losses.

The scaling approach for the component stagnation pressure losses is first developed. The bypass ratio is an order of magnitude larger at windmill than at the cruise condition [2], so that the approximation $m˙2A≈m˙b$ can be made. Then, because the mean stagnation temperature remains constant through the stage, it follows from Eq. (5) that Φ may be expressed in terms of the mean Mach number, $M¯$ and stagnation pressure, $p¯T$ at any station by Display Formula

(11)$Φ=γ12AA2Ap¯Tp¯T,2AM¯F−12(γ+1γ−1)$

where $F$ is defined as

$F=F(M¯)=1+12(γ−1)M¯2$

Similarly, the dynamic pressure at any station takes the form Display Formula

(12)$ρ¯c¯2=γp¯TM¯2F−γγ−1$

In what follows, $M¯14′$ is taken as the independent variable and all parameters of interest are expressed in terms of this quantity. From Eqs. (8) and (12), it follows that Display Formula

(13)$p¯T,2Ap¯T,VI=1+12γZrsM¯VI2FVI−γγ−1$

while the analogous use of Eq. (10) yields Display Formula

(14)$p¯T,VIp¯T,14′=1+12γζM¯14′2F14′−γγ−1$

Equations (13) and (14), when used in Eq. (11), yield two expressions for Φ in terms of and $M¯VI$ and $M¯14′$; equating these expressions shows that Display Formula

(15)$AVIMVIFVI−12(γ+1γ−1)=(1−μ¯)A14M¯14′F14′−12(γ+1γ−1)1+12γζM¯14′2F14′−γγ−1$

Equation (15) is a nonlinear relationship that, for a given value of $M¯14′$, can be solved numerically to determine $M¯VI$. Here, the bisection method is employed for this purpose. The corresponding value of Φ is then found to be Display Formula

(16)$Φ=γ12AVIA2A[1+12γZrsM¯VI2FVI−γγ−1]−1M¯VIFVI−12(γ+1γ−1)$

The fractional loss across the rotor and splitter follows from Eq. (13) and is given by Display Formula

(17)$ΔpT,rsp¯T,VI=12γZrsM¯VI2FVIγγ−1$

Making use of Eqs. (9) and (14), the fractional loss across the vane can be expressed as Display Formula

(18)$ΔpT,vp¯T,VI=12γZvM¯14′2F14′−γγ−11+12γζM¯14′2F14′−γγ−1$

###### Fan Rotational Speed.

Next, a scaling approach is developed for the windmill rotational speed, expressed in nondimensional form as $ΩRt/a¯T,2A$. As was shown earlier, the fan speed parameter, λ, is fixed for a given configuration from which it follows that

$ΩRt/a¯T,2A=λc¯RE/a¯T,2A$

Making use of the fact that the mean stagnation temperature is preserved through the stage, the above expression can be written in terms of the mean bypass-region Mach number at rotor exit (station RE), $M¯RE$, as Display Formula

(19)$ΩRt/a¯T,2A=λM¯RE[1+12(γ−1)M¯RE2]−12$

It is, therefore, necessary to determine the quantity $M¯RE$. This is accomplished using Eq. (11). Thus, neglecting changes in the stagnation pressure between stations RE and VI, and applying Eq. (11) at both locations, it is evident that

$ARE,bM¯REFRE−12(γ+1γ−1)=AVIM¯VIFVI−12(γ+1γ−1)$

where ARE,b is the area of the bypass region at rotor exit (station RE). For a known value of $M¯VI$, the above equation can be solved numerically using bisection to determine the corresponding value of $M¯RE$, and $ΩRt/a¯T,2A$ follows from Eq. (19).

###### Results.

The procedure described earlier was applied to the configuration of Ref. [2]. The value of λ is taken to be 0.951, the average over the range of Φ in Fig. 2. Similarly, using Tables 1 and 2, we take the blockage and loss coefficients to be the mean values over the range of Φ. Specifically, we assume that $μ¯=0.253$, Zrs = 0.189, ζ = 0.158, and Zv = 0.309. For prescribed values of $M¯14′$, corresponding values of $M¯VI, M¯RE$, and Φ are determined and the results are shown in Fig. 11(a). We observe that $M¯VI$ and $M¯RE$ are of comparable value, and that while they remain moderate over the range 0 < Φ < 0.32, the magnitude of $M¯14′$ grows rapidly for $Φ≳0.25$. In particular, the flow becomes choked in a one-dimensional sense (i.e., $M¯14′=1$) when Φ = 0.32.

The stagnation pressure loss fractions across the rotor-splitter combination and stator are illustrated in Fig. 11(b), and follow the qualitative trends exhibited by $M¯VI$ and $M¯14′$. Specifically, the rotor-splitter loss remains less than 2%, while the vane loss increases rapidly with Φ, assuming values of nearly 10% for large Φ. The analogous loss fractions determined from the previous simulations [2] are also shown in Fig. 11(b) and comparison of these results with the present model indicates very good agreement. In particular, the rotor-splitter loss prediction based on the model is nearly exact, while that for the stator is slightly larger than the simulation results. The stator loss obtained from the model exhibits the same rapid increase with Φ as the simulation results and the engine test data of Ref. [2]. The reason for this behavior is directly related to the choking that occurs in the virtual throat formed in the vane passage owing to the large flow separation at windmill.

Finally, the nondimensional fan rotational speed, $ΩRt/a¯T,2A$, is illustrated in Fig. 11(c). The values of this quantity increase monotonically with Φ and are in good agreement with the engine data of Ref. [2], which are also plotted in Fig. 11(c). The model results are somewhat larger than the engine data, and the agreement could be improved by allowing for the stagnation pressure loss across the splitter. However, the discrepancy is small and the present approximation suffices for design purposes.

###### Application Procedure.

The preceding discussion illustrates that, with a knowledge of the loss coefficients, Zrs, ζ, and Zv, reliable estimates of the component losses and rotational speed can be obtained. The manner in which these coefficients can be deduced, based on a single numerical simulation at any one windmill condition, is now described; a flowchart illustrating the overall procedure is depicted in Fig. 12.

Beginning with the system geometry, as illustrated in Fig. 1, a representative value of the flow parameter, Φ, is chosen together with free-stream conditions, thus defining a throughflow rate. The fan rotational speed corresponding to this throughflow rate is then estimated using the asymptotic theory given here or the numerical model developed previously [2]. This permits a numerical simulation to be carried out. It should be noted that a more accurate estimate of the fan rotational speed can be obtained by incorporating into the simulation an iteration loop to drive the rotor work to zero [8].

The simulation results can then be postprocessed as described earlier to determine the stator blockage from which the stator loss coefficients, ζ and Zv, are obtained. Postprocessing of the simulation results also yields the rotor-splitter loss coefficient, Zrs and the fan speed parameter, λ. Making use of the loss coefficients, the losses at arbitrary windmill conditions, corresponding to different Φ-values, can then be determined using the scaling method, resulting in curves such as those in Fig. 11(b) for the components. The overall system loss then follows from summing the component losses. In a similar manner, use of the fan speed coefficient provides the rotational speed at arbitrary windmill conditions, corresponding to different values of Φ.

## Conclusion

The objective of the present study was to gain greater physical insight into the nature of windmilling turbofan flows and to employ this insight to develop methods to scale the internal losses under these conditions. An asymptotic model was used in conjunction with simulation results to demonstrate that the intrastage flow profiles take on a self-similar form when appropriately scaled. This self-similarity was then used to deduce a basis for scaling the fan rotational speed and stagnation pressure loss across the rotor and splitter. A similar approach was adopted for the stator flow, where it is shown that the blockage of the vane passage is invariant with throughflow. It was then shown that a characteristic velocity defined using the mass flow rate and unblocked area is the appropriate velocity scale and that the stator stagnation pressure loss scales with the associated dynamic pressure. A model for the component losses based on the scaling approach was developed and found to be in good agreement with Ref. [2]. Since the stator blockage is rather substantial, higher throughflow rates are found to result in choking, which is also consistent with the previous observations.

The ability of the scaling approach developed in the present study for the fan rotational speed as well as the component and stage losses with good accuracy demonstrates its suitability for use in design prediction models. As turbofan designs drive toward lower values of pressure ratio, the traditional approach of determining windmilling performance based on legacy engine data becomes increasingly less applicable and must be used with circumspection, if at all. The advantage that the methods developed in the present study offer is that, notwithstanding their simplicity, they are based on first principles and can, therefore, be employed in the development of propulsion system designs that lie outside the existing design space.

## Acknowledgements

The author is grateful to Dr. W. K. Lord for insightful discussions on this topic. It is also a pleasure to acknowledge Professor Z. S. Spakovszky's comments on an earlier version of the paper that led to several improvements in its presentation.

## Nomenclature

• A =

duct cross-sectional area

• Cp =

specific heat at constant pressure

• c =

absolute flow velocity

• k =

angular speed to axial velocity ratio

• $m˙$ =

mass flow rate

• M =

Mach number

• pT =

stagnation pressure

• r =

radial coordinate

• $R$ =

gas constant

• Rb =

fan trailing edge tip radius

• Rs =

splitter leading edge tip radius

• Rt =

fan leading edge tip radius

• TT =

stagnation temperature

• x =

axial coordinate

• Zrs =

overall loss coefficient across rotor and splitter

• Zv =

overall loss coefficient across stator

• α =

absolute angle

• β =

relative angle

• γ =

specific heat ratio

• ζ =

vane passage loss coefficient

• λ =

fan speed parameter

• μ =

stator wake blockage

• Φ =

flow parameter

• θ =

circumferential coordinate

• Θ =

stagnation temperature similarity variable

• Π =

stagnation pressure similarity variable

• ρ =

density

• Ω =

fan rotational angular speed

## Appendices

###### Appendix: Asymptotic Model for Rotor Exit Flow

In this Appendix, we illustrate the development of the asymptotic model for the rotor exit flow, which motivates the scaling for the parameters of interest. The rotor exit velocity triangle [2] implies that Display Formula

(A1)$cθ=cx tan β−rΩ$

It then follows from Eq. (1a) that Display Formula

(A2)$cθ=c¯x[−(kr−tan β)+εF(r)]$

where $k=Ω/c¯x$. Comparison of Eq. (A2) with Eq. (1b) shows that

$kr−tan β=O(ε)$

Since the absolute flow entering the rotor is swirl-free, application of the Euler turbine equation across the rotor, in conjunction with Eqs. (1c) and (A2), yields, upon simplification Display Formula

(A3)$εQ(r)=M̂2γ−1kr[(kr−tan β)−εF(r)]$

where γ is the ratio of specific heats and $M̂=c¯x/a¯T,2A$, with aT representing the stagnation sonic speed. Next, denoting by Cp the specific heat at constant pressure and taking the rotor flow as being isentropic, it follows from Crocco's theorem that [2]

$CpdTTdr=cxdcxdr+cθrddr(rcθ)$

Upon making use of the scaling in Eqs. (1a)(1c), it is observed that the last term is small and can be dropped from the above equation. This enables it to be integrated, yielding Display Formula

(A4)$Q(r)=M̂2γ−1[F(r)+C]$

where C is a constant of integration that is determined using the constraint in Eq. (2), as is now shown.

Combining Eqs. (A3) and (A4), it is found that Display Formula

(A5)$εF(r)=(1+kr)−1[kr(kr−tan β)−εC]$

In preparation for the analysis to follow, the integrals Display Formula

(A6)$In=∫RsRbkn−1rn1+krdr; (n=1,2,3)$
Display Formula
(A7)$J=∫RsRbkr2 tan β1+krdr$
are defined. It is evident that, in general, J must be evaluated by numerical means. However, the integrals In can be determined in closed form and are given by Display Formula
(A8)$In=(−1)nk2ln1+kRb1+kRs−∑m=1n(−1)mmkm−2(Rbm−Rsm)$

Next, making use of Eqs. (A5) and (2), it is found that Display Formula

(A9)$εC=(I3−J)/I1$

It now remains to determine the quantity $M̂$ (or equivalently, $c¯x$) and the function F(r), which is accomplished using the prescription of the bypass duct mass flow rate and the constraint of zero work across the rotor as expressed by Eqs. (3a) and (3b). With the assumption of isentropic flow and the scaling in Eqs. (1a)(1c), it can be shown that Display Formula

(A10)$ρ=ρ¯T,2AS1γ−1[1+ε{Q(r)−(γ−1)M̂2F(r)(γ−1)S}]$

where $S=1−12(γ−1)M̂2$ and ρT denotes the density at the stagnation condition. Making use of Eqs. (1a) and (A10) in Eq. (3a) and gathering terms of like order, the O(1) approximation yields Display Formula

(A11)$M̂[1+12(γ−1)M̂2]1γ−1=m˙bρ¯T,2Aa¯T,2AARE,b$

where $ARE,b=π(Rb2−Rs2)$ represents the bypass portion of area at rotor exit (station RE). Equation (A11) can be solved numerically for $M̂$ using Newton iteration.

Applying the same procedure to Eq. (3b), the leading order contribution, which in this case is O(ε), yields the requirement that Display Formula

(A12)$∫RsRbrQ(r)dr=0$

The O(ε) contribution of Eq. (3a) requires that

$∫RsRb{Q(r)+(γ−1)[1+12(γ+1)M̂2]F(r)}rdr=0$
which is automatically fulfilled in view of Eqs. (2) and (A12). Thus, provided Eq. (A12) holds, Eqs. (3a) and (3b) are satisfied, correct to O(ε2).

Now, upon making use of Eqs. (A4), (A5), and (A9) in Eq. (A12), it is found that (I3 − J)(I1 + I2) = 0. From Eq. (A8), it is evident that $I1+I2=12(Rb2−Rs2)>0$, whence

$I3−J=0$

Employing the definition of J in Eq. (A7) and the expression in Eq. (A8) for I3 in the above equation, the condition that k must satisfy can, therefore, be written explicitly as Display Formula

(A13)$k3(Rb3−Rs3)−12(Rb2−Rs2)+1k(Rb−Rs)−1k2ln(1+kRb1+kRs)−∫RsRbkr2 tan β1+krdr=0$

Equation (A13) can be solved numerically to determine k, and, since $M̂$ is known, it follows that the windmill rotational speed $Ω=a¯T,2AM̂k$ can be determined.

## References

Daggett, D. L. , Brown, S. T. , and Kawai, R. T. , 2003, “ Ultra-Efficient Engine Diameter Study,” National Aeronautics and Space Administration, Washington, DC, Technical Report No. NASA/CR 2003-212309.
Prasad, D. , and Lord, W. K. , 2010, “ Internal Losses and Flow Behavior of a Turbofan Stage at Windmill,” ASME J. Turbomach., 132(3), p. 031007.
Zachos, P. K. , 2013, “ Modelling and Analysis of Turbofan Engines Under Windmilling Conditions,” J. Propul. Power, 29(4), pp. 882–890.
Dufour, G. , Carbonneau, X. , and García Rosa, N. , 2013, “ Nonlinear Harmonic Simulations of the Unsteady Aerodynamics of a Fan Stage Section at Windmill,” ASME Paper No. GT2013-95485.
Dufour, G. , García Rosa, N. , and Duplaa, S. , 2015, “ Validation and Flow Structure Analysis in a Turbofan Stage at Windmill,” Proc. Inst. Mech. Eng., A, 229(6), pp. 571–583.
García Rosa, N. , Dufour, G. , Barènes, R. , and Lavergne, G. , 2015, “ Experimental Analysis of the Global Performance and the Flow Through a High-Bypass Turbofan in Windmilling Conditions,” ASME J. Turbomach., 137(5), p. 051001.
Dufour, G. , and Thollet, W. , 2016, “ Body Force Modeling of the Aerodynamics of the Fan of a Turbofan at Windmill,” ASME Paper No. GT2016-57462.
Gunn, E. J. , and Hall, C. A. , 2016, “ Loss and Deviation in Windmilling Fans,” ASME J. Turbomach., 138(10), p. 101002.
Anderson, B. A. , Messih, D. , and Plybon, R. C. , 1997, “ Engine-Out Performance Characteristics,” 13th International Symposium on Air Breathing Engines, Chattanooga, TN, Sept. 7–12, ISABE Paper No. 97-7216.
Braig, W. , Schulte, H. , and Riegler, C. , 1999, “ Comparative Analysis of the Windmilling Performance of Turbojet and Turbofan Engines,” J. Propul. Power, 15(2), pp. 326–333.
Walsh, P. P. , and Fletcher, P. , 2004, Gas Turbine Performance, Blackwell Science, Oxford, UK.
Greitzer, E. M. , Tan, C. S. , and Graf, M. B. , 2007, Internal Flow: Concepts and Applications, Cambridge University Press, Cambridge, UK.
Copyright © 2018 by ASME
View article in PDF format.

## References

Daggett, D. L. , Brown, S. T. , and Kawai, R. T. , 2003, “ Ultra-Efficient Engine Diameter Study,” National Aeronautics and Space Administration, Washington, DC, Technical Report No. NASA/CR 2003-212309.
Prasad, D. , and Lord, W. K. , 2010, “ Internal Losses and Flow Behavior of a Turbofan Stage at Windmill,” ASME J. Turbomach., 132(3), p. 031007.
Zachos, P. K. , 2013, “ Modelling and Analysis of Turbofan Engines Under Windmilling Conditions,” J. Propul. Power, 29(4), pp. 882–890.
Dufour, G. , Carbonneau, X. , and García Rosa, N. , 2013, “ Nonlinear Harmonic Simulations of the Unsteady Aerodynamics of a Fan Stage Section at Windmill,” ASME Paper No. GT2013-95485.
Dufour, G. , García Rosa, N. , and Duplaa, S. , 2015, “ Validation and Flow Structure Analysis in a Turbofan Stage at Windmill,” Proc. Inst. Mech. Eng., A, 229(6), pp. 571–583.
García Rosa, N. , Dufour, G. , Barènes, R. , and Lavergne, G. , 2015, “ Experimental Analysis of the Global Performance and the Flow Through a High-Bypass Turbofan in Windmilling Conditions,” ASME J. Turbomach., 137(5), p. 051001.
Dufour, G. , and Thollet, W. , 2016, “ Body Force Modeling of the Aerodynamics of the Fan of a Turbofan at Windmill,” ASME Paper No. GT2016-57462.
Gunn, E. J. , and Hall, C. A. , 2016, “ Loss and Deviation in Windmilling Fans,” ASME J. Turbomach., 138(10), p. 101002.
Anderson, B. A. , Messih, D. , and Plybon, R. C. , 1997, “ Engine-Out Performance Characteristics,” 13th International Symposium on Air Breathing Engines, Chattanooga, TN, Sept. 7–12, ISABE Paper No. 97-7216.
Braig, W. , Schulte, H. , and Riegler, C. , 1999, “ Comparative Analysis of the Windmilling Performance of Turbojet and Turbofan Engines,” J. Propul. Power, 15(2), pp. 326–333.
Walsh, P. P. , and Fletcher, P. , 2004, Gas Turbine Performance, Blackwell Science, Oxford, UK.
Greitzer, E. M. , Tan, C. S. , and Graf, M. B. , 2007, Internal Flow: Concepts and Applications, Cambridge University Press, Cambridge, UK.

## Figures

Fig. 2

Variation of fan speed parameter, λ with flow parameter, Φ. Both models and data indicate that λ is independent of Φ.

Fig. 3

Circumferentially averaged profiles of (a) axial velocity and (b) stagnation temperature at rotor exit (station RE), scaled by a¯T,2A and T¯T,2A, respectively, based on the previous simulations [2]. The dependence of the profiles on the flow parameter, Φ, is evident.

Fig. 1

Notional meridional section of engine, illustrating the stations used in the present study (adapted from Ref. [2])

Fig. 5

Nondimensional radial profiles at vane inlet: (a) velocity, (b) absolute flow angle, (c) stagnation temperature, and (d) stagnation pressure. The profiles show that flow self-similarity continues to hold downstream of the rotor.

Fig. 4

Radial profiles of the normalized, circumferentially averaged (a) axial and tangential velocity and (b) stagnation enthalpy change at rotor exit, obtained from the numerical simulations of [2]. The symbols correspond to those of Fig. 3. The collapse of the profiles demonstrates the underlying self-similarity. Results of the asymptotic theory (– – –) and the model of [2] (–⋅–⋅) display good agreement with the simulations.

Fig. 7

Illustration of the wake blockage at vane exit for Φ = 0.194 (left), Φ = 0.246 (center), and Φ = 0.313 (right). Contours of the normalized velocity, |v|/max|v|, are employed to visualize the blockage, which is observed to be essentially invariant with Φ.

Fig. 8

Spanwise variation of the wake blockage at vane exit, indicating the invariance of this quantity with flow parameter, Φ. The symbols correspond to those of Fig. 3.

Fig. 6

Schematic illustration of the vane-to-vane flow in the stator. The large negative incidence results in the formation of an extended separation zone.

Fig. 10

Spanwise profiles of the unblocked circumferentially averaged velocity, c14′, normalized by the vane inlet area-averaged velocity, c¯VI (——), and by the unblocked vane exit area-averaged velocity, c¯14′ (– – –). The symbols correspond to those of Fig. 3. The normalization based on c¯14′ is seen to collapse the profiles.

Fig. 9

Spanwise profiles of the unblocked circumferentially averaged absolute flow angle, α14′, illustrating invariance with the flow parameter, Φ. The symbols correspond to those of Fig. 3.

Fig. 12

Flowchart illustrating a procedure to estimate the fan speed parameter and component loss coefficients

Fig. 11

Flow parameter dependence of (a) bypass duct Mach numbers, (b) nondimensional stagnation pressure losses, and (c) nondimensional fan rotational speed. The lines represent results obtained using the present the models. The symbols in (b) denote simulation results, while those in (c) correspond to test data.

## Tables

Table 1 Rotor-splitter stagnation pressure loss coefficient
Table 2 Vane blockage and stagnation pressure loss coefficient

## Errata

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