Trailing Edge Loss of Choked Organic Vapor Turbine Blades ^†

Abstract

The present study reports the outcome of an experimental study of organic vapor trailing edge flows. As a working fluid, the organic vapor Novec 649 was used under representative pressure and temperature conditions for organic Rankine cycle (ORC) turbine applications characterized by values of the fundamental derivative of gas dynamics below unity. An idealized vane configuration was placed in the test section of a closed-loop organic vapor wind tunnel. The effect of the Reynolds number was assessed independently from the Mach number by charging the closed wind tunnel. The airfoil surface roughness and the trailing edge shape were evaluated by experimenting with different test blades. The flow and the loss behavior were obtained using Pitot probes, static wall pressure taps, and background-oriented schlieren (BOS) optics. Isentropic exit Mach numbers up to 1.5 were investigated. Features predicted via a simple flow model proposed by Denton and Xu in 1989 were observed for organic vapor flows. Still, roughness affected the downstream loss behavior significantly due to shockwave boundary-layer interactions and flow separation. The new experimental results obtained for this organic vapor are compared with correlations from the literature and available loss data.

1. Introduction

The trailing edge loss is typically a main contribution to the total loss of transonic turbine cascades [1], and good knowledge of the underlying mechanisms is important for improving turbine efficiencies.

Although trailing edge flows are highly complex, as reviewed by Sieverding and Manna [2], the flow pattern at a supersonic trailing edge is now mainly well documented in the literature. Figure 1 illustrates the structure of the supersonic trailing edge flow with its typical shock pattern at the base region. Immediately behind the trailing edge is a base pressure region characterized by relatively low velocity at nearly uniform pressure. This region is bounded by shear layers. In the shear layers, the velocity increases rapidly from the low value in the base region to the supersonic flow in the free stream. The flows from the suction and pressure surfaces meet downstream of the trailing edge, and this occurs in the confluence region at the downstream end of the base pressure region.

Since the groundbreaking study [3] by Sieverding et al. in 1979, several researchers have investigated trailing edge flows in detail. Whereas, in [3], viscous shear stresses in the trailing edge region were considered predominant, shock patterns and vortex shedding were examined by Motallebi et al. [4], Sieverding and co-workers [5,6,7], Currie and Carscallen [8], and Melzer and co-workers [9,10]. Although the Sieverding correlation [2,3] is frequently used, other correlations for predicting the base pressure have also been proposed in the literature [11,12]. Energy separation phenomena in the wake were discussed by Gostelow et al. [13], and the role of coherent structures for trailing edge losses was investigated by Gohl et al. [14].

Although this brief literature survey demonstrates the richness of details in trailing edge flows, in 1989, Denton and Xu [15] proposed a relatively simple model for predicting the mixed-out loss far downstream of a choked turbine cascade. Their model was based on the fundamental balance equations and has been considered for compressible perfect gas flows. Due to its general approach, it can be adapted to non-ideal gas dynamics by using appropriate equations of state for real gases (see the following section). Hence, the Denton and Xu model might provide useful relations for flows in turbines for organic Rankine cycle (ORC) power systems where little experimental work is reported in the literature.

Due to the low speed of sound, organic vapor flows in turbines are usually transonic and supersonic. Because of the complex molecular structure, significant non-ideal compressible flow phenomena can occur [16]. Although some profile loss data for organic vapor flows have been recently published [17,18], relatively little is known regarding trailing edge flow details and losses. Wheeler and co-workers published experimental data regarding the role of the thermophysical properties of the working fluid on losses including trailing edge flows for smooth airfoils [19,20]. Recently, it was shown in [21] through high-fidelity computational fluid dynamics (CFD) methods and experimental data that the boundary-layer state and, hence, the roughness of the blades are relevant to trailing edge vortex shedding and losses for organic vapor flow.

The topic of the present work is the impact of the Mach number and roughness on the loss behavior and reliability of literature loss correlations, especially with a look at trailing edge organic vapor flow phenomena. In this context, the simple Denton and Xu model might help to explain some aspects of the complex compressible flow phenomena. This model has not been applied to an organic vapor trailing edge flow so far, and its reliability to non-ideal gas flows has not been assessed. The novelty of this study concerning the existing literature is given by the working fluid used in this study. Experimental results for the organic vapor flow under mild non-ideal thermodynamic conditions will be reported, which can be compared with the available literature data for perfect gases.

2. Denton and Xu Model

In this section, the loss model proposed by Denton and Xu [15] is briefly reviewed. Their loss model applies to choked turbine cascades, where a Mach number of unity is reached at the throat, i.e., Ma_t = 1.

2.1. Staggered Trailing Edges

A cascade of staggered thick plates, illustrated in Figure 2, is the configuration considered by Denton and Xu. The flow is choked, and the general conservation equations for mass and momentum yield

$${\displaystyle \frac{p_{o2\infty }}{p_{o1}}}={\displaystyle \frac{F\left(1\right)}{F\left({\mathit{Ma}}_{2\infty }\right)}}{\displaystyle \frac{S\mathit{cos}\beta -t-{\delta }^{*}}{S\mathit{cos}(\beta -\delta )}}$$

(1)

$$\dot{m^{\prime }}{(c}_t-c_{2\infty }\mathit{cos}\delta )-\rho c_t^2{\delta }^{**}+p_t\left(S\mathit{cos}\beta -t\right)+p_{bp}t=p_{2\infty }S\mathit{cos}\beta$$

(2)

$$(p_{ss}-p_{2\infty })S\mathit{sin}\beta =\dot{m^{\prime }}\mathit{sin}\delta$$

(3)

The above equations are valid for a plane far downstream of the cascade, denoted by subscript 2∞. More common in cascade experiments are measurements at a downstream plane closer to the trailing edge, denoted by subscript 2 in Figure 2. The displacement thickness of the boundary layer, δ*, increases the effective blade thickness t. The momentum thickness, δ**, does not directly affect the loss but the base pressure, p_bp. The dimensionless mass flow function, F, can be expressed in the case of a perfect gas as a function of the Mach number Ma and isentropic exponent γ = c_p/c_v, see [22].

2.2. Real Gas Modification

To apply the basic ideas of Denton and Xu to real gas flows, as required for organic vapor and non-ideal compressible fluid dynamics, a different approach than the mass flow function F must be chosen. In non-ideal compressible flows, no universal mass flow functions exist; instead, the expansion process depends on the specific inlet or stagnation state [16].

Figure 3 illustrates the scheme for evaluating the total pressure ratio (Equation (1)) and the other flow quantities (Equations (2) and (3)) for an expansion of a real gas from state 1 to state 2. For a given state with p₁ and T₁, the constant total specific enthalpy h_o1 can be evaluated for a choked station (subscript 1) by adding the kinetic energy with the speed of sound a₁ to h₁. The exit Mach number Ma₂ and the pressure ratios p_o2/p_o1 and p₂/p_o1 are related through h_o1 = h_o2 = h(p_o2, T_o2). The condition of a constant mass flux through planes 1 and 2 fixes the total pressure ratio using the densities ρ₁ and ρ₂, respectively. Hence, the total pressure ratio p_o2/p_o2 of Equation (1) is only a function of the exit Mach number Ma₂ and the effective blockage, as in the case of a perfect gas. To conclude, real gas dynamics do not substantially change the architecture of the Denton and Xu model; they affect the constitutive equations but not the salient model features.

2.3. Discussion of the Denton and Xu Model

For the special case of an unstaggered cascade (i.e., for parallel plates with finite thickness), the flow deviation vanishes, i.e., δ = 0. Neglecting the boundary-layer thickness effects, it follows that the mixed-out stagnation pressure is completely determined by the downstream Mach number Ma_2∞ for a given effective blockage, see Equation (1). Applying the momentum Equation (2), it follows that the base pressure p_bp and the total pressure ratio and, hence, the loss can be exactly determined without involving any viscous effects [15]. However, this holds only for the mixed-out loss far downstream of a choked cascade, denoted by the subscript 2∞. For a staggered cascade with a non-vanishing flow deviation δ, an additional relation is required to specify the base pressure value p_bp because the above set of equations, Equations (1)–(3), is not complete. For this, Sieverding’s correlation [2,3] or other relations using the suction-side pressure p_ss [12,15] might be applied to obtain the base pressure. Blade boundary layers can easily be included in the above analysis. The displacement thickness δ* adds to the blade thickness, leading to a slightly higher blockage ratio. The momentum thickness δ** affects the base pressure through Equation (2).

Equations (1)–(3) hold rigorously only for a control plane far downstream denoted by subscript 2∞, at which mixed-out flow quantities exist. Frequently, the engineer is more interested in flow quantities obtained by a downstream plane, denoted by subscript 2, much closer to the trailing edge of the blades, see Figure 2. Since the Denton and Xu model is only able to relate the mixed-out loss at a station far downstream for a choked blade row, experimental work, and measurements are still needed to get a clearer picture of the trailing edge loss contribution for engineering purposes.

3. Results

The experimental method and procedure continued a previous study [17]. To assess the impact of the Reynolds number and roughness on the loss, modifications of the former experiments were needed, which are described in the following.

3.1. Test Section and Configurations

A single-blade, two-passage vane configuration with a simplified blade was used as the test object, as in [17]. The high-speed test section is shown in detail in Figure 4, and some data are listed in

Table 1. Technical data of the test configuration.

Chord	C	61.2 mm
Spacing	S	40.0 mm
Stagger angle	β	60°
Blade height (span)	H	35.0 mm
Inlet channel width	B	36.5 mm

. The single airfoil and the stagger angle of β = 60° corresponded to a configuration used by Sieverding et al. [23]. A similar approach was utilized by Deckers and Denton [24], and the configuration had proven its value for tip-gap flow investigations with air [25]. In the present study, the tailboards of the test section were fixed to enable profile pressures at the airfoil corresponding to a typical representative distribution (see Section 4).

During the test campaign, the central airfoil was changed. Three different airfoils were investigated in the present study, see Figure 5. They differed regarding their actual roughness level and their trailing edge radii. The measured average roughness parameters and the trailing edge radius r_TE of the three airfoils, called “smooth”, “rough”, and “sharp”, are listed in

Table 2. Details of the investigated airfoils.

Type	Ra [µm]	Rz [µm]	Rt [µm]	Rp [µm]	rTE [mm]
Smooth	0.3	1.3	1.8	0.6	1.2
Rough	18.0	126.3	143.4	57.4	1.2
Sharp	0.3	1.3	1.8	0.6	<0.2

. The smooth surface was even for the highest Reynolds number and hydraulically smooth in terms of the condition R_t/2.58 < 100 μ/(ρ c).

The nominal airfoil design corresponded to the type “smooth”, manufactured conventionally. The rough airfoil was manufactured via three-dimensional metal printing, with a prescribed rough surface structure but with the same profile as the nominal smooth airfoil. The conventionally manufactured sharp airfoil had a smooth surface. Still, the trailing edge had a sharp shape, ending in a tiny trailing edge radius, see Figure 5. The nominal trailing edge radius of the airfoil was r_TE = 1.2 mm and characterized by a circular shape.

3.2. Flow Cases and Thermodynamic Data

The idealized vane configuration shown in Figure 4 was placed in the test section of a closed-loop organic vapor wind tunnel (called CLOWT), enabling flow investigations with the organic vapor Novec 649 at elevated pressure and temperature levels. Details about the wind tunnel CLOWT can be found elsewhere [26], and some thermodynamics of Novec 649 are discussed in [18]. The inflow turbulence level, obtained by a single hot wire probe, for the present test section, was in the order 1 up to 6%, increasing with the Mach number. The corresponding turbulence macro-length scale was about 1 mm.

The average density level in the test section was controlled by the total mass filled in the wind tunnel (charging). Two different cases were selected in the present study, called “high-density HD” and “low-density LD”.

Table 3. Flow cases and thermodynamic data.

Physical Quantities	Flow Case	HD	LD
Pressure	p [bar]	2.3	1.6
Temperature	T [K]	383	378
Density	ρ [kg/m3]	25	17
Speed of sound	a [m/s]	94.6	95.6
Compressibility factor	Z [−]	0.92	0.94
Isentropic exponent	γ [−]	1.04	1.04
Fundamental derivate	Г [−]	0.92	0.95

lists the main operation conditions and thermodynamic data for the two flow cases. The speed of sound, a, was nearly the same for the two density levels. The actual relationship between the compressor running speed n and the isentropic exit Mach number Ma_2s is shown in Figure 6. For low speeds, a nearly linear relation between n and Ma_2s existed, but at higher speeds, nonlinear behavior could be observed. This behavior was very similar for both density levels. The main difference between the two flow cases, HD and LD, was their different Reynolds number levels. For all flow cases and velocities, the Reynolds number Re, based on the airfoil’s chord and exit velocity, exceeded 1.1 × 10⁶ (low-density case at incompressible flow conditions). Due to the high density of the organic vapor, the maximum Reynolds number was in the order Re = c₂ Cρ₂/µ₂ = 3.67 × 10⁷. The values for the isentropic exponent γ, the compressibility factor Z, and the fundamental derivative of gas dynamics Г remained similar for the two flow cases. The value of Г < 1 indicated that mild non-ideal compressible flow conditions were present in the test section [16,26].

3.3. Instrumentation and Data Reduction

The stagnation pressure p_o and temperature T_o were measured in the settling chamber of the wind tunnel upstream of the test section and controlled by heating and cooling of the entire test facility. The inventory mass of the wind tunnel defined the average density. The nearly isentropic flow through the contraction zone of the wind tunnel to the test section was checked by measuring the inlet total pressure p_o1 through an upstream Pitot probe, see Figure 4. The deviation between p_o and p_o1 was within the experimental uncertainty level of about 0.5%.

The mass flow rate $\dot{m}$ was controlled by the variable running speed n of the centrifugal compressor and recorded by a mass flow device placed in the return of the wind tunnel, where, essentially, incompressible flow conditions existed.

The profile pressure distribution around the airfoil, including the base pressure p_bp and the suction-side pressure p_ss at the trailing edge, and further static pressure values were measured using several static wall pressure taps, shown in Figure 5. Miniaturized Pitot tubes measured upstream and downstream total pressures p_o1 and p_o2 (at a far-downstream distance of 3.2C, mixed-out conditions could be expected after [27], i.e., p_2∞ and p_o2∞). For evaluating the thermodynamic properties and flow variables based on pressure and temperature measurements, the constitutive equations provided by REFPROP [28] were utilized.

The shock pattern around the airfoil in the test section was visually observed using background-oriented schlieren optics (BOS). Although BOS pictures are typically not characterized by such high resolution as in conventional schlieren optics, the great advantage of BOS is that only a single optical access to the test object is necessary. One casing wall of the test section shown in Figure 4 was transparent, and the background wall of the test section was equipped with a random dot pattern. A high-speed camera recorded the flow past the airfoil, with a maximum frame rate of 3 kHz. This frame rate was not sufficient to resolve the vortex shedding at the trailing edge in detail, and, hence, the BOS pictures correspond to average flow fields.

The relative uncertainty level of the pressure measurements was in the order ∆p/p = 0.5% up to 2% depending on their absolute values. The uncertainty was the result of precision errors (at a confidence level of 95%) and bias errors caused by the unknown hydrostatic pressure due to potential condensation in the pressure lines (see reference [26] for a further discussion about measurement errors due to condensation in pressure lines). Although the relative temperature measurement uncertainty was less than ∆T/T = 0.1%, a significant temperature drift of about 1–3 K occurred during a measurement campaign. Due to the large time constant of the temperature control method, the drift and temperature scattering of stagnation conditions during a measurement campaign could not be avoided. Taking the pressure and temperature uncertainty levels and the uncertainty levels guaranteed by REFPROP into account, the resulting uncertainty level of the Mach number was about ∆Ma/Ma = 5%.

4. Experimental Results

In this section, the outcome of the measurements is reported. The flow data based on pressure measurements are presented, and the flow field observations made by BOS are shown and discussed.

4.1. Choking Mach Number

For compressible flow through cascades, the choking Mach number Ma_1ch is important [29]. The actual inflow Mach number Ma₁ is plotted against the achieved isentropic exit Mach number Ma_2s in Figure 7. Due to the independent high-accuracy mass flow records in the return of the wind tunnel, the Mach numbers could be obtained with low uncertainty (with error bars comparable with the symbol size in Figure 7).

As predicted by theory, a certain value of the choking Mach number Ma_1ch = 0.550 + 0.005 was noticed for the actual idealized vane configuration. The tendency to a slightly lower choking Mach number for the rough airfoil might be attributed to the thicker boundary layer δ*, but the observed deviations remained within the actual uncertainty level, and no definite conclusion is possible. In terms of wind tunnel compressor running speed n, choking was achieved for Novec 649 at n ≈ 50 Hz. Since the Denton and Xu model applies only to choked cascades, it should be considered only for operation points obtained at n > 50 Hz.

4.2. Profile Mach Number Distributions

Based on the static pressure measurements, see Figure 5, the profile Mach number distributions were obtained for the high- and low-density cases for the smooth and the rough airfoils. The experimental data for a constant compressor running speed n = 70 Hz (corresponding to an isentropic exit Mach number Ma_2s ≈ 1.5) together with the prediction of a two-dimensional prediction obtained by computational fluid dynamics (CFD) are plotted in Figure 8.

Since the absolute values of the Mach numbers scattered slightly for each test run due to the temperature drift and the density variations, see Figure 6, it was necessary to normalize the Mach number distribution by assuming a reference value. The profile Mach number distributions were similar for the rough and smooth airfoils, and no significant differences for the two Reynolds number levels were observed. The latter finding supported the frequently made observation that the Reynolds number dependency vanishes for Re > 10⁷ [27]. That roughness had little effect on the airfoil Mach number distributions, which was reported in the literature [30] for air, too. However, that does not necessarily imply that the trailing edge losses would not be affected by roughness (see following discussion).

4.3. Total Pressure and Energy Loss

Using the total pressure measurements, the total pressure loss coefficient

$$L={\displaystyle \frac{p_{o1}-p_{o2\infty }}{p_{o1}}}$$

(4)

was obtained for the flow cases and the airfoils. The behavior of L as a function of the Mach number is shown in Figure 9. As expected, the total pressure loss L increased with the Mach number Ma_2s.

In principle, three different flow regimes can be distinguished in Figure 9. For low up to high subsonic Mach numbers, nearly incompressible flow occurred, and the roughness only slightly increased the profile loss. The smooth airfoil with the sharp trailing edge had a lower profile loss in this regime, probably due to the lower trailing edge loss (see Section 5). At Mach numbers higher than 0.7, a strong rise in the losses can be observed. This corresponds to the transonic flow regime, where compressibility effects increase the profile loss. For supersonic Mach numbers, the increase in loss was weaker and reached a relatively stable high level for the airfoils with thicker trailing edges. The sharp edge airfoil exhibited a local loss maximum for an exit Mach number of about 0.9 corresponding to choked flow conditions.

Interestingly, there was a clear tendency for higher losses for the rough airfoils for Ma_2s > 0.9 and a tendency to substantially lower losses for the sharp (smooth) airfoil.

Although the total pressure loss coefficient L is a suitable quantity to assess the aerodynamic performance of transonic cascades [31], the behavior of the energy loss coefficient

$$\zeta ={\displaystyle \frac{h_{o2}-h_{o2s}}{h_{o1}-h_2}}$$

(5)

is of interest, too. Based on the pressure data, the energy loss coefficient ζ was evaluated using the relationship provided by Horlock [32]. The result for the present idealized vane configuration is shown in Figure 10.

For high subsonic Mach numbers, 0.6 < Ma_2s < 0.9, the data scattered significantly. In this flow regime, the energy loss was relatively high, and for supersonic Mach numbers, the energy loss coefficient decreased. Interestingly, for a just choked flow condition (at a Mach number of about 0.9), there was a tendency to a maximum of the energy loss coefficient ζ for the airfoils. This can be explained by the occurrence of a shock at the throat region and the triggered flow separation in the downstream channel region. As in the case of the total pressure loss coefficient L, see Figure 9, there was a clear tendency of rough airfoils to higher losses for the energy loss coefficient ζ in Figure 10 within a certain Mach number range. The loss obtained for the sharp airfoil was significantly lower at high subsonic and high supersonic flow but reached a similar level to the airfoils with thick trailing edges in transonic flow conditions.

No significant Reynolds number effects were observed for the airfoils at supersonic Mach numbers, which can be attributed to the overall high Reynolds number level for all investigated density and velocity levels.

4.4. Flow Field Analysis

To gain more insight into the flow field, BOS pictures were obtained. Figure 11 shows BOS pictures obtained for the smooth airfoil at the high-density level.

For Ma_2s = 0.88, the configuration was choked, and two normal shocks occurred close to the two narrowest channel passages (see the top picture in Figure 11). At Ma_2s = 1.03, the shock at the suction side moved downstream (middle picture), and for supersonic Mach numbers (bottom picture in Figure 11), the typical fishtail shock pattern occurred and might be compared with the sketch shown in Figure 1. The observed shock pattern for the supersonic flow past a rough airfoil was very similar to the smooth airfoil at Ma_2s ≈ 1.5.

In Figure 12, a corresponding BOS picture is shown for the rough airfoil. Due to some improvements in the optical setup, the BOS picture quality could be increased for the test campaigns with the rough airfoil (here, the rough airfoil is denoted by a black filling color, whereas white color is used for the smooth airfoil). There was no significant difference between the shock pattern shown in Figure 12 for the rough airfoil and the bottom picture of Figure 11 illustrating the supersonic flow past a smooth airfoil. Fishtail angles and the location of the shock reflection at the suction side were the same.

In the case of transonic flow, a strong sensitivity to the roughness was observed at the trailing edge. This is illustrated using Figure 13 and Figure 14. The normal shocks looked very similar under just choked conditions for the rough and smooth airfoils, as shown in Figure 13. The normal shocks can be identified as dark, perpendicular lines, reaching the airfoil surfaces in Figure 13. Due to the high Reynolds number level in the order Re = 3.6 × 10⁷, the boundary layer was very thin, and the shock lines hit the suction sides of the airfoils in Figure 13.

Serious flow separation due to a shockwave boundary-layer interaction was observed for a slightly higher Mach number in the case of a rough airfoil. This is illustrated in Figure 14. On the suction side of the rough airfoil, massive flow separation can be observed in Figure 14. The observations regarding the shockwave boundary-layer interaction (SWBLI) for the organic vapor Novec 649 can be compared with photography from the literature [33], illustrating SWBLI in perfect gas flows. The typical λ-like pattern can be recognized in Figure 14. Some secondary shocks downstream of the separation point occurred, too.

The flow separation in the case of rough airfoils led to more dissipative wake flows, and this might explain the higher losses observed for transonic Mach numbers in Figure 9 and Figure 10. In the case of smooth airfoil, no massive flow separation was observed in the BOS pictures (see the middle image in Figure 11). As stated in [33], “the boundary-layer velocity distribution is perhaps the most important factor” for flow separation downstream. This might explain why roughness on the airfoil led to a flow separation downstream of the shock, whereas the flow past the smooth airfoil did not separate, and the turbulent region remained tiny; no substantial turbulent wake flow was created.

4.5. Base Pressure

The importance of the base pressure p_bp for the trailing edge losses is well documented in the literature [1,2,3,4,5,6,7,8,9,10,11,12]. Still, only limited data about real gas trailing edge flows are available. In the present study, base pressure values p_bp were measured utilizing a static pressure hole close to the airfoil’s trailing edge, see Figure 5. Due to technical restrictions, only time-averaged pressure values could be obtained. Furthermore, it should be noted that the finite size (diameter 0.5 mm) of the hole and its location, which was not exactly on the airfoil’s trailing edge in midspan, could introduce some systematic errors. The base pressure coefficient

$$C_{p,bp}={\displaystyle \frac{p_2-p_{bp}}{p_{o1}-p_2}}$$

(6)

against the freestream Mach number Ma₂ is plotted in Figure 15. In the present study, the behavior of the base pressure was very similar for all flow cases. For higher Mach numbers, the base pressure coefficient increased substantially. No qualitative difference was observed for smooth and rough blades. However, the few data points obtained at supersonic Mach numbers do not permit a definite statement. In Figure 15, the freestream Mach number Ma₂ was obtained using the method proposed by Sieverding and Heinemann [6], utilizing, as the reference static pressure p_2,ss, the averaged values close to the trailing edge (see Figure 5). It should be noted that there are other conventions and definitions for the base pressure coefficient in use [27], leading to other absolute values for the base pressure coefficient.

The new experimental data for the base pressure ratio p_bp/p_o1 of the flow of the organic vapor Novec 649 under the conditions listed in Table 3 are plotted in Figure 16 against the static exit pressure ratio p₂/p_o1.

In addition to the real gas data, the single data point obtained in [25] for air is provided in Figure 16. Furthermore, the prediction of the Sieverding correlation [2,3] for the actual airfoil trailing edge angle condition is shown. The new organic vapor data agreed well with this popular correlation, and the air data point followed it. However, it has to be remarked that this good agreement was achieved using the static exit pressure value p₂ obtained at a station downstream of the trailing edge, where, typically, traverse measurements for loss investigations would be executed in cascade experiments. This station corresponded to the pressure taps shown in Figure 5 on the right side of the airfoil.

In the case of using the suction-side pressure p_2,ss, the agreement between the new data and the Sieverding correlation was a little bit poorer in the case of high subsonic and transonic flow. That observation underlines that selecting the “correct” static exit pressure is crucial for applying the correlation, as mentioned earlier in [12].

It is instructive to consider alternative correlations for the base pressure and compare their predictions with the new data obtained for a real gas. The base pressure correlation due to Carrière [11] is valid only for base pressure Mach numbers Ma_bp > 1.6. That value was not achieved during the present organic vapor experiments (the maximum base pressure Mach number was about Ma_bp = 1.5). Hence, the Carrière method is not further considered in the present discussion.

An alternative base pressure correlation was proposed by Bölcs and Sari in [12]. This approach utilizes the so-called recompression point Mach number Ma_R. As stated in [12], “due to the small size of the training edge […], it is not possible to measure directly the recompression pressure p_R in a cascade.” Bölcs and Sari proposed an indirect way, using the method of characteristics and the impingement point of the trailing edge shock on the suction side of the airfoil. However, the accurate determination of Ma_R was impossible for the actual configuration with its instrumentation and limited spatial resolution of the pressure distribution measurements.

5. Discussion and Interpretation

The above section demonstrated that compressible trailing edge flows are other than simple and that a large variety of phenomena can occur. On the other hand, the basic considerations of Denton and Xu [15], although dealing with an oversimplified configuration, might still support the interpretation of the observations. In the following, the experimental results are discussed within the framework of their theory, briefly explained in Section 2, and a comparison with literature correlations and loss data is made.

5.1. Mixed-Out Total Pressure Ratio and Losses

The model of Denton and Xu requires mixed-out flow variables, obtained at a plane far downstream of the blade (denoted by 2∞). Using the available data for choked flow, the total pressure ratio p_o2∞/p_o1 and the isentropic total pressure loss coefficient

$$Y_{s\infty }={\displaystyle \frac{p_{o1}-p_{o2\infty }}{p_{o1}-p_{2\infty }}}$$

(7)

are plotted against the static pressure ratio p_2∞/p_o1 in Figure 17 and Figure 18, respectively. The agreement with the theory (computed for unstaggered plates without boundary-layer thickness effects or flow separation or deflection) is reasonable in Figure 17 and Figure 18.

There was no systematic impact of the roughness detectable, as predicted by the theory. However, the value of knowledge of the mixed-out flow variables far downstream of the test section for turbine designers might be limited in practice. Furthermore, it might be argued that the Denton and Xu model repeats a very general consequence of the compressible flow relations, which applies to any one-dimensional gas dynamical problem. Indeed, the results shown in Figure 17 and Figure 18 would apply to any choked straight channel flow with a kind of internal obstacle. Hence, the outcome of the above plots should not be overestimated for turbine design tasks.

5.2. Analysis of the Observed Loss Behavior

The Denton and Xu model [15] permits an interpretation of the observed loss behavior in the case of thick trailing edges, for which the plate cascade in Figure 2 is a reasonable approximation. This is illustrated using Figure 19, where the observed total pressure loss for the smooth and rough airfoils is plotted against the isentropic exit Mach number (see Figure 9). In Figure 19, the nearly incompressible flow range is covered, too. For the smooth airfoil, the subsonic, non-choked flow data points lie on a simple line. The loss of the smooth airfoil increased substantially after choking for Ma_2s > 0.8 (see Figure 7, too). After a strong jump, the loss mainly stabilized at a high level for high Mach numbers Ma_2s > 1.2 (with only a moderate increase rate).

Interestingly, the simple theoretical model of Denton and Xu similarly predicted a substantial increase (depending on the effective blockage) within the Mach number range 0.7 < Ma_2s < 1. In the case of a rough blade, the thicker boundary layer and the separation zone increased the effective blockage ratio, directly leading to a higher maximum loss value. The behavior illustrated in Figure 19 indicated that the strong loss increase for a choked cascade is the result of a very fundamental gas dynamical process. The main features of this loss increase can be explained without the need to introduce viscosity or dissipation. This was the exact main statement of Denton and Xu in [15], and it can be supported by the present real gas experiments. Viscosity and dissipation are of secondary importance for the strong loss increase after choking, and it might be relevant for stabilizing the loss level after that rise. The very fundamental gas dynamical relations govern the loss behavior in this flow regime. This indicates that the impact of real gas properties is probably limited in the transonic flow regime. However, the simple one-dimensional flow analysis was not able to cover all salient features of the loss behavior for the entire Mach number range. In particular, the simple model failed to predict the substantial loss plateau for higher Mach numbers.

5.3. Comparison with Literature Data

For the high-density flow past the smooth airfoil, loss results from a high-fidelity CFD analysis were already published in [17] for Ma_2s about 1.5. In addition to the total loss coefficient ζ, the decomposition of the profile loss coefficient proposed by Denton [1]

$${\zeta }_{Den}=C_{p,bp}{\displaystyle \frac{2r_{te}}{S\mathit{sin}\beta }}+{\displaystyle \frac{2{\delta }^{**}}{S\mathit{sin}\beta }}+{\left({\displaystyle \frac{{\delta }^{*}+2r_{te}}{S\mathit{sin}\beta }}\right)}^2$$

(8)

was considered in [17], too. In this decomposition, it is assumed that the trailing edge loss results from the base pressure and the finite thickness contributions. A purely trailing edge loss part, neglecting the boundary-layer thickness contribution, would be given by the expression

$${\zeta }_{te}=C_{p,bp}{\displaystyle \frac{2r_{te}}{S\mathit{sin}\beta }}+{\left({\displaystyle \frac{2r_{te}}{S\mathit{sin}\beta }}\right)}^2$$

(9)

Using the actual base pressure data, the trailing edge contribution ζ_te to the profile loss could be estimated. No massive flow separation was observed for the highest exit Mach number flow, and, hence, the boundary-layer thickness values δ* and δ** were assumed to be much smaller than the trailing edge radius r_te for the nominal airfoil. This assumption enabled a comparison with the available literature data. The outcome of this comparison is listed in

Table 4. Comparison of loss data obtained for an organic vapor at an isentropic Mach number Ma_2s≈1.5.

Loss Quantity	Method	Value
Total loss coefficient ζ	CFD [17]	8.1%
Total loss coefficient ζ	Experiment	9.1 ± 0.6%
Denton profile loss ζDen	CFD [17]	5.5%
Trailing edge loss ζte	CFD [17]	2.9%
Trailing edge loss ζte	Experiment	3.6 ± 0.6%
Trailing edge loss ζte	Traupel’s correlation	2.86%
Trailing edge loss ζte	D&T data [27]	2.8%

.

The prediction of the Traupel loss correlation [2] is also listed in Table 4. The experimental value of about ζ = 9.1 ± 0.6% obtained for the smooth airfoil at Ma_2s = 1.55 and high density was in reasonable agreement with the prediction ζ = 8.1% of a recent high-fidelity CFD simulation [17]. The computed Denton loss expression ζ_Den could not be compared directly with experimental data because the detailed boundary-layer information was missing. However, neglecting the boundary-layer thickness contributions due to the high Reynolds number level and the absent flow separation at the high Mach number, the expression ζ_te could be compared. Experimentally, a higher value ζ_te = 3.6 ± 0.6% was obtained due to the higher value of the experimental base pressure coefficient C_p,bp. The prediction ζ_te = 2.86% of Traupel’s correlation, as quoted in [1], was in excellent agreement with the CFD analysis. Interestingly, the data and method provided by Dejc and Trojanovskij [27] would lead to a similar value of ζ_te = 2.8%.

It is instructive to plot the new trailing edge loss data obtained for the organic vapor Novec 649 at a high Mach number with the available literature data in a diagram, see Figure 20. Here, the open red circles correspond to data points evaluated from the published work from Dejc and Trojanovskij (see, for instance, reference [27] and other internal work from their research group). In addition to their experimental data points, their recommended linear trailing edge loss correlation [27] is plotted as a dotted red line. Dejc and Trojanovskij recommend the use of their simple linear relationship only for moderately thick trailing edges because nonlinear phenomena would occur for thick ones. Furthermore, the trailing edge loss contribution, as quoted by Kacker and Okapuu [34], for axial entry blades is plotted as a solid red line in Figure 20 for the sake of comparison. The red literature data and correlations are in fair agreement, despite their different Mach number levels covering Ma_2s = 0.4 up to 1.2 (and even higher). The new experimental data obtained for the sharp and thick trailing edges are plotted as black symbols in Figure 20. Furthermore, the high-fidelity CFD result from [17] is shown in Figure 20 as a black dot.

The new trailing edge loss data for the organic vapor flow fit reasonably within the literature data point distribution, although the scattering of the loss is substantial. The CFD prediction is in excellent agreement with the correlations suggested by Kacker and Okapuu [32] and Dejc and Trojanovskij [27], which predict similar trailing edge losses in this range of thickness levels. The new slightly higher experimental trailing edge loss for a normalized thickness of about 0.125 is within the scattering level of the data. In any case, the frequently quoted statement of Denton [1], that trailing edge losses can contribute to one-third (or even more) of the profile loss, could be supported by the experimental and numerical data for the organic vapor flow. That observation underlines the importance of trailing edge flows for turbine design.

6. Summary

An experimental study of organic vapor trailing edge flows was conducted using the organic vapor Novec 649 at mild non-ideal compressible flow conditions for an idealized turbine vane configuration [35]. The flow and the loss behavior were obtained using pneumatic probes and background-oriented schlieren (BOS) optics. Isentropic exit Mach numbers up to 1.5 were investigated. The effect of the Reynolds number was assessed independently from the Mach number by charging the closed wind tunnel. The impact of the airfoil surface roughness and the trailing edge shape was assessed by conducting experiments with different airfoils.

It was found that roughness led to systematically higher losses for supersonic flow. The BOS analysis indicated that this could be, at least for a certain Mach number range, attributed to massive flow separation, leading to higher profile losses.

For predicting the base pressure, the popular Sieverding correlation proved its reliability for organic vapor flows, characterized by values of the fundamental derivative of gas dynamics below unity.

It was found that some features of the simple Denton and Xu loss model were still applicable even for organic vapor flows characterized by values of the fundamental derivative of gas dynamics below unity. Still, for quantitative loss predictions, the simple Denton and Xu flow model is insufficient.

The new trailing edge loss data agreed reasonably with the available literature data and loss correlations, but further research is necessary to obtain more precise information about the impact of the Mach number and the underlying mechanisms, especially in combination with roughness.