2.1 Quantification of loss

To quantify the profile loss at a turbomachine component, different definitions of loss coefficients have been established for different applications. Denton^{(Reference Denton2)} emphasises three commonly used definitions based on:

1. 1. energy

2. 2. entropy

3. 3. total pressure

Figure 1. Exemplary enthalpy–entropy diagram for an adiabatic compression and expansion at a stationary blade row. (a) Compressor. (b) Turbine.

For adiabatic cases, the energy loss coefficient is a measure of the utilisation of the kinetic energy. Setting up the first law of thermodynamics in Equation 2 based on Fig. 1, the understanding of the definition of the energy loss coefficient in Equation 3a and 3b according to Denton^{(Reference Denton2)} is facilitated. The loss of kinetic energy and of enthalpy can be regarded synonymously at an adiabatic flow.

(2) \begin{equation}\frac{{c_1^2}}{2} = {\rm{\Delta }}{h_{12}} + \frac{{c_2^2}}{2}= {\rm{\Delta }}{h_{12}} + \frac{{c_{2s}^2}}{2} - {h_{loss}}\end{equation}

(3a) \begin{equation}\text{Compressor:}\ {\zeta _{h,C}} = \frac{{c_{2s}^2/2 - c_2^2/2}}{{c_1^2/2}} = \frac{{{h_2} - {h_{2s}}}}{{{h_{t1}} - {h_1}}} = \frac{{{h_{loss}}}}{{{h_{t1}} - {h_1}}}\end{equation}

(3b) \begin{equation}\text{Turbine:}\ {\zeta _{h,T}} = \frac{{c_{2s}^2/2 - c_2^2/2}}{{c_2^2/2}} = \frac{{{h_2} - {h_{2s}}}}{{{h_{t2}} - {h_2}}} = \frac{{{h_{loss}}}}{{{h_{t2}} - {h_2}}}\end{equation}

A second way of loss quantification is the entropy loss coefficient by which the loss is directly related to the generation of entropy as in Equation 4. Assuming air as an ideal gas, the entropy generation can be calculated by Equation 5.

(4) \begin{equation}{h_{loss}} = \mathop \int \limits_{{s_1}}^{{s_2}} Tds \approx {T_2} \cdot {\rm{\Delta }}{s_{12}}\end{equation}

(5) \begin{equation}{\rm{\Delta }}{s_{12}} = {c_p} \cdot \ln \frac{{{T_{t2}}}}{{{T_{t1}}}} - R \cdot \ln \frac{{{p_{t2}}}}{{{p_{t1}}}}.\end{equation}

The resulting entropy loss coefficient is defined in Equations 6a and 6b^{(Reference Denton2)}.

(6a) \begin{equation}\text{Compressor:}\ {\zeta _{s,C}} = \frac{{{T_2} \cdot {\rm{\Delta }}{s_{12}}}}{{{h_{t1}} - {h_1}}}\end{equation}

(6b) \begin{equation}\text{Turbine:}\ {\zeta _{s,T}} = \frac{{{T_2} \cdot {\rm{\Delta }}{s_{12}}}}{{{h_{t2}} - {h_2}}}\end{equation}

Finally, the most common definition of the loss coefficient is given by the total pressure loss coefficient in Equations 7a and 7b^{(Reference Denton2)}. It results from the energy loss coefficient, assuming an incompressible flow.

(7a) \begin{equation}\text{Compressor:}\ {\zeta _{p,C}} = \frac{{\rho c_{2s}^2/2 - \rho c_2^2/2}}{{\rho c_1^2/2}} = \frac{{{q_{2s}} - {q_2}}}{{{q_1}}} = \frac{{{p_{t1}} - {p_{t2}}}}{{{p_{t1}} - {p_1}}}\end{equation}

(7b) \begin{equation}\text{Turbine:}\ {\zeta _{p,T}} = \frac{{\rho c_{2s}^2/2 - \rho c_2^2/2}}{{\rho c_2^2/2}} = \frac{{{q_{2s}} - {q_2}}}{{{q_2}}} = \frac{{{p_{t1}} - {p_{t2}}}}{{{p_{t2}} - {p_2}}}\end{equation}

To determine the mentioned loss coefficients for a test object, primarily the values of total and static pressure have to be determined upstream and downstream of the test object. Assuming ${T_{t2}} = {T_{t1}}$ for adiabatic cases, the temperature information in the wake, which is necessary for the energy and entropy loss coefficient, can be calculated from the pressure data and the inlet total temperature. Thus, the measurement of the total temperature in the wake is not obligatory in experimental setups, although it could deliver helpful additional information, as shown later on.

Denton^{(Reference Denton2)} has shown that the energy and entropy loss coefficients have to be approximately equal. The justification of the entropy loss coefficient is according to Denton^{(Reference Denton2)} that the summation of the created entropy in each blade row is valid to be assumed as the entropy created in the whole machine, whereas values of total pressure or total enthalpy depend on its relative frame, either rotating or stationary. Moreover, all three mentioned definitions of the loss coefficient deliver the same result for incompressible flows under adiabatic conditions.

2.2 Averaging methods

For the calculation and comparison of the former mentioned loss coefficients, the values of total and static pressure as well as total and static temperature, if desired or necessary, have to be determined for the inflow Section 1 and the outflow Section 2. Regarding the investigation of a two-dimensional blade row, the inflow pressure at Section 1 is typically homogeneous, whereas the outflow field at Sections 2a and 2b is characterised by the wake, as illustrated in Fig. 2. In this paper, cases without inlet distortion are discussed, but the averaging techniques presented below can be applied similarly for non-uniform inlet flow parameter.

Figure 2. Illustration of wake measurement planes at a linear cascade (airfoil not to scale)

To quantify the loss, the local loss coefficient can be determined according to the discrete distribution of pressure and temperature along the pitch. This may give information about height and width of the wake peak. However, in order to compare the loss of different cascades or different operating points appropriately, there are several established averaging techniques for the characterising parameters of the wake typically used^{(Reference Pianko and Wazelt14)}.

According to Cumpsty and Horlock^{(Reference Cumpsty and Horlock7)}, the appropriate averaging method depends on the context of evaluation and the intended purpose. The first discussed example is the area-weighted average, which is only valid for static pressure information, interpreting $F = \overline p \cdot A$ as the net force on the outlet area A ^{(Reference Cumpsty and Horlock7)}. Nevertheless, it is commonly used for the total pressure or the total temperature as well for practical reasons, as its determination is straightforward. This can be justified in several cases, for example if separation regions are negligible^{(Reference Cumpsty and Horlock7)}. Accordingly, the area-weighted loss coefficient is determined by the averaged flow quantities ${\rm{\Omega }} \in \left\{ {{p_2}\;,{p_{t2}},{T_2},{T_{t2}}} \right\}$ as shown in Equation 8

(8) \begin{equation}{\overline {\rm{\Omega }} ^a} = \frac{1}{t}\mathop \int \limits_0^t {\rm{\Omega }}du.\end{equation}

A second common method is the mass flow-weighted average. The previously discussed loss definitions are based on enthalpy considerations, which are defined per unit mass.^{(Reference Cumpsty and Horlock7)} By the mass flow-weighted average, the consideration of a mass deficit in a boundary layer as a reduced enthalpy flux instead of a direct loss is possible. Hence, it is valid for total temperature information and can be transferred to total pressure distributions, if its variations in the wake are comparably low^{(Reference Cumpsty and Horlock7)}. The resulting mass flow-weighted loss coefficient is defined by the averaged flow quantities ${\rm{\Omega }}$ in Equation 9

(9) \begin{equation}{\overline {\rm{\Omega }} ^m} = \frac{{\frac{1}{t}\mathop \int \nolimits_0^t {\rm{\Omega }} \cdot {\rho _2}{c_{x,2}}du}}{{\frac{1}{t}\mathop \int \nolimits_0^t {\rho _2}{c_{x,2}}du}}.\end{equation}

Particularly for two-dimensional cascade testing, Cumpsty and Horlock^{(Reference Cumpsty and Horlock7)} emphasise another averaging method, which is the mixed-out average according to Amecke^{(Reference Amecke8)}. The hypothetical mixed-out state at infinity in Section 2 $\infty $ of static and total pressure as well as the two-dimensional flow direction is calculated by Amecke^{(Reference Amecke8)} analytically under consideration of mass conservation in Equation 10 and momentum conservation in axial flow direction x in Equation 11 as well as in pitch-wise direction u in Equation 12, each set up between Sections 2a or 2b respectively and Section 2 $\infty $ (cf. Fig. 2).

(10) \begin{equation}\;{\dot m_{2\infty }} = {\dot m_2}\;\qquad \Leftrightarrow\qquad\qquad\ \ {\rho _{2\infty }} \cdot {c_{x,2\infty }}\;\; = \;\;\frac{1}{t}\mathop \int \limits_0^t {\rho _2} \cdot {c_{x,2}}du\end{equation}

(11) \begin{equation}{P_{x,2\infty }} = {P_{x,2}}\qquad \Leftrightarrow\qquad {\rho _{2\infty }} \cdot c_{x,2\infty }^2 + {p_{2\infty \;\;}} = \;\;\frac{1}{t}\mathop \int \limits_0^t {\rho _2} \cdot c_{x,2}^2 + {p_2}du\;\end{equation}

(12) \begin{equation}{P_{u,2\infty }} = {P_{u,2}}\qquad \Leftrightarrow\qquad {\rho _{2\infty }} \cdot {c_{x,2\infty }} \cdot {c_{u,2\infty }}\;\; = \;\;\frac{1}{t}\mathop \int \limits_0^t {\rho _2} \cdot {c_{x,2}} \cdot {c_{u,2}}du.\end{equation}

Using this approach, averaged values can be understood as synonymous with the data in the mixed-out state. The shown integrals can be determined straightforward from the wake in experimental or numerical investigations while utilising the trapezoidal rule for spatially discretised data. The solution of the three equations depending on the known integrated wake data leads to the three unknown parameters in the assumed mixed-out state ${c_{x,2\infty }}$ , ${c_{u,2\infty }}$ and ${p_{2\infty }},$ applying ${T_{t,2\infty }} = {T_{t1}}$ . Amecke^{(Reference Amecke8)} noted the corresponding energy conservation additionally, but assumed a negligible temperature variation in the wake and showed that this makes the energy conservation obsolete. By means of the results of these equations, the loss coefficients ${\overline \zeta ^A}$ can be determined according to the above-mentioned definitions^{(Reference Amecke8)}.

One of the major advantages of the Amecke averaging is that the determined flow values in the mixed-out state and, therefore, the determined averaged loss coefficients are theoretically independent of the axial location of measurement downstream of the cascade. This means that the wake traverse at Section 2a delivers the same mixed-out loss coefficient as the measurement in Section 2b, as no change in mass, momentum or energy occurs between the two areas. However, a drawback of the Amecke approach is that it is not valid anymore, if temperature gradients inside the wake are high, i.e. in cases of heated or cooled blades. Additionally, the existing approach is two-dimensional and, thus, not valid for field traverses with three-dimensional flow phenomena.

2.3 Overcoming the limits for non-adiabatic cases

Many experimental and numerical investigations of turbomachine components meet the assumption of an adiabatic structure. Triggered by the intended investigation of the interaction of heat transfer and aerodynamic loss, the established evaluation methods had to be revisited. Deriving methods for the loss determination at non-adiabatic test cases is an essential step to enable the thermodynamic assessment of thereby occurring loss mechanisms.

2.3.1 Effects to the loss coefficient

Dealing with non-adiabatic cases in a turbomachinery component may have different reasons that have to be evaluated specifically. In the case of a compressor, a heat flux through the casing is often not avoidable but might be of negligible order at least for low-pressure compressors. Regarding a turbine, blades and casing are often actively cooled. The heat flux is comparably high and achieved by an additional effort. Evaluating the loss coefficient by the above-mentioned equations, the cooling effort would be counted as subtracted loss, which may impede its correct interpretation. The same argument applies accordingly for the active heating of engine components. For this special case, the thermodynamics of a non-adiabatic compression or expansion with heat addition is revealed in Fig. 3. The effects of a cooling are analogous.

Figure 3. Enthalpy–entropy diagram for a non-adiabatic compression and expansion. (a) Compressor. (b) Turbine.

It should be considered that the heat application and the aerodynamic loss generation are commutative in thermodynamic regard. From this point of view, it is not relevant, if the addition of heat is considered before the aerodynamic loss, or the other way around, or if both processes are considered in parallel. Based on this assumption, the modified changes of state are evaluated as follows.

The amount of heat flux can be calculated as the local increase of total enthalpy in the wake according to Equation 13 while a constant specific heat capacity ${c_p}$ is assumed for low temperature variations. Considering a homogeneous heat flux over the entire blade, this is equal to the total induced heating power per air mass flow over the entire blade.

(13) \begin{equation}\frac{q}{{\dot m}} = {\rm{\Delta }}{h_t} = {c_p} \cdot \left( {\overline {{T_{t2}}} - {T_{t1}}} \right).\end{equation}

As a consequence of the heat transfer, the energy conservation is modified according to Fig. 3 in Equation 14.

(14) \begin{equation}\frac{{c_1^2}}{2} + \frac{q}{{\dot m}} = {\rm{\Delta }}{h_{12}} + \frac{{c_2^2}}{2}= {\rm{\Delta }}{h_{12}} + \frac{{c_{2s}^2}}{2} - {h_{loss}} + \underbrace {\frac{q}{{\dot m}} - {\rm{\Delta }}{h_q}}_{\frac{{{\rm{\Delta }}c_{2q}^2}}{2}}.\end{equation}

The above-introduced energy loss coefficient for the adiabatic case can be interpreted as a measure of the utilisation of the kinetic energy, as shown in Equation 3. For the non-adiabatic case, it is obvious that the heat application implies a flow acceleration by ${\rm{\Delta }}{c_{2q}}$ . Hence, the loss of enthalpy is not equal to the loss of kinetic energy for non-adiabatic cases. This is a very important fact, to keep in mind for the consideration of a suitable measure of loss.

Heading back to the explanation of Denton^{(Reference Denton2)}, a deviation from isentropic flow inevitably involves a reduction of efficiency. In that respect, the addition of heat unavoidably leads to an increase of entropy according to the Rayleigh line^{(Reference Strutt15)}. Thus, the heating of a flow is never lossless, even under otherwise ideal conditions. Therefore, the evaluation of the transformation of heat into kinetic energy is not fully reasonable, as it physically can never occur completely. More meaningful for investigations in turbomachinery is the state of static enthalpy. Hence, the extended enthalpy loss coefficient is defined in Equation 15 with respect to Fig. 3.

(15a) \begin{equation}\text{Compressor:}\ {\zeta _{hq,C}} = \frac{{{h_2} - {h_{2s}} - {\rm{\Delta }}{h_q}}}{{{h_{t1}} - {h_1}}}\end{equation}

(15b) \begin{equation}\text{Turbine:}\ {\zeta _{hq,T}} = \frac{{{h_2} - {h_{2s}} - {\rm{\Delta }}{h_q}}}{{{h_{t2}} - {h_2}}}\end{equation}

For typical investigations, the total and static quantities at inlet Section 1 and outlet Section 2 are known. The more complex process in between includes a loss of total pressure due to the polytropic change of state as well as by the application of heat. In parallel, both processes underlie a creation of entropy. As indicated above, it does not matter which part of the loss is caused by one of the two overlapping effects in order to find a suitable calculation method for an enthalpy loss coefficient. Eventually, the real outlet conditions in Section 2 are related to the process under ideal conditions which is isentropic compression or expansion to state 2s and isobaric application of heat to state 2q, as illustrated in Fig. 3. From these considerations the increase of static enthalpy due to the application of heat can be calculated by Equation 16.

(16) \begin{equation}{\rm{\Delta }}{h_q} = {h_{2q}} - {h_{2s}} = {c_p} \cdot \left( {{T_{2q}} - {T_{2s}}} \right)\end{equation}

The static temperature and flow velocity for the theoretical isentropic case are determined by Equations 17 and 18 with ${T_{2s}} = {T_1} \cdot {\left( {{p_2}/{p_1}} \right)^{\frac{{\kappa - 1}}{\kappa }}}$ .

(17) \begin{equation}{T_{2s}} = {T_{t1}} - \frac{{c_{2s}^2}}{{2{c_p}}}\end{equation}

(18) \begin{equation} \Rightarrow {c_{2s}} = \sqrt {2{c_p} \cdot \left( {{T_{t1}} - {T_{2s}}} \right)}\end{equation}

The mass flow conservation in Equation 19 leads to the theoretical flow velocity after heat addition in Equation 20, provided that the theoretic areas are ${A_{2q}} = {A_{2s}}$ and that ${p_{2q}} = {p_{2s}}$ due to the isobaric addition of heat.

(19) \begin{equation}{\rho _{2s}} \cdot {c_{2s}} = {\rho _{2q}} \cdot {c_{2q}}\end{equation}

(20) \begin{equation} \Rightarrow {c_{2q}} = \frac{{{T_{2q}}}}{{{T_{2s}}}} \cdot {c_{2s}}\end{equation}

Finally, the theoretical static temperature after heat addition can be determined by Equation 21 and resolved by the quadratic formula in Equatoin 22.

(21) \begin{equation}{T_{2q}} = {T_{t2}} - \frac{{c_{2q}^2}}{{2{c_p}}} = {T_{t2}} - {\left( {\frac{{{T_{2q}}}}{{{T_{2s}}}}} \right)^2} \cdot \frac{{c_{2s}^2}}{{2{c_p}}}\end{equation}

(22) \begin{equation} \Rightarrow {T_{2q}} = - \frac{{{c_p}T_{2s}^2}}{{c_{2s}^2}} \pm \sqrt {2c_{2s}^2 \cdot {c_p}{T_{t2}} + c_p^2T_{2s}^2} \cdot \frac{{{T_{2s}}}}{{c_{2s}^2}}\end{equation}

As it is obvious that a negative temperature is not reasonable, the positive solution in Equation 22 must be chosen. Hence, ${T_{2q}}$ and ${T_{2s}}$ can be applied in Equation 16 to determine ${\rm{\Delta }}{h_q}$ and, finally, ${\zeta _{hq}}$ from Equation 15. Furthermore, Equation 14 to 22 are valid analogously for the active cooling of a blade with a negative heat transfer ${\rm{\Delta }}{h_q}<0$ . The entropy loss coefficient can be adapted similarly, leading to approximately equal results as was already shown for the adiabatic definition.

By means of this new definition, which is referred to as the extended enthalpy loss coefficient below, it is targeted to evaluate the aerodynamic performance of an engine component under non-adiabatic conditions independently from detailed thermal aspects.

2.3.2 Effects to averaging techniques

As the area-weighted average is only affected by the distribution of the variables for the loss coefficient themselves, the average calculation is independent from the adiabatic or non-adiabatic point of view. The mass flow-weighted average is indirectly influenced by a heat transfer as the mass flow density can vary as a consequence, which does not require a modified definition as well. Concerning the total pressure loss coefficient, it can be discussed, if it is reasonable to use the density distribution ${\rho _2}$ at the outflow or rather the density ${\rho _1}$ at the inlet for the calculation of the mass flow-weighted average, as the justification for the total pressure loss coefficient is the assumption of an incompressible flow. The authors decided to use $\rho = {\rho _2}$ for adiabatic and non-adiabatic cases for three reasons:

1. 1. The inlet density ${\rho _1}$ can only vary due to, in the presented investigations, undesirable local or temporal variations in the experimental set-up and is otherwise constant. The intended correction of local mass flow variations in the wake is not fully possible assuming an incompressible flow, as only velocity variations are taken into account.

2. 2. The total pressure loss coefficient is nowadays an established value for the evaluation of the aerodynamic performance, for incompressible flows as well as for obviously compressible flows. This is not wrong in the first instance, as long as the interpretation is based on the introduced stagnation pressure rather than on energetic arguments. Nevertheless, the density at outflow can vary considerably from the density at inflow. Thus, the mass flow-weighting for the average at outflow should be based on the correct density distribution in the wake.

3. 3. A heat transfer over the blade row or entire component can have a distinctive impact on the density at outflow. This would be ignored if the weighting was performed by means of the inflow density, independent from the issue, whether the flow is compressible or incompressible.

Finally, the averaging approach of Amecke^{(Reference Amecke8)} is based on the neglected temperature variation in the wake, as mentioned above. To keep the basic idea valid, the method has to be extended. Apart from temperature gradients in the wake, also the third, span-wise dimension of the flow cannot be evaluated by the method of Amecke, which could be useful, particularly for wake traverses in regions of secondary flow. Besides, the utilisation of the third dimension of the flow can also improve the accuracy of single wake traverses at mid-span, as small yaw angles of the inflow cannot always be avoided. Thus, the equation for the third dimension of the momentum conservation is noted in Equation 23 as well as the energy conservation in Equation 24 which was already utilised by Kost.^{(Reference Kost16)}

(23) \begin{equation}{P_{z,2\infty }} = {P_{z,2}}\qquad\Leftrightarrow\qquad\quad \ {\rho _{2\infty }} \cdot {c_{x,2\infty }} \cdot {c_{z,2\infty }} = \frac{1}{t}\mathop \int \limits_0^t {\rho _2} \cdot {c_{x,2}} \cdot {c_{z,2}}du\end{equation}

(24) \begin{equation}{E_{2\infty }} = {E_2}\qquad\Leftrightarrow\qquad {\rho _{2\infty }} \cdot {c_{x,2\infty }} \cdot {c_p} \cdot {T_{t2\infty }} = \frac{1}{t}\mathop \int \limits_0^t {\rho _2} \cdot {c_{x,2}} \cdot {c_p} \cdot {T_{t2}}du\end{equation}

Taking the integrated values as known from the wake traverse utilising the trapezoidal rule, the five equations in Equations 10, 11, 12, 23 and 24 can be solved to the five unknown, averaged parameters ${c_{x,2\infty }}$ , ${c_{u,2\infty }}$ , ${c_{z,2\infty }}$ , ${p_{2\infty }}$ and ${T_{t2\infty }}$ assumed in the theoretic mixed-out state. The resulting averaged loss coefficient ${\overline \zeta ^{mo}}$ is referred to as the complete mixed-out average below. The presented demonstration is the first application of this extended averaging approach directly compared to the approach of Amecke^{(Reference Amecke8)} for adiabatic and non-adiabatic cases.

3.1 Test facility

The high-speed cascade wind-tunnel of the Institute of Jet Propulsion is shown schematically in Fig. 4, giving basic information on this special test facility in addition. It is an open-loop test facility with an open test section located inside a pressure tank. This enables the variation of the density inside the test facility and thus, the adjustment of Reynolds number and Mach number independently from each other. Therefore, it is ideally suitable for the investigation of basic aerodynamics on linear compressor or turbine cascades.

Figure 4. Schematic of the high-speed cascade wind tunnel at the ISA

The unique test facility was put into service in 1955 under coordination of Schlichting^{(Reference Schlichting17)} at the Deutsche Forschungsanstalt für Luftfahrt in Braunschweig, Germany. The entire wind-tunnel was moved in 1985 to the Institute of Jet Propulsion of the Bundeswehr University Munich while only minor constructional changes were made^{(Reference Sturm and Fottner18)}. Since a major revision of the HGK in 2017, it is driven by a 1.8MW high voltage asynchronous motor with a frequency converter, see Fig. 4. By means of a mechanical transmission, it drives a three-stage axial compressor providing a maximum pressure ratio of 2.74. A bypass system enables lower mass flow rates in the test section than allowed by the compressor map by preventing the compressor from stalling. Depending on the variable channel height, this system provides a Mach number range from $Ma=0.1$ to $Ma=1$ . Behind a diffuser a flap-controlled water cooler allows for a temperature control between 298 and 333K. In order to realise a homogenous flow field, a honeycomb mesh is installed inside the settling chamber, before the flow is accelerated in the nozzle to the desired Mach number at the test section. Optionally, various turbulence generators can be mounted inside the nozzle to achieve a defined turbulence level. By a rotatable ring at the section of measurement, the inflow angle towards the cascade can be adjusted.

For wake traverses by various types of probes, a four-axis traversing device is installed next to the test section. Beside the axis in flow direction and an axis perpendicular to that in blade span direction, the vertical axis can be aligned parallel to the cascade inlet plane. The fourth axis is a rotating axis in blade span direction enabling the alignment of the probe parallel to the main outflow direction.

3.2 Specific cascase set-up

The cascade under consideration here is based on a compressor vane, which is provided to the ISA by GE Aviation. For the presented test set-up, the center blade of the cascade is equipped with a heating system by means of an internal cycle of hot oil. Concerning the planned investigations, it is not primarily targeted to receive a homogeneous heat flux distribution on the blade, as necessary for measurements of the heat transfer coefficient. The heat transfer was measured at this cascade by a different set-up described by Aberle et al.^{(Reference Aberle, Bitter, Hoefler, Carretero Benignos and Niehuis19)}

In order to evaluate the performance of the cascade, wake traverses were conducted by means of an in-house designed five-hole probe with integrated thermocouple for a simultaneous measurement of pressure and temperature data. The thermocouple is placed within a Kiel-shroud on top of the five-hole probe head, as shown in Fig. 5. The probe was orientated parallel to the blades, such that temperature and pressure data are acquired synchronously at the same pitch-wise direction. By traversing the probe to in total 63 discrete positions along one pitch length downstream of the center blade, the wake can be characterised. The distance of adjacent measurement points is thereby between 0.25mm directly in the wake and 4mm in the distortion-free stream. At each measurement position, the pressure balancing of each pressure tap is monitored in order to ensure a reliable steady state measurement. Additionally, each measurement value is an average of several seconds of measurement to account for slight temporal variations of the flow parameter. A prerequisite is furthermore the continuous monitoring of the inflow parameters to provide a constant inflow.

Figure 5. Illustration of cascade and probe alignment for wake traverses (airfoil not to scale)

The pressure data of the operating point at the HGK is acquired by multiple 98RK-1 rackmount pressure scanners. A specific pressure module is chosen depending on the required pressure range to minimise measurement uncertainties. The inlet total temperature, however, is determined as a mean of four PT100 temperature sensors, one at each wall of the wind tunnel. With these four temperature values, the homogeneity of the inflow temperature can be furthermore monitored. For the presented tests, no turbulence generator is used. The remaining turbulence intensity is lower than 1%. As the investigation of inlet distortions is not targeted in the scope of the presented tests, a homogeneous inflow is ensured within the investigated range.

As to the five-hole probe data, the pressure values of the five-hole probe were measured by a PSI 9116 pressure scanner. Data from the type T class 1 thermocouple mounted on the probe was acquired by an NI ethernetRIO 9147 system utilizing an NI 9211 module. Both were mounted inside the tank of the HGK to keep tube and cable lengths low.

Beside the variation of inflow Reynolds and Mach number, the blade temperature was varied from cold state, which equals the inflow total temperature of 303.15K, to different degrees of hot state up to about 415K, as this is the order of magnitude of the application relevant temperature gradient. The corresponding heating power is determined by means of the temperature difference at the heating medium. Extensive insulations keep conductive losses low. The airfoil wakes were investigated for high-speed $\left( {M{a_1} \gg 0,3} \right)\;$ as well as low-speed operating points $(M{a_1}<0,3)$ . Additionally, several field traverses were conducted.

3.3 Measurement uncertainty

The estimation of measurement uncertainties gives an order of magnitude of the reliability of the results. Starting with the acquirement of raw data, the pressure values were determined by an accuracy of 3.8Pa for low inflow Mach numbers and by 17.2Pa for medium and high inflow Mach numbers according to the manufacturer’s calibration of the pressure scanners. This accuracy applies for the inflow pressure measurement in the same way as for measurements in the outflow by means of the five-hole probe. The inlet total temperature is determined by an accuracy of 0.5K, including the entire measurement chain with sensors and acquisition device. The outflow temperature, determined by a thermocouple type T on the five-hole probe, is moreover influenced by the recovery factor which has to be calibrated properly. This was conducted inside the test facility providing an implicit calibration of the measurement chain including thermocouple and acquisition module. The remaining measurement uncertainty can be numbered to 0.5K.

Based on the measurement uncertainty of the raw data, a Gaussian error propagation according to Barlow^{(Reference Barlow20)} was conducted up to the evaluated five-hole probe results utilising: ${\epsilon}_{f} =\sqrt{\sum \left(\frac{\partial f}{\partial {\rm \Omega }} {\epsilon}_{{\rm \Omega }} \right)^{2} } $ . The calibration of the five-hole probe, including the recovery factor at the temperature sensor, leads to further uncertainties according to the residuals of the calibration polynomials which are added at this point to the error of the processed results. The error propagation is continued up to the integration of the wake data. As the utilised trapezoidal integration is based on a sectional linear interpolation of the measurement points, the wake curve can be interpreted as sectionally differentiable. Consequently, a corresponding error can be determined for each position along the wake. For the integration of the flow quantities in the wake, linear error propagation has to be applied: $\; {\epsilon }_{I} =\mathop{\sum }\nolimits_{i=1}^{N} \frac{\partial I}{\partial {\rm \Omega }_{i} } {\epsilon }_{{\rm \Omega }_{i} } $ . It can be assumed that further errors of discretisation are low due to the high resolution of measurement points within the wake. The further data processing, however, underlies Gaussian error propagation. With this process, the resulting measurement uncertainty for the averaged loss coefficient is dependent on the operating point, the averaging technique and, finally, the definition of the loss coefficient. In Table 1 the cases with the respective highest and lowest absolute error values concerning the presented results at high temperatures are shown.

4.1 Comparison of loss coefficients

In Fig. 6 the discussed definitions of the loss coefficient for wake traverses at the cascade’s mid-span at medium inflow Mach number and high Reynolds number at different heating levels are shown. All results are based on an identical data set. The heating power is normalised to the air mass flow per passage for a better comparability of different inflow Mach and Reynolds numbers.

At first sight, it is obvious that the energy and entropy loss coefficients show a distinct, almost linear progress with increased heating power per passage mass flow while the total pressure loss coefficient as well as the extended enthalpy loss coefficient only show a slight elevation with increased heating power. In approaches that consider the temperature elevation in the wake, it can be furthermore observed that the inclination of the extended enthalpy loss coefficient is higher than the one of the total pressure loss coefficient. This indicates that losses caused by thermal mixing are not adequately considered utilising the total pressure loss coefficient.

Table 1 Measurement uncertainty of averaged loss coefficients (absolute values)

Figure 6. Comparison of different definitions of the loss coefficient, shown for different averaging techniques; operating point: medium Mach number, high Reynolds number. (6a) Area-weighted average. (6b) Mass flow-weighted average. (6c) Amecke average. (6d) Complete mixed-out average

This progress of the loss coefficient with increasing heating power is reasonable, taking the Reynolds number dependency of the loss coefficient into account. A higher temperature of the blade surface leads to a locally higher flow temperature inside the boundary layer while a higher air temperature means a higher viscosity according to Sutherland^{(Reference Sutherland21)} and, thus, a lower local Reynolds number on the blade surface. Provided that the loss coefficient is decreasing with increasing Reynolds number per chord length due to the displacement of boundary layer transition as was shown by Mayle,^{(Reference Mayle22)} it is reasonable that a higher blade temperature with a lower local Reynolds number leads to a higher loss.

Comparing the different averaging techniques, the area- and mass flow-weighted average follow similar distributions while the mass flow-weighted average is always lower than the area-weighted average. This is conformable to the theory, as the loss at boundary layers is overrated by the area-weighted average. Cumpsty and Horlock^{(Reference Cumpsty and Horlock7)} have already shown that for the evaluation of enthalpy information, the usage of the area-weighted average is only justified, if the deviation to the more reasonable mass flow-weighted average is low which is not the case for the presented experiments here.

Furthermore, it is noteworthy that with the Amecke average as well as with the complete mixed-out average, the energy and entropy loss coefficient fit nearly exactly together, whereas for the area-weighted average and the mass flow-weighted average both distributions are only similar. Thus, the mixed-out averaging techniques lead to physically more coherent results, as the energy and entropy loss coefficient always have to be approximately equal according to Denton^{(Reference Denton2)}.With the Amecke average, also the extended enthalpy loss coefficient is approximately equal to the energy and entropy loss coefficient. This is due to the fact that the temperature gradient is ignored by the approach of Amecke and the extension term ${\rm{\Delta }}{h_q}$ in Equation 14a is zero. While for both mixed-out averaging techniques, the extended enthalpy loss coefficient is lower than the total pressure loss coefficient, the offset between both is almost constant with the Amecke average. As this difference is getting lower with increased heating power, using the complete mixed-out average, the fact that the Amecke average can only be a reasonable method for non-adiabatic wake-traverses, if the impact of the temperature gradient is low, is underscored. Hence, the complete mixed-out average is in focus of the following evaluations.

Figure 7. Comparison of different definitions of the loss coefficient; operating point: low Mach number, low Reynolds number. (7a) Complete mixed-out average: Full range; (7b) Complete mixed-out average: Zoomed-in.

The tendencies pointed out in Fig. 6 for different averaging techniques can analogously be shown for other operating points. Comparing the definitions of the loss coefficient in the case of a low inflow Mach number and low Reynolds number, as depicted in Fig. 7, the difficulty of the usage of the so far established energy and entropy loss coefficient at non-adiabatic test articles is obvious. The loss of energy is interpreted as extensively high while the local values of the loss in the wake raise up to 500%. It is needless to say that a loss higher than 100% is not reasonable. This confirms that the extension of the definition of the enthalpy loss coefficient is necessary for its correct interpretation. The distinct increase of the energy loss coefficient is only due to the misinterpreted energy addition as loss. However, by applying the extended enthalpy loss coefficient, a sound assessment of the aerodynamic loss behaviour is enabled, leading to a similar order of magnitude as the total pressure loss coefficient.

Table 2 Overview of suitable combinations of loss coefficient definitions and averaging techniques for a non-adiabatic flow

In contrast to the measurements at higher inflow Mach number, the extended enthalpy loss coefficient is higher than the total pressure loss coefficient at high blade temperatures. It is the other way around for the cold blade, although, as to the theory, all loss coefficients have to deliver equal results for adiabatic and incompressible cases. The results indicate that a further rise of the temperature gradient between blade and flow would cause a further increase of the profile loss and might also lead to an intersection of the curves of both loss definitions for the medium inflow Mach number. Additionally, it can be assumed that the difference between both loss definitions correlates with the influence of compressibility, beside the consideration of heat transfer. If the extended enthalpy loss coefficient is higher than the total pressure loss coefficient, the impact of the heat addition on the loss is higher than effects of compressibility. Anyway, it can be useful to consider both values for the interpretation of aerodynamic data.

4.2 Rating of loss coefficients for non-adiabatic cases

In all, it turned out that different definitions of the loss coefficient and different averaging techniques cannot be evaluated fully independently from each other. To give an overview, of the former presented results to the suitability of different approaches, the respective combinations are rated in Table 2. The main assessment criterion for the presented methods is their thermodynamic validity. Additionally, their robustness, which is highly correlated to their measurement uncertainty, is an important measure of the specific suitability.

It was shown above that the energy loss coefficient is not a suitable measure, if a temperature increase is present, as this is erroneously counted as loss. For low-temperature variations, the Amecke average can be a suitable measure, accepting that the temperature increase is fully neglected. The same arguments apply for the entropy loss coefficient. As to the total pressure loss coefficient and the extended enthalpy loss coefficient, the area-weighted average is only suitable, accepting too, that the mass deficit in the wake is not accounted for. This, however, is considered by the mass flow-weighted average. As well, the complete mixed-out average leads to reasonable results, whereas the Amecke average is only suitable, if the temperature increase is consciously not to be considered.

Beside theoretical considerations on the reasonability of the discussed methods, their robustness might be an additional argument which is summarised in Table 3. This applies particularly for experimental data, as a loss definition is not necessarily suitable, if a low change of the input data due to measurement uncertainties effects a high change in the resulting loss coefficient. It was found that all definitions of the loss coefficient which imply a temperature measurement are more sensitive to measurement uncertainties than methods based on pressure data only. For the presented operating points, the pressure measurements are more accurate in relation to the measured value, than measurements of the total temperature. Hence, the higher the temperature information is weighted for the respective loss coefficient, the less robust is its reliable calculation. As to the mixed-out averaging techniques, the solution of the conservation equations leads to comparably complex mathematic results, effecting that the measurement uncertainties sum up to higher orders. This means, that although the complete mixed-out averaged extended enthalpy loss coefficient might be physically the most reasonable measure, a more robust measure, such as the mass flow averaged total pressure loss coefficient, might give more reliable results, if measurement uncertainties are high. In this concern, the acceptable uncertainty is highly dependent on the operation point and on the intended purpose of the evaluation. With measurement uncertainties within the typically occurring range, the presented ranking is consistently applicable and supports as a guideline in choosing the respective most suitable measure.

Table 3 Robustness of loss coefficient definitions and averaging techniques

4.3 Transfer to field traverses

The previously presented results were gained from wake traverses at the cascade’s mid-span while focussing on the investigation of the profile loss. The conduction of field traverses over the entire blade span often deals with superimposed endwall losses or, in special cases, as well with leakage losses. Details on these loss mechanisms are also shown by Denton^{(Reference Denton2)}. The secondary flow near the endwalls typically cause high losses. Here, the gradients in total pressure are mostly higher than for wake traverses at mid-span, because these are only influenced by the profile loss. Therefore, the question about the reasonability of the mass flow-weighted average for total pressure data arises. Moreover, it was already shown that the area-weighted average of the loss parameter is not reasonable in this case as well. The implementation of the complete mixed-out average compared to the Amecke average, however, enables the evaluation of field traverses by a further, promising method. Similar as for single wake traverses, it could be shown that energy and entropy loss coefficient fit best by this approach.

In Fig. 8 the extended enthalpy loss coefficient is compared to the total pressure loss coefficient for field traverses at two operating points. The field includes initial parts of secondary flow at both blade ends but does not represent the endwall losses entirely. From these measurements, it is obvious that both presented loss coefficients for field traverses are rising with increased heating power, similar as for a single wake measurement. Furthermore, the gradient of the extended enthalpy loss coefficient seems to be higher for the field traverse than for the single wake traverse. In contrast to that, this is not indicated by the total pressure loss coefficient.

Figure 8. Completed mixed-out averaged loss coefficients for field traverses and single wake traverses at mid span, each for a cold and a hot blade at similar blade temperatures. (8a) Medium Mach number, high Reynolds number. (8b) Low Mach number, low Reynolds number.