This section describes the measurement techniques utilized during the experimental campaign. The pressure drop across the nozzle is measured using a KuliteTM XT190 transducer (Leonia, NJ, USA), which has a full scale of 5 psi. The outlet pressure is the ambient pressure that is read by a FisherTM (Drebach, Germany) model 104 barometer with an average uncertainty of 50 Pa calibrated in the LAT n° 024. In order to phase-average the hot-wire measurements at the frequency of the injected disturbance, the pressure in injector duct 1 is used as a trigger signal and is measured using a KuliteTM XT190 transducer with a full scale of 25 psi. The transducer’s maximum uncertainty is 0.05% of its full scale. The measuring signals are acquired using a National Instrument data acquisition board (PCI 6052E) with a range of ±10 V.
Temperatures are measured using thermocouples: the nozzle supply air temperature is measured using a T type, whereas the injector ducts and ambient temperature are measured using a K type. Temperature uncertainty after calibration is 0.3 °C.
At the traversing planes, a 5-hole probe and a hot-wire anemometer are used.
2.3.2. Hot-Wire
The hot wire used in this experimental campaign is a slanted single-sensor probe with a slanting angle of 45° that is connected to a DISA55M system. The wire diameter of ~5 µm guarantees a very high dynamic response (~30 kHz) in the constant temperature configuration, making this probe suitable for measuring the turbulence structures of both low- and high-speed flows. The probe is mounted on four stepping motors (
Figure 1): two control the traversing position, one controls the yaw angle, and one controls the pitch angle.
To keep the wire temperature constant in the presence of an incoming flow, the hot-wire Wheatstone bridge regulates the supply voltage. This well-established response relationship is described by King’s law:
where
is the mean average of the supply voltage, once corrected to account for temperature drifts;
is the cooling velocity; and
,
, and
are the calibration constants obtained after a least-square regression of the calibration data. In King’s law, the hot wire is aligned perpendicularly to the main flow so that the cooling velocity
corresponds to the flow velocity
. In the case of a slanted single-wire probe, this condition is achieved by placing the probe at a pitch angle of −45° and a yaw angle of 0°. In this way, the cooling velocity depends only on the normal component of the flow velocity, and the other velocity components are null; in all other cases, one would need to know the angular sensitivity of the probe, which is normally still unknown at this stage of the calibration. Another constraint for determining King’s law is that the turbulence intensity of the calibrating jet is small (below 5%) because, as shown later, the velocity also depends on the Reynolds tensor components. Both these conditions were properly matched in the calibrations performed for this study.
The methodology applied in this work requires careful discussion, as specific actions are taken to improve the results’ reliability. First, King’s law is split into three different voltage ranges (
Figure 3) to improve the accuracy of the interpolation. Second, the anemometer output signal
is corrected to account for possible temperature drifts during the calibration using Equation (2), as suggested by Bruun [
22]:
is the constant temperature of the wire (set at 493 K), is the temperature at which the voltage zero is performed (at rest conditions), and is the temperature of the incoming air.
The fluctuating component of the cooling velocity is derived by decomposing King’s law [
23]. It depends on the King’s coefficients and the mean velocity:
where
is the fluctuating velocity component and
is the root mean square of the hot-wire voltage measurements.
In real applications, the actual velocity could not be perpendicular to the wire. In this case, the other velocity components impact the cooling velocity of the hot wire in a non-linear way. The influence of these velocity components on the cooling velocity is well described by Jorgensen’s law [
23]:
where
is the normal component,
is the tangential component, and
is the binormal component with respect to the wire (
Figure 1).
and
are the two angular calibration coefficients to be defined through an aerodynamic calibration. It must be emphasized that in the definition of King’s law, only the normal component is not null. In fact, the change in the reference system from the hot wire to the nozzle yields:
In the literature, the coefficient
is available for standard probes [
24]. In this paper’s application, the choice is to carry out a calibration to also determine the
coefficient by varying the pitch angle. For this purpose, the hot-wire yaw angle is set to zero, that is,
= 0, and the pitch angle is varied within the calibration range of ±45° every 5°. This is possible after verifying that the dependence of
on the yaw angle is negligible. The
coefficient can be computed according to Equation (6) by imposing the yaw angle
.
The last angular calibration coefficient
can be calculated, as shown in Equation (7), by imposing different combinations of the yaw and pitch angles. For each pitch angle, the yaw is moved within the range of ±120° every 5°.
Once the calibration coefficients are defined, the probe can be applied in an unknown flow field to reconstruct its velocity components and the turbulence content. The flow velocity can be decomposed into its three components in the
x-y-z reference system that can be correlated to the hot-wire reference system components using Equation (8), where
is the rotation of the yaw motor.
Therefore, the cooling velocity can be related to the velocity components in the nozzle reference system by substituting Equation (8) into Equation (4):
The A coefficients depend on the yaw angular position and the slanted angle, as well as the calibration coefficients and . One single equation is not enough to solve the flow field due to the presence of three unknowns. To obtain reliable results, the system is overdetermined using 13 equations, calculated by varying the probe yaw angle within the range of ±120° every 20°. This set of angles is chosen after proper validation of the procedure, considering different ranges and angle steps. The overdetermined problem is resolved using a Matlab iterative script and the lsqnonlin function. The first estimation for the velocities is a vector [0; Q; 0], where Q is computed using King’s law.
This procedure is valid in the case of low-turbulence intensity flows because the effective cooling velocity depends on the fluctuating components. By expressing each velocity component in terms of mean (denoted with upper-case letters) and fluctuating (denoted with lower-case letters) components, after calculating the mean value (denoted with an overline), Equation (9) becomes:
This equation does not allow for the separation of the time mean components and the Reynolds stresses. Equation (10) could be rewritten in terms of both mean and Reynolds stress coefficients, considering the largest velocity component as the one in the axial direction (
U2):
After approximating Equation (11) at the first order, the mean velocity and the mean of its fluctuating component can be derived. The assumption is that all other components are much lower than
so series expansion can be applied. By further neglecting third-order terms or greater, according to Buresti and Di Cocco [
25], the mean effective velocity can be expressed as:
considering that
By subtracting Equation (10) from Equation (12), properly squared, and neglecting third-order terms,
The use of these equations is justified by the fact that the turbulence intensity in this application does not exceed 20% [
25], making Equations (12) and (14) consistent. In this way, all components of the Reynolds stress tensor can be estimated.
All coefficients (
A: Equation (9);
R: Equation (12); and
Z: Equation (14)) are taken from [
24,
25], with the only difference being the different reference systems.
However, at the first traversing position, the assumption of being the dominant velocity is not correct everywhere, especially in the vortex core. If the tangential velocity becomes dominant, this issue can be easily solved by rotating the reference system: after estimating the velocity direction by resolving the over-constrained system of equations, the central hot-wire rotation is updated to reflect the actual velocity direction. In this way, again becomes the main velocity component, and the previous equations can be applied. To make this possible, it is necessary to carry out measurements over a wider range of motor rotations to account for possible changes in the reference system. Therefore, for each point, measurements are carried out within the range of ±160°, with ±40° being the maximum/minimum angles expected.
In the case where is the dominant component, the previous equations are used again but with changes in the procedure from Equation (11) on.
Ultimately, the procedure for resolving the flow field with the hot wire involves the following steps, as shown in the flowchart in
Figure 4:
- 1.
The acquired voltages are used to compute and by applying King’s law.
- 2.
Assuming low-turbulence content and Q = [0, , 0], the mean flow field is solved by iterating Equation (9) over 13 sets of rotations.
- 3.
The first prediction of the velocity vector is used to update the rotational range and solve the Reynolds tensor (Equation (14)). The updated rotational range is centered on the nearest multiple of 20° relative to the measured yaw angle, encompassing 13 rotations spaced every 20° within the range of ±120° with respect to that closest multiple of 20°.
- 4.
Considering that the mean velocity depends on the Reynolds stress tensor components (Equation (10)), the assumption made in step 2 is now relaxed. The mean velocity components are recalculated using a least-square regression on Equation (10), utilizing the Reynolds stresses computed in step 3.
- 5.
With the new mean velocities, the cycle is repeated, starting from step 3 until convergence.
A Monte Carlo simulation is carried out to perform an uncertainty quantification of the HW measurements. Starting from the measuring uncertainties, these are propagated through the whole calibration process. Uncertainties vary depending on the flow regime, with high uncertainties associated with low speed (see
Figure 3). The extended uncertainties at a 95% confidence interval are listed in
Table 3 for the flow quantities under study.
Figure 4.
Hot-wire flowchart.
Figure 4.
Hot-wire flowchart.