Fast-Response Five-Hole Probe Data Acquisition and Processing

In order to accurately quantify the effects of turbulence on a standard five-hole probe it is necessary to acquire accurate measurements of the total and static pressure in a turbulent flow and this will be accomplished using a fast-response five-hole probe. A FR5HP provides measurements of the instantaneous pressure at each of the five pressure ports. These five pressures can be processed using the method outlined in Section 4.2.2 but a few additional

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factors must be accounted for. This section will provide details regarding the fast-response probe used in this dissertation and the data acquisition and processing steps that were used.

4.2.8.1 Fast-Response Five-Hole Probe Data Acquisition

A fast-response five-hole probe from Aeroprobe Corp., model P-E05C02S-SX-S, was used for this study. Figure 4.9 shows an image and schematic of the fast-response five-hole probe. The probe had a hemispherical tip geometry with a diameter of 0.094-in. and a length of 9.5-in. Unlike a standard five-hole probe the fast-response probe had five high-frequency response pressure transducers embedded near the tip of the probe, approximately 1.5-in. downstream from the probe tip. The transducers had a maximum range of ±1.0 psi. The quoted maximum frequency response of this fast-response probe was 3.7 kHz. Voltages from the five pressure transducers were acquired using a National Instruments SCXI system which is capable of simultaneously acquiring and conditioning measurements from up to 48 separate channels using simultaneous sample and hold. Signal conditioning included applying a gain of 5 to the voltages and then passing the signals through a low-pass 8th order Bessel filter with a cutoff frequency of 3 kHz. The voltages from the five transducers were sampled at a rate of 10 kHz for 10 seconds. Each transducer was referenced to the tunnel freestream static pressure measured upstream of the model. A calibration for each transducer was supplied by Aeroprobe and was used to convert the measured voltages to pressures.

4.2.8.2 Fast-Response Five-Hole Probe Distortion Correction

A FR5HP is capable of a high frequency response because the pressure transducers are mounted near the tip of the probe. The maximum frequency response would be obtained by placing the transducer flush with the surface of the probe tip. Unfortunately, this would require a

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large probe tip reducing the spatial resolution of the probe. A compromise is achieved by placing the transducers a short distance away from the probe tip and connecting the transducer to the tip through a short length of metal tubing. Even a short length of small diameter tubing can have an effect on the pressure sensed by the transducer as significant attenuation or amplification and phase lag may occur between the probe tip and transducer.53 This problem has been extensively studied theoretically and experimentally.

Bergh and Tijdeman54 derived an analytical transfer function that accurately models the distortion of the pressure signal from the probe tip to the transducer. The distortion is a function of the tube length and diameter, the volume of the transducer, the properties of the fluid in the tube and the frequency of the pressure fluctuations. With the tubing geometry of common FR5HPs, the distortion can result in significant attenuation of the pressure signal but also typically exhibits one or more resonant frequencies at which the pressure signal is amplified. If the internal geometry of the tubing and transducer system is accurately known, the expression derived by Bergh and Tijdeman54 can be used to correct the measured pressure for the distortion. Johansen et al.53 used Bergh and Tijdeman’s54 analytical result to estimate the influence of uncertainty in the tubing geometry. They demonstrated that with the level of uncertainty common in the tubing geometry of FR5HP probes, which may only be on the order of ±0.01 to ±0.1 mm, leads to unacceptable errors in the transfer function necessitating a method of experimentally determining the transfer function for a given probe.

Johansen et al.53 determined the experimental transfer function by placing the FR5HP near a loud speak driven by a function generator. A reference pressure signal was acquired by placing a microphone near the FR5HP. The pressure attenuation was determined by the ratio of the FR5HP measured pressure to the microphone measurement, and the phase lag was

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determined by the lag of the FR5HP relative to the microphone. The attenuation and phase lag were determined as a function of frequency of a sinusoid pressure wave generated by the loud speaker. Once the transfer function was known the following process was used to correct the measured pressures for distortion. Let 𝑃𝑚 and 𝑃𝑐𝑜𝑟 represent the distorted measured pressure and the corrected pressure, respectively. Then define 𝑃̃𝑚 and 𝑃̃𝑐𝑜𝑟 as the Fast Fourier Transforms (FFT) of 𝑃_𝑚 and 𝑃_𝑐𝑜𝑟, respectively. The distorted and corrected pressures are related by the transfer function (TF) which is a function of frequency (𝑓) and can be expressed using complex notation.

𝑃̃

_𝑚

𝑃̃

_𝑐𝑜𝑟

= 𝑇

𝐹

= 𝑃̃

𝑟𝑎𝑡𝑖𝑜

(𝑓)𝑒

𝑖𝜙(𝑓) Eq. 4.32

Where 𝑃̃_{𝑟𝑎𝑡𝑖𝑜} represents the magnitude of the FFT of the experimentally determined transfer function and represents the pressure attenuation, and 𝜙(𝑓) is the argument of the FFT of the transfer function and represents the phase lag. The magnitude of 𝑃̃𝑐𝑜𝑟 can be determined by:

| 𝑃̃

_𝑐𝑜𝑟

| =

| 𝑃̃

𝑚

|

𝑃̃

_{𝑟𝑎𝑡𝑖𝑜}

(𝑓)

Eq. 4.33

Where | 𝑃̃𝑚| represents the magnitude of the FFT. The phase of 𝑃̃𝑐𝑜𝑟 is given by:

𝑎𝑟𝑔( 𝑃̃

_𝑐𝑜𝑟

) = 𝜃

_𝑐𝑜𝑟

= 𝜃

_𝑚

− 𝜙(𝑓)

Eq. 4.34 The operator “arg” represents the argument of the given variable. 𝜃_𝑐𝑜𝑟 and 𝜃_𝑚 are the phases of the corrected pressure and measured pressure, respectively. The corrected pressure is determined by combing the magnitude and phase information.

𝑃̃

_𝑐𝑜𝑟

= | 𝑃̃

_𝑐𝑜𝑟

| ∙ 𝑒

𝑖𝜃_{𝑐𝑜𝑟 Eq. 4.35}

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After an experiment, the time history of pressure measured be each of the five ports was corrected for distortion. The experimentally generated transfer function was supplied by Aeroprobe and the pressure measurements for each transducer were corrected for distortion using Aeroprobe’s Acoustic Recovery Software.55

Figure 4.11 shows the experimental data used to create the transfer function for port 1 of the FR5HP. The pressure ratio in Fig. 4.11a shows a resonant peak centered at approximately 1200 Hz where the pressure is amplified by a factor of approximately 5.6, and Fig. 4.11b shows that the distorted signal lags behind the correct signal. The reason for this is that it takes a finite amount of time for the pressure signal to travel from the pressure port to the transducer so that the measured signal lags the true pressure fluctuation. Figure 4.12 compares a portion of the time history of the pressure measured by port 1 in a turbulent flow before and after application of the distortion correction. It can be seen throughout the entire portion of the time history that the distorted signal contained larger peaks and generally lagged slightly behind the corrected signal. It is also interesting to note that there is a strong periodic contribution to this signal and the frequency of the periodic portion of the signal increases significantly during the second half of the time history shown, and the amplification of the pressure signal is significantly larger during the second half of the time history.

4.2.8.3 Fast-Response Five-Hole Probe Total Pressure Correction in Turbulent Flows

The FR5HP requires a correction to the total pressure to account for turbulent fluctuations. If the length scale of the turbulence is larger than the probe tip, then at each instant the probe can be treated as if it were in a steady laminar flow and the steady calibration discussed above can be applied. It can then be assumed that the FR5HP measures the correct instantaneous total and static pressure as well as the instantaneous total velocity.

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𝑃

_𝑡

(𝑡) = 𝑃

_𝑠

(𝑡) +1

2𝜌(𝑈

2

(𝑡) + 𝑉

2

(𝑡) + 𝑊

2

(𝑡))

Eq. 4.36 Using the conventional Reynolds decomposition, the instantaneous velocities U, V and W can be written in terms of the time averaged and fluctuating components.

𝑃

_𝑡

(𝑡) = 𝑃

_𝑠

(𝑡) +1

2𝜌 ((𝑈̅ + 𝑢

′

(𝑡))

2

+ (𝑉̅ + 𝑣

′

_(𝑡))

2

_{+ (𝑊̅ + 𝑤}

′

_(𝑡))

2

₎

_{Eq. 4.37}

Time averaging the instantaneous total pressure, measured by the FR5HP, will result in a total pressure that is biased to a higher value due to the nonlinear relationship between total pressure and velocity.

𝑃

_𝑡,𝑚

̅̅̅̅̅̅ = 𝑃̅ +

_𝑠

1

2𝜌(𝑈̅

2

+ 𝑉̅

2

+ 𝑊̅

2

) +

1

2𝜌(𝑢′̅̅̅̅ + 𝑣′

2

̅̅̅̅ + 𝑤′

2

̅̅̅̅̅ )

2 Eq. 4.38 The subscript “m” indicates the measured and uncorrected quantity. It should be pointed out that average total pressure measured by the FR5HP is equivalent to the total pressure measured by a chamfered Pitot probe, see Eq. 2.6. The average total pressure measured by the FR5HP or the chamfered Pitot probe can be corrected if the turbulent dynamic pressure 𝑞̅ , Eq. 2.8, is known. ′ Recall that the FR5HP measures the correct instantaneous total velocity and flow angle which implies that probe measures the correct instantaneous components of the velocity vector. Therefore, when the instantaneous velocity components are time averaged, the correct values for 𝑈̅, 𝑉̅and 𝑊̅ are obtained as well as the correct turbulent fluctuations. The measurements of the turbulent fluctuations from the FR5HP can then be used to correct the total pressure.

𝑃̅̅̅̅̅ = 𝑃

_𝑡,𝑐

̅̅̅̅̅̅ −

_𝑡,𝑚

1

2 𝜌( 𝑢′̅̅̅̅ + 𝑣′

2

̅̅̅̅ + 𝑤′

2

̅̅̅̅̅)

2 Eq. 4.39 Where 𝑃̅̅̅̅̅ is the correct time-average total pressure defined in Eq. 2.4 and repeated here. _𝑡,𝑐

𝑃

_𝑡,𝑐

̅̅̅̅̅ = 𝑃̅ +

_𝑠

1

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4.2.8.4 Summary of Processing Fast-Response Five-Hole Probe Data

The steps for processing the fast-response five-hole probe data can be summarized with the following steps.

1. Acquire time history of voltages from each port.

2. Apply the voltage to pressure conversion using the pressure transducer calibrations supplied by Aeroprobe Corp.

3. Apply the distortion correction to each port using Aeroprobe’s Acoustic Recovery Software55 which implements the process described in Section 4.2.8.2. This process provides corrected time histories of the port pressures 𝑃₁₋₅(𝑡).

4. For each instant in time apply the five-hole probe calibration process described in Section 4.2.2 to determine 𝑃_𝑡(𝑡), 𝑃_𝑠(𝑡), 𝑈(𝑡), 𝑉(𝑡) and 𝑊(𝑡).

5. Calculate the time-averaged total pressure, static pressure, velocity components and any desired turbulence statistics. In addition to calculating 𝑢′̅̅̅̅, 𝑣′2 ̅̅̅̅ and 𝑤′2 ̅̅̅̅̅ the 2 FR5HP measurements can be used to calculate the remaining quantities of the Reynolds Stress Tensor56 as well as correlations of velocity and pressure fluctuations. 6. Correct the time-averaged total pressure using the method described in Section

4.2.8.3.

4.2.8.5 Processing a Fast-Response Probe as a Standard Probe

The measurements of a fast-response five-hole probe are an important aspect of this dissertation because they allow for accurate measurements of the total and static pressure, but these measurements also provide an additional opportunity to study the effects of turbulence contaminating the port pressure measurements of a standard five-hole probe. This can be done by processing the measurements of a FR5HP as if it were a Std5HP. Recall that the relevant

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difference between a FR5HP and a Std5HP is where the averaging occurs. With a Std5HP the port pressures are pneumatically averaged prior to being measured, whereas with a FR5HP the averaging occurs in post processing after the instantaneous flow quantities have been measured. If the FR5HP measured time history of port pressures is first averaged prior to applying the probe calibration then the FR5HP data can be processed as if it were a Std5HP. The relevant steps are as follows:

1. Acquire time history of voltages from each port.

2. Apply the voltage to pressure conversion using the pressure transducer calibrations supplied by Aeroprobe Corp.

3. Apply the distortion correction to each port using Aeroprobe’s Acoustic Recovery Software55 which implements the process described in Section 4.2.8.2. This process provides corrected time histories of the port pressures 𝑃₁₋₅(𝑡).

4. Calculate time-average port pressures 𝑃̅̅̅̅̅̅. This step simulates the pneumatic ₁₋₅ averaging effect of the urethane pressure lines on the standard five-hole probe.

5. Process the averaged port pressures using apply the five-hole probe calibration process described in Section 4.2.2.

The results produced from this procedure will contain errors due to the turbulent velocity fluctuations contaminating the measured pressure at each port. The method will prove useful because isolates the effects of turbulence on the port pressures measured by a Std5HP.

In document The effects of turbulence on the measurements of five-hole probes (Page 76-83)