The dynamic response of a pressure transducer for measurements in water

Abstract

Dynamic pressure measurements are indispensable in the field of fluid mechanics. Attaching tubing as a transmission line to the pressure transducer is often unavoidable but significantly reduces the usable bandwidth of the measurement system. Complex fluid-wall interactions and potential outgassing of air are present within systems with water-filled tubes. Comprehensive studies aiding researchers in selecting suitable transmission line parameters (i.e., material, length, and diameter) are not available. A simple calibration apparatus is designed for the frequency response characterization of multiple pressure transducers simultaneously applying a pressure step. The setup is thoroughly characterized and a detailed description is provided to optimize the bandwidth. A piezoresistive pressure transducer attached to water-filled tubes, as commonly used in hydrodynamic experiments, is characterized in the low-frequency range (i.e., $f \le {300}$ Hz). Tube-related effects, such as length, diameter, and material are investigated. The impact of entrapped air within the tubing is analyzed. The feasibility of substituting water with silicone oil to fill the tubes is explored. To optimize the usable bandwidth of the pressure measurement system for dynamic applications, it is essential to maintain short tubing that is as rigid as possible and free from entrapped air. Pressure wave propagation speed is reduced by two orders of magnitude in elastic transmission lines made of silicone. Pressure corrections through dynamic calibration are challenging due to the system’s sensitivity to various parameters affecting the dynamic response.

1 Introduction

Pressure measurements represent one of the most applied and established techniques in fluid dynamics. The pressure is measured utilizing a transducer that converts applied loads on a membrane into a measurable electrical quantity. A static calibration with known pressure input is performed to determine the sensor sensitivity. A dynamic calibration is required to assess the dynamic characteristics of the transducer. A universal approach does not exist as the calibration apparatus and procedures must be uniquely engineered to meet the specific measurement needs. Many different calibration approaches have been developed to calibrate transducers within an appropriate pressure and frequency range, utilizing different pressure transmitting medium (e.g., gas or liquid) (Hjelmgren 2002). The Instrumentation, Systems, and Automation Society (ISA) provides guidelines for the dynamic calibration of pressure transducers (ISA 2002).

Pressure transducers achieve best frequency response when flush-mounted. The natural frequency is usually large enough for most hydrodynamic experiments, providing a flat frequency response for the range of relevant frequencies. However, flush-mounted sensors are often not feasible. Space constraints and machinability of the model require tubing as transmission lines to connect the transducer with the pressure tap. The tubing affects the system’s dynamic response, leading to signal distortion (i.e., frequency-dependent signal amplification and attenuation as well as phase lag) (Bergh and Tijdeman 1965). Additionally, often the membrane of piezoresistive pressure transducers is embedded within a housing cavity. This further lowers the usable frequency of the sensor-tube system (Walter 2002). The severity of the distortions introduced by the transmission line depends on tubing length, tubing diameter, cross-sectional changes (i.e., change in diameter due to coupling of multiple tubes), tubing material, working fluid, and temperature (Pemberton 2010; Moens 1878; Korteweg 1878; Kutin and Svete 2018).

Unlike dynamic pressure calibrations in air, there is limited research for pressure sensor-tube configurations tested in water. In particular, there is limited research exploring the applicability of common, affordable piezoresistive pressure transducers. Crucial differences exist for measurements with water-filled tubes relevant to obtain meaningful measurement data. The elasticity of the tube wall becomes important due to the low compressibility of water (Moens 1878; Korteweg 1878). Especially in flexible silicone tubing, a drastic reduction in propagation speed is observed, leading to a non-negligible time delay. Furthermore, the system’s resonance frequency is vastly overestimated applying organ pipe and Helmholtz resonator formulations not considering the reduction in propagation speed.

Air bubbles may be present within the transmission line or sensor cavity due to unsuccessful system purging or outgassing of air caused by an increase in temperature (Wylie and Streeter 1978). Gas bubbles lower the resonance frequency of the system by increasing the volumetric compliance (i.e., change in bubble volume caused by pressure changes) of the system (Anderson and Englund 1971; Doebelin 1990). Furthermore, damping is increased (Hurst and VanDeWeert 2016). Anderson and Englund (1971) provide an experimentally verified formula to compute the resonant frequency of the system with liquid-filled tubes based on the system’s overall compliance (i.e., transducer-, liquid in the transducer-, and liquid in the tube compliance) valid for metal diaphragm pressure gauges. However, the transducer compliance is rarely provided by manufacturers. Obtaining the exact value through measurements is challenging and often not feasible. Kutin and Svete (2018) provide a mathematical derivation to take into account the compliance of the connecting tube that describes the relative change of the internal cross-sectional area due to a pressure change. Compliance effects are proportional to the bulk modulus of the fluid $\rho _0 c^2$ and therefore impact liquid-filled systems but are generally negligible when working with gaseous media. In most cases, a dynamic calibration becomes inevitable to characterize the dynamic response of the measurement system consisting of transducer and tubing.

The frequencies of interest for hydrodynamic experiments are commonly well below 1 ${\text {kHz}}$. The bandwidth requirement of the pressure measurement system depends on the underlying phenomena to be investigated. Accurate readings up to a few hundred  ${\text {Hz}}$ are required to truthfully capture dynamic stall phenomena (Wei et al. 2023), boundary layer wall pressure fluctuations (Ciappi et al. 2009), or pressure pulsations in hydrodynamic turbomachineries (Ke et al. 2022). Two different excitation methods exist for pressure transducer calibration (i.e., periodic and aperiodic). When working with water-filled tubes, the periodic excitation is commonly generated by vibrating a liquid column using an electrodynamic shaker (Hurst and VanDeWeert 2016; Lally 1991). The cut-on frequency of such calibration setups is greater than zero (approximately 10 ${\text {Hz}}$) due to the inability to generate a signal at lower frequencies. Calibration is performed by testing multiple single frequencies or via a continuous frequency sweep. Aperiodic calibration setups generate a pressure step through rapid pressure equalization between two pressure reservoirs. For calibration in water it is achieved by a membrane burst (Webster and Nimunkar 2020), via fast opening device (Smith 1964), or switching between two pressure reservoirs using a scanivalve (Conger and Ramaprian 1993). The bandwidth of the calibration setup depends on the rise time of the pressure step. Shorter rise times correlate with an increase in the frequency bandwidth as more Fourier terms are required to compose the steep signal (Damion 1994). The lower frequency limit is determined by the capability of the calibration apparatus to maintain the final pressure level at a constant level (Bean 1994). Regardless of the excitation method, knowledge of the true pressure is essential. Commonly, a reference transducer with known characteristics and a flat frequency response over the frequencies of interest is utilized.

The current study presents a setup to dynamically calibrate pressure transducers by measuring their step response, operating with water as working media (Sect. 3.1). An aperiodic excitation is selected to capture the response at low frequencies essential for hydrodynamic measurements. It encompasses an analysis of the tank acoustics of the chosen calibration vessel (Sect. 3.1.1), determination of the system’s bandwidth and repeatability of measurements (Sect. 3.1.2), assessment of data scatter due to tube material and sensor-tube connection (Sect. 3.1.3), and a linearity check of the test transducer to be calibrated (Sect. 3.1.4). Section 3.2 discusses tube-related effects. The effect of tube length on the dynamic response of the pressure measuring system is examined in Sect. 3.2.1 using silicone tubing. Section 3.2.2 provides results obtained with different tube diameters using tubing made of stainless steel. The measurements with varying tube length and diameter are analyzed in the time domain to investigate their effect on the pressure wave speed (Sect. 3.2.3). Tube elasticity effects are discussed in Sect. 3.2.4. The final section (Sect. 3.3) examines fluid effects, encompassing the impact of entrapped air in the tubing (Sect. 3.3.1) and exploring the option of filling the tubing with silicone oil instead of water (Sect. 3.3.2). Many aspects relevant to dynamic pressure measurements in water facilities (i.e., water tunnels, flumes, or towing tanks) are addressed, with a focus on system integration. The application-oriented approach reveals the capability of an affordable pressure sensor type to be used for pressure measurements over a wide frequency range if the integration is performed appropriately.

2 Setup, instrumentation, and signal processing

This chapter describes the pressure sensors and test rig utilized to characterize the dynamic response of the pressure measurement system. Piezoresistive pressure transducers are connected to water-filled tubes similar to the experimental setup in common hydrodynamic experiments (Erm 2003; Conger and Ramaprian 1994; Schmidt et al. 2017). The transducer-tube system investigated balances affordability with ease of system integration. Connecting tubes of varying diameters is accomplished through direct insertion into one another, ensuring leak tightness. Although this method may not be optimal, it serves as a quick solution frequently encountered in experimental setups. The aim is to reveal common issues that affect the dynamic response of a pressure measurement system. Section 2.1 gives detailed information regarding the two different pressure sensors employed. Section 2.2 describes the calibration vessel with all its components and the pressure sensor locations. Details regarding the signal processing to obtain the frequency response function (FRF) are provided in Sect. 2.3. The temperature during all measurements ranged between 18 and 23$^{\circ }$C.

2.1 Pressure sensors

A piezoresistive sensor with metal diaphragm (Kulite XTME-190 L-25A) serves as the reference sensor. It is suitable for application in any liquid or gas compatible with stainless steel. The small size of the sensing element allows for a flush-mounted installation with the calibration tank inner wall. The sensor’s pressure range is up to 1.7 bar absolute pressure. It exhibits a combined nonlinearity, hysteresis, and repeatability of 1% full scale output (FSO). The sensor’s natural frequency is 75 kHz. The manufacturer specifies an error of 5% and a rise time of 23 ${\upmu \text {s}}$ at 20% of its natural frequency. Thus, a flat frequency response for the frequencies of interest ($f \le {300}\,{\text {Hz}}$) is ensured with the flush-mounted installation. The output is considered the true wall pressure and serves as the reference signal. The sensor is utilized in conjunction with a signal conditioner from Kulite (KSC-2).

The sensor to be characterized is a piezoresistive, temperature-compensated (0–50$^{\circ}$C), unamplified, differential pressure sensor (Honeywell 26PCCFA6D). It is suitable for use with wet or dry media. The delicate sensing element requires placement within a small cavity in a housing, which is further equipped with a straight port to connect tubing. Thus, a true wall-mounted installation is not possible. The sensor has a range of ± 1 bar with a linearity of 0.25 % (FSO). The response time is max. 1 ms. No information is provided by the manufacturer regarding the natural frequency of the sensor. The Honeywell sensor is commonly utilized in the water towing tank facility at Technische Universität (TU) Berlin because of its small size and affordability. A custom-made amplifier is utilized for signal conditioning.

2.2 Calibration vessel and data acquisition

A cylindrical steel pressure vessel with a maximum operating pressure of 5.5 bar serves as the test rig (Fig. 1). The inner surface is galvanized and allows filling with water. The vessel has an inner diameter of 240 $\text {mm}$, a height of 260 $\text {mm}$ measured from the center of the convex bottom, and a wall thickness of 3 $\text {mm}$. The vessel’s lid is made of an aluminum plate with a thickness of 10 $\text {mm}$. The lid is tightened to the vessel using four clamps. A rubber gasket is placed between the lid and vessel to ensure a tight seal. A hole with a diameter of 30 $\text {mm}$ is placed at the center of the tank’s lid to enable the implementation of a membrane. An aluminum membrane with a thickness of 23  ${\upmu {\text {m}}}$ is chosen as the diaphragm material. The membrane is fixed using a 3D printed flange made of polylactide (PLA) that is screwed onto the lid with the addition of an O-ring seal to prevent leaking. A feeding line is attached to the lid for pressurized air supply. It consists of a hose, a pressure regulator with a ball valve, and a manometer used to regulate the pressure inside the tank. After pressurizing the tank, the ball valve is closed and the membrane is ruptured by a falling strut. The strut is made of stainless steel, has a length of 200 $\text {mm}$, a diameter of 10 $\text {mm}$, and weighs 137 g. The tip of the strut is conical. A linear guide is used to drop the strut from a height of 160 $\text {mm}$ centered onto the aluminum membrane. This method ensures a reliable and consistent membrane burst in successive measurements, which improves data repeatability.

Fig. 1

Experimental setup: Principle sketch and 3D CAD model

A total of seven pressure sensors (one Kulite and six Honeywell) are installed and positioned 93 $\text {mm}$ above the bottom of the tank. The Kulite sensor is flush-mounted (reference sensor) and one Honeywell sensor is wall-mounted. It is referred to as wall-mounted and not flush-mounted because the design (i.e., sensor housing) does not allow a true flush-mounted installation where the sensor membrane aligns with the inner wall surface. The remaining five Honeywell pressure sensors are each attached to different tubing. One sensor is connected to a flexible silicone tube that is connected to the calibration vessel via a hypodermic needle. The hypodermic needle has an inner diameter of 0.7 $\text {mm}$, an outer diameter of 1.3 $\text {mm}$, a length of 13 $\text {mm}$, and is flush-mounted with the inner surface of the vessel. It enables the attachment and quick exchange of different silicone tubing with an inner diameter of 1 $\text {mm}$ to connect the pressure sensor with the pressure port. The remaining four Honeywell sensors are connected to stainless steel tubing with a length of 300 $\, \text {mm}$ of different inner diameters and wall thicknesses. The steel tubes are welded to the calibration vessel and flush-mounted with the inner surface of the calibration apparatus. The tube inner diameters are $d_i = {1}$ mm, 2 $\,\text {mm}$, 4 $\,\text {mm}$, and 8 $\,\text {mm}$. The corresponding wall thicknesses are $s = {0.25}{}$ mm, 0.50 $\,\text {mm}$, 1 $\text {mm}$, and 2 $\,\text {mm}$, yielding a constant inner diameter to wall thickness ratio of $\frac{d_i}{s} = 4$. A short adapter piece is used for mounting purposes to connect each individual pressure sensor port with the corresponding tubing. Because the inner diameters of the tubes are different, a unique adapter is used for each tube. All adapter pieces are made of a short soft-PVC tube material with an inner diameter of 3 $\,\text {mm}$ and an outer diameter of 5 $\,\text {mm}$ that fits onto the pressure port. On the other side, the test tubing is attached either by direct insertion or via a custom-made attachment (SLA printed or silicone tube reinforced with a hypodermic needle). The sensors’ cavities and tubing are carefully filled with water. A medical syringe with a thin needle is used to fill the sensor cavities of the differential pressure sensors to prevent any entrapped air pockets. The reference side of all differential pressure sensors is connected via tubing to ambient pressure obtained from a vessel with venting holes isolated against external disturbances. A pressure calibrator (KAL 84) can be connected to the reference side, enabling a static pressure calibration to determine each sensor’s sensitivity value.

All pressure signals are acquired with a data acquisition system by National Instruments. It consists of a cDAQ-9188 mainframe equipped with analog input modules (NI 9223). All channels are synchronized, and data are acquired at a sampling rate of $f_s = {100\,}\textrm{kHz}$. The high sampling frequency is selected to enable improved investigation in the time domain (e.g., determination of the time delay between two signals). Triggered data acquisition with a pre-buffer of 500 samples is initiated based on the signal of the flush-mounted reference transducer (Kulite). The total measurement time for each test run is 3 s. For post-processing purposes, the data are truncated to a total length of 60000 samples (i.e., 0.6 s) sufficient to capture the dynamic effects of the system.

2.3 Signal processing

Each transducer’s output signal p(t) is normalized with the steady-state initial ($p_0$) and final ($p_f$) pressure values according to Eq. 1.

$$\begin{aligned} p^{*}(t) = \frac{p(t)-p_0}{p_f-p_0} \end{aligned}$$

(1)

The main objective of the signal processing is the determination of the frequency response function (FRF). It is expressed in the frequency domain and contains the steady-state information of the magnitude and phase at various frequencies. A flush-mounted sensor (Kulite) with a flat frequency response up to a few $\textrm{kHz}$ is used to measure the true reference pressure $p_\text {ref}$. The signal of the piezoresistive pressure sensor to be characterized is denoted by $p_{\text {tube}}$. Both signals are transformed from the time to the frequency domain via a fast Fourier transform (FFT) algorithm. The frequency response function (FRF) is derived by computing the ratio of the signals in the frequency domain (Eq. 2). The amplitude ratio is obtained by calculating $|\text {FRF}|$, and the phase with $\arctan (\text {FRF})$. A Bode plot is utilized to present the results of the amplitude ratio and phase lag at the corresponding frequencies.

$$\begin{aligned} \text {FRF} = \frac{\mathcal {F}[p_\text {tube}]}{\mathcal {F}[p_\text {ref}]} \end{aligned}$$

(2)

One constraint encountered when applying a fast Fourier transform (FFT) is the requirement for a discrete, finite-time signal that is periodically extended. This requirement is not met by the step signal generated by the experimental setup in the current investigation. To circumvent this problem, the step response signal is converted into a periodic signal by padding a sinusoidal downslope to the end of the pressure–time signal (Popp 1999). The downslope needs to be sufficiently long to preserve the dynamic characteristics of the system. It introduces a bias to the low-frequency content that is dependent on the length of the downslope. A downslope 1000 times longer than the rise time of the input step signal is found to have a negligible effect on the frequency spectrum and is utilized for all calculations. Alternatively, to apply the FFT algorithm on the step signals, they can be transformed using the Nicolson’s ramp method (Nicolson 1973) or the method developed by Gans and Nahman (1982) to generate a time-finite signal that is periodically continued.

The post-processing routine is validated using generic data with arbitrary coefficients. The output $Y(s)$ of a second-order linear system (i.e., mass-spring-damper system) is generated in response to a unit step input signal $X(s) = \frac{1}{s}$ using the transfer function described in Laplace space (see Eq. 3) where $\omega _n$ is the natural frequency and $\xi$ is the damping ratio. The time signals x(t) and y(t) are obtained via inverse Laplace transform and used in the post-processing routine to obtain the FRF. No deviations are detected in the frequency domain between the analytic solution and the FRF obtained via the post-processing routine used in the current study.

$$\begin{aligned} H(s) = \frac{Y(s)}{X(s)} = \frac{\omega _n^2}{s^2 + 2 \xi \omega _n s + \omega _n^2} \end{aligned}$$

(3)

A disadvantage of the FFT approach is the bias due to the Gibbs phenomenon that is inherent to the method applied (Paniagua and Dénos 2002). The Gibbs phenomenon describes the oscillatory (ringing) artifact in the neighborhood of a signal discontinuity (or large gradient) in the time domain that is reconstructed from a series of harmonic functions (Canuto et al. 1988). Consequently, the computation of the FRF, and thus also the correction of the measurand to obtain the true wall pressure, will never be exact. These effects become particularly relevant in experiments where the rise time is in the range of nanoseconds (e.g., shock tube experiments). The bias due to the Gibbs phenomenon for the current investigation is found to be negligible. The rise time of the negative pressure step is 1 ms or greater. It is equal or greater compared to the sensor response time of both sensor types such that the signal is assumed to be sufficiently smooth.

3 Results

The results section is divided into three subsections. Section 3.1 provides detailed information of a thorough setup characterization. A well-designed setup can be utilized for sensor calibration up to a few hundred ${\text {Hz}}$, sufficient for the majority of hydrodynamic experiments conducted in water. Multiple tube parameters (i.e., length, diameter, material) relevant for dynamic measurements with water-filled tubes are discussed in Sect. 3.2. The effect of an air bubble within the water-filled tubing on the dynamic response of the transducer-tube system is addressed in Sect. 3.3. Additionally, the usability of silicone oil as a substitute for water is explored to avoid outgassing of air. The results provided within this article are utilized to reveal transducer-tube-system-related effects with liquid-filled tubes. The experimental approach elucidates critical factors to be considered essential for dynamic pressure measurement in water. These findings serve as a guide for researchers, aiding in the avoidance of common pitfalls encountered when conducting dynamic pressure measurements in hydrodynamic facilities.

3.1 Setup characterization

The experimental setup is examined for the intended purpose of dynamically calibrating pressure sensors with liquid as the working fluid. A water-filled vessel with attached reference and test transducer is pressurized. A negative pressure step is generated by bursting an aluminum membrane leading to pressure equalization with the environment. Several tests are conducted to characterize the pressure step response. A leakage check is conducted before all measurements by ensuring that no pressure drift occurs in the pressurized vessel before bursting the membrane. The calibration vessel’s acoustic response is analyzed in Sect. 3.1.1. Resonant frequencies and wave propagation are examined theoretically based on the pressure vessel dimensions. The bandwidth (i.e., range of excited frequencies) via the negative pressure step is analyzed in Sect. 3.1.2. Section 3.1.3 discusses the impact of tubing quality and sensor-tubing connection required to link the transducer with the tubing. The linearity of results when varying the amplitude of the pressure step is investigated and discussed in Sect. 3.1.4. The information provided shall help other researchers to quickly adapt a similar calibration apparatus with which liquid-filled transducer-tube configurations are examined over a broad range of frequencies to study unsteady phenomena in water facilities.

3.1.1 Tank acoustics

The calibration setup is devoted to calibrate frequencies below 300 ${\text {Hz}}$. Higher frequencies are not sufficiently excited due to the capabilities of the setup as described in the following section (Sect. 3.1.2). When the membrane bursts, a pressure wave propagates through the vessel with speed of sound. The vessel is divided into two sections with different fluids (i.e., air and water) separated by the free surface. The first resonant frequency in each section is calculated. The upper part filled with air closely resembles that of a Helmholtz resonator as it contains a main cavity with a small opening. The resonant frequency of the Helmholtz resonator $f_{\text {HR}}$ is determined using Eq. 4. For calculations the speed of sound in air is assumed as $c_{\text {air}} =~{340}\;\textrm{ms}^{-1}$. The area of the neck opening ($A_{\text {neck}}$) scales with the diameter of the hole that is covered by the aluminum membrane ($d_{\text {membrane}} = {0.03}\,\text{m}$). The volume of the cavity filled with air is determined by the vessels inner diameter ($d_{\text {vessel}} = {0.24}\,\text{m}$) and the cavity height ($h_{\text {cavity}} = {0.009}\,\text{m}$). End corrections are applied using Eq. 5 to calculate the effective neck length ($l_{\text {eff}}$) according to Kinsler (2000). The actual neck length measured from the bottom of the aluminum lid to the top of the 3D printed PLA flange holding the membrane in place is $l_{\text {neck}} = {0.017}\,\text{m}$. Using these parameters, the Helmholtz resonant frequency with end correction applied yields $f_{\text {HR}} = {413}$ Hz. The frequency is sufficiently high to avoid interference with transducer calibration at low dynamic pressure frequencies (i.e., $<~{300}\,\text {Hz}$).

$$\begin{aligned} f_\text {HR} = \frac{c_\text {air}}{2\pi }\sqrt{\frac{A_\text {neck}}{V_\text {cavity} l_\text {eff}}} \end{aligned}$$

(4)

$$\begin{aligned} l_\text {eff} = l_\text {neck} + \frac{8}{3\pi } \frac{d_{\text {membrane}}}{2} \end{aligned}$$

(5)

For the section filled with water, the first resonant frequency is calculated using Eq. 6, applicable for an open-closed tube. This assumption is valid because the tank floor is rigid and the acoustic impedance of the free surface is low. The fill level of water is $h_{\text {water}} =~{0.251}\,\text{m}$. The concave tank floor is assumed flat for simplicity. Assuming a speed of sound in water of $c_{\text {water}} = {1500}\,{\textrm{ms}}^{-1}$, the corresponding first resonant frequency is $f_{\text {water}} \approx {1494}\,$ Hz. This frequency is sufficiently large to not interfere with the dynamic pressure calibration.

$$\begin{aligned} f = \frac{c}{4 h} \end{aligned}$$

(6)

Further, the cut-on frequency of the first spiral mode is determined. In order for a mode to propagate, Eq. 7 must be satisfied, where $\omega$ represents the angular frequency and $r_{\text {tank}}$ the tank’s inner radius. Solving this equation utilizing the lowest zero value of the Bessel function $\alpha _{mn} = 1.8412$ yields $f > {3663}\,{\text {Hz}}$ for the first spiral mode to be cut-on. This frequency is sufficiently large to guarantee plane wave propagation in the vessel and justifies the circumferential positioning of the pressure sensors.

$$\begin{aligned} \frac{\omega r}{c_\text {water}} > \alpha _\text {mn} \end{aligned}$$

(7)

3.1.2 Frequency bandwidth

The frequency bandwidth of the setup is examined. It is determined by comparing the measured true pressure signal (i.e., signal of flush-mount Kulite sensor) to an ideal step (i.e., Heaviside function). The step response in the form of a Heaviside function renders a perfect and abrupt jump that can only be achieved in theory and not in any physical setup. Figure 2 shows results in the time domain with discernible difference between experiment and theoretical limit.

Fig. 2

Time signal of Heaviside function and flush-mount reference transducer

The signals are transformed into the frequency domain via a Fourier transform, and the power spectral density is calculated (Fig. 3). Thirty data sets acquired over a time span of two weeks are presented in the form of mean value and its standard deviation. The water level for these measurements is set to 9 mm below the lid. A threshold value is introduced to assess the quality of the step response. The frequency value at which the power of the experimental step is reduced by $30\%$ (i.e., equivalent to a reduction of 1.5 dB) compared to the perfect theoretical jump is determined. Experimental results showed that this threshold of $\text {PSD}_{\text {p}_{\text {ref}}} / \text {PSD}_{{\textsc {Heaviside}}} = 0.7$ is a conservative estimation below and around which the frequencies are sufficiently resolved by the pressure step (i.e., signal-to-noise ratio at acceptable level). The mean bandwidth of these measurements is $f_{\text {bw,mean}} = {143}\,{\text {Hz}}$ with a standard deviation of $\Delta f_{\text {bw,std}} = {17}\,{\text {Hz}}$. Additionally, a peak in the frequency spectrum of the measurement data is observed at approximately 250 ${\text {Hz}}$. It may indicate the presence of a Helmholtz resonant frequency. The resonant frequencies may be overestimated with the calculations in Sect. 3.1.1.

Fig. 3

Bandwidth determination of the pressure step, $p = {0.7}\,{\textrm{bar}}$, water level 9 mm below lid. Shaded areas represent the standard deviation obtained from 30 measurements

The bandwidth of the calibration setup provided herein depends on the rise time of the pressure step. The rise time is affected by the time it takes to equalize the pressure within the vessel after the membrane bursts. This time is driven by three parameters that can be controlled within the experiments: the remaining amount of air contained within the vessel, pressurization level, and membrane burst behavior. An increased amount of air results in an increase in the rise time because the bulk of compressed gas must leave the vessel for pressure equalization after the burst of the membrane. A reduction in air in the vessel is desired. It is achieved by increasing the water level. However, the water cannot be filled up to an arbitrary level, which depends on the pressurization level of the vessel. A limit exists where the air exit velocity becomes large enough to entrain water particles during the pressure equalization. Water splashing is the consequence. It is undesirable because it changes the water level within the vessel for consecutive measurements, introduces additional measurement noise by generating surface waves, and increases the risk of damaging measurement equipment. A water fountain of up to approximately 2 m was set-off when the water level was not set appropriately and the tank was pressurized at 0.9 bar. Thus, a compromise between a fast rise time and a repeatable pressure step without water splashing has to be found. A stiff circular floating plate atop the free surface is inserted to reduce the amount of air contained in the vessel. Its outer diameter is 238 mm (i.e., 2 mm smaller compared to the vessel’s inner diameter). The water level is increased until a distance of 2 mm between floating plate and lid was achieved, thus reducing the amount of air inside the vessel to a minimum while avoiding that lid and floating plate touch. The bandwidth is increased to $f = {266}\,$Hz and no water is splashing. Additionally, installing the device enhances the data quality of the recorded signals by mitigating the generation of surface waves. While surface waves remain detectable within the setup, their impact on data quality is minimal.

The membrane burst behavior plays a crucial role in the quality of the pressure step. Depending on the pressure level within the vessel, a membrane must be chosen that does not spontaneously burst when pressurizing the tank, but at the same time, it shall burst entirely to keep the rise time minimal and concomitantly the bandwidth high. Different aluminum membranes are utilized for different tank pressure levels. Poking the aluminum membrane with a blunt object works as a quick and simple actuation method. However, it results in a reduced mean bandwidth of 89 ${\text {Hz}}$ and compromises the repeatability with an increased standard deviation of 34 ${\text {Hz}}$ compared to the measurements carried out with a falling strut. Further improvements are possible to increase the bandwidth of the pressure step if necessary. The hole that is covered by the aluminum membrane can be increased to enable faster pressure equalization. Due to the increased surface area of the membrane, the stresses within are increased, necessitating the selection of an appropriate membrane that does not spontaneously burst.

3.1.3 Tubing and sensor-tubing connections

The fluid dynamic effects of tubing, especially their length, diameter, and material, are discussed in detail in Sect. 3.2. This section describes the effect of material quality and setup design on the obtained results. Model and test rig design often require that pressure tap and pressure sensor must be bridged with a piece of tubing. Common tubing materials are silicone and stainless steel. If high temperatures are not reached in the experiment, silicone tubing is a common standard because it is inexpensive and simplifies the integration into the test rig. However, the quality of silicone tubes may fluctuate with vast deviations from the nominal values as provided by the manufacturer. Large deviations in the outer diameter were observed on one and the same spool of tubing of 100 m length. The outer diameter fluctuated from its nominal outer diameter value of 3 mm down to 2 mm, which is a reduction of $33.3\%$. These fluctuations may not always be so drastic and depend on the individual source, received batch, and tolerance class. These fluctuations have a direct consequence on the dynamic response of the transducer-tube system. Additionally, knowledge of the tube’s E-modulus is essential to calculate the propagation wave speed in tubes (see Sect. 3.2.3 for more details).

This detail is often not provided by the manufacturer of silicone tubes, yielding a parameter with great uncertainty. Figure 4 shows the amplitude ratio and phase lag of two different silicone tubes manufactured by different companies. The nominal values for tube length ($l =$ 300 mm), inner diameter ($d_i =$ 1 mm), and wall thickness ($s =$ 1 mm) are identical. Even though the E-modulus remains unknown, noticeable differences exist between the tubes in terms of visual and tactile characteristics, with one being notably softer than the other. The magnitude of the pressure step is identical in both measurements. Comparing the response of both signals, it is observed that both tubes resonate at a similar frequency around 20 ${\text {Hz}}$. silicone tube 1 reaches a maximum amplitude ratio of approximately $1.3$ , whereas the value for silicone tube 2 is approximately $1.7$. The behavior of the system using silicone tube 1 deviates from that of a typical second-order system, exhibiting pronounced higher harmonics not observed in the configuration with silicone tube 2. Moreover, the use of silicone tube 1 results in an increased phase delay. These measurements already demonstrate the sensitivity of the system when the tubes are filled with water. Such aspects are typically not a concern when dynamic pressure measurements are conducted in air.

Fig. 4

Frequency response of test transducer connected to silicone tubes obtained from two different manufactures with identical nominal values ($l =$ 300 mm, $d_i =$ 1 mm, $s =$ 1 mm)

Often, an adapter piece has to be implemented to merge two counterparts of different diameters to connect the pressure transducer with the tubing. In case a smaller tube diameter is fitted into one with a larger diameter, as must be done in this study, a short hypodermic needle is utilized to stiffen the joint section and to ensure leak tightness (see Fig. 5a). For the experiments with larger diameter steel tubing (see Sec 3.2.2), a SLA 3D printed connector is used in combination with a short soft-PVC tube to connect to the pressure transducer (see Fig. 5b). These connections contribute to further fluctuations with direct consequences on the dynamic response of the system. It affects the total length of the system, introduces additional cross-sectional changes, and results in alterations in elasticity due to the use of different materials. For example, a shift of approximately 3 ${\text {Hz}}$ to lower frequencies accompanied by a reduction in the amplitude ratio of 0.14 is observed for the first resonance peak when retracting the hypodermic needle by 1 mm from the transducer port as depicted in Fig. 5a compared to just inserting it into the port. Therefore, measurement quantities are expected to show deviations solely due to the change of the configuration (i.e., exchanging the tube and reattaching the sensor), likewise in the studies presented within this article but also in experimental measurement campaigns.

Fig. 5

Exemplary transducer-tube connections: a small diameter silicone tubing and b large diameter steel tubing

3.1.4 Linearity

Pressure measurement systems consisting of tubing and a pressure sensor are often treated and approximated as second-order linear systems. Measurements are carried out at different pressure levels to check the linearity of the pressure sensors used for this study. The response of the piezoresistive test transducer attached to a silicone tube of 300 mm length, inner diameter of 1 mm, and wall thickness of 1 mm is shown in Fig. 6. The amplitude ratio shows a dependence on the input pressure. With increasing input magnitude (i.e., pressurization level), a decrease in amplitude ratio is observed. It reveals that damping is dependent on the pressure input level and that damping increases with increasing magnitude. Additionally, the resonant frequency shifts to smaller values. Phase delays are present but less pronounced. Similar trends are observed when a stainless steel tube with identical dimensions to the silicone tube is utilized (not shown). Similar tests are carried out by attaching the Kulite transducer, which is mechanically different (metal membrane), to the same tube using a machined aluminum connector as the tube-sensor connection. The cavity within the aluminum connector is intentionally spacious to facilitate system de-airing. The system with Kulite transducer employed behaves linearly without any dependence on the magnitude of the input pressure. The pressure-dependent nonlinear response of the Honeywell test transducer may point to the inability to properly de-air the transducer cavity despite all efforts (Hurst and VanDeWeert 2016). Correcting measurement data via FRF obtained through dynamic calibration poses challenges because it necessitates an in situ sensor calibration with extensive dynamic calibration data to capture the amplitude-dependent effects.

Fig. 6

Frequency response of test transducer connected to silicone tube ($l =$ 300 mm, $d_i =$ 1 mm, $s =$ 1 mm) obtained at four different pressurization levels

3.2 Tube effects

With the fully characterized setup, the influence of tube properties on the dynamic response of the system are investigated in this section. The effect of tube length is assessed in Sect. 3.2.1. The impact of varying tube diameter is addressed in Sect. 3.2.2, followed by the effect of tube material elasticity in Sect. 3.2.4. The propagation speed of the pressure signal is investigated in Sect. 3.2.3 for silicone and stainless steel tubing.

3.2.1 Tube length

The test transducer is characterized for five different tube lengths. A silicone tube with an inner diameter and wall thickness of 1 mm is utilized to bridge the distance between the pressure tap and sensor. A total of five tube lengths ranging from 50 mm to 1000 mm are tested. Additionally, the results obtained with a wall-mounted Honeywell sensor (denoted as 0 mm) are included (pink dashed dotted line). Figure 7 shows the measurement results in the time domain for the reference and test transducers. The flush-mount reference sensor is demarcated by a black dotted line. No discernible differences between both transducers without connecting tubes is observed. All other five configurations with different tube length show under-damped characteristics. The occurrence of an overshoot and ringing characteristic is observed. With increasing tube length, the corresponding decay time increases, and the ringing frequency decreases. The delay time increases with tube length. It is further elaborated in Sect. 3.2.3 to determine the pressure wave propagation speed.

Fig. 7

Time signal of flush-mount reference transducer and test transducer connected to silicone tubes of different length ($d_i =$ 1 mm, $s =$ 1 mm)

The amplitude ratio and phase lag of all configurations are presented in Fig. 8. The response of the wall-mounted test transducer is analyzed. It reveals a flat frequency response up to approximately ${150}$ Hz for both the amplitude ratio and phase lag. At higher frequencies, deviations from unity for the amplitude ratio and from zero for the phase lag are apparent. These deviations appear to be random and do not show any clear resonance phenomena as would be expected from a Helmholtz resonator. These findings agree well with the $-$1.5 dB bandwidth range of the calibration system with a frequency of $f_{bw,\text {mean}} \approx {140}$ Hz. It is concluded that deviations at higher frequencies are caused by the poor signal-to-noise ratio. A flush-mounted sensor of the Honeywell type can thus be utilized for unsteady measurements of up to at least ${100}$ Hz without the need to correct measurement results in amplitude or phase.

Results with silicone tube of different length attached to the test transducer are compared (Fig. 8). Each configuration shows a distinct resonance peak and amplification of the amplitude around its resonant frequency. With increasing tube length $L$, the system’s resonance frequency is shifted to lower values while the maximum value of the amplitude ratio is reduced, indicating increased damping effects. The occurrence of higher harmonics is detected. The amplitudes are attenuated with increasing frequency. When tubing is attached, the system acts as a low-pass filter, effectively attenuating frequencies beyond the resonant frequency. At higher frequencies above $f \approx {200}\,$ Hz, the frequency response obtained with the Fourier transform shows an erratic behavior for all measurements, likely caused by a poor signal-to-noise ratio due to the system’s bandwidth. Concomitantly, the phase lag is increased with increasing tube length. At the resonant frequency (i.e., the frequency at which the maximum is reached in the amplitude ratio) the phase lag is $\phi \approx {90}\,^{\circ }$. However, it is not observed exactly at $\phi = {90}\,^{\circ }$ as would be the case for a true second-order system. The phase lag increases drastically for frequencies around the natural frequency and higher values, except for the configurations with the shortest (50 mm) and no tubing (0 mm). This is in agreement with the bandwidth of the calibration apparatus.

Fig. 8

Frequency response of test transducer connected to silicone tubes of different length ($d_i =$ 1 mm, $s =$ 1 mm)

These results yield valuable insights for experimental consideration. With the detection of input pressure-dependent nonlinearities (as discussed in Sect. 3.1.4), coupled with observed fabrication deviations in tube material and challenges in repeatability with sensor-tubing connections (Sect. 3.1.3), correcting results becomes unrealistic. Assuming that maximal $30\%$ of the system’s natural frequency shall be considered for dynamic investigations to keep amplitude deviations and phase distortions below approximately $10\%$ (Hurst et al. 2015), the usable frequency range is rather small. For example, for meaningful results it yields a maximum frequency of $f = {9}\,{\text {Hz}}$ for a tube length of $l =$ 300 mm.

3.2.2 Tube diameter

The effect of tube diameter is investigated. Stainless steel tubes with a length of 300 mm are attached to the calibration vessel. The ratio of inner diameter ($d_i$) to wall thickness ($s$) is identical for all tubes (i.e., $d_i/s = 4$). Four different tube inner diameters are tested with $d_i =$ 1 mm, 2 mm, 4 mm, and 8 mm. Even though tube diameters of $d_i =$ 2 mm, $d_i =$ 4 mm, and 8 mm have no practical application for the intended purpose of surface pressure measurements in water (White 2009), these configurations are valuable to qualitatively assess the effect of shear during wave propagation. Shear effects may influence propagation speeds as discussed in Sect. 3.2.3.

Fig. 9

Time signal of flush-mount reference transducer and test transducer connected to stainless steel tubes of different inner diameter ($l =$ 300 mm and $d_i/s = 4$)

The time history of all four configurations, including the signal of the flush-mount reference sensor, is shown in Fig. 9. An increase in ringing frequency with increase in diameter is identified. The corresponding amplitude ratio and phase lag is shown in Fig. 10, confirming the observations made in the time domain. The increase in tube diameter alleviates the effects of viscous damping. Additionally, the resonant frequencies are shifted to higher frequency values when increasing the diameter. Larger tube diameters appear to be the preferred choice to increase the usable frequency range for dynamic pressure measurements. However, for pressure measurements conducted on a model’s surface, usually a small hypodermic needle is implemented. It minimizes induced effects caused by flow tripping and mitigates biases attributed to spatial averaging comparing the pressure tap diameter to pressure wavelength. However, the second argument for water-based experiments is of secondary importance as the attenuation due to spatial averaging is small, approximately 0.1% at a frequency of 10 kHz (Wulff, 2006). Connecting the hypodermic needle of the pressure tap to the tubing introduces cross-section changes that affect the dynamic response. Especially the damping is expected to be influenced substantially as a small amount of fluid has to move a much larger fluid column after the cross-section change. In the current setup tested, these effects are excluded by inserting each respective tube flush into the vessel’s surface.

As observed in the measurements conducted with silicone tubes (Sect. 3.2.1), higher harmonics are detected in the amplitude ratio plot indicating deviations from a second-order system. The presence of higher harmonics has a notable effect on the phase response of all configurations. A decrease in phase delay is observed at frequencies around higher harmonics, showing the formation of multiple local minima and maxima. It indicates complex system interactions that deviate vastly from a second-order linear system

Fig. 10

Frequency response of test transducer connected to silicone tubes of different length ($l =$ 300 mm and $d_i/s = 4$)

3.2.3 Propagation speed

Boundary conditions (i.e., walls) greatly affect the wave speed in fluids. In cylindrical tubes, Eq. 8 is derived by considering the pressure wave generation and propagation induced by the waterhammer effect (Wylie and Streeter 1978). The equation takes into account the fluid and tube material properties as well as tube dimensions. $K_{\text {l}}$ stands for the liquid’s bulk modulus and $\rho _{\text {l}}$ for the density of the liquid. The E-modulus of the tube is denoted by $E_{\text {t}}$ , whereas $d_i$ and $s$ describe the tube inner diameter and wall thickness, respectively.

$$\begin{aligned} a = \sqrt{\frac{ \frac{K_{\text {l}}}{\rho _{\text {l}}} }{1+\left( \frac{K_{\text {l}}}{E_{\text {t}}} \right) \left( \frac{d_i}{s} \right) }} \end{aligned}$$

(8)

Two limits to Eq. 8 exist (i.e., rigid wall and elastic wall). In the limit of rigid walls (i.e., $E_{\text {t}} \rightarrow \infty$) the second term in the denominator vanishes, resulting in Eq. 9, well-known for sound speed in unbounded water (i.e., $a_{\text {water}} = {1483}\, {\text{ms}}^{-1}$) where $K_{\text {l}} = {2.2}\,\text{GPa}$ and $\rho _{\text {l}} = {1000}\, {\text{kg}} m^{-3}$.

$$\begin{aligned} a = \sqrt{\frac{K_{\text {l}}}{\rho _{\text {l}}}} \end{aligned}$$

(9)

For elastic walls, $E_{\text {t}}$ is much smaller compared to the bulk modulus of the liquid $K_{\text {l}}$, such that the 1 in the denominator of Eq. 8 becomes negligible, leading to Eq. 10 also known as the Moens–Korteweg equation. It is often applied to calculate the pulse wave speed in arterial blood flow (Parker 2021).

$$\begin{aligned} a = \sqrt{\frac{ \frac{K_{\text {l}}}{\rho _{\text {l}}} }{\left( \frac{K_{\text {l}}}{E_{\text {t}}} \right) \left( \frac{D}{s} \right) }} = \sqrt{\frac{E_{\text {t}} s}{\rho _{\text {l}} D}} \end{aligned}$$

(10)

The applicability of Eq. 8 for pressure measurements with water-filled tubes is assessed. Pressure wave speed is extracted from the data provided in the time domain from the previous sections presented in Figs. 7 and 9. The sampling rate of 100 kHz allows for an analysis in the time domain. The time delay between the flush-mount reference sensor and sensor with tubing is determined. Accurate determination is challenging as dispersive behavior of the pulse wave is detected in the experiments. High-frequency content is observed before the bulk of the signal contributes to a rise of the signal (i.e., the pressure step). These higher frequencies have been observed by other researchers in air- and liquid-filled tubes due to frequency dispersion (Tijdeman 1975; Holmboe and Rouleau 1967). The time delay for this study is obtained by extracting the time instances at which the flush-mounted reference transducer and transducer connected to tubing reach a value of $p^* = 0.1$. This describes the time instance at which the majority of the signal is rising, essentially capturing the transfer of signal information at group velocity (Brillouin 1960). The tube length is then divided by the time difference $\Delta t_{\text {bulk}}$ to obtain the group velocity of the signal (i.e., propagation speed of the main information of the signal). The obtained values from the experiments presented in Secs. 3.2.1 and 3.2.2 are tabulated in Table 1 and compared to the pressure wave velocity calculated via Eq. 8. Theoretical values in the last column are obtained with $K_{\text {l}} = 2.2$ GPa, $\rho _{\text {l}} = {1000\,\text {kg}\text {m}}^{-3}$, as well as the tube dimension $d_i$ and $s$ identical to those in the experiment as tabulated. Additionally, the tube system is treated as a lumped system including the connector piece and sensor port. The fractional contribution to the total pressure wave travel time in each component is taken into account by considering their respective dimensions (i.e., lengths, inner diameter, and wall thickness) and corresponding E-modulus.

Findings for silicone tubing are discussed first. The E-modulus of silicone tubes is often not provided by the manufacturer. Estimating the values is challenging as it depends on various parameters such as material composition (e.g., polymers, filler, additives) and processing conditions (e.g., temperature, extrusion rate). Therefore, minimum (0.517 MPa) and maximum (62.1 MPa) values of the E-modulus for silicone rubber are taken from MatWeb (2024) and used for calculation. The measured group velocity is compared to the theoretical results obtained via Eq. 8. The values from the experiment fall within that range, validating the applicability of Eq. 8 when used with silicone tubes. The wave speed is largely reduced due to the elasticity of the tube material. Delay times cannot be neglected if tubing length becomes large. Furthermore, group velocities extracted from the experimental data show a decrease in velocity with an increase in tubing length. This is not predicted by Eq. 8. These deviations are likely caused by end-effects (i.e., tube connections on either side), variations in tube diameter and wall thickness, measurement inaccuracies, and the ability to detect the characteristic point at which the bulk of the signal is rising.

Similar to the calculations with silicone tubing, a range of values for the E-modulus is taken in which stainless steel usually is found (190 to 210 GPa). Since the ratio of inner diameter to wall thickness is kept constant for all steel tubes, the wave velocity is theoretically identical among all configurations. Deviations, as tabulated in the last column of Table 1, are introduced by the tube-transducer connectors used. The extracted group velocity from the steel tube experiment shows a vast discrepancy to the theoretically obtained values. Extracted from the experiment, the speed of sound ranges from 99 ${\textrm{m}}s^{-1}$ for the smallest inner diameter of 1 mm to 403 ${\textrm{m}}s^{-1}$ for the largest inner diameter of 8 mm. A trend is observed in which the group velocity increases with increasing inner diameter, contradicting the prediction of Eq. 8 (last column of Table 1). Additionally, the values in the experiment are lower by a factor of approximately 3 to 14. The applicability of Eq. 8 to stainless steel tubing is not assured. The inadequacy of the equation is twofold. First, connections such as adapters introduce a discontinuity in the material properties. Especially the tube’s E-modulus is significantly reduced locally by inserting flexible tubing or 3D printed fittings. Moens (1878) deliberately introduced flexible material in large steel tubing to measure the wave train propagation alongside the tube generated by the waterhammer effect using a cardiograph. He stated that the speed of the pulse wave is greatly affected by the flexible inserts. Utilizing less invasive fittings reduced the decelerating effect on the wave pulse. Additionally, viscosity is neglected to derive Eq. 8. Experimental verification of this formula for steel tubing is obtained via waterhammer experiments with tubing of much larger diameter. Having tubing of small diameter increases the effect of viscosity. The importance of shear effects has been theoretically and experimentally validated for tubes filled with air (Kirchhoff 1868; Weston 1953; Rayleigh 1945; Tijdeman 1975). It reveals a frequency dependency that is dependent on the shear wave number, causing dispersion in cylindrical tubes. The group velocity of the acoustical signal is reduced for narrower tubes. Fuller and Fahy (1982) describe the dispersive behavior for liquid-filled tubes with elastic walls. This frequency dispersion is not captured by Eq. 8 as the effect of viscosity is neglected in the derivation.

Table 1 Pressure wave propagation speed for different tube configurations. Measured group velocity vs. theoretically determined wave velocity using Eq. 8

Another possible contribution to the reduction in sound speed in liquids is the presence of free air uniformly distributed within the solution. When small air bubbles are present in water, they effectively act as springs, causing a delay in the transport of information. Even small amounts of non-dissolved air can significantly impede the propagation speed (Wylie and Streeter 1978). According to Leighton (1994), the sound speed can drop to less than 5% of its value in bubble-free medium. Outgassing may occur, forming free air bubbles, as temperature rises. In this study, it is precluded that outgassing took place as the temperature in the lab was nearly constant. The silicone tube was regularly inspected for free air bubbles. Prior to filling the calibration vessel, tubes, and transducer cavities with water obtained from the in-house osmosis system, the water was left in the laboratory sufficiently long to ensure equalization with ambient temperature. Findings from experiments conducted with various tube diameters reveal a notable impact on the speed of sound. If the primary factor influencing the reduction in propagation speed were caused by free air bubbles, any change in the fractional amount of free air within the system would uniformly impact the propagation speed across all tubes of different diameter.

3.2.4 Tube elasticity

Other than in tubes filled with air, the tube walls cannot be treated as rigid anymore when filled with liquid such as water. Pronounced fluid-wall interactions are the consequence and cannot be neglected. The propagation speed of the pressure wave is greatly influenced as discussed in Sect. 3.2.3. Measurements are conducted using a silicone tube of 300 mm length, inner diameter of 1 mm, and wall thickness of 1 mm. After conducting the measurements, the tube is fully submerged in epoxy resin to affect the wall stiffness. These results are compared to those obtained in a stainless steel tube with a similar tube length and inner diameter. Figure 11 shows the data in the time domain of all three configurations in addition to the flush-mounted reference sensor. The ringing frequency is increased and damping decreased as the tube wall’s stiffness increases. Additionally, when examining the time instance at which each signal begins to increase from zero, it becomes apparent that the time delays are influenced, as discussed in the previous section (Sect. 3.2.3). Stiffer tubing material increases the velocity of pressure wave propagation in accordance to Eq. 8.

Fig. 11

Time signal of flush-mount reference transducer and test transducer connected to tubing with a length of $l =$ 300 mm but different elasticity

The corresponding amplitude ratio and phase lag plots are presented in Fig. 12. Because the nominal inner diameter of all three configurations is identical, the respective resonant peaks are in close proximity as observed in the amplitude ratio plots. However, with stiffer walls, the resonant peak is shifted to higher frequencies and damping is reduced. Additionally, Fig. 12 shows the phase response of all three configurations. The phase lag is larger the more elastic the wall is. Submerging the silicone tube in epoxy resin decreases the phase lag. The phase lag is comparable to the results obtained with steel tubing up to the second harmonic occurring at approximately 50 ${\text {Hz}}$. For higher frequencies, the phase lag increases abruptly, whereas the increase in phase lag with steel tubing is less pronounced. As these results suggest, stiff tubes (i.e., stainless steel) are the preferred choice in a setup with water-filled tubes, not only because of enhanced performance with respect to the usable bandwidth for dynamic measurements, but also because it enhances the information transfer resulting in higher propagation speed. This reduction in time delay consequently minimizes phase lag in the signal.

Fig. 12

Frequency response of test transducer connected to tubing with a length of $l =$ 300 mm but different elasticity

3.3 Fluid effects

Enclosed air within water-filled tubes influences the response of the pressure measurement system (Sect. 3.3.1). It changes the sensor compliance, thus lowering the resonant frequency and bandwidth of the measurement system (Anderson and Englund 1971). Properly de-airing a system is difficult. Vacuum-fill methods or systems with bleed holes exist but are impractical for pressure measurement systems with thin tubing. Additionally, residual air will always be present within the system when employing the vacuum-fill approach because a perfect vacuum cannot be achieved in practice. Using a syringe as described by Jentzsch et al. (2021) offers greater applicability, but it is not entirely foolproof either. Tiny cavities within the pressure senor housing may remain unreachable and pockets of air could persist regardless of the filling technique owing to the surface tension of the water. Even if the de-airing of the transducer-tube system were assumed to be perfect, air bubbles could still form due to temperature effects. An increase in water temperature may lead to outgassing of dissolved air. Especially in narrow tubes (i.e., $d_i =$ 1 mm), it generates air bubbles large enough to create a discontinuity in the water-filled tube. Detecting the air bubble in a running experiment is difficult because visual inspection is often not possible due to the integration of the tubing within the model or the usage of steel tubing. The effect of an air bubble present within the transmission line is discussed in Sect. 3.3.1. The use of silicone oil as a viable solution to avoid degassing is explored in Sect. 3.3.2.

3.3.1 Enclosed air bubble

An air bubble of 9 mm length is purposely introduced into silicone tubing of 150 mm length. Although such an extreme scenario of extended size may be rare in typical experiments, it nonetheless offers valuable insights. Introducing smaller air bubbles was tested showing decreased severity on the signal distortion. Measurements are conducted using a silicone tube of $l =$ 150 mm, $d_i =$ 1 mm, and $s =$ 1 mm. The results are compared to the reference case of the solely water-filled tube. Six consecutive measurements are conducted subsequent to the introduction of the air bubble. The time domain data is presented in Fig. 13. In comparison with the results obtained with the tube entirely filled with water, the amplitude of the ringing is dampened and distorted, indicating the presence of additional nonlinearities. After each test run, the air bubble is scattered into many smaller ones. As a result, the system’s response more and more resembles that of the state where no air is present in the tube. The group velocity remains constant across all six consecutive measurements as the splitting of the air bubble has no effect on the time delay of the pressure signal.

Fig. 13

Time signal of flush-mount reference transducer and test transducer connected to silicone tubing ($l =$ 150 mm, $d_i =$ 1 mm, $s =$ 1 mm). Introduction of an air bubble extending 9 mm in length and conducting six consecutive measurements

Amplitude ratio and phase lag are presented in Fig. 14. The resonant peak shifts to lower frequencies and increased damping is observed when air is present within the transmission line, as reported in the literature (Webster and Nimunkar 2020). The lowest resonant frequency with the least amplification is observed in the initial measurement, where the air bubble was still apparent as a single bubble before the pressure step was initiated. After each measurement and subsequent splitting of the air bubbles into multiple smaller ones, the first resonant peak detected in the amplitude ratio plot approaches that of a completely water-filled system (blue line). For higher frequencies, especially at the second higher harmonic, a distortion of the signal is apparent, not allowing for clear peak identification. Similar trends are observed for the phase lag. The largest lag is observed in the first measurement until the occurrence of the first resonant peak. In between approximately 50 ${\text {Hz}}$ and 100 ${\text {Hz}}$ (i.e., occurrence of the second distorted resonance peak), the trend is reversed. Within this frequency range, the phase lag is nearly constant. Additionally, the phase lag in that frequency range is largest for the configuration without any air bubble. For higher frequencies the phase lag decreases further.

Fig. 14

Frequency response of a test transducer connected to silicone tubing ($l =$ 150 mm, $d_i =$ 1 mm, $s =$ 1 mm). Introduction of an air bubble extending 9 mm in length and conducting six consecutive measurements

Despite the presence of an air bubble of substantial size within the transmission line, the transducer-tube configuration still permits dynamic measurements up to approximately 15 ${\text {Hz}}$. Ensuring short tubing length even in the presence of non-dissolved air may in many instances still yield satisfactory results across a wide range of frequencies. The time delay is not affected either when a silicone tube is used for the experiments, indicating that the pressure wave propagation speed is unaffected. This may be different in tubing with increased rigidity. However, the universality of these results is in question owing to the nature of the experiment. A strong pressure pulse is initiated with a wavefront strong enough to scatter the bubbles subsequent to each measurement. Such splitting does not occur in measurements where pressure changes are more benign, potentially affecting the findings presented within. Therefore, these measurements must be treated with caution. Utilizing an electrodynamic shaker with lower pressure amplitudes (as is the case in most hydrodynamic experiments) might be the preferred choice to analyze these effects in more detail.

During additional tests (data not shown for brevity), the Kulite transducer is attached to silicone tubing as described in Sect. 3.1.4. No pressure-dependent dynamic response is observed when the sensor cavity and transmission line are properly de-aired. By intentionally introducing a small air bubble into the silicone tube, a dependency on the input pressure becomes apparent. This observation strengthens the previous statement made, suggesting that the input pressure-dependent nonlinear response of the Honeywell transducer stems from the inability to completely remove air from the sensor cavity.

3.3.2 The use of silicone oil

The utility of silicone oil as an alternative pressure transmitting medium is explored to avoid outgassing within water-filled tubing. Even though silicone oil is composed of silicon, oxygen, carbon, and hydrogen, it has a lower solubility for gases compared to water. Degassing, if apparent, occurs delayed, is slower, and less pronounced. Jentzsch et al. (2021) describe a de-airing process for pressure sensors, using silicone oil as a transmission fluid. Only the sensor cavity is filled with oil. The reduced surface tension of the oil enhances the filling process in the sensor cavity. In addition to preventing a decrease in the natural frequency caused by entrapped air, this method also diminishes sloshing effects in the transmission line during surge experiments due to the compressibility of remaining air. These experiments are conducted in a towing tank where the sensor-tube system experiences sinusoidal accelerations concurrent with the motion profile of the carriage train. Filling the complete transducer-tube system was not tested. Wulff (2006) uses silicone oil in the cavity of a recessed-mounted pressure sensor to prevent fluid leaking when changing inserts of the setup.

Fig. 15

Time signal of flush-mount reference transducer and test transducer connected to silicone tubing ($l =$ 150 mm, ${d_i}$ = 1 mm, ${s =}$ 1 mm) completely filled with water or silicone oil (50 cST)

Within this study, a transducer-tube system is filled with silicone oil (50  cST). Both the sensor cavity and the tube (silicone tube of 150 mm in length, inner diameter of 1 mm, and outer diameter of 3 mm) are filled. Results are presented in Figs. 15 and 16, comparing a transducer-tube system completely filled with water (blue line) and the other with silicone oil (orange line). Severe damping of the pressure signal is observed in the time domain (Fig. 15) without the occurrence of a ringing frequency when filling the tube entirely with oil. The system response exhibits characteristics typical of a first-order system. Additionally, the dead time is increased for the configuration filled with oil. This could be attributed to the smaller bulk modulus of the silicone oil versus water (refer to Eq. 8). The exact bulk modulus of the oil is not known as it is not provided by the manufacturer. However, depending on formulation and temperature, it can range between 1 to 3 GPa, whereas water at 20$\,^{\circ}$C has a bulk modulus of about 2.2 GPa. If frequency dispersion occurs, the group velocity may be further reduced as high-frequency content is effectively damped out, as depicted in Fig. 16. The amplitude ratio is reduced by 10% at a frequency of 2 ${\text {Hz}}$. Additionally, increased phase lag at low frequencies is observed. At 2 ${\text {Hz}}$, the phase lag is approximately $15^{\circ }$, whereas it is negligible for the water-filled tube. The decreased bandwidth of the system does not allow for dynamic investigations, and filling the entire tube with silicone oil (50  cST) is not a viable alternative.

Fig. 16

Frequency response of test transducer connected to silicone tubing (${l =}$ 150 mm, ${d_i =}$ 1 mm, ${s =}$ 1 mm) completely filled with water or silicone oil (50  cST)

A different silicone oil (0.65 cST) with reduced surface tension is also tested (not presented). Due to its aggressive properties, it reacts adversely with the silicon membrane of the piezoresistive test transducer, causing damage to the sensor. To avoid damaging the sensor, its sensor cavity is filled with water, and only the tube is filled with the silicone oil. When filling the silicone tubing, it elongated rather quickly (from 300 mm to 310 mm within 15 min), thus affecting consecutive measurements. However, damping was not as pronounced as with silicone oil of 50 cST. Ringing frequencies are observed, and the usable bandwidth is increased compared to a system filled with 50 cST but yet smaller compared to the water-filled system. Even though not applicable with silicone tubing and a pressure transducer with a silicon membrane, it could render a potential solution for liquid-filled systems using metal tubing in which degassing of air is a problem.

4 Conclusion

A simple yet effective apparatus for characterizing and calibrating pressure sensors connected to water-filled tubes is presented. The bandwidth of a well-designed calibration vessel approaches approximately 300 ${\text {Hz}}$, which suffices for sensor-tube characterization relevant to most hydrodynamic measurements.

The dynamic response of cost-effective piezoresistive pressure sensors commonly used in hydrodynamic experiments is investigated. Sensor-dependent nonlinearities are revealed, presumably caused by the inability to entirely remove all air from the transducer’s cavity. The frequency response varies with the magnitude of the step input signal (i.e., the pressurization level) making measurement corrections challenging. Using tubing to bridge the distance from the pressure tap to the sensor reduces the usable flat frequency response range. The natural frequency of the system decreases as the length of the tube increases or the inner diameter decreases.

The wave propagation speed in water-filled tubes is greatly affected by the tube elasticity. In experiments using elastic silicone tubing with an inner diameter and wall thickness of 1 mm, a propagation speed of approximately 50 ${\textrm{m}}s^{-1}$ is measured. The propagation speed in elastic tubes is in agreement with the Moens–Korteweg equation. Although stiff stainless steel tubing increases the propagation speed, it is reduced by an order of magnitude compared to theoretical values. Possible reasons for this contradiction include the adapter pieces (i.e., short soft-PVC tube) used to couple the pressure sensor and tubing. Additionally, viscous effects not captured by the general equation derived by the Moens (1878) and Korteweg (1878) may become significant at small tube diameters. Furthermore, viscous effects in narrow tubes appear to cause a frequency dispersion that is rarely mentioned in the literature.

De-airing a sensor-tube system poses challenges, as entrapped air pockets often persist or develop over time due to temperature changes and degassing effects. Enclosed air in the tubing lowers the natural frequency of the system and introduces additional nonlinearities. Filling the entire tubing with silicone oil (50 cST) as an alternative to avoid degassing is not recommended because it significantly dampens the response, substantially reducing the usable frequency range.

The small, cost-effective, piezoresistive pressure transducers are suitable for dynamic pressure measurements but care must be taken in the transducer-system integration. Ideally, the sensor is wall-mounted to expand the range of flat frequency response. If tubing must be used, stiff and short tubing (i.e., stainless steel) should be utilized. The assumption that the pressure signal propagates with the full theoretical speed of sound is not applicable in narrow tubes and the real propagation speed is at least one order of magnitude smaller.

Based on the findings presented, a checklist crucial for pressure transducer integration is compiled to guide the setup of a pressure measurement system for underwater measurements:

General considerations:

• Install surface-mounted pressure transducers

• Consider using absolute pressure transducers

• Integrate pressure taps with inner diameter of 0.7 to 1 mm

• Keep the water temperature constant during the measurement campaign

Transmission line considerations:

• Use tubing if surface-mounted transducer integration cannot be facilitated

• Keep tubing as short as possible

• Use stiff tubing (i.e., stainless steel)

• Use tubing with an increased inner diameter (e.g., 2 to 4 mm)

• Use stiff connectors to mount transducer and transmission line

De-airing transducer cavity and transmission line:

• Fill the transducer cavity with water before submergence

• Use a syringe with a needle of sufficient length to de-air the transducer cavity

• Ensure that the syringe is free of any trapped air

• Thoroughly de-air the tubing underwater before mounting it to the transducer

• If the transducer-tube system moves with the model, align the longitudinal axis of the tubing and transducer membrane area vector perpendicular to the direction of motion

• Consider filling the transducer cavity with silicone oil (50 cST) but keep in mind the damping effects of the oil

• The water used for de-airing must be at the same temperature or warmer than the body of water into which the transducer-tube system is submerged