## 1 Introduction  
Flow rate sensors are commonly employed in hydraulic engineering since knowing the flow rate is critical for detecting leaks and monitoring the wear of system components in both stationary and mobile hydraulic systems. Additionally, in the emerging field of predictive maintenance, knowing flow rates and pressure levels allows for calculating power, efficiency, and potential losses. There are two main issues with current volumetric flow sensors: First, most must be installed directly into a pipe, often using turbines or rotors, which disrupt the flow and increase system complexity. Second, the physical components of these sensors possess inertia, making them ineffective for measuring flows above a particular frequency. Pump pulsation measurements, for instance, become problematic because they rely on flow rate sensors. The number of displacement units and the pump's rotational speed influence the fundamental frequency of pump pulsations. Pumps also produce pulsations at harmonic frequencies, many of which exceed the capabilities of most state-of-the-art sensors. These limitations highlight the need for new flow measurement methods. An ideal flow sensor would be non-invasive and accurately measure transient flow conditions. Ideally, it would operate as a soft sensor, relying solely on signals from other sensors, particularly pressure signals, often available in fluid power systems, and are non-invasive. The concept of calculating flow rates from pressure data is not new. Analytical methods for determining volumetric flow rates have been available since the 1900s. One of the most well-known formulas is the Hagen-Poiseuille law (HP) [1], which calculates flow rates based on the friction-induced pressure drop within a pipe. However, this law is limited to steady, laminar flows and neglects the effects that become significant at higher frequencies. The so-called [PERSON] effect [2] alters the velocity profile at higher frequencies. Another significant contribution is by [PERSON] et al., who created a method for measuring unsteady flow rates and velocities based on pressure differences, expressed through the transfer function between the mean flow velocity and the pressure gradient [3]. Hydraulic systems are often modeled analogously to electrical circuits, using resistances, inductors, and capacitances [4]. This approach is typically accurate and straightforward for many applications. However, at higher frequencies, the system behavior changes due to transient effects. For instance, the characteristic impedance, propagation operators, and wall shear stress within pipes have all been shown to depend on frequency [5]. Recent advances in transient flow rate determination include work by [PERSON] et al., who presented a method for arbitrary transients in laminar pipe flow, starting from an initially steady state. This method relates the flow rate to the pressure-gradient history without making assumptions about velocity profiles [6]. In 2008, [PERSON] et al. introduced an indirect method that links flow rate to the centerline velocity history [7]. [PERSON] et al. [8] enhanced frictional modeling, reducing errors in flow rate calculations for the pressure-time method, a technique commonly used to measure discharge in hydraulic machinery. [PERSON] et al. proposed a method in 2019 for time-resolved transient flow rate calculation based on differential pressure measurements [9]. Their approach utilized the kinetic energy to predict specific laminar flow conditions well in real-time, relying only on two coefficients. [PERSON] et al. focused on unsteady turbulent pipe flow, validating their method through the transient response of the velocity field due to external perturbations, using a pressure step function as a test case [10]. Further investigations into turbulence were conducted by [PERSON] et al. in 2022, studying the transition from turbulent to laminar flow without an increase in bulk velocity. The observed laminarization was explained using a derived mathematical model [11]. [PERSON] et al.