Pressure Field Estimation from 2D-PIV Measurements: A Case Study of Fish Suction-Feeding

Abstract

Particle image velocimetry (PIV) flow measurements are common practice in laboratory settings in a wide variety of fields involving fluid dynamics, including biology, physics, engineering, and medicine. Dynamic fluid pressure is a notoriously difficult property to measure non-intrusively, yet its variation is a driving flow force and critical to model correctly. Techniques have been developed to estimate the pressure from velocity and velocity gradient measurements. Here, we highlight a novel application of boundary conditions when applying such pressure estimation techniques based on two-dimensional PIV data; the novel method is especially relevant to problems with complex boundary conditions. As such, it is demonstrated with PIV measurements of in vivo fish suction-feeding, which represents a challenging flow environment. Suction-feeding is a common method for capturing prey by aquatic organisms. Suction-feeding is a complex fish–fluid interaction governed by various hydrodynamic forces and the dynamic behavior of the fish (motion and forces). This study focuses on estimating the pressure within the flow field surrounding the mouth of a Bluegill sunfish (Lepomis macrochirus) during suction-feeding utilizing two-dimensional PIV measurements. High-speed imaging was used for measurements of the fish kinematics (duration and amplitude). Through the Poisson equation, the pressure field is estimated from the PIV velocity measurements. The boundary conditions for the pressure field are determined from the integral momentum equation, separately for three phases of the suction-feeding cycle. We demonstrate the utility of the technique with this case study on fish suction-feeding by quantifying the pressure field that drives the flow towards the buccal cavity, a feeding mechanism known to be dominated by pressure spatial variations over the feeding cycle.

1. Introduction

The spatiotemporal evolution of the velocity field is directly related to the pressure field, which is both governed by the Navier–Stokes equations coupled with fluid properties such as density and viscosity. Thus, accurate modeling and parameterization of the fluid dynamics of a specific flow phenomenon requires knowing the pressure distribution within the flow domain. The pressure field in any given flow is especially useful when calculating forces and their effect on the flow field. Current pressure measurement techniques in fluid dynamics can be broadly categorized into two types: intrusive and non-intrusive methods. Intrusive methods involve direct contact with the fluid, typically through pressure transducers such as gauges, strain gauges, or load cells. While these instruments can offer high accuracy, their measurements are often confined to near-wall regions (e.g., boundary layers) or require inserting a probe into the flow, such as with Pitot tubes, which can disturb the flow and alter the results. In contrast, non-intrusive techniques estimate the pressure field indirectly using optical flow measurement methods, such as particle image velocimetry (PIV), to obtain velocity fields from which pressure can be inferred. These approaches enable visualization and analysis of pressure distribution throughout the flow domain without physical interference. However, their accuracy is sensitive to experimental noise and errors, especially with respect to the numerical differentiation and integration utilized when estimating pressure in a non-intrusive manner. The reliability of non-intrusive methods depends significantly on the quality of the measurements and accuracy of the derived velocities, as well as the robustness of the pressure reconstruction algorithm. In order to estimate the pressure field in a non-intrusive manner, the incompressible Navier–Stokes equation is as follows:

$$\rho {\displaystyle \frac{D\overrightarrow{V}}{Dt}}=-\nabla p+\rho \overrightarrow{g}+\mu {\nabla }^2\overrightarrow{V}$$

(1)

where $\rho$ is density, p is pressure, $\overrightarrow{g}$ is gravity, $\mu$ is viscosity, and $\overrightarrow{V}$ is the velocity. In Equation (1), $\rho {\displaystyle \frac{D\overrightarrow{V}}{Dt}}$ is the inertia force, $\nabla p$ is the pressure gradient or pressure force, $\rho \overrightarrow{g}$ is the body/gravitational force, and $\mu {\nabla }^2\overrightarrow{V}$ is the viscous force. There are two main approaches to solving the pressure field based on experimental data: (i) utilizing measured velocities in time and space to integrate the Navier–Stokes equations to solve for the pressure field, and (ii) utilizing the Poisson equation to solve for the pressure gradients and integrating them to obtain the pressure field. In both cases, a measured velocity field is used to estimate the other terms in the relevant equation (either Navier–Stokes or Poisson) in order to solve for the unknown pressure field; it also requires integration of the pressure gradients. In the first case, one is required to measure the spatial and temporal variations in the velocity, whilst in the second, it is restricted to quasi-steady state solutions. For both solutions, we are required to know the boundary conditions, and for the first case, the initial conditions (at t = 0), too. The first method provides a more accurate result as it does not assume steady-state; however, it requires the utilization of time-resolved PIV or using triple/quadruple cameras [1] to allow the instantaneous measurement of acceleration (i.e., direct measurement of the velocity gradient over time: $D\overline{V}/Dt$ [2]. Using 3D-PIV, such as Tomo-PIV, requires high-speed cameras to resolve the temporal terms and is relatively high-cost. One can extract the pressure field from 3D-PTV data directly. Unlike PIV, which is an Eulerian technique, PTV is a Lagrangian-based technique. Thus, direct calculation of the force (i.e., pressure gradient) along a trajectory path does not require mathematical manipulations. Using PTV, it is possible to estimate the forces (i.e., pressure) along the particle’s trajectory. However, transforming the pressure estimates onto a fixed grid introduces large errors due to interpolating the Lagrangian measurements. This step cannot be bypassed when spatial gradient calculations of the pressure are needed. For example, to calculate the force distribution or lift.

This manuscript focuses on pressure field estimation from 2D-PIV measurements in the context of flow–organism interactions, such as those generated during suction feeding in fishes. Because 2D-PIV measurements are widely accessible and commonly used, developing effective methods to extract pressure information from them is important for advancing studies of organismal motion. Here, we introduce a novel approach for defining boundary conditions when solving for the pressure field based on 2D-PIV data, specifically tailored to applications involving moving biological boundaries.

2. Materials and Methods

For 2D, assuming an incompressible quasi-steady state, ignoring the gravitational contribution due to the small volume, and by incorporating the continuity equation and applying divergence on the Navier–Stokes equation, the pressure can be expressed via the Poisson equation:

$${\nabla }^{2}p=-\rho \left\{{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2+2\left({\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial v}{\partial x}}\right)+{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2\right\}$$

(2)

where u is the streamwise (i.e., horizontal) velocity, v is the normal (i.e., vertical) velocity, x is the horizontal coordinate, and y is the vertical coordinate of a Cartesian system. The validity of the assumption of steady flow depends on the specific flow phenomena. To solve Equation (2) for the pressure, boundary conditions need to be determined. Commonly, two types of boundary conditions are applied: namely, Dirichlet (pressure) or Newman (pressure derivative) conditions that typically assume a solid body at the boundary. These boundary conditions are needed for both pressure field estimation approaches: Navier–Stokes or Poisson. When the body is in motion, the pressure and its derivatives are not constant. Assuming constant pressure can potentially generate a relatively large error. When estimating the pressure field during fluid–organism interaction, involving motion commonly associated with flexible bodies, we cannot use boundary conditions suitable for solid surfaces. Alternatively, we define the boundary conditions of an arbitrary control volume within the flow phenomenon. We choose the PIV field of view as the control volume domain. Because the PIV field of view is rectangular, we need to define the boundary conditions on its four sides. While this control volume (CV) is somewhat arbitrary, it is well defined by the 2D-PIV data. In order to account for the motion of the organism and its impact on the flow, we suggest calculating the pressure at the boundaries (i.e., boundary conditions) based on the integral momentum equation. This choice allows us to determine the pressure at every time step and at every point in the domain as a function of the surrounding fluid velocities, thus bypassing the need to define an ad hoc value for the pressure. The drawback lies within the measurement error of the specific PIV system [3]. The integral momentum equation for a non-inertial CV is as follows:

$${\displaystyle \frac{d}{dt}}\iiint \rho \overrightarrow{V} d𝒱+∯\rho \left(\overrightarrow{V}·\widehat{\mathcal{n}}\right)\overrightarrow{V} dA=∯-p\widehat{\mathcal{n}} dA+\iiint \rho \overrightarrow{g} d𝒱+\iint \stackrel{=}{\tau } dA$$

(3)

where $\rho \overrightarrow{V} d𝒱$ is the rate of change in momentum within the control volume, $\rho \left(\overrightarrow{V}·\widehat{\mathcal{n}}\right)\overrightarrow{V} dA$ is the momentum flux, $p\widehat{\mathcal{n}} dA$ is the pressure, $\rho \overrightarrow{g} d𝒱$ is the body force, and $\stackrel{=}{\tau } dA$ is the viscous force. This equation enables calculation of the pressure on the boundaries of the CV from the measured velocity field. Assuming Newtonian, incompressible, steady-state conditions with gravity in the vertical direction, the pressure in the x- and y-directions on the CV boundaries, for the x-direction, is calculated with the following equation:

$${\int }_1^2{p}_x dA= -\mu {\int }_1^2{\displaystyle \frac{\partial u}{\partial y}}dA+ \rho {\int }_1^2{u}^2 dA$$

(4)

and for the y-direction, the following equation is used:

$${\int }_1^2{p}_y dA= \rho g𝒱-\mu {\int }_1^2{\displaystyle \frac{\partial v}{\partial x}}dA+ \rho {\int }_1^2{v}^2 dA$$

(5)

where $𝒱$ is volume. The points of intersection at the corners of the boundaries provide cross-validation of the pressure calculation by comparing their values from the horizontal (x) and vertical (y) components, which should be the same; in all cases (see Section 2.1 and Section 3), the values at the corners matched when computed independently from x and y components, which provides a partial validation on the boundary condition methodology. The pressure is computed over all four boundaries of the PIV field of view in both the horizontal and vertical directions. Once the boundary conditions are determined, the Poisson equation is solved using an iterative method (see Gurka et al. [4] for further details on the numerical method).

2.1. Case Study

We exemplify the technique by applying it to observations of suction-feeding, which is a complex phenomenon that can demonstrate the robustness of this method. There are three basic techniques for fish to capture prey: manipulation, ramming, and suction [5]. Suction-feeding is the most common technique for prey capture and is used by the majority of bony fish (Osteichthyes) [6]. It is a versatile technique that enables the capture of a wide range of prey, from hard-shelled organisms to mobile, evasive prey [7]. This feeding mechanism involves quickly expanding the mouth cavity, which reduces the pressure inside the mouth compared to the surrounding water pressure. This lower pressure inside the mouth causes water to flow inward, drawing prey into the buccal cavity with fast-moving water [8,9]. This pressure difference yields forward acceleration of the water towards the fish’s mouth. Once the mouth closes again, the gills open to allow water to flow out, ensuring the capture of the prey [10]. Free-swimming prey are generally pushed away by the flow generated by the approaching predator. During suction, the fish attempts to eliminate the stagnation point at the snout tip through the suction phase, often combined with swimming or protrusion [11].

There are three dominant hydrodynamic forces acting on prey during suction-feeding: drag, pressure, and inertia [12]. The hydrodynamic forces are associated with the spatial and temporal gradients in flow velocities; these forces depend on the magnitude of flow variations and the volume of prey, regardless of its shape [13]. The success of suction-feeding was noted to rely directly on the rate of expansion and closure of the buccal cavity to form suction and enable the capture of prey [5,13,14,15,16]. Peak water velocities associated with suction-feeding have been found to occur shortly after the buccal cavity begins opening, and the mouth gradually decelerates the flow during closing [17,18,19]. Rapid opening of the mouth generates a rapid water flow in the direction of the fish, which coincides with a low-pressure zone within the mouth that draws the prey into the fish’s mouth [13,14,18]. Increased fluid speeds during suction-feeding relate to a drop in pressure inside the buccal cavity, which suggests that the force exerted on surrounding water is directly related to the magnitude of the buccal pressure [20]. In addition to these fluid–fish interactions, results from other studies have shown that parameters such as swimming speed, muscle function, and jaw morphology also contribute to prey capture efficiency [5,7,16,21]. The spatial distribution of pressure is important to understand how pressure variations evolve through the suction-feeding process, as they govern some of the hydrodynamic forces exerted during this action and may contribute to the success of the predator in capturing the prey and vice versa. Flow measurements were performed surrounding the exterior of the head during in vivo suction-feeding. We quantify the pressure field in the vicinity of the buccal cavity through analysis of velocity field measurements made using PIV.

Suction-feeding in fish can be considered as a dynamic system, with moving flexible solid boundaries at a relatively low Reynolds number. We use Bluegill sunfish (Lepomis macrochirus) to measure the pressure during suction-feeding. The Reynolds number is 1420 defined as $Re={\displaystyle \raisebox{1ex}{$u{L}_1$}\!\left/ \!\raisebox{-1ex}{$\upsilon $}\right.}$, where u is the ram speed (the fish speed in respect to the flow, which was estimated to be on the order of 10 mm/s), and L₁ is the characteristic length, defined as the distance from the snout to the base of the tail fin, respectively, of the swimming fish, and ν is the kinematic viscosity of the fluid. The Womersley number is 10.8, defined as ${\alpha }^2={\displaystyle \frac{\omega L_2^2}{\upsilon }}={\displaystyle \frac{\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$TTPG$}\right.\times {PG}^2}{\upsilon }}$, where ω is the angular frequency of the pulse, and L₂ a characteristic length associated with ω. Following Krishnan et al. [22], we consider suction as a single-pulse event, in which the angular frequency is the time it takes for the fish to fully open its mouth (time to peak gape, TTPG), $\omega =2\pi /TTPG$, and the characteristic length is the peak gape diameter (PG) [7].

2.1.1. Test Species

Five Bluegills were used in this study with lengths ranging from 15.2 to 17.8 cm. Bluegills were chosen for their history of positive results with similar experimental practices and ease of training [17]. Due to their abundance in the region, the fish were collected at Hackler Pond on the Coastal Carolina University campus. They were housed in a 1135 L aquaculture tank kept at a constant temperature of ~22 °C [19,20]. All fish maintenance and experimental procedures used in this research followed a protocol (#2022.02) approved by the IACUC (Institutional Animal Care and Use Committee) at Coastal Carolina University. During experimentation, each Bluegill was transferred to an experimental tank (208 L glass aquarium) for the flow experiments (see Section 2.1.3). It was equipped with a divider that sectioned the experimental tank into a holding section and experimental section, ensuring a repeatable position for the Bluegill to be reset while the bait, Canadian Nightcrawlers, were attached to a threaded rod in the experimental section (see Figure 1). This divider or trap door to the experimental section also required the fish to align its body perpendicular to the light sheet as it swam through the door towards the prey, which helped ensure the fish’s mouth was properly oriented with respect to the light sheet when suction feeding.

2.1.2. Kinematic Measurements

The suction-feeding mechanism is partially governed by the rate of capture (i.e., time between opening and closing); therefore, both the flow field characteristics and buccal cavity kinematics of the specimen were measured. High-speed imaging enables the measurement of the speed, acceleration, and duration of the exterior of the fish’s head during suction-feeding. A Photron SA3 high-speed camera (Photron Limited, Tokyo, Japan) was used with a 50 mm Nikon lens that operated at 1000 frames per second (fps) at a resolution of 1024 × 1024 pixels. As the Bluegill swam through the opening in the experimental tank, as shown in Figure 1, a high-speed video was recorded. The process was repeated for all five fish multiple times. The video images were processed using Kinovea software (2025.1.1) [23] to calculate the average peak gape, velocity of gape opening, and duration of the suction-feeding process for all individual subjects. These parameters were computed by tracking the motion of six points placed on the mouth (three on the upper jaw and three on the lower jaw) (see Figure S1 in Supplementary Materials). The average peak gape (h) was found to be 15.7 mm. Using the distance traveled over the time period, the average speed of gape opening was 778 mm/s, and the average duration of the suction-feeding cycle was 0.065 s.

2.1.3. Flow Measurements

PIV is a non-intrusive optical flow measurement technique [3]. PIV experiments were performed in the experimental tank with a CCD camera, laser, synchronizer, data acquisition system, and optics. The CCD camera (TSI Inc., Shoreview, MN, USA) was an 8 Megapixel double-exposure camera equipped with an 80 mm Nikon lens (Nikon, Tokyo, Japan) and was positioned approximately 1.6 cm away from the tank perpendicular to the tank walls, as shown in Figure 1. The CCD was oriented at either 80° or 90° relative to the light sheet to better visualize the suction-feeding process. At 80°, this angular view required a cos(10°) correction to the projected velocity components, which was performed during the post-analysis. Hollow glass spheres, ~10 microns in diameter and with a specific gravity of 1.05, were used as tracer particles. The laser was a dual-head Nd:YAG Quantell Evergreen (Lumibird Photonics, Bozeman, MT, USA) with 200 mJ/pulse, and the optics included a 45° mirror, a spherical lens, and a cylindrical lens that were used to form a ~1 mm light sheet through the center of the experimental section of the tank that intersected with the threaded rod holding the bait ([24]; Figure 1). This rod setup ensured that when the fish approached the bait and performed suction-feeding, the light sheet was aligned with the fish’s mouth. The synchronizer allowed the laser and camera to illuminate and capture images simultaneously in quick succession to provide image pairs taken with a 10 ms time gap. Data acquisition at 8 Hz began when the divider in the experimental tank was removed, allowing the fish to swim through an opening between the holding and the experimental sections of the tank. Data acquisition continued until the suction feeding process was complete. Because the feeding cycle is 0.065 s (~15 Hz) on average and the acquisition of image pairs occurs at a much slower frequency of 8 Hz, multiple repetitions of the experiments were needed to obtain a small set of quality image pairs to cover the feeding cycle. The experiments resulted in 36 image pairs during the feeding cycle that could be used for further analysis. The images were analyzed using interrogation windows of 64 × 64 pixel² with a magnification ranging from 44 to 65 pixels per mm. On average, the analyzed maps each contained ~6000 flow velocity vectors. The magnitude of the average pixel displacement was 2.30 pixels in the x-direction and 1.25 pixels in the y-direction. The pixel displacements were filtered using a 3-standard-deviation global validation filter and a local 5 × 5 median validation filter [25]. The result of this post-analysis was ensembles of 2D velocity vector maps of two components of the velocity field (horizontal and vertical). Subsequently, the velocity maps were sorted based on the acquired images into the following phases: (1) the fish’s mouth is closed, prior to prey capture; (2) the fish’s mouth is open, during prey capture; (3) the fish’s mouth has closed and is retreating.

There are several sources of uncertainty when estimating pressure from PIV data. The accuracy depends on the numerical scheme, the quality of the PIV images, the PIV correlation analysis, and the experimental uncertainties. The estimated error for the PIV velocity measurements is about 2% [3]. The error associated with the numerical scheme used to compute the pressure field was estimated by Gurka et al. [4] to be about 2%, assuming a sufficient sample size, which is not fully achieved here due to the limitation of working with live organisms. The error resulting from the fish motion, and associated misalignments, is challenging to quantify. However, because the fish motion manifests itself in the PIV measurements, these experimental uncertainties are largely accounted for in the error associated with the PIV velocity measurements, which accounts for the correlation analysis, outliers, sub-pixel interpolation, large gradients, and 3D effects [26].

3. Results and Discussion

The results of applying the boundary condition estimation methodology on the 2D PIV measurements performed in the case study of fish suction-feeding are presented in this section. First, the flow field measurements are shown, followed by estimation of the pressure field during the suction feeding process, as divided into the three aforementioned phases.

3.1. Flow Field

An example mosaic of the suction-feeding cycle assembled from instantaneous velocity maps for a single fish is depicted in Figure 2. The fish approaches the prey in Figure 2A, captures the prey in Figure 2B, and retreats in Figure 2C. The flow induced in the tank is only the result of the fish’s motion, as the water is initially quiescent. The flow patterns vary with the phase of suction-feeding. As the Bluegill swims towards the prey, the flow follows the direction of the fish’s movement. During the capturing of the prey, the velocity vectors point towards the mouth of the Bluegill in the region surrounding its mouth. Once the prey is captured and the fish is moving out of the field of view, the flow is pointed in the direction of the fish’s movement due to the residual water circulation.

We examine the velocity vertical profiles for each phase relatively close to and far from the mouth. A sample of the instantaneous horizontal (u) and vertical (v) velocities averaged spatially over the x-direction within the rectangular region shown in Figure 2, representing the flow close (blue) and far (white) from the buccal cavity, is shown in Figure 3. During phase 1 (Figure 3A), as the fish approaches the prey, there is almost no vertical variation in the horizontal velocity profile. The slight non-uniformity in the horizontal velocity profile is due to the presence of the fish displacing the fluid as it swims towards the prey. A similar trend is observed for the vertical velocity profile. During phase 2 (Figure 3B), there is a strong gradient in the horizontal velocity in the vertical profile close to the mouth as a result of the fish’s rapid approach and its buccal opening towards the prey; these velocity gradients are associated with pressure variations. The velocity has also changed direction and is directed toward the fish. Far from the mouth, the flow stays relatively uniform. In the vertical velocity profiles, there are similar trends, with even larger gradients portraying the upward movement into the mouth. During phase 3 (Figure 3C), near the mouth, the horizontal velocity exhibits large gradients while the vertical velocity shows only small vertical gradients, likely due to the influence of the free surface and the fish’s motion during prey capture. Closer to the mouth, both horizontal and vertical velocities appear to be gradually returning toward a more uniform profile.

3.2. Pressure Field

The estimated pressure gradients in the horizontal and vertical directions for each phase of the suction-feeding process for the mosaic series demonstrated in Figure 2 are shown in Figure 4A–F. Recall that in order to estimate the pressure field from the PIV data, we have followed the same approach as in Gurka et al. [4] while defining the boundary conditions differently. The boundary conditions (see Equations (3)–(5)) were applied to the 2D-PIV data to solve for the pressure field (see Section 2). Figure 4G–I shows an example of the instantaneous pressure field distribution for the single fish mosaic shown in Figure 2.

Figure 4A–I demonstrate how the pressure gradient varies over the suction-feeding cycle. During phase 1, the pressure gradients are near-zero (Figure 4A,D), which is consistent with quiescent flow and a stagnation point at the mouth as the Bluegill is approaching the prey with its mouth closed. However, during phase 2, localized regions of strong positive and negative gradients are formed (Figure 4B,E). The development of large gradients is indicative of suction towards the mouth. During the final phase, the pressure gradient field becomes fragmented but still larger than phase 1 as the flow field recovers from the disturbance (Figure 4C,F). Figure 4G–I demonstrates how the pressure field evolves during the suction-feeding cycle. Consistent with the pressure gradient distributions, during phase 1, the pressure field is relatively uniform (Figure 4G). Phase 2, in which the suction-feeding is occurring, causes the formation of significant spatial variations in the pressure field near and surrounding the mouth (Figure 4H). A high-pressure region surrounding the mouth is observed, consistent with the larger pressure gradients during phase 2. Phase 3 reflects remnants of the flow from the suction feeding process as it decays back to quiescent conditions. The entire flow around the buccal is dominated by this strong momentum (i.e., velocity of flow due to suction), causing widespread areas of lowered fluid pressure and spatial variability (Figure 4I). It is noteworthy to indicate that the pressure gradient in the vertical direction is large compared to the pressure variations in the horizontal direction due to hydrostatic pressure variations that are significant in water. In order to quantify the pressure field during suction-feeding, for each of the 36 pressure fields estimated, we calculate the average pressure surrounding the mouth and the maximum pressure in the same area for each of the three phases, where the average is taken over 10.9 mm in the horizontal direction and 10.2 mm in the vertical direction directly in front of the buccal cavity. These results are shown in Figure 5.

Each point in Figure 5 represents one of the 36 instantaneous pressure fields clustered based on phase, while each color represents one of the five different fish. The average pressure during phase 1 is generally smaller (closer to zero) compared to the other two phases, considering all the fish collectively, and the maximum pressure is also smaller, both of which reflect the baseline condition of the water prior to suction-feeding. During phase 2, when the fish’s mouth is open and actively drawing the prey into the mouth, the pressure magnitude in this area is at its largest. It is also apparent that different fish induce different suction forces on the flow, as expected, but pressure magnitude on-average is 2–3 times higher in phase 2. Assuming that during phase 1 the state of the flow in the fish’s vicinity is mostly quiescent, and the pressure results solely from the fish motion and the hydrostatic pressure, one can infer that the fish is capable of forming a relatively large pressure gradient in the flow (two times higher compared to the surrounding fluid) that is sufficient to draw prey into the buccal cavity solely by the rapid movement of the buccal cavity and its acceleration [5,13]. The three phases show differing distributions of maximum pressures, where phase 1 has a cluster of points near zero while phase 2 shows a wider variety of values. Pressure for the same fish reduces between phases 2 and 3, indicative of phase 3 being a flow decay state, which shows more point clusters than phase 2. Figure 5A,B also show that the pressure is generally larger in magnitude in phase 3 than in phase 1, consistent with residual pressure changes from the suction feeding and its return to flow conditions from phase 1. Collectively, these results provide important insights into the magnitude of pressure fields during the suction-feeding cycle, demonstrating the effectiveness of the method used to estimate the pressure field.

4. Conclusions

This study presented a new technique to estimate boundary conditions for pressure field computations from 2D-PIV flow measurements in complex flow fields with moving boundaries. The technique was demonstrated by computing the pressure field around the buccal cavity of a Bluegill during suction-feeding. To calculate the pressure field, the Poisson formulation of the Navier–Stokes equations was applied to 2D-PIV data, assuming quasi-steady-state, incompressible flow. To solve the Poisson equation, an iterative method was used, where the boundary conditions were defined through the integral momentum equation applied along the boundaries of the imaged flow field. The spatial distribution of the pressure field was phase-averaged for three phases within the suction-feeding cycle. The motion of the buccal cavity governed the pressure field, where large pressure gradients were formed during suction-feeding, causing an overall increase in the pressure of the flow field surrounding the buccal cavity. The pressure increased by approximately a factor of 2–3 during the feeding process. This test demonstrated the feasibility of estimating the pressure field using 2D-PIV flow measurements in a complex, biologically dynamic system with moving boundaries. This approach addresses the challenge highlighted in the introduction of estimating pressure in flows involving moving organisms, providing an alternative method to quantify pressure distributions without intrusive measurements. Further validation of the method in other similarly complex flows is needed, as well as application of the technique to time-resolved 3D-PIV data.