# An open 3D CFD model for the investigation of flow environments experienced by freshwater fish

Ali Hassan Khan, Karla Ruiz-Hussmann, Dennis Powalla, Stefan Hoerner,

Maarja Kruusmaa, Jeffrey A Tuhtan

► To cite this version:

Ali Hassan Khan, Karla Ruiz-Hussmann, Dennis Powalla, Stefan Hoerner, Maarja Kruusmaa, et al. An open 3D CFD model for the investigation of flow environments experienced by freshwater fish. Ecological Informatics, 2022, 69, pp.101652. 10.1016/j.ecoinf.2022.101652. hal-04206503

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https://hal.science/hal-04206503v1

Submitted on 13 Sep 2023

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# An open 3D CFD model for the investigation of flow environments experienced by freshwater fish

Ali Hassan Khan<sup>a, *</sup>, Karla Ruiz Hussmann<sup>b</sup>, Dennis Powalla<sup>b</sup>, Stefan Hoerner<sup>b</sup>,
Maarja Kruusmaa<sup>a</sup>, Jeffrey A. Tuhtan<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Freshwater ecosystems

Fish

CFD

Numerical modelling

Model validation

## ABSTRACT

Computational fluid dynamics (CFD) provides a powerful numerical tool to simulate and study many of the complex fluid-body interactions experienced by freshwater fish. However, major gaps remain in the application of CFD to study the fluid-body interactions of fish, including the absence of an openly available reference body geometry, the lack of a detailed study on suitable numerical methods and a deficit of available velocity laboratory measurements for model calibration and validation. To address these gaps, we provide a set of numerical models based on the open-source CFD toolkit OpenFOAM. The contributions of this work are two-fold: First, to provide a validated openly available numerical setup using a realistic fish model geometry including laboratory velocity measurements. Second, to determine the best-performing turbulence models and near-wall treatments using Reynolds-Averaged Navier-Stokes (RANS) numerical simulations. Finally, we conclude with a critical evaluation of the effects and trade-offs of resolving or modelling the boundary layer (BL) in numerical studies of fish-shaped bodies.

## 1. Introduction

Freshwater ecosystems host one-third of all vertebrate species, and are experiencing a prolonged and rapid decline (Reid et al., 2019). The negatively compounding impacts of climatic and anthropogenic change are reducing freshwater vertebrate populations at more than twice the rate of terrestrial or marine populations (Tickner et al., 2020). To mitigate these negative impacts, advanced integrated modelling approaches are needed. Such methods have shown promising preliminary results when exploring physical hydrological processes and their ecological and socioeconomic interactions (Li et al., 2021). Currently, large-scale physical river processes can be simulated due to advances in computational power, improved algorithms and the rapid growth of high-quality remote sensing data to calibrate and validate numerical simulations (Monegaglia et al., 2018; Vanzo et al., 2021). However, riverine fish habitats naturally encompass a broad spectrum of physical flow conditions (Belletti et al., 2017). Furthermore, the sensory ecology of freshwater fish remains at a nascent stage, where major gaps persist in our knowledge of the hydrodynamic stimuli present in natural flows

(Mogdans, 2019). This is expressed as a persistent lack of established thresholds for abiotic autecological parameters associated with lotic fish habitats, including the flow velocity and water depth (Smialek et al., 2019). Methods to simulate fish-flow interactions under realistic conditions are therefore urgently needed to study, understand, predict and support sustainable freshwater fish populations.

Fish sense the surrounding flow field with their highly specialized lateral line sensory system (Dijkgraaf, 1963). This "touch at a distance" sensing modality allows fish to perceive minute changes in the pressure, velocity and acceleration fields (van Netten and McHenry, 2014). It allows them to orient themselves into the flow, gain information about their spatial environment and plays vital roles in feeding, spawning, migration and predator avoidance (Mogdans, 2019). Measurements and simulations of fish-flow interaction have been used to predict physical habitats (García-Vega et al., 2021) and have the potential to update and improve large-scale fish community distribution models (Cyterski et al., 2020). A fish's highly streamlined body shape provides minimal resistance while swimming (Lucas et al., 2020) and efficiently modulates the detection of hydrodynamic stimuli in the underwater environment

<footnote><sup>a</sup> Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia</footnote>

<footnote><sup>b</sup> Institute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke University Magdeburg, Germany</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: ali.khan@taltech.ee (A.H. Khan), karla.ruiz@ovgu.de (K.R. Hussmann), dennis.powalla@ovgu.de (D. Powalla), stefan.hoerner@ovgu.de (S. Hoerner), maarja.kruusmaa@taltech.ee (M. Kruusmaa), jeffrey.tuhtan@taltech.ee (J.A. Tuhtan).</footnote>
<table><caption>**Table 1.**
Dimensions and specifications of the equipment used during physical experiments, including the fish-shaped body, laboratory flume and laser Doppler anemometer.</caption><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>Dimensions of the fish-shaped body (length x width x height)</td><td>366.5 × 49.2 × 75.5 mm</td></tr><tr><td>Specifications of the laboratory flume</td><td></td></tr><tr><td>Dimensions (length(x) x width(y))</td><td>10.0 × 1.2 m</td></tr><tr><td>Water level</td><td>0.68 m</td></tr><tr><td>Mean velocity</td><td>0.5382 m/s ± 0.0116 m/s</td></tr><tr><td>Mean turbulence intensity</td><td>3.1646% ± 0.1798%</td></tr><tr><td>Reynolds number</td><td>680,000</td></tr><tr><td>Froude number</td><td>0.21</td></tr><tr><td>Specifications of the laser Doppler anemometer</td><td></td></tr><tr><td>Model</td><td>Dantec Flow Explorer DPSS<br>300 2D</td></tr><tr><td>Laser type</td><td>Continuous laser</td></tr><tr><td>Wavelength</td><td>Horizontal laser: 532 nm<br>Vertical laser: 561 nm</td></tr><tr><td>Nominal measurement distance</td><td>485 mm with a 500 mm front lens</td></tr><tr><td>Measuring volume (length(x) x width(y) x height(z))</td><td>0.14 × 0.14 × 2.30mm</td></tr><tr><td>Software</td><td>BSA Flow Software</td></tr></tbody></table>

(Oteiza et al., 2017). This makes the study of fish-flow interactions a topic of increased interest beyond applied sensory ecology. Indeed, recent technological advances in miniature underwater sensing now include artificial lateral lines (Kottapalli et al., 2014), which can be used for field measurements in rivers and fish passage structures (Tuhtan et al., 2016) as well as in underwater robotics (Liu et al., 2020).

Previous works have shown that fish-flow interactions can be evaluated using computational fluid dynamics (CFD), focusing predominantly on fish swimming kinematics, thrust, drag and the development of complex vortex structures in the wake region (Adkins and Yan, 2006; Macia et al., 2020; Owsianowski and Kesel, 2008). Considering fish sensing, a limited number of studies have investigated the velocity and pressure fields around fish-shaped bodies. A notable contribution to fish sensing using CFD was the pioneering work of Windsor et al. (2010), which applied CFD to study the sensing range of blind Mexican cave fish, estimated as 0.2 fish body lengths. This was done by correlating observations of live fish swimming behavior to the flow field experienced by a CFD simulation of a revolved NACA 0013 profile. A comparison of two-dimensional (2D) and three-dimensional (3D) body geometries by Rapo et al. (2009) showed that 2D CFD simulations were not capable of accurately representing the perturbing effects of the fish-shaped body from a dipole stimulus. This study simulated a flat plate, and highlighted the importance of the numerical treatment used in the boundary layer,

and its impacts on the simulation results of the near-body velocities and strain rates. Herzog et al. (2017) carried out CFD simulations of the flow velocity and pressure fields from a small vibrating sphere interacting with a high-definition 3D scan of the cephalic lateral line from a common carp, (Leuciscus idus). The authors of this study recommended that future studies resolve the boundary layer to ensure realistic simulations of the low velocities common to the near-body flow field.

Despite the existing studies using CFD for fish sensing research, substantial gaps remain for the generalized investigation of the fluid-body interactions fish experience outside of the laboratory. Specifically, the absence of an openly available reference numerical model hinders the cross-comparison of numerical studies on fish sensing. This impedes urgently needed improvements to the attraction flows used in fish migration structures (Schütz et al., 2021), which can be investigated with large-scale CFD simulations (Gisen et al., 2017). The significant contribution of this work is the open CFD benchmark study it provides. Future researchers can either make direct use of the model itself, or base new CFD simulations on the best practices established in this work to improve our capacity to represent, understand and predict the flow-body interactions experienced by freshwater fish species.

A critical evaluation of suitable turbulence models and associated boundary layer treatments also remains absent from existing numerical studies of fish-flow interactions, and must be evaluated to ensure robust and repeatable CFD simulation results. This work is the first to assess the effects of boundary layer and turbulence modelling on the stream-wise and lateral velocity components around a fish-shaped body using the Reynolds-averaged Navier-Stokes (RANS) approach. Furthermore, we provide a critical comparison of three different RANS turbulence models as well as modelled and resolved boundary layers. All numerical simulations were carried out on a static fish-shaped body in a rectangular flume, similar to the gliding phase of a freely swimming fish. Planar velocity measurements using a Laser Doppler Anemometer (LDA) were obtained with a physical model situated in a open channel laboratory flume. The LDA measurements were used for numerical model calibration and compared for each turbulence model, considering either modelled or resolved boundary layers.

## 2. Experimental setup

The laboratory setup was designed for 2D LDA measurements of the flow field around a rigid 3D printed model of a brown trout swimming during the gliding phase of a swimming gait cycle. A summary of the physical model setup and equipment is provided in Table 1. The fish-shaped body geometry is based on the brown trout, (Salmo trutta), a

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Overview of the numerical and the experimental domains; (Left) fish-shaped body, spatial distribution of measurement points around the physical model and at the upstream boundary of the CFD model. (Right) Inlet velocity and turbulence distributions at inlet patch (red rectangle in the left panel) measured with LDA and later applied as a mapped inlet condition in the numerical model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_5.png" /></figure>
<table><caption>**Table 2.**
Summary of the OpenFOAM models investigated in this work.</caption><thead><tr><th>Numerical solver version</th><th>OpenFOAM-v2012</th></tr></thead><tbody><tr><td><strong>Solver characteristics</strong></td><td>Incompressible, transient</td></tr><tr><td><strong>Solver algorithm</strong></td><td>SIMPLE</td></tr><tr><td><strong>Temporal discretization</strong></td><td></td></tr><tr><td>Accuracy</td><td>First order</td></tr><tr><td>Time step</td><td>10<sup>-03</sup> (adjustable)</td></tr><tr><td><strong>Spatial discretization*</strong></td><td></td></tr><tr><td>Mesh type</td><td>Hexahedral</td></tr><tr><td>Max cell size</td><td>0.0125 m</td></tr><tr><td>Min cell size</td><td>0.001 m</td></tr><tr><td>Total number of cells</td><td>N<sub>flne</sub> = 4.2 M</td></tr><tr><td><strong>Turbulence models</strong></td><td>k &minus; &epsilon;, k &minus; &omega; SST, Spalart Allmaras</td></tr><tr><td><strong>Wall treatment</strong></td><td>Wall functions (unresolved)<br>Calculated (resolved)</td></tr><tr><td><strong>Convergence criteria</strong></td><td></td></tr><tr><td>Residuals</td><td>U = 10<sup>-04</sup> P = 10<sup>-05</sup></td></tr><tr><td>Relaxation factors</td><td>U = 0.7, P = 0.6</td></tr><tr><td><strong>Simulation time</strong></td><td>&asymp; 384 CPUh</td></tr></tbody></table>

* corresponding to the final mesh grid chosen after completing the grid sensitivity analysis.

common freshwater rheophile fish species. A detailed overview of the body geometry is provided in a separate work on the development of a bio-inspired robotic fish (Abbaszadeh et al., 2021) which has an identical geometry to the fish-shaped body investigated in this work. All physical experiments were conducted in the open-channel laboratory flume at the Otto-von-Guericke University (OVGU) in Magdeburg, Germany. The flume is 10 m long, 1.2 m wide and the mean water depth for all experiments was fixed at 0.68 m. The LDA system was placed underneath the flume, with optical access from the bottom to a 0.6 m long, 0.53 m wide and 0.26 m high measurement volume. All measurements were taken with a mean streamwise velocity of 0.5382 m/s, corresponding to a fully turbulent flow with a Reynolds number of $Re = 6.8 \times 10^5$ after Eq. 1:

$$Re = \frac{\rho \cdot U \cdot d_h}{\mu} \qquad (1)$$

where $\rho$ is the density of water, $U$ is the streamwise mean velocity, $d_h$ is the hydraulic diameter of the flume and $\mu$ the dynamic viscosity. The hydraulic diameter was calculated for open channel flows (Eq. 2), following Surek and Stempin (2017), where $b$ is the flume width and $h$ is the mean water depth:

$$d_h = \frac{4 \cdot b \cdot h}{b + 2 \cdot h} \qquad (2)$$

The Froude number was calculated as $Fr = 0.21$, (Eq. 3), assuming a gravitational constant of $g=9.81$ m/s², indicating that the flow is subcritical, and near-surface perturbations in the flume will therefore propagate in the upstream direction.

$$Fr = \frac{U}{\sqrt{g \cdot h}} \qquad (3)$$

The LDA velocity point measurements were carried out at 253 locations with increasing sample density in the vicinity of the body of the fish (Fig. 1). Measurement locations included the undisturbed section upstream of the fish-shaped body, as well as the wake region immediately after the tail fin. In the vicinity of the fish-shaped body the closest measurement probes were located at a distance of 3 mm, and at the tail 2.5 mm from the body surface. At each location, the planar 2D velocity ($U_x$ and $U_y$) was measured, and the turbulence intensity was calculated. The measurement were grouped into the head, body and tail regions of interest, to compare the performance of the different wall treatments and turbulence models. In order to ensure reproducibility over multiple days of measurement, a grid of 18 points in the upstream section was measured at the beginning of each experiment. The mean streamwise

velocity of the laboratory flume was 0.538 m/s, resulting in a standard deviation of 0.0116 m/s. At each location, 2000 ($U_y$) to 10,000 ($U_x$) measurements were recorded over a maximum time interval of 200 s. The LDA data was post-processed and stored with the commercial software BSA Flow using a Dantec Flow Explorer DPSS 300 2D, which allowed for the acquisition of raw data at a single measurement location. The same equations applied by the commercial software were used to compare the results from each of the numerical simulation setups. The software calculates the signal quality for each measurement based on the signal to noise ratio (SNR), which was used to qualitatively assess all LDA measurements before comparison with the CFD models. The lowest SNR value of 70.1% was obtained for LDA points close to the surface. Additional information regarding the LDA velocities, turbulence intensity and data validity are provided in the supplementary material, and the LDA measurements are included as part of this work's open data repository.

The upstream flow boundary immediately preceding the fish-shaped body used a mapped inlet condition and is indicated with a red outline in the left panel of Fig. 1. This was due to the non-symmetric velocity and turbulence intensity distributions within the laboratory flume, as illustrated in the right panels of Fig. 1. The planar LDA measurement point locations also served as the reference coordinates for the CFD probes, which retrieved the simulated model velocity at the closest cell to the given LDA measurement location.

### 3. Numerical model

Numerical modelling was performed using the OpenFOAM toolkit, an open source collection of C++ libraries and applications for continuum mechanics and multi-physics simulations (Jasak and Uroić, 2020). OpenFOAM was chosen because it is well-established in both the academic and commercial CFD communities, and provides a freely available model setup for ongoing and future studies. An overview of the numerical setup established in OpenFOAM environment for current case study is illustrated in Table 2, stating the range and selection of parameters. For the current case study, RANS modelling was preferred due to the trade-off between robustness and computational efficiency, where the turbulence is modelled, in contrast to more advanced approaches which partially or fully resolve turbulence effects, such as Large-Eddy Simulations (LES) or Direct Numerical Simulations (DNS).

A single-phase simulation was selected as the planar velocity measurements can be assumed to be largely unaffected by perturbations induced by the free surface at the LDA measurement locations. This assumption was further validated after comparison with the LDA point measurements, and are presented in the results section of this work. A segregated, single-phase algorithm (pimpleFoam) was used to solve the general momentum equation for incompressible, unsteady flow. It implements both the Pressure-Implicit-Split-Operator (PISO) and Semi-Implicit-Method-Of-Pressure Linked Equations (SIMPLE) algorithms for the treatment of the velocity and pressure fields. An advantage of using the SIMPLE algorithm is that it can be applied to simulations with high Courant-Friedrichs-Lewy (CFL) numbers (Deng and Tang, 2002). Due to the complexity of the problem, the CFL number in this study was set to a maximum of 20. All model setups evaluated in this work, required the use of an adjustable time step to preserve the CFL number under unsteady flow conditions. Within each time step, multiple outer loops of iterations were run for the pressure-momentum correction, with a residual tolerance of $10^{-05}$ for pressure and $10^{-04}$ for the velocity.

#### 3.1. Geometry and mesh

The simulation domain was uniformly discretized along the x (streamwise), y (lateral) and z (vertical) directions. And the volume of domain was constrained to $1850 \times 800 \times 600$ mm³, to reduce the computational cost to the greatest extent feasible (Fig. 1). The model domain was defined using a hybrid mesh using two separate open source
<table><caption>**Table 3.**
Overview of the numerical model setups investigated in this work as combinations of near-wall treatments and RANS turbulent models.</caption><thead><tr><th rowspan="2">Near-wall<br>treatment</th><th colspan="3">RANS turbulence model</th></tr><tr><th>Standardk - ε</th><th>k - ω SST</th><th>Spalart<br>Allmaras</th></tr></thead><tbody><tr><td>Boundary Layer<br>Unresolved</td><td>BL modelled<br>Wall treatment<br>through<br>wall functions<br>(30&lt;y+&lt;300)</td><td>BL modelled<br>Wall treatment<br>through<br>wall functions<br>(30&lt;y+&lt;300)</td><td>-</td></tr><tr><td>Boundary Layer<br>Resolved</td><td>BL fully resolved<br>(y+&lt;1)</td><td>BL fully resolved<br>(y+&lt;1)</td><td>BL fully<br>resolved (y+&lt;1)</td></tr></tbody></table>

meshing tools. The OpenFOAM utility *blockMesh* was first applied to generate the flume domain, with the exception of a rectangular block cavity which contained the fish geometry. This first mesh also encompasses all of the domain's outer boundaries. The *blockMesh* utility allows for the control of the grid with a block-oriented, structured meshing approach using hexahedral elements, but it is generally unsuitable for complex geometries (Greenshields, 2021). The second mesh region contained the fish-shaped body geometry, and was generated with the open source version of *cfMesh* (Juretić, 2015). The body geometry was imported as an .stl file in a Cartesian hexahedral mesh and then recursively refined into a structured mesh around fish body surface, including the boundary mesh layers. Three near-body mesh layers were added around fish body, starting from the first layer having a thickness of 1 mm and an expansion ratio of 1.5. An advantage of this tool is the improved handling of complex geometries. In a final meshing step, both meshes were merged and stitched together. For more complex geometries, e.g. more detailed shapes of real fish with fins, a tetrahedral mesh can be generated in the cavity and merged with the structured block oriented hexahedral mesh generated with *blockMesh*. An advantage of our approach is that it uses only open source mesh generation tools which are largely automated, making it especially suitable for future fish-shaped bodies with different geometries.

## 3.2. Boundary conditions and initial values

In the numerical simulations carried out in this work, all walls were considered with a slip boundary condition, except the fish body, which was assigned a no-slip boundary condition. The effects of the free surface were therefore neglected in this work. Accounting for the free surface would require computationally expensive multi-phase calculations without a significant change in the flow field around the fish-shaped body. Even a simplified but computationally efficient volume-of-fluid model would require the calculation of additional terms for the air phase, and at a finer spatial discretization of the phase interface at the free surface. Furthermore, the temporal discretization required to satisfy

the condition that CFL < 1 would increase substantially to ensure numerical stability. Considering a trade-off between computational cost and the need for only near-body velocity and pressure fields, a single phase flow model was employed in this study.

The inlet boundary included an outer sub-patch, which was assigned a uniform velocity of 0.55 m/s, as well as an inner sub-patch, which was mapped using the LDA measurements (Fig. 1). The mapping was made using a second order polynomial derived from the LDA velocity vectors $U_x$ and $U_y$, interpolated as the inner sub-patch using the expressions-based boundary conditions utility in OpenFOAM-v2012. It is important to note that $U_z$ was not included in our study, and was considered to

<table><caption>**Table 4.**
Calculation of discretization error for current study.</caption><thead><tr><th>Parameter</th><th>Drag coefficient (c<sub>d</sub>)</th></tr></thead><tbody><tr><td>N<sub>1</sub>, N<sub>2</sub>, N<sub>3</sub>, N<sub>4</sub>, N<sub>5</sub></td><td>47K, 107K, 700K, 4.2M, 5.5M</td></tr><tr><td>y<sub>1</sub><sup>1</sup>, y<sub>2</sub><sup>1</sup>, y<sub>3</sub><sup>1</sup>, y<sub>4</sub><sup>1</sup>, y<sub>5</sub><sup>1</sup></td><td>3.9, 2.7, 1.4, 0.7, 0.5</td></tr><tr><td>φ<sub>1</sub>, φ<sub>2</sub>, φ<sub>3</sub>, φ<sub>4</sub>, φ<sub>5</sub></td><td>0.0417, 0.0367, 0.0320, 0.0282, 0.0285</td></tr><tr><td>P<sub>ave</sub></td><td>1.49</td></tr><tr><td>φ<sub>ext</sub><sup>21</sup>, φ<sub>ext</sub><sup>32</sup>, φ<sub>ext</sub><sup>43</sup>, φ<sub>ext</sub><sup>54</sup></td><td>0.0467, 0.0414, 0.0358, 0.1116</td></tr><tr><td>e<sub>a</sub><sup>21</sup>, e<sub>a</sub><sup>32</sup>, e<sub>a</sub><sup>43</sup>, e<sub>a</sub><sup>54</sup></td><td>11.99%, 12.80%, 11.87%, 1.06%</td></tr><tr><td>GCI<sub>fine</sub><sup>21</sup></td><td>14.98%</td></tr><tr><td>GCI<sub>coarse</sub><sup>32</sup></td><td>16.00%</td></tr><tr><td>GCI<sub>fine</sub><sup>43</sup></td><td>14.84%</td></tr><tr><td>GCI<sub>fine</sub><sup>54</sup></td><td>5.30%</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Evolution of drag ($c_d$), friction drag ($c_f$) and pressure drag ($c_p$) coefficients for increasing mesh resolution and number of cells (abscissa in logarithmic scale).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 2.** Mesh discretization of the flume domain from coarse to fine (N1–N5), with cell sizes ranging from 0.1 m to 0.01 m, including step-wise mesh refinement regions and boundary layers around the fish surface.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 4. Left: Velocity measurements and numerical probes around fish body in three regions: head (red), body (green) and tail (blue). Gray points further away from body surface correspond to LDA measurement locations not used in the mesh sensitivity analysis. Right: Convergence of the streamwise and lateral velocity components in the head, body and tail regions over a simulation time of 3 s. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Normalized streamwise velocity profiles at three different regions along the fish-shaped body for all turbulence models and LDA measurements. At some locations, the k − ω SST model overlaps the Spalart Allmaras model and is not visible.</figcaption></figure>

be negligible. The inlet boundary was kept 0.1 m upstream of the head of the fish, to most closely match the mapped inner sub-patch interpolated from the LDA experiments. This short distance allowed for a close matching of the flow pattern directly upstream of the body. This choice was justified based on the comparison of the mapped inlet boundary condition with a uniform inlet condition placed 0.4 m distance to the fish, where it was observed that the mapped inner sub-patch resulted in an average reduction of the deviation in the streamwise velocity of 7.3%. Considering pressure, a Neumann boundary condition ( ∇ P = 0) was specified at the inlet patch and a Dirichlet boundary condition (P = 0) was applied at the outlet patch. The side and bottom walls were specified with Neumann boundary conditions ( ∇ P = 0) for the pressure.

As velocity and turbulence intensity (TI) were acquired from the LDA experiments, an average turbulence intensity field from the experiments (TI = 3.07%) was also implemented at the inlet patch implicitly by assigning the turbulence kinetic energy (k) and dissipation using (ε) and (ω), respectively. Mathematical expressions used for the calculation of turbulent kinetic energy (k) and dissipation rate (ε) and (ω) were

adapted from Launder and Sharma (1974):

$$
k = \frac{3}{2} (U \times TI)^2, \quad \varepsilon = c_{\mu}^{\frac{3}{2}} k^2 l^{-1}, \quad \omega = \sqrt{k} / l \quad (4)
$$

where, U is the velocity, c_μ is the model constant with a value of 0.09 and l is the turbulent length scale of 0.22 h₀, where h₀ ≈ 0.1 m is the height of fish-shaped body.

Different wall functions for k, ε, ω and μt depending upon the type of mesh discretization (e.g. for setups where the boundary layer was either resolved or unresolved) were applied as initial conditions. A detailed table of the numerical model setups is provided in the supplementary material. The roughness at the surface of the fish model and walls was considered to be uniform with a roughness height of 100 × 10⁻⁰⁶, and remained as the default value for all model setups evaluated in this study. This roughness value was chosen as it is commonly applied in CFD studies of hydraulically smooth surfaces (Adams et al., 2012). The adaptation of the surface roughness allows for a fine tuning of the flow separation zone. However for this study, it was not required to be
<figure><img src="image_10.png" /><figcaption>Fig. 6. Contour plots of the absolute difference in streamwise velocity between LDA measurements and the standard $k-\epsilon$ (left), $k-\omega$ SST (center) and Spalart Allmaras (right) turbulence models.</figcaption></figure>

adjusted as the default value was found suitable when compared to the LDA measurements. It is also interesting to point out that previous research on the roughness at the surface of trout has found that the presence of scales and mucus did not substantively alter the boundary layer (Gorb et al., 2017).

### 3.3. Turbulence models

The RANS approach used in this work allows for a sufficient representation of the physical model flow conditions, but required further assessment to determine the best performing turbulence model. The turbulence models evaluated in this study were based on the Boussinesq hypothesis, in which the Reynolds stresses are proportional to the mean rates of deformation (Schmitt, 2007). Using Reynolds decomposition, the Navier-Stokes momentum equation is given as (Batchelor, 2000):

$$ \rho \frac{\partial u_i}{\partial t} + \rho \frac{\partial}{\partial x_j} (u_i u_j) = - \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j} (2\mu S_{ij} - \rho \overline{u_i u_j}) \quad (5) $$

where $u$ is velocity, $t$ is time, $p$ is pressure, $\mu$ is the dynamic viscosity, $S_{ij}$ is the mean strain rate tensor, $u'$ is the fluctuating velocity component and $\overline{u_i u_j}$ are the mean velocity gradients. The Boussinesq approach fundamentally relates the Reynolds stresses to the mean velocity gradients which are expressed as:

$$ -\rho \overline{u_i u_j} = \mu_i \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{2}{3} \left( \rho k + \mu_i \frac{\partial u_i}{\partial x_k} \right) \delta_{ij} \quad (6) $$

where $\delta_{ij}$ is the Kronecker delta with orthogonal coordinate indices $i$ and $j$. An advantage of modelling the Reynolds stresses with the Boussinesq hypothesis is the decrease in computational cost associated with using a turbulent viscosity $\mu_t$. The Boussinesq hypothesis fundamentally assumes $\mu_t$ to be a scalar quantity, and the turbulent kinetic energy (k) produced within the flow is then estimated as:

$$ k = \frac{1}{2} \overline{u_i u_j} \quad (7) $$

Three well-established RANS turbulence models, $k-\epsilon$, $k-\omega$ SST
<figure><img src="image_11.png" /><figcaption>Fig. 7. Box-plots of the absolute difference between the time-averaged velocities ($U_x$ and $U_y$) for the three turbulence models, compared to the LDA measurements. The box-plot indicates the inter-quartile range, mean (green triangle), median (orange line) and outliers are shown as black circles. The fill colour indicates either resolved or modelled boundary layers. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

and Spalart-Allmaras models were selected based on their general suitability for the case study. They were compared with the experimental LDA measurements to determine the best-performing model for flow around fish-shaped bodies. A summary of the turbulence models and near-wall treatments investigated in this work is provided in Table 3.

### 3.3.1. Standard $k-\epsilon$ model

The standard $k-\epsilon$ model is one of the most widely used for RANS simulations. It is a two-equation model which solves the turbulent kinetic energy $k$ and turbulent dissipation $\epsilon$, approximating turbulence in the averaged flow field as calculated by the RANS approach (Jones and Launder, 1972). The $k-\epsilon$ model is known to perform well far from surfaces, where pressure gradients tend to be small (Bardina et al., 1997; Launder and Sharma, 1974) and thus its performance was not anticipated to be the best considering the near-body flows of interest in this study. There are multiple extensions of the standard model available, including the realizable $k-\epsilon$ model and Re-Normalisation Group (RNG) $k-\epsilon$ model (Yakhot and Orszag, 1992). Differences between the standard model and its extensions concern the calculation of model constants for the turbulent viscosity ($\mu_t$) and the inclusion of different scales of motion. This results in improved predictions of jet spreading rates, a stronger capacity to capture the mean flow around structures, and for complex flows including rotation and boundary layers under high adverse pressure.

### 3.3.2. $k-\omega$ SST model

To resolve some of the issues when applying the standard $k-\epsilon$ model, the hybrid turbulence model $k-\omega$ SST model was also chosen in this work. It is a two-equation model which combines the advantages of the $k-\omega$ model (Wilcox, 1988) and $k-\epsilon$ model. The $k-\omega$ SST model is especially well-suited for wall-bounded and unbounded flows (Menter, 1993), where the transport of shear stress is included in the turbulent viscosity to improve the prediction of flow separation on smooth surfaces with adverse pressure gradients. Menter (1994) added a turbulent prediction limiter to avoid over-prediction of turbulent kinetic energy in stagnation regions without influencing the shear layers. Due to the importance of the near-wall flow field for fish sensing, this model was chosen as it incorporates a blended function which determines the position and activates the required turbulence model. The function becomes zero far from the wall, applying $k-\epsilon$, and remains unity within the boundary layer where $k-\omega$ is used. In the present study, all model constants were executed with the default values presented in Menter (1994).

### 3.3.3. Spalart-Allmaras model

The Spalart-Allmaras model is a single equation approach which implements linear eddy viscosity. The model was first introduced by Spalart and Allmaras (1994) and developed for modelling airfoils with adverse pressure gradients, which are geometrically similar to fish-shaped bodies. In this model, the turbulence kinetic energy is not readily available. Instead, while estimating the Reynolds stresses the last term in Eq. 6 is ignored and the kinematic eddy viscosity is calculated through the use of closure functions (Spalart and Rumsey, 2007).

## 3.4. Boundary layer and near-wall treatment

The flow inside the flume is fully turbulent ($Re_{channel} = 6.8 \times 10^5$) and strongly influenced by the presence of the fish-shaped body, primarily due to the large velocity gradients resulting from the no-slip condition on the body surface. As the distance from the surface increases, turbulence increases due to the increased production of turbulent kinetic energy, generated principally due to the presence of large gradients in the mean streamwise and lateral velocities. Modelling the flow field accurately in the vicinity to the fish-shaped body thus required special attention to the combinations of turbulence models and near-wall treatments, as summarized in Table 3. Furthermore, a comprehensive overview table of the wall functions used in conjunction with the investigated turbulence models and near-wall treatments is provided in the supplementary material to this work.

The exploration of the sensing range of a fish through numerical modelling requires simulations of the near-body flow fields. Close to the surface of fish, the flow is dominated by viscous effects, where the near-body velocity depends on the distance from the surface of the fish, the fluid density, viscosity and shear stress. The viscous forces inside the viscous region of the boundary layer dominate over the inertial forces, creating a no slip boundary condition ($U = 0$) at the surface of the fish's body. This region is extremely thin, and the mean velocity is assumed to increase linearly with increasing radial distance from the body surface (wall). The dimensionless wall distance ($y^+$) was calculated as:

$$y^+ = \frac{y u_{\tau}}{\mu} \qquad (8)$$

where $y$ is the absolute distance from the wall, $u_\tau$ is the frictional velocity and $\mu$ is dynamic viscosity. Simulating the boundary layer can be challenging for biologically inspired geometries such as the fish-shaped body investigated in this work, as it depends on the finest regions of the mesh.
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Comparison of the 2D velocity fields at the midsection plane $z = 0$, showing the LDA measurements and best-performing RANS simulation using a resolved boundary layer. Left: LDA measurements. Center: *Spalart Allmaras* turbulence model. Right: Contour plot of the dynamic pressure field around the fish-shaped body from the *Spalart Allmaras* model at $z = 0$. The pressure values are reported as P-P0, where P0 corresponds to the hydrostatic pressure at the stagnation point of the body.</figcaption></figure>

Specifically, the near-body mesh must adequately capture the geometry, and the chosen simulation approach depends on the Reynolds number and wall roughness. There are two general numerical simulation approaches used to simulate flow around complex geometries. The first is a more simplified approach where the boundary layer is modelled using a wall function ($30 < y^+ < 300$). A more sophisticated approach resolves the boundary layer with a fine mesh discretization ($y^+ < 1$), which generally improves the accuracy but is correspondingly more computationally demanding. As one of the research objectives of this work was to address the lack of a comparison of near-wall treatments for CFD studies on fish-shaped bodies, we compared the unresolved and resolved boundary layers in conjunction with the three RANS turbulence models $k-\epsilon$, $k-\omega$ SST and *Spalart Allmaras*. To accomplish this, it was necessary to create two different types of mesh to achieve $y^+$ values within the desired range of $y^+ < 1$ for the resolved, and $30 < y^+ < 300$ for a modelled boundary layer (Table 3).

## 4. Mesh and time sensitivity analyses

Mesh and time sensitivity analyses were performed to ensure the consistency of the numerical simulation performance and in preparation for the numerical model comparison with the LDA measurements. The numerical settings were first verified through a mesh refinement process (from coarse to fine) to identify the appropriate spatial discretization, as presented in Fig. 2. The mesh sensitivity results for the five different resolutions tested are summarized in Table 4. Subsequently, a time sensitivity analysis was also performed to determine the time for a fully developed flow field at which the streamwise and lateral near-body velocities for each of the head, body and tail regions stabilize.

The mesh sensitivity analysis was carried out with a setup employing the $k-\omega$ SST turbulence model. For the mesh sensitivity analysis, the drag coefficient was chosen as an evaluation metric, because it considers the pressure and shear forces over the entire surface of the fish-shaped body, providing an integrated overview of both the velocity and pressure fields as a single value. Mathematical expression for the calculation of drag coefficient is as follows (Heddleson et al., 1957):
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** 3D isosurfaces of the dynamic pressure around the fish-shaped body. The illustration neglects the effect of the sting, providing a qualitative depiction of the highly complex pressure surface topology in the vicinity of the body. The pressure values are reported as P—P0, where P0 corresponds to the hydrostatic pressure at the stagnation point of the body.</figcaption></figure>

$$c_d = \frac{2F_d}{\rho U^2 A} \qquad (9)$$

Where $F_d$ is the drag force on the reference area A of fish (which is approximately 0.01448 m² for current fish model) moving with velocity U inside a fluid of density $\rho$. The mesh sensitivity analyses followed the ASME criteria (Celik et al., 2008) using five different base-mesh cell sizes: 0.1 m, 0.05 m, 0.025 m, 0.0125 m and 0.01 m with a global refinement ratio r of 10 ($r = h_{course}/h_{fine}$ where h = mesh cell size). The y⁺ values for all meshes were restricted to remain in the viscous range (i.e up to y⁺ = 5). This was due to the functional requirements of the resolved wall functions, and was required to achieve comparable near-body velocities and pressure fields. These quantities are of particular interest for future studies on the spatial extents of the active sensory space that includes the velocity, acceleration and pressure fields in the immediate vicinity of a fish-shaped body. The numerical uncertainty was assessed using the grid convergence index (GCI) method. First, the approximate relative error ($e_d^{ji}$) in the numerical model results between two successive mesh grids was evaluated. Afterwards, the grid convergence index (Roache, 1994) between consecutive meshes was calculated. For each of the five mesh sizes, the drag coefficient ($c_d$), which is composed of the sum of friction ($c_f$) and pressure drag ($c_p$), was calculated at the surface of the fish model for comparison. Table 4 represents the mean variable $\phi_n$ – in our case the drag coefficient – which denotes its value on the nth grid and the local order of accuracy $p_a$ of the method for each respective mesh, which was calculated as:

$$p_a = \frac{1}{\ln(r_{ji})} \left| \ln \left| \frac{\varepsilon_{kj}}{\varepsilon_{ji}} \right| + q(p_a) \right| \qquad (10)$$

where $\varepsilon_{kj} = \phi_k - \phi_j$ and $\varepsilon_{ji} = \phi_j - \phi_i$.

$$q(p_a) = \ln \left[ \frac{r_{ji}^{p_a} - s}{r_{kj}^{p_a} - s} \right] \qquad (11)$$

$$s = 1.\operatorname{sgn}(\varepsilon_{kj}/\varepsilon_{ji}) \qquad (12)$$

The extrapolated values of the variable were calculated as:

$$\phi_{est}^{ji} = \frac{(r_{ji}^{p_a} - \phi_i) / (r_{ji}^{p_a} - 1)}{1.5 \operatorname{sgn}(\varepsilon_{kj}/\varepsilon_{ji})} \qquad (13)$$

And the approximate relative error ($e_d^{ji}$) between each consecutive mesh was calculated along with the grid convergence index (GCI) using following expressions:

$$e_d^{ji} = \left| \frac{\phi_i - \phi_j}{\phi_i} \right| \qquad (14)$$

$$GC_{fine}^{ji} = \frac{1.25 e_d^{ji}}{r_{ji}^{p_a} - 1} \qquad (15)$$

The relative difference in simulated results between consecutive meshes (i.e. for refinements between meshes of $\Delta x = 0.1$ and $0.01$), had a maximum value of 31.65%. The average order of accuracy $p_{ave}$, defined as the lowest order term in the truncation error, was found to be 1.49. This value is in good agreement with the formal order of the numerical schemes applied, which were chosen to be first order. Therefore, the mesh sizes used in this case study lie within the asymptotic range. The results of the mesh sensitivity analyses based on the drag coefficient are presented in Fig. 3, which illustrate the drag coefficient convergence occurring between meshes N4 and N5 with a 1.06% reduction of the relative percentage error. To optimize numerical model efficiency and reduce the computational time, the N4 mesh was selected based on the mesh sensitivity analyses for all further model setups used in this work.

The time sensitivity analyses were performed for three distinct near body regions around the model geometry corresponding to the head, body and tail regions as shown in Fig. 4 and are denoted by the red, green and blue points, respectively. Considering a fish’s lateral line sensing system, the bulk flow intensity and its direction are encoded by the upstream (head) flow, swimming kinematics are regulated by the forces acting laterally (body) and the energy consumption used during swimming is reflected in the momentum deficit present in the wake (tail). For this reason, the time sensitivity as well as the results for the numerical model setups are presented for these three regions in the remainder of this work. The streamwise and lateral velocities were monitored over a 3 s interval. The flow field becomes fully developed after 1.5 s, as indicated in the gray hatched region of Fig. 4. The velocities $U_x$, $U_y$ corresponded to locations where the LDA measurements were obtained and were ensemble averaged for all points within each of the three regions at each time step. In the head region, the velocities were averaged from 13 points at each time step, 36 for the body and 18 in the tail region (the locations of all points is shown in Fig. 4). As expected, the convergence of $U_x$ occurred in the head region first, followed by the body and finally the tail region. The delayed convergence of the streamwise $U_x$ velocity in the tail region is due to the flow separation. Similar behavior was observed for the $U_y$ velocity. A time step sensitivity analysis was also conducted for the velocity field by using three different time steps, i.e. $\Delta t = 0.1, 0.01$ and $0.001$ resulted in negligible differences.
# 5. Results and discussion

## 5.1. Turbulence model assessment

The normalized streamwise velocity profiles for all three turbulence models were plotted against the LDA profiles in the near-body region, and are shown in Fig. 5. The profiles in the head region, beginning 5 mm upstream of the body, exhibited the largest relative error of 20% when compared with the LDA measurement, decreasing rapidly to 0.4% as the profile extended into the freestream.

All turbulence models performed nearly equally at the stagnation point, which features an especially rapid velocity gradient (Fig. 6). Laterally along the fish-shaped body, it can be seen that the $k-\epsilon$ model failed to adequately capture the adverse velocity profile at the head. The $k-\omega SST$ and Spalart Allmaras models performed nearly equally at the head with minor exceptions. Similar trends were observed in the body region profile, where the $k-\omega SST$ and Spalart Allmaras models had a 6% deviation, whereas the $k-\epsilon$ model exhibited a 23% deviation from the LDA measurements.

The tail region includes boundary layer separation and vortex shedding, and therefore represents one of the most challenging regions for accurate numerical simulation. Within the tail region, the Spalart Allmaras model using a resolved boundary layer predicted the streamwise velocity profile with the highest overall accuracy, both near the body surface and extending into the freestream flow. In addition to the near-body velocity profiles, contour plots of the absolute difference in streamwise velocity for all three turbulence models and LDA measurements are presented in Fig. 6. The results provide a spatially distributed quantitative assessment of regions with higher and lower absolute errors. In the vicinity of the fish body, a significant difference of up to 0.15 m/s in the absolute streamwise velocity was observed when using the $k-\epsilon$ model. In contrast, the $k-\omega SST$ model exhibited far lower deviations with a maximum of 0.1 m/s, and the Spalart Allmaras had a maximum of 0.05 m/s absolute deviation from the LDA measurements.

## 5.2. Resolved and modelled boundary layer comparison

To assess the impact of modelling or resolving the boundary layer, a comparative study between the subset of LDA measurements points within the maximum distance of 0.0139 m at the surface of fish were chosen for evaluation. The absolute difference between the time-averaged velocities from the CFD models and the LDA measurements was obtained for both streamwise ($U_x$) and lateral velocity ($U_y$), at the probe locations within the range. It is worth pointing out that the Spalart Allmaras model only operates with a resolved boundary layer case. The results of the near-wall treatments for all turbulence models are given in Fig. 7 as box-plots. From the plots, it was found that considering the streamwise velocity, the Spalart Allmaras model had the lowest observed absolute difference but there were no substantive differences observed across model setups considering the lateral velocity. Overall the resolved boundary layer cases showed less absolute difference in velocity as compared to the modelled one for streamwise velocity component, whereas it was marginal in case of lateral velocity component.

## 5.3. Validation with LDA measurements

Among the three turbulence models, the Spalart Allmaras model performed best overall, closely followed by the $k-\omega SST$ model, which was found to have both streamwise and lateral velocity errors with similar distributions. A comparison of the velocity contour plots using the Spalart Allmaras model with resolved boundary layer as compared to the experimental LDA data is shown in Fig. 8. The CFD model and LDA measurements are in good agreement, especially considering the strong velocity gradients in the upstream head region. The experimental results had a slightly larger time-averaged mean freestream velocity of 0.52 m/s, whereas the numerical model had an average of 0.50 m/s. The spatial

distribution of the high velocity region around the body found in the LDA contour plot illustrates a broader spatial extent than the numerical model. This may have been caused by wall effects in the laboratory flume. The highly turbulent tail region shows that the CFD model accurately captured the flow separation, which occurred at 2/3 of the total body length. The wake region shows good agreement in the streamwise velocity between both the measured and simulated results.

# 6. Summary and conclusions

This work was motivated by the lack of an openly available laboratory measurements and fish body model. Addressing this gap will improve the ability of researchers to systematically study turbulent flow fields around fish-shaped bodies. The major contributions of this work are the laboratory LDA velocity measurements, and the critical evaluation of the most suitable turbulence model considering the boundary layer numerical treatment.

In this study, RANS turbulence models were selected due to the trade-off between robustness and computational efficiency. Correspondingly, we chose the standard $k-\epsilon$, the $k-\omega SST$ and Spalart Allmaras models. The numerical model was validated with LDA measurement data, where it was found that a resolved boundary layer and Spalart Allmaras had the lowest overall error, followed by $k-\omega SST$ and finally $k-\epsilon$ with respect to the streamwise velocity. Although minor differences were found, the three turbulence models performed nearly equally well for the near-body, lateral averaged velocity. In this study, the Spalart Allmaras performed the best over the whole length of the fish-shaped body, when compared with the $k-\omega SST$ and $k-\epsilon$ models. The average deviations of the streamwise velocity when compared with the LDA measurements were 7.8%, 11.4% and 15.17%, respectively, in the near-surface region. Thus the Spalart Allmaras model was found to be the most suitable RANS turbulence model for future CFD studies on the active sensory space of fish.

Streamwise velocity gradients are used by the fish's superficial neuromast flow sensing system, and previous works have shown that flow-sensitive receptors are highly concentrated in the head region (Ristroph et al., 2012). The density of sensory receptors was found to have high densities around the eye socket, with a maximum receptor density at 10% of the total body length, decreasing rapidly and tailing off at 20%, where sparsely located lateral canal neuromasts remain. Motivated by these biological observations, future analyses of the near-body flow field should also include an assessment of the spatial sensitivity of velocity gradients and pressure.

The boundary layer around the surface of fish was resolved to a thickness of 0.014 m, following the work of Yanase and Saarenrinne (2016) who measured it on a swimming trout with PIV. The absolute difference between the near-body streamwise velocity for modelled and resolved boundary layers was consistently lower for the resolved boundary layer setup (21%–52.5%), when comparing the $k-\epsilon$ and $k-\omega SST$ models. Considering the lateral velocity, the absolute differences between modelled and resolved setups were found to be marginal. This is partially due to the presence of wall effects in the laboratory flume, which resulted in a significant reduction of the lateral velocity component. The Spalart Allmaras turbulence model resolved boundary layer case exhibited the least absolute difference among the three turbulence models tested in this study. Based on these findings, we recommend resolving the boundary layer to ensure an accurate representation of the near-body streamwise and lateral velocities, which is in agreement with the recommendations from Rapo et al. (2009) and Herzog et al. (2017).

The pressure gradients around the fish body were not the major focus of this work. This was primarily because the LDA velocity measurements served as the measured reference for model tuning and validation. However, since the canal-based lateral line system does indeed rely on the near-body pressure distribution, it was also of interest to plot the pressure isosurfaces as shown in Fig. 9. As first illustrated by Kogan et al. (2015), the pressure isosurfaces provide an interesting view of the
highly complex “touch at a distance” sensing modality. It is worth pointing out that the pressure isosurfaces can be easily recovered from CFD models. Interesting observations may be obtained where the fish-shaped body is positioned downstream of obstacles or near walls, to further study how changes in the near-body pressure field relate to a wider range of freshwater fish body geometries and realistic flow environments.

Future works are ongoing to investigate the effects of synthetic turbulence of known length and correlation time scales on the pressure and velocity fields in the boundary layer of fish-shaped bodies. We will also study additional body geometries from nine different common European freshwater fish species. These studies will include bottom-oriented, weak and strong swimmers, and the results of the ongoing research will provide new insights as to how a fish's body geometry interacts with a broader range of turbulent flows. These insights can be used to improve our basic understanding of how fish may use or avoid flow fields which they commonly encounter in Nature during feeding, for predator avoidance, as well as for spawning and migration. The findings and recommendations in this work are meant to inspire and encourage a more rapid adoption of CFD to improve our understanding of freshwater fish's advanced sensing abilities as well as the complex flows they inhabit and rely on for survival. Although our initial contribution is a single fish model, we are optimistic that a far wider range of freshwater “CFD fish” are soon to follow.

## 6.1. Data availability

Additional data supporting the findings of this study and supplementary materials including the open numerical model and measurement data are available from the Open Science data repository of the Otto-von-Guericke University Magdeburg at doi:10.24352/UB.OVGU-2022-001

## Declaration of Competing Interest

None.

## Acknowledgements

The research work presented in this paper has received funding from European Union Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Actions, Grant Agreement No. 860800. Laboratory measurements were conducted as part of the RETERO project, sponsored by the German Federal Ministry of Education and Research Grant no. 161L0152A. Jeffrey A. Tuhtan's contribution was funded by the Estonian Research Council Grant PRG1243 and Maarja Kruusmaa was funded by EXCITE supported by the Estonian Centre of Excellence in IT, funded by the European Regional Development Fund Project nr 2014-2020.4.01.15-0018. The authors are grateful to M. Yanneck Kiiski for the design and 3D printing of the fish geometry and the CAD model of the laboratory flume.

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