# On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental and Modeling Configuration

#### 2.1.1. Governing Equations and Configurations Studied

_{s}

_{,max}is the maximum close packing concentration (close to 0.635 for non-cohesive spherical solid particles). Equations (3)–(5) are as follows:

#### 2.1.2. The Release Phenomenon

_{p}), the sediment release volumes (V

_{d}), the initial concentrations of the mixture (C

_{m}), and the imposed ambient current (U

_{c}). Villaret [1] and Boutin [2] determined the falling time (time lapse between the release and the impact of the sediment cloud on the bottom), the up- and downstream radius of the cloud, the maximum height of the density current, and the front velocity of the density current (Table 1).

#### 2.1.3. Design of Numerical Experiments

_{inj}) are imposed, assuring the right mixture discharge. Indeed, it was observed in this previous work that the Poiseuille-type profile is well suited to the experimental conditions. The values of the channel inlet velocity and the release are shown in Table 1. Two sand diameters are considered, namely D

_{p}

_{1}= 90 µm (sand 1) and D

_{p}

_{2}= 160 µm (sand 2). A uniform mesh of 1401 nodes on the horizontal and 61 vertical nodes were used, and the time step was fixed to 0.001 s. Data analyses were performed over the simulation outputs for every 0.1 s. This work is focused on the averaged motion, and the outputs timestep seems to be adapted to this characterization. It will serve for the POD analysis hereafter.

_{p}

_{1}(sand 1) and D

_{p}

_{2}(sand 2), assumed in this work as reference cases, were previously presented by Guillou et al. [3]. They studied the evolution of the sediment cloud as well as solid velocity fields in still water for both cases. It was found that the two-phase model reproduces two counter-rotating vortices, as observed experimentally by Villaret [1] and Boutin [2].

_{p}

_{1}and D

_{p}

_{2}, three steps are observed during a sediment release: the convective descent of the sediment cloud, the sediment cloud impact on the bottom, and the propagation of density currents up- and downstream until the equilibrium under the ambient current influence. For results obtained within an ambient current, only cases simulated for the sediment particle diameter of 90 µm and the ambient current going from 10 to 25 [cm/s] (D

_{p}

_{1}-U

_{c}

_{10}, D

_{p}

_{1}-U

_{c}

_{15}, D

_{p}

_{1}-U

_{c}

_{20}and D

_{p}

_{1}-U

_{c}

_{25}) will be studied.

#### 2.2. Proper Orthogonal Decomposition

#### 2.2.1. Theoretical Aspects

_{max}) of spatio-temporal modes, as in the following Equation (11):

_{1C}, will be applied onto the solid volume fraction fields snapshot collection, with ${\alpha}^{\prime}\left(\overrightarrow{x},{t}_{i}\right)=\alpha \left(\overrightarrow{x},{t}_{i}\right)-\overline{\alpha}\left(\overrightarrow{x}\right)$. In the second case, two component vectorial POD

_{2C}will be applied onto fluctuating solid phase velocity fields $\overrightarrow{{u}^{\prime}}\left(\overrightarrow{x},{t}_{j}\right)=\overrightarrow{u}\left(\overrightarrow{x},{t}_{j}\right)-\overrightarrow{\overline{u}}\left(\overrightarrow{x}\right)$, and this leads to spatial eigenfunctions within two-component vectors. To represent these eigenfunctions, it is necessary to calculate a pseudo-vector field equal to the rotational of the two-component vectors field. The choice of the size M of the snapshots collection decomposed by POD technique (cf. Table 1) will be discussed in the following.

_{1C}and POD

_{2C}, we, respectively, obtain $E\left({\alpha}_{s}^{\prime}\right)$ [ ] and $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ [m

^{2}/s

^{2}]. The average root mean square error percentage highlights the relative difference between an original set of realizations and a reconstructed one.

#### 2.2.2. About the Size of the Snapshot Collection Decomposed by POD

_{p}

_{2}, dt is 0.1 s and Ns = 400 (total time = 40 s). The size of the sequence decomposed, with dt = 0.1 s, has been selected equal to M = 40 (short collection with a total time of 4 s) and M = 400 (long collection). Considering dt = 0.2 s, new snapshot collections can be extracted from the initial one and, thus, M is, respectively, equal to 20 (4 s) and 200 (40 s).

_{max}is the number of POD spatio-temporal modes used in the reconstruction). In both cases, for a fixed total time sequence, no significant differences are observed in spite of the change in the sampling frequency. Both decompositions, based on $\left({\alpha}_{s}^{\prime}\right)$ and $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$, lead to a similar accuracy of reconstruction for a given k

_{max}(cf. Figure 3). The average root mean square error obviously decreases with the increase in k

_{max}. Note that it can be observed that, for a given k

_{max}, the average root mean square error is lower for the POD decomposition based on $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$. These results verify that the good choice of the snapshot collection decomposed by POD is not strictly linked to the snapshot dt. The POD results presented in the following have been obtained from snapshot sequences computed with dt = 0.1 s, which seems to be sufficient to lead to suitable decomposition with a short or a long sequence of realizations. The parameters (M, dt) are defined according to the steps of the release phenomenon that will be studied.

## 3. POD Results for a Water–Sediment Mixture Release

_{1C}is applied to the fluctuating solid volume fraction fields and a two-component vectorial decomposition POD

_{2C}is applied to the fluctuating solid phase velocities fields. Reference cases D

_{p}

_{1}(for D

_{p}= 90 µm) and D

_{p}

_{2}(for D

_{p}= 160 µm), respectively, present water–sediment mixture release without ambient current configurations. POD decompositions are performed on different sized snapshot collections for case D

_{p}

_{1}(M = 15; 40; 75; 100; 150; 200, and a total sequence time going from 1.5 s to 20 s) and D

_{p}

_{2}(M = 13; 40; 75; 100; 150; 200; 250; 300; 350; 400, and a total sequence time going from 1.3 s to 40 s).

_{p}

_{1}and D

_{p}

_{2}cases) to apply POD decomposition and to compare the results extracted seems suitable (for the release phenomenon going on until the sediment transport).

_{f}). Two different processes are involved in this phenomenon: on the one hand the density effect of the sediment–water mixture, like a saline water plume, and, in the other hand, the terminal settling velocity of individual particles. The time required for the plume to reach the bottom depends on both processes. Table 2 shows T

_{f}obtained by Villaret [1] and Boutin [2], the results obtained in the present work, and by Farout-Fréson [28]. Note that Farout-Fréson [28] proposed a model based on the averaged form of the hydrodynamic biphasic equations, coupled to a sediment transport equation with a specific numerical settling velocity. This model was calibrated on the experimental campaigns of Villaret [1] and Boutin [2].

_{p}

_{1}and D

_{p}

_{2}, convective descent of the sediment cloud is characterized by a falling time (T

_{f}) of 1.5 s and 1.3 s (cf. Table 2) observed in the instantaneous field series. The experimental fall time was extracted from snapshots taken of the generated turbid plume with a synchronized camera. This implies an important uncertainty of 0.5 s. Thus, numerical results presented in this paper can be considered as satisfactory despite being superior to that of the upper limit observed for experimental results. It can be seen that the reduction in the falling time due to a greater D

_{p}turns out to be slight.

#### 3.1. POD of Snapshots Collections Describing Falling Time

_{f}/0.1 was chosen as equal to 15 for D

_{p}

_{1}(which corresponds to time T

_{f}= 1.5 s) and 13 for D

_{p}

_{2}(T

_{f}= 1.3 s).

_{p}

_{1}and POD

_{1C}(i.e., for D

_{p}

_{1}and POD

_{2C}) and also the first spatial eigenfunctions extracted from the difference compared to the mean field, in other words the eigenfunctions linked to the most energetic eigenvalues.

_{2C}(cf. Figure 4b) present strong and compact couples of counter-rotating structures or POD modes associated to the appearance and growing of symmetrical vortices during the falling of the water–sediment mixture. The number of the POD modes observed in an eigenfunction ${\Phi}_{k}\left(\overrightarrow{x}\right)$ increases with the index k. Similarly, the eigenfunctions obtained from the POD

_{1C}decompositions exhibit aligned and piled POD modes (cf. Figure 4a). This arrangement is, obviously, strictly associated to the convective descent of sediment. The eigenfunctions extracted for the D

_{p}

_{2}case are not shown here. The spatio-temporal modes extracted from this snapshot collections have broadly similar characteristics to the D

_{p}

_{1}case. However, the increase in D

_{p}, leading to the enhancement of the particles’ weight, implies the elongation in the form of the POD modes.

_{1C}applied to a snapshot collection corresponding to the falling time, $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$ has a circular shape, while $\left({a}_{3}\left(t\right),{a}_{4}\left(t\right)\right)$ seems to have a temporal pattern close to T

_{f}/2.

_{2C}case, the characteristic time of the phase space projections $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$ seem to be a slightly longer that in the POD

_{1C}case. It can be inferred that the dynamics of the phenomenon have a small difference for the solid volume fraction and the solid volume velocities behaviors.

_{1C}and POD

_{2C}.

_{p}has an impact on the amount of total energy extracted (cf. Table 3). On the one hand (POD

_{1C}), the D

_{p}impact on the dynamics of the volume fraction is slight (a reduction of 7%), and on the other hand (POD

_{2C}), its impact on the velocity field reaches 30% of augmentation. Moreover, note that the distribution of the contributions of the first four values remains, with very slight differences, the same for the two particle diameters. This confirms that the dynamics of these phenomena are weakly impacted by the modification of the diameter of the particles. Thus, the particle size and its weight seem to have a modest impact on the solid volume fraction distributions during the sediment descent stage; however, this impact is most marked on the level of agitation of the solid phase velocities.

#### 3.2. POD for Snapshot Collection Describing All the Release Phenomenon

_{p}

_{1}(D

_{p}

_{2}) until 200, with dt = 0.1 s; in other words, from a short duration of 1.5 s (1.3 s) to a duration of 20 s, describing the three steps of the release phenomenon. For D

_{p}

_{2}, complementary decompositions have been made for values of M until from M = 400 (a duration of 40 s). Figure 5 shows the evolution of the total energy (E) captured by POD decompositions according to the number of snapshots M decomposed for ${\alpha}_{s}^{\prime}$ and for $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ for D

_{p}

_{1}and D

_{p}

_{2}.

_{1C}and POD

_{2C}applied on snapshot collections obtained for D

_{p}

_{1}and M = 200 and a duration of 20 s (cf. Figure 6 and Figure 7 for D

_{p}

_{1}).

_{p}

_{1}) to 63% (for D

_{p}

_{2}). Thus, the particle’s size and its weight seem to have an impact on the complexity of the behavior compared to the mean solid volume fraction.

_{2C}decomposition based on the parameter $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ and 15–18 in the POD

_{1C}decomposition based on $\left({\alpha}_{s}^{\prime}\right)$. It is clear that, for this specific set-up, it would be possible to obtain a reduced-order model covering the different steps of the sediment release.

_{2C}applied to $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ and the D

_{p}

_{1}case, the total kinetic energy contained by the two initial counter-rotating vortices increases until a maximum value of 198 reached M = 100, corresponding to a total collection time of ~10 s (cf. Figure 5b). This growth observed during (and just beyond) beyond the falling time implies an enhancement in the level of agitation of the solid phase velocities linked to the sediment descent. For snapshot collections with a size M > 100, $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ slowly decreases with M values, to reach 180 at 20 s. This implies that the level of agitation of the flow during the entire event, and more particularly during the formation of a turbidity current, become weaker. It is be reasonable to associate it with the mixing and dilution of the sediment. Thus, it is not surprising to observe (cf. Figure 5a) for POD

_{1C}applied to $\left({\alpha}_{s}^{\prime}\right)$ that the energy curve decreases for D

_{p}

_{1}from 1.19 to 0.86 (for a duration of 10 s) and 0.63 (for 20 s).

_{p}on the evolution of $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ according to the size (M) of the decomposed snapshot collection? Additional decompositions resulted in D

_{p}

_{2}and M growing up to 400. Note that for M > 200 the snapshot collections contain a lot of data relative to the turbidity current step. The slight decrease observed for D

_{p}

_{1}is most drastic and earliest for the largest particle diameter case D

_{p}

_{2}: in this case $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ decreases from 198 (for a duration of 5 s) to 133 (for 20 s) and 86 (for 40 s). It seems that, as observed for the vertical descent phase of the release, the marked influence of the particle size and weight is also present during the passive transport of the sediment leading to a resistance opposed to the motion of the solid phase. In an opposite way, for the $E\left({\alpha}_{s}^{\prime}\right)$ curves (cf. Figure 6a) the trend observed for D

_{p}

_{1}and D

_{p}

_{2}becomes different at around M = 100. Indeed, while $E\left({\alpha}_{s}^{\prime}\right)$ continues to decrease for D

_{p}

_{1}, for the largest particle diameter case D

_{p}

_{2}we observe an augmentation until ~1.1 for 20 s, and then a stabilization until 40 s. This trend is observed for POD

_{2C}applied to inlet snapshot collections including the third step of the release phenomenon, characterized by a long and complete collapse of sediment, concentrated at the bottom (cf. Figure 2b). Indeed, as observed by Nguyen et al. [5], the radius of the particle bed grows and reaches nearly 4 m for a total time of 20 s in both D

_{p}cases. However, the collapsing of the sediment mixture differs until time t > 15 s.

_{p}

_{1}case for POD

_{2C}applied on the difference compared to the mean solid volume velocities snapshots (cf. Figure 7a) reveal well-defined POD modes, which also support the existence of a low complexity of the dynamics. In the same way, the temporal coefficient reveals a marked temporal organization (cf. Figure 7d,e). The marked oscillations observed during the first seconds are smoothed after the falling time. The phase space projection $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$ present a harmonious shape. For D

_{p1}, the footprint of falling and collapse of the water–sediment mixture is present on the four most energetic POD

_{2C}eigenfunctions. They put to the fore strong counter-rotating structures associated to the appearance of the initial counter-rotating vortices due to sediment falling and water–sediment mixture collapse. All these eigenfunctions present POD modes stretched on the bottom. Indeed, the opposition of the quiet water to the motion implies their stretched shape. For k ≥ 3 other compact, counter-rotating, and symmetrical POD modes of ${\Phi}_{k}\left(\overrightarrow{x}\right)$ evolve parallel to the bottom, above the flat POD modes previously described. Their size decreases with the increasing index.

_{1C}also reveal well-defined compact and strong POD modes. Temporal coefficient also reveals a marked temporal organization (cf. Figure 6d,e) (as will be noticed in the following, for the snapshot collection describing all the release phenomenon, the contribution of the first eigenvalues is less marked than in the falling time snapshot collection case. The three, or even the first four, eigenvalues can have contributions of the same order. Also, drawing other curves $\left({a}_{1}\left(t\right),{a}_{3}\left(t\right)\right),\left({a}_{1}\left(t\right),{a}_{4}\left(t\right)\right),\left({a}_{12}\left(t\right),{a}_{3}\left(t\right)\right)$, … may be relevant. As can be observed (cf. Figure 6e), the phase space projection $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$ for the two particle diameters studied in this work have different shapes. However, it can be observed that there is a similarity between $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$—D

_{p}

_{1}and $\left({a}_{1}\left(t\right),{a}_{3}\left(t\right)\right)$—D

_{p}

_{2}.). Note that the oscillations fade faster to reach constant values, thus, the dynamics of the solid volume fraction seem to reach a stability ahead of the dynamics of the solid volume velocity. In the D

_{p}

_{1}case (cf. Figure 6a), the two most energetic eigenfunctions, namely ${\Phi}_{1}\left(\overrightarrow{x}\right)$ and ${\Phi}_{2}\left(\overrightarrow{x}\right),$ show central POD modes (labelled I and II) strongly linked to the water–sediment mixture falling; they also present a symmetrical couple of modes dealing with the deposit of the water–sediment mixture after its collapse and the transport of the particle cloud. ${\Phi}_{3}\left(\overrightarrow{x}\right)$ seems to be also marked by collapse and the deposit of water–sediment mixture. POD modes labelled III and IV (cf. Figure 6a) are linked to the sediment collapse, while the other ones are linked to the density current. For D

_{p}

_{1}, one layer of strong symmetrical POD modes on the bottom, linked to the progressive passive transport of density, appears for ${\Phi}_{k}\left(\overrightarrow{x}\right)$ with k ≥ 5 and k ≥ 6 for D

_{p}

_{2}. In the case of D

_{p}

_{1}, the POD modes have regular contours, while for D

_{p}

_{2}they seem to be weaker and less regular. For D

_{p}

_{2}, the presence of a larger particle diameter D

_{p}implies that the POD modes (I and II) linked to the deposit of the water–sediment mixture are also present in the third eigenfunction.

_{p}

_{1}and D

_{p}

_{2}. The mean field presents two main counter-rotating and symmetrical structures (MS). The interaction with the surface leads to the appearance of flat structures (FS) along the bottom. The MS structure on the right turns in the clockwise sense (as observed in the mean solid volume fraction velocity field, cf. Figure 7a), inducing the mixing of the flow and, thus, a water–sediment mixture resuspension. The right flat structure (FS) turns in the anti-clockwise sense and induces the retention of the water–sediment mixture. All of the most energetic eigenfunctions present the two layers of POD modes previously identified. For D

_{p}

_{1}, the main structures (MS) overcome and overtake the coherent creeping structures (FS) (cf. Figure 7b). As a result, the cloud outline does not show any crushing (cf. Figure 6b). Note that the particle cloud was identified as the place where the solid volume fraction becomes different to 0 during the mixture release. The cloud outline is the front of the particle cloud. In the opposite way, for D

_{p}

_{2}, the lower layer extends almost as far as the upper one (cf. Figure 6c) and the density front of the particle cloud outlines a crushing in its extremity. Thus, it is not surprising to observe the external POD modes collapsing due to the resistance of the bottom to transport of sediments (more difficult density transport). This slump is due to the slowdown in the front density observed in the turbidity snapshots (cf. Figure 2b). Thus, the influence of the size and the weight of the particles due to a larger D

_{p}induces to POD modes more marked by the settling of the water–sediment mixture after the sediment collapse.

_{1C}, we associate the base parameter $\left({\alpha}_{s}^{\prime}\right)$ to the duration covered by the inlet snapshots collection, which needs to be sufficient to clearly observe the settling in the concentration field (cf. Figure 2). POD

_{2C}is associated with the two-dimensional base parameter $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$, and it presents the advantage of revealing the trends based on shorter snapshots collections than in the POD

_{1C}because the inlet data contains the settling velocity data.

#### 3.3. Impact of the Ambient Current

_{p}

_{1}will be studied. POD

_{1C}and POD

_{2C}will be applied to determine the differences between the mean solid volume fraction fields and the mean solid phase velocities fields for cases within an ambient current going from U

_{c}= 10 cm/s to U

_{c}= 25 cm/s (D

_{p}

_{1}-U

_{c}

_{10}, D

_{p}

_{1}-U

_{c}

_{15}, D

_{p}

_{1}-U

_{c}

_{20}, D

_{p}

_{1}-U

_{c}

_{25}).

_{c}on the total energy extracted by both decompositions applied in this work for sand 1. Figure 9 and Figure 10, respectively, present eigenfunctions obtained from the $\left({\alpha}_{s}^{\prime}\right)$ and $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ snapshot decompositions for the cases D

_{p}

_{1}-U

_{c}

_{10}and D

_{p}

_{1}-U

_{c}

_{25}.

_{c}= 0 cm/s) to 10 m for (U

_{c}= 25 cm/s). The upstream radius linearly decreases from 4 m to nearly 0 because the propagation velocity of the front of the density current on the bottom decreases with U

_{c}. Thus, for the cases with ambient velocity, the number of calculation steps (Ns) necessary to observe that the sediment cloud is out of the computation domain, but goes down with increasing ambient velocity (cf. Table 1). To analyze and compare the water–sediment mixture release phenomenon, the size M of the collection of decomposed snapshots is chosen to be equal to Ns.

_{c}. As previously noticed, for U

_{c}= 0 cm/s and POD

_{2C}decomposition is based on the parameter $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ and 10 modes (compared to the M = 200 modes extracted), in other words 5% of M. This ratio evolves until it reaches 7.9% with the increasing U

_{c}. The presence of the ambient current leads to a modest increment in the number of eigenvalues necessary to reach 95% of the total energy extracted. A similar trend is observed for results extracted by POD

_{1C}(cf. Table 5).

_{c}values less than U

_{c}= 15 cm/s. Then, for strong ambient currents, we observe a modification in the evolution, as $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ reaches a marked augmentation of nearly 124%. The level agitation of the solid volume phase velocities suddenly increases beyond a threshold placed between 15 and 20 cm/s. It can be inferred that, below this threshold, the resistance opposed to the strength of Uc limits the development of the agitation. Note that the evolution of $E\left({\alpha}_{s}^{\prime}\right)$ shows more marked differences. Before the threshold we observe a clear decrease, then a similar augmentation of ~124% is also reached for U

_{c}= 20 cm/s.

_{f}) and the water–sediment mixture release diameter when the collapse occurs, as measured by Nguyen [4] for sand 1 and the different values of U

_{c}studied in this work. This author noticed an augmentation of both values with the increase in U

_{c}; however, this augmentation is clearly less marked when the ambient current reaches 20 cm/s.

_{p}

_{1}-U

_{c}

_{10}, the stretching of upstream POD modes of the eigenfunctions extracted by POD

_{2C}stays slight (Figure 10a). The increase in the U

_{c}value leads to a decrease in the size of the POD modes, their gradual stretching and distortion and their displacement in the left direction. In other words, as shown in Figure 10, according to the strength of the ambient current the dynamics of the solid volume fraction can exhibit: a regime within a marked spatial organization and POD modes placed on both sides of the water–sediment mixture injection point (D

_{p}

_{1}-U

_{c}

_{10}), or a regime showing a heavy loss of the spatial organization of the eigenfunctions and all the POD modes placed on the left of water–sediment mixture injection point (D

_{p}

_{1}-U

_{c}

_{25}). The eigenfunctions extracted by POD

_{2C}for different values of U

_{c}, and not presented here, highlight that the transition between the two regimes occurs at a value between 15 and 20 cm/s. Obviously, this shape of the POD modes reflects the presence of shear observed on turbidity snapshots (cf. Figure 2c,d). Note that at the early steps of time, turbidity isocontours have regular shapes; later, when sediments evolve on the bottom, turbidity presents irregular contours. With the increase in the U

_{c}value, the shearing and the irregularity of the turbidity contours appears at an earlier time after the sediment injection. A similar impact of an ambient current can be also observed on the eigenfunctions obtained from POD

_{1C}decomposition of the difference compared to the mean solid volume fraction snapshots. Indeed, for D

_{p}

_{1}-U

_{c}

_{10}(cf. Figure 9), POD modes of density transport placed downstream the point of injection of the water–sediment mixture are slightly distorted, and the deformation is more marked for upstream POD modes. These collapses and distortions are due to the resistance of the bottom to the density transport against the direction of the ambient current. It can be also shown that the POD modes placed upstream have more important intensities compared to the downstream ones. Obviously, this intensity distribution reflects that sediment is pushed by the ambient current to the left. Changes from U

_{c}= 10 cm/s to U

_{c}= 25 cm/s imply an amplification of the loss of the spatial organization of the solid phase volume fraction, in other words the destruction of the coherency of the particle’s clouds.

_{p}

_{1}-U

_{c}

_{10}(i.e., D

_{p}

_{1}-U

_{c}

_{25}). As previously noticed, in all the eigenfunctions, the POD modes have been displaced to the left, pushed by the ambient current. For U

_{c}= 25 cm/s, the higher value of ambient current studied in this paper, all the POD modes observed in both sets of eigenfunctions are strictly placed on the left of the water–sediment mixture injection point.

_{p}

_{1}-U

_{c}

_{10}(cf. Figure 9c and Figure 10c) have strong similarities with the features found without the ambient current. A different behavior is noted for U

_{c}= 25 cm/s, similar to the one previously shown for the D

_{p}

_{2}-U

_{c}

_{0}case. A lower layer of flat creeping structures (FS) extends almost as far as the main and upper structures (MS) (cf. Figure 10d). Thus, the downstream front of the particle cloud outline shows a crushing linked to the resistance of the bottom against the sediment transport. Note that the sediment plume observed in Figure 2d (t = 14 s) has a very low concentration.

## 4. Conclusions

_{p}

_{1}case.

_{p}

_{2}cases, the POD modes show an elongated shape compared to the D

_{p}

_{1}cases. An analysis extended to the entire release phenomenon put to the fore that the particle size and its weight affect the passive transport of the sediment. Indeed, for D

_{p}

_{2}cases, the eigenfunctions extracted by POD present POD modes stretched on the bottom due to the enhancement of the opposition of the quiet water to the solid phase motion, compared to the D

_{p}

_{1}case. The evolutions of the total energy information extracted by both POD decompositions reveal some information about the influence of the particle diameter. While the impact on $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ appears for the decomposition of the short snapshot collection limited to the falling time, the impact on $E\left({\alpha}_{s}^{\prime}\right)$ appears later for longer snapshots collections covering the settling in the concentration field. Indeed, the POD

_{2C}presents the advantage of revealing the trends earlier than POD1C because the settling velocity date is intrinsic to the base parameter $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$.

_{2C}highlight the presence of a main structure layer placed over a second single layer of flat and creeping structures. The interaction of these two layers is different according to the particle size, and it seems to be linked to the behavior of the water–sediment mixture transport, and more particularly to the shape of the front of the particle cloud outline. Indeed, a water–sediment mixture collapse is observed for D

_{p}

_{2}and, conversely, this crushing is not observed for the D

_{p}

_{1}case.

_{p}

_{1}snapshots. According to the strength of U

_{c}, the total energies and the eigenfunction patterns extracted by both POD decompositions evolve. Indeed, the regime of the release phenomenon changes. For a weak ambient current, POD patterns are displaced to the left but show regular contours. For a strong ambient current, we observe an impact on the spatial organization of the POD pattern identified in the eigenfunction extracted by both decompositions. The decrease in their size and their gradual stretching and distortion reveal the loss of their spatial organization. Their complete displacement downstream of the injection point is also observed. The transition between the regime for U

_{c}15 and 20 cm/s is clearly identified in the curves of total energies.

_{1C}(i.e., POD

_{2C}) was applied to collections of 2D snapshots with one component (i.e., two components). A 3D case could be considered, but it would need to use a different solver. In future developments, we would like to use OPENFOAM coupled with SEDFOAM to solve the full 3D case as we carried out in the case of the flow around tidal turbines [29,30]. Those simulations are sensitive and time consuming. They will be the objective of future works. In future work, further investigation could be developed for the D

_{p}

_{2}case with different ambient current velocities to determine if the discrepancies observed stay or become more marked compared to the D

_{p}

_{1}case.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Villaret, C.; Claude, B.; Du Rivau, J.D. Etude Expérimentale de la Dispersion des Rejets par Clapage; LNHE, EDF: London, UK, 1998. [Google Scholar]
- Boutin, R. Dragage et Rejets en Mer. Les Produits de Type Vase; Presses de l’ENPC: Paris, France, 2000; 307p. [Google Scholar]
- Guillou, S.; Chauchat, J.; Pham Van Bang, D.; Nguyen, D.H.; Nguyen, K.D. Simulation of the dredged sediment’s release with a two-phase flow model. Bull. Perm. Int. Assoc. Navig. Congr.
**2011**, 142, 25–33. [Google Scholar] - Nguyen, D.H.; Lévy, F.; Pham Van Bang, D.; Guillou, S.; Nguyen, K.D.; Chauchat, J. Simulation of dredged sediment releases into homogeneous water using a two-phase model. Adv. Water Resour.
**2012**, 48, 102–112. [Google Scholar] [CrossRef] - Sirovich, L. Turbulence and the dynamics of coherent structures. Q. Appl. Math.
**1987**, 5, 561–590. [Google Scholar] [CrossRef] - Holmes, P.; Lumley, J.L.; Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems, and Symmetry; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar] [CrossRef]
- Cizmas, P.G.; Palacios, A.; O’Brien, T.; Syamlal, M. Proper-orthogonal decomposition of spatio-temporal patterns in fluidized beds. Chem. Eng. Sci.
**2003**, 58, 4417–4427. [Google Scholar] [CrossRef] - Yuan, T.; Cizmas, P.G.; O’Brien, T.O. A reduced model for a bubbling fluidized bed based on proper orthogonal decomposition. Comput. Chem. Eng.
**2005**, 30, 243–259. [Google Scholar] [CrossRef] - Brenner, T.A.; Fontenot, R.L.; Cizmas, P.G.; O’Brien, T.O.; Breault, R.W. Augmented proper orthogonal decomposition for problems with moving discontinuities. Powder Technol.
**2010**, 203, 78–85. [Google Scholar] [CrossRef] - Reddy, S.R.; Freno, B.A.; Cizmas, P.G.; Gokaltun, S.; MacDaniel, D.; Dulikravich, G.S. Constrained reduced-order models based on proper orthogonal decomposition. Comput. Methods Appl. Mech. Engrg.
**2017**, 321, 18–34. [Google Scholar] [CrossRef] - Haghgoo, M.Z.; Bergstrom, D.J.; Spiteri, R.J. Analyzing dominant particle-flow structures inside a bubbling fluidized bed. Int. J. Heat Fluid Flow
**2019**, 77, 232–241. [Google Scholar] [CrossRef] - Li, X.; Hu, B.X. Proper orthogonal decomposition reduced model for mass transport in heterogeneous media. Stoch. Environ. Res. Risk. Assess
**2013**, 27, 1181–1191. [Google Scholar] [CrossRef] - Li, X.; Chen, X.; Hu, B.X.; Navon, M.I. Model reduction of a coupled numerical model using proper orthogonal decomposition. J. Hydrol.
**2013**, 507, 227–240. [Google Scholar] [CrossRef] - Vermeulen, P.T.M.; Heemink, A.W.; Te Stroat, C.B.M. Reduced models for linear groundwater flow models using empirical orthogonal functions. Adv. Water Resour.
**2004**, 27, 57–69. [Google Scholar] [CrossRef] - Vermeulen, P.T.M.; Heemink, A.W.; Valstar, J.R. Inverse modeling of groundwater flow using model reduction. Water Resour. Res.
**2005**, 41, W06003. [Google Scholar] [CrossRef] - Polansky, J.; Wang, M. Proper Orthogonal Decomposition as a technique for identifying two-phase flow pattern based on electrical impedance tomography. Flow Meas. Instrum.
**2017**, 53, 126–132. [Google Scholar] [CrossRef] - Olbrich, M.; Bar, M.; Oberleithner, K.; Schmelter, S. Statistical characterization of horizontal slug flow using snapshot proper orthogonal decomposition. Int. J. Multiph. Flow
**2021**, 134, 103453. [Google Scholar] [CrossRef] - Munir, S.; Israr, M.; Heikal, M.R.; De Sercey, G. Identification of dominant structures and their flow dynamics in the turbulent two-phase flow using POD technique. J. Mech. Sci. Technol.
**2015**, 29, 4701–4710. [Google Scholar] [CrossRef] - Munir, S.; Farooq, U. POD based on vorticity: Application in a two-phase slog flow. J. Fluids Eng.
**2022**, 144, 041501. [Google Scholar] [CrossRef] - Munir, S.; Azis, A.R.A.; Heikal, M.R.; Israr, M. Combination of linear stochastic estimation and proper orthogonal decomposition: Application in two-phase slug flow. J. Braz. Soc. Mech. Sci. Eng.
**2023**, 45, 112. [Google Scholar] [CrossRef] - Nguyen, D.H.; Guillou, S.; Hien, L.T.T.; Nguyen, Q.H. Proper orthogonal decomposition of a very high concentrated release of sediment in water: Spatio temporal patterns. IJRTE
**2019**, 8, 2347–2353. [Google Scholar] [CrossRef] - Guillou, S.; Thiébot, J.; Chauchat, J.; Verjus, R.; Besq, A.; Nguyen, D.H.; Pouv, K.S. The Filling Dynamics of an Estuary: From the Process to the Modelling. In Sediment Transport in Aquatic Environments; Manning, A., Ed.; InTech: London, UK, 2011; pp. 125–146. ISBN 978-953-307-586-0. Available online: https://www.intechopen.com/chapters/20914 (accessed on 3 October 2011).
- Barbry, N.; Guillou, S.; Nguyen, K.D. Une approche diphasique pour le calcul du transport sédimentaire en milieux estuariens. Comptes Rendus De L’académie Des Sci.
**2000**, 328, 793–799. [Google Scholar] [CrossRef] - Chauchat, J.; Guillou, S. On turbulence closures for two-phase sediment-laden flows models. J. Geophys. Res.
**2008**, 113, C11017. [Google Scholar] [CrossRef] - Nguyen, K.D.; Guillou, S.; Chauchat, J.; Barbry, N. A two-phase numerical model for suspended-sediment transport in estuaries. Adv. Water Resour.
**2009**, 32, 1187–1196. [Google Scholar] [CrossRef] - Lundgren, T. Slow flow through stationary random beds and suspensions of spheres. J. Fluid. Mech.
**1972**, 51, 273–299. [Google Scholar] [CrossRef] - Graham, A.L. On the viscosity of suspensions of solid spheres. Appl. Sci. Res.
**1981**, 37, 275–286. [Google Scholar] [CrossRef] - Farout-Fréson, I.; Sergent, P.; Lefrançois, E.; Datt, G. Modèle numérique de clapage–phase de chute. In Proceedings of the IXèmes Journées Nationales Génie Civil—Génie Côtier, Brest, France, 12–14 September 2006; pp. 179–186. [Google Scholar] [CrossRef]
- Khaled, F.; Guillou, S.S.; Méar, Y.; Hadri, F. Impact of blockage ratio on the transport of sediments in the presence of a hydrokinetic turbine: Numerical modelling of the interaction sediments-turbine. Int. J. Sediment Res.
**2021**, 36, 696–710. [Google Scholar] [CrossRef] - Khaled, F.; Guillou, S.S.; Méar, Y.; Hadri, F. Numerical investigation of the local impact of hydrokinetic turbine on sediment transport—Comparison between two actuator models. Eur. J. Mech. B Fluids
**2023**, 102, 31–45. [Google Scholar] [CrossRef]

**Figure 1.**Illustrative sketch of the release phenomenon: (

**a**) description of the sediment release experiments, Rd and Ru are, respectively, downstream and upstream radius of the sediment cloud front; (

**b**) water–sediment mixture dumping without ambient current (D

_{p1}); (

**c**) water–sediment mixture dumping with ambient current (D

_{p1}-U

_{c20}).

**Figure 2.**Instantaneous turbidity C

_{m}[g/L] fields’ evolutions obtained by simulations: (

**a**) case D

_{p}

_{1}; (

**b**) case D

_{p}

_{2}; (

**c**) case D

_{p}

_{1}-U

_{c}

_{10}; (

**d**) D

_{p}

_{1}-U

_{c}

_{25}. N represents the number of the snapshot. The first image of each case represents the turbidity distribution at falling time. In the cases without velocity (D

_{p}

_{1}, D

_{p}

_{2}), the sediments settle and move on the bottom in both directions until the total sedimentation of the particles, whereas in the cases with a current (D

_{p}

_{1}-U

_{c}

_{10}, D

_{p}

_{1}-U

_{c}

_{25}), the sediment plume is transported in the left direction until the limit of the simulation domain.

**Figure 3.**Average root mean square error percentage according to k

_{max}, the number of POD spatio-temporal modes used in the snapshot reconstruction. The duration time of a sequence is equal to 0.1 M. The results shown in this figure are extracted by POD

_{1C}and POD

_{2C}for D

_{p}

_{2}particles applied to snapshot collections with a short duration time of 4 s (

**a**) and a longer one of 40 s (

**b**).

**Figure 4.**Results obtained for the D

_{p}

_{1}case for POD applied to a snapshot collection corresponding to the falling time T

_{f}= 1.5 s (dt = 0.1 s and M = 15). (

**a**) Mean solid volume fraction field and POD

_{1C}spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained from the difference compared to the mean solid volume fraction $\left({\alpha}_{s}^{\prime}\right)$; (

**b**) mean solid phase velocity field and POD

_{2C}spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained from the difference compared to the mean solid volume velocities $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$. Dark colours represent positive values and clear colors represent negative values; (

**c**) POD

_{1C}temporal orthogonal coefficients ${a}_{j}\left(t\right)$ (

**d**) POD

_{2C}temporal orthogonal coefficients ${a}_{j}\left(t\right)$; (

**e**) phase space projections of temporal coefficients obtained through the POD

_{1C}for D

_{p}

_{1}and D

_{p}

_{2}cases; (

**f**) phase space projections of temporal coefficients obtained through the POD

_{2C}for D

_{p}

_{1}and D

_{p}

_{2}cases.

**Figure 5.**(

**a**) Total POD

_{1C}energy ($E\left({\alpha}_{s}^{\prime}\right)$ [ ]) and (

**b**) total POD

_{2C}energy ($E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ [m

^{2}/s

^{2}]) according to the M number of snapshots decomposed for D

_{p}

_{1}and D

_{p}

_{2}. Here, dt is equal to 0.1 s, and the duration of the size collection is equal to Mdt.

**Figure 6.**(

**a**) Mean solid volume fraction field and POD

_{1C}spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained from the D

_{p}

_{1}case for the difference compared to the mean solid volume fraction $\left({\alpha}_{s}^{\prime}\right)$ and M = 200 (dark colours represent positive values and clear colors represent negative values). (

**b**) Particle’s cloud outline (solid line) superimposed to the four first eigenfunctions for M = 200. (

**c**) Particle’s cloud outline (solid line) superimposed to the four first eigenfunctions for the D

_{p}

_{2}case M = 200. (

**d**) POD

_{1C}temporal orthogonal coefficients ${a}_{j}\left(t\right)$. (

**e**) Phase space projections of temporal coefficients obtained through the POD

_{1C}for the D

_{p}

_{1}and D

_{p}

_{2}cases.

**Figure 7.**(

**a**) Mean field and POD

_{2C}spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained from the D

_{p}

_{1}case for the difference compared to the mean solid volume velocities $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ and M = 200 (with POD

_{2C}we obtain dimensionless spatial eigenfunctions; however, we have represented the rotational of each eigenfunction. Dark colors represent positive values and clear colors represent negative values). (

**b**) Outline obtained for the solid volume fraction velocities (solid line) superimposed to the four first eigenfunctions for the D

_{p}

_{1}case M = 200. (

**c**) Outline obtained for the solid volume fraction velocities (solid line) superimposed to the four first eigenfunctions for the D

_{p}

_{2}case M = 200. (

**d**) POD

_{2C}temporal orthogonal coefficients ${a}_{j}\left(t\right)$. (

**e**) Phase space projections of temporal coefficients obtained through the POD

_{2C}for D

_{p}

_{1}and D

_{p}

_{2}cases (as observed for the phase space projection $\left({a}_{1}\left(t\right),{a}_{2}\left(t\right)\right)$ (cf. Figure 7e), $\left({a}_{1}\left(t\right),{a}_{3}\left(t\right)\right)$ and $\left({a}_{2}\left(t\right),{a}_{3}\left(t\right)\right)$ also present similar shapes for both particle diameters. This finding suggests that, despite the differences observed elsewhere related to the impact of the diameter of the particles, the difference compared to the mean solid volume velocities maintain a similar dynamic).

**Figure 8.**Total POD energy for sand 1 according to the ambient current: $E\left({\alpha}_{s}^{\prime}\right)$ [ ] extracted from solid volume fraction snapshots collection and total kinetic energy $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$[m

^{2}/s

^{2}]. Both values are dimensionless by the values obtained without an ambient current: 0.63 for POD

_{1C}and 180 for POD

_{2C}.

**Figure 9.**Mean solid volume fraction field and POD1C spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained for the difference compared to the mean solid volume fraction (${\alpha}_{s}$’). Dark colors represent positive values and clear colors represent negative values: (

**a**) D

_{p}

_{1}-U

_{c}

_{10}(M = 200); (

**b**) D

_{p}

_{1}-U

_{c}

_{25}(M = 140); (

**c**) particle’s cloud outline (solid line) superimposed to the four first eigenfunctions for D

_{p}

_{1}-U

_{c}

_{10}and (

**d**) for D

_{p}

_{1}-U

_{c}

_{25}.

**Figure 10.**Mean solid phase velocity field and POD

_{2C}spatial dimensionless eigenfunctions ${\Phi}_{k}\left(\overrightarrow{x}\right)$ obtained for the difference compared to the mean solid volume velocities $\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$. To represent these eigenfunctions, it is necessary to calculate a pseudo-vector field equal to the rotation of the two-dimensional vector field. Dark colors represent positive values and clear colors represent negative values: (

**a**) D

_{p}

_{1}-U

_{c}

_{10}(M = 200); (

**b**) D

_{p}

_{1}-U

_{c}

_{25}(M = 140); outline obtained for the solid volume fraction velocities (solid line) superimposed to the four first eigenfunctions for (

**c**) D

_{p}

_{1}-U

_{c}

_{10}and (

**d**) for D

_{p}

_{1}-U

_{c}

_{25}.

**Table 1.**Testing conditions of the simulated cases and nomenclature: T

_{f}is the falling time defined from simulations, Ns is the size of the complete collection of snapshots outputted from simulation for each configuration studied, M the number of snapshots used in the POD decomposition (≤Ns), W

_{inj}is the injection velocity from the recipient, D

_{p}the sediment particle diameter (90 μm or 160 μm), μ the dry density of the solid (2650 kg/m

^{3}), Cm is the concentration of the mixture (450 g/L), V

_{r}is the volume of dumped material (60 L), and U

_{c}the ambient velocity [2].

Cases | D_{p}_{1} | D_{p}_{1}
-U_{c}_{10} | D_{p}_{1}
-U_{c}_{20} | D_{p}_{1}
-U_{c}_{15} | D_{p}_{1}-U_{c25} | D_{p}_{2} |
---|---|---|---|---|---|---|

Ns | 200 | 200 | 159 | 183 | 140 | 400 |

T_{f} (s) | 1.5 | 2.1 | 2.6 | 2.4 | 2.8 | 1.3 |

W_{inj} (m/s) | 0.79 | 0.79 | 0.79 | 0.79 | 0.79 | 0.89 |

D_{p} (μm) | 90 | 90 | 90 | 90 | 90 | 160 |

V_{r} (L) | 60 | 60 | 60 | 60 | 60 | 60 |

U_{c} (cm/s) | 0 | 10 | 20 | 15 | 25 | 0 |

Size of the decomposed collection | M = 15–200 (dt = 0.1 s) | M = 200 (dt = 0.1 s) | M = 159 (dt = 0.1 s) | M = 183 (dt = 0.1 s) | M = 140 (dt = 0.1 s) | M = 15–400 (dt = 0.1 s) |

Falling Time for the Case D_{p}_{1} | Falling Time for the Case D_{p}_{2} | |
---|---|---|

Experimental results (Villaret [1] and Boutin [2]) | 0.95 (0.45–1.45) | 0.72 (0.22–1.22) |

Numerical results in this work | 1.5 | 1.3 |

Numerical results—bi esp2D (Farout-Fréson [28]) | 1 | 0.7 |

**Table 3.**Energy contribution of the modes for POD applied to snapshot collection limited to the falling time.

D_{p}_{1} M = 15 (1.5 s) | D_{p}_{2} M = 13 (1.3 s) | |
---|---|---|

$E\left({\alpha}_{s}^{\prime}\right)$ | 1.27 | 1.19 |

$E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ | 98 | 140 |

${\lambda}_{1}/E\left({\alpha}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ and ${\lambda}_{2}/E\left({\alpha}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 44, 28 | 42, 29 |

${\lambda}_{1}/E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ and ${\lambda}_{2}/E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 68, 23 | 64, 24 |

$({\lambda}_{1}+{\lambda}_{2}+{\lambda}_{3})/E\left({\alpha}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 80 | 79 |

$({\lambda}_{1}+{\lambda}_{2}+{\lambda}_{3})/E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 96 | 94 |

**Table 4.**Energy contribution of the modes for POD applied to a snapshot collection describing the release phenomenon.

D_{p}_{1} M = 200 (20 s) | D_{p}_{2} M = 200 (20 s) | |
---|---|---|

$E\left({\alpha}_{s}^{\prime}\right)$ | 0.63 | 1.03 |

$E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ | 180 | 133 |

$({\lambda}_{1}+{\lambda}_{2}+{\lambda}_{3})/E\left({\alpha}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 43 | 63 |

$({\lambda}_{1}+{\lambda}_{2}+{\lambda}_{3})/E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)\left[\mathrm{\%}\right]$ | 62 | 64 |

Number of eigenvalues necessary to reach 95% of $E\left({\alpha}_{s}^{\prime}\right)$ | 15 | 18 |

Number of eigenvalues necessary to reach 95% of $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ | 10 | 10 |

**Table 5.**Energy contribution of the modes for POD applied to a snapshot collections describing the density current propagation for cases within current ambient.

D_{p}_{1} M = 200 (20 s) | D_{p}_{1}-U_{c}_{10} M = 200 (20 s) | D_{p}_{1}-U_{c}_{25} M = 140 (14 s) | |
---|---|---|---|

$E\left({\alpha}_{s}^{\prime}\right)$ | 0.63 | 0.59 | 0.8 |

$E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ | 180 | 184 | 224 |

Number of eigenvalues necessary to reach 95% of $E\left({\alpha}_{s}^{\prime}\right)$ | 15 | 18 | 14 |

Number of eigenvalues necessary to reach 95% of $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$ | 10 | 12 | 11 |

(Number of eigenvalues necessary to reach 95% of $E\left({\alpha}_{s}^{\prime}\right)$)/M [%] | 7.5 | 9 | 10 |

(Number of eigenvalues necessary to reach 95% of $E\left({u}_{s}^{\prime},{w}_{s}^{\prime}\right)$)/M [%] | 5 | 6 | 7.9 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Santa Cruz, A.; Nguyen, D.H.; Guillou, S.S.
On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current. *Water* **2023**, *15*, 3284.
https://doi.org/10.3390/w15183284

**AMA Style**

Santa Cruz A, Nguyen DH, Guillou SS.
On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current. *Water*. 2023; 15(18):3284.
https://doi.org/10.3390/w15183284

**Chicago/Turabian Style**

Santa Cruz, Alina, Duc Hau Nguyen, and Sylvain S. Guillou.
2023. "On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current" *Water* 15, no. 18: 3284.
https://doi.org/10.3390/w15183284