Fast Prediction of Solute Concentration Field in Rotationally Influenced Fluids Using a Parameter-Based Field Reconstruction Convolutional Neural Network

Abstract

Many high-performance fluid dynamic models do not consider fluids in a rotating environment and often require a significant amount of computational time. The current study proposes a novel parameter-based field reconstruction convolutional neural network (PFR-CNN) approach to model the solute concentration field in rotationally influenced fluids. A new three-dimensional (3D) numerical solver, TwoLiquidMixingCoriolisFoam, was implemented within the framework of OpenFOAM to simulate effluents subjected to the influence of rotation. Subsequently, the developed numerical solver was employed to conduct numerical experiments to generate numerical data. A PFR-CNN was designed to predict the concentration fields of neutrally buoyant effluents in rotating water bodies based on the Froude number (Fr) and Rossby number (Ro). The proposed PFR-CNN was trained and validated with a train-validation dataset. The predicted concentration fields for two additional tests demonstrated the good performance of the proposed approach, and the algorithm performed better than traditional approaches. This study offers a new 3D numerical solver, and a novel PFR-CNN approach can predict solute transport subjected to the effects of rotation in few seconds, and the PFR-CNN can significantly reduce the computational costs. The study can significantly advance the ability to model flow and solute transport processes, and the proposed CNN-based approach can potentially be employed to predict the spatial distribution of any physical variable in the lentic, ocean, and earth system.

1. Introduction

Wastewater effluents, riverine plumes, and cooling water are often discharged into large-scale water bodies, and the prediction of the relevant mixing and solute transport processes is important for assessing the relevant environmental and ecological effects and is also of significant importance in hydrodynamic, coastal, and offshore engineering [1,2,3,4]. For example, nutrients, sediment, and contaminants are often carried by rivers into large lakes, estuaries, or coastal oceans [5,6,7,8], and improper disposal of effluents significantly jeopardizes the ecology and environment of the receiving water bodies [9,10,11]. In addition, the incoming flows for many water-supply reservoirs are directed into storage in the form of turbulent jets, and a better understanding of the momentum and entrainment processes is necessary for predictions of the stratification and circulation of the general body of water, and these predictions are required for water quality analysis and optimal design of intakes [12]. The mixing and transport characteristics of effluents in small water bodies are typically governed by inertia and buoyancy [13], but the Coriolis effect for large water bodies is evident and cannot be neglected [12]. Furthermore, the fluids in a water treatment plant are also often influenced by rotation. To date, the accurate modeling of the mixing properties of effluent-driven solute transport subjected to the effects of rotation is still not properly handled and thus requires further investigation.

The existing jet-plume and solute-transport models can generally be classified into either near-field or far-field models. The term “near-field” refers to the region where the spreading and dilution characteristics are primarily governed by the local properties of the effluents [6]. Examples of well-known near-field models include CORMIX [14], VISJET [15], and JETLAG [16]. The term “far-field” refers to the region beyond the near field, and examples of popular far-field models include Delft3D, the Princeton Ocean Model, and the Regional Ocean Model System. Most of these models have adopted some significant simplifications or schematizations to reduce the computational costs. For instance, most near-field models use highly simplified versions of the mass and momentum conservation equations and assume horizontally homogeneous or other simple conditions for the receiving water environment [6]. These models become invalid for complex conditions. Most far-field models have adopted the hydrostatic assumption and ignored the velocity gradients in the near-field and spatial variations in turbulent diffusivity and viscosity. These assumptions or simplifications often lead to significant misunderstandings of the mixing and transport processes of water effluents.

In recent decades, the modeling of effluent-driven solute transport using fully three-dimensional (3D) computational fluid dynamics (CFD) models is becoming a mainstream practice. The commercial CFD software FLUENT is one of the most widely used packages for jet and plume simulations [17,18,19]. For example, M. Hosseini et al. [20] present a large eddy simulation investigation of heat transfer from impinging axisymmetric jets at moderate Reynolds numbers. M.S. Khan [21] presents a study on the large eddy simulation of turbulent confined jet using a commercial computational fluid dynamics code. Xiao and Tang [22] reported numerical simulations of multiple tandem jets in cross flow. Zhang et al. [23] successfully simulated vertical buoyant wall jets discharged into a linearly stratified environment, and Lou et al. [24] modeled two coalescing turbulent forced plumes in linearly stratified fluids using FLUENT. The popularity of OpenFOAM, an open-source CFD platform, has increased in recent years primarily due to its open-source nature, can be readily modified, and has abundant sub-modules, such as turbulence closures and numerical schemes, and it has been widely used in ocean engineering research [25,26,27]. Kheirkhah Gildeh et al. [28] developed a new solver based on the standard solver pisoFoam in OpenFOAM and simulated turbulent buoyant wall jets in stationary ambient water. Yan and Mohammadian [10] implemented a buoyancy-corrected k-ε turbulence model in OpenFOAM and utilized the modified pisoFoam solver to simulate vertical buoyant jets subjected to lateral confinement. Zhang et al. [29] performed large eddy simulations of 45° inclined dense jets using the solver twoLiquidMixingFoam within the OpenFOAM platform. The same solver has been further employed to study buoyant jets discharged from a rosette-type multiport diffuser [30], multiple inclined dense jets discharged from moderately spaced ports [31], multiple vertical buoyant jets in stationary ambient water [32], and inclined plane jets in a linearly stratified environment [33]. These models within the framework of OpenFOAM allow for both structured and unstructured meshes with different levels of local refinements, and the governing equations or numerical algorithms in the modeling system can be easily modified and are thus capable of providing high-performance numerical predictions for the spreading and dilution characteristics of water effluents. However, there are two challenges that limit the wider spread usage of these models. First, these high-performance models do not consider the influence of rotation, while the effects of rotation on hydrodynamics are evident for large-scale water bodies. Second, these models are very computationally expensive and are not affordable in many practical engineering applications.

Data-driven approaches based on modern machine learning (ML) algorithms for water-related problems have become a topic of significant research due to both academic and practical interests [34,35,36,37,38,39,40] and have recently been employed to complement CFD models. The existing studies either used ML algorithms to improve hydrodynamic models or directly developed surrogates for the numerical models. The former approach uses ML algorithms to develop new sub-models or numerical schemes for CFD models. More specifically, new turbulence closures or reduced-order models have been developed using ML approaches, and their applications showed that these models can provide satisfactory predictions and significantly reduce computational costs [41,42]. The other approach developed ML models as surrogates for numerical models. These models were typically trained on numerical simulations, and a key merit of this approach is that the established ML-based models can predict target variables using input variables without using the CFD models. ML-based surrogate models have been successfully established to predict the flow field of a curved open-channel flow [43], density-driven solute transport [44], the spatial distribution of flow depth in fluvial systems [45], etc. In recent years, data-driven models based on ML algorithms have also been proposed for effluent-driven solute transport [30,33,46], but none of them have considered the Coriolis effect. Furthermore, most of these studies on water effluents only predicted certain characteristic parameters instead of the complete concentration fields [30,33] or reconstructed the concentration fields by combining pointwise data instead of directly reproducing the concentration fields [46].

The convolutional neural network (CNN) algorithms have been shown to be powerful tools for dealing with problems in the field of ocean engineering [47,48,49]. A significant advantage of the CNN algorithm compared to traditional ML algorithms (such as artificial neural network and genetic programming) is that CNNs can better extract local features in image-like data and eliminate unnecessary computational costs thanks to the convolutional and pooling layers [44]. Examples of successful applications of CNNs in ocean and water-related problems include the classification of water pixels in satellite images [50], prediction of sea surface temperature [51], velocity field estimation on density-driven solute transport [44], modeling of two-dimensional (2D) unsteady flows around a circular cylinder [52], downscaling of daily precipitation and temperature [53], automatic water stage measurements [54], prediction of turbulent flow statistics past bridge piers in large-scale rivers [55], etc. Although these previous studies based on CNNs have made significant contributions, they have not focused on concentration field modeling of effluent-driven solute transport in rotating fluids. This gap in the literature prohibits the application of ML approaches in modeling of the mixing and transport processes of water effluents in large-scale water bodies, which are commonly subjected to the Coriolis effect.

The main purpose of this study was to develop a parameter-based field reconstruction CNN (PFR-CNN) for fast solute concentration field prediction in rotationally influenced fluids. As an explanatory example, we focused on horizontal neutrally buoyant effluents. More specifically, we attempted to construct a PFR-CNN to predict the concentration fields of horizontal neutrally buoyant effluents subjected to the effects of rotation. To prepare ground truth data for PFR-CNN development, a new numerical solver was implemented and utilized. To achieve this, we implemented a new numerical solver named TwoLiquidMixingCoriolisFoam within the framework of OpenFOAM, and the solver can simulate two mixing fluids with the Coriolis effect being considered. The established numerical model was subsequently utilized to perform simulations for different scenarios, which were used as training and validating cases. The outputs of the numerical solver were then processed to generate a training-validation dataset for the PFR-CNN establishment. The purposely built PFR-CNN in this study can predict the concentration fields based on the dimensionless governing parameters. After being trained and validated, the PFR-CNN was further employed to carry out predictions for two additional unseen cases, i.e., testing cases. The PFR-CNN predictions and numerical data were then compared to evaluate the generalization capability of the established PFR-CNN. Another ML algorithm, multi-gene genetic programming (MGGP), was also utilized for the same purpose, and the set of results was used as a reference for the evaluation of the PFR-CNN performance. The results in this study showed that the CNN predictions matched the ground truths very well, and the performance of PFR-CNN was better than the traditional approach, demonstrating that the proposed PFR-CNN is an effective tool for predicting effluent-driven solute transport in rotating fluids.

The remaining parts of this paper are organized as follows. Section 2 describes the problem, the experimental setup, the numerical models, the ML algorithms, processing of data, and performance metrics. Section 3 and Section 4 present and discuss the results, respectively. Section 5 summarizes the study and draws key conclusions.

2. Methodology

2.1. Statement of Problem

Figure 1 presents a schematic of a momentum jet (a horizonal neutrally buoyant effluent) in a rotating fluid with an initial effluent diameter D, source flow rate Q₀, and density ρ_j. The effluent discharges horizontally into the ambient fluid with uniform density ρ_j and rotating speed ω. In the initial region, the ambient water is entrained into the jet due to the shear layer and momentum transfer between the jet and the ambient fluid. For a momentum jet ejected into a non-rotating environment, the jet path is straight. However, for an effluent in a rotating environment, the transport pattern is altered by the rotation effects. Instead of being straight, the trajectory of a momentum jet in a rotating environment is typically in a spiral form. At the very early stage, the jet is dominated by the initial momentum and travels very rapidly in the initial direction, so the deflection of the path is not very obvious. The distance at which the turning of the jet becomes evident is hereafter denoted as y_r. The trajectory gradually reaches the terminal location, which is located at a distance of x_t in the original transverse and of y_t in the original streamwise direction from the nozzle’s surface.

The characteristics of a momentum jet in a non-rotating system are primarily governed by the discharge volume flux, Q, and kinematic momentum flux, M, which are defined as [13,56]:

$$Q=\frac{\pi }{4}{D}^2{U}_j;\hspace{1em}M=U_{j}Q$$

(1)

where U_j is the initial jet velocity.

In the presence of rotation, the effluent-formed flow pattern is affected by the Coriolis force. Therefore, the momentum effluent in a rotating environment is subjected to the combined effects of the discharge volume flux, kinematic momentum flux, and Coriolis force. These effects can be described by two dimensionless parameters: the Froude number (Fr) and Rossby number (Ro).

Fr is the ratio of the inertial forces to the gravitational forces and can be expressed as

$$Fr=\frac{u}{\sqrt{gD}}$$

(2)

where u indicates velocity, g indicates gravitational acceleration, and D is the nozzle diameter.

The Rossby number is the ratio of the velocity of a system to the product of the Coriolis parameter and the length scale of the motion, which can be expressed as

$$Ro=\frac{u}{Lf}$$

(3)

where L is the length scale and f is the Coriolis frequency.

2.2. Experiments and Considered Cases

Preliminary laboratory experiments were conducted to provide general observations and determine appropriate scenarios. The PIV-LIF (Particle Image Velocimetry-Laser Induced Fluorescence) system was initially planned to measure the concentration fields, but we found it difficult to set up the LIF facilities and obtain satisfactory data in the rotating system. Thus, simple dye tests were finally conducted (Supplementary Material), and we primarily relied on numerical models to prepare data for the development of data-driven models. Considering that the main purpose of this study was to evaluate the performance of data-driven algorithms and the data-driven models were developed based upon data generated by numerical models, these preliminary experimental runs were believed to be adequate.

The experiments were conducted in a 0.74 m × 0.74 m water tank on a rotating table at the University of Ottawa, and the experimental configurations are shown in Figure 2. The inner diameter of the nozzle was 0.4 cm. The center of the nozzle surface was kept at an elevation of 10.5 cm from the bottom of the tank, was located in the middle of the tank in the x direction, and was at a distance of 16 cm in the y direction from the closer side face of the tank. The water depth was kept at 15 cm. The flow rate was adjusted by the pump and valve, and the rotating speed was controlled by a switch for the turning table. The turning table rotated horizontally around the center of the tank. A camera was mounted on the tank to take records, and its location relative to the tank was kept fixed. The distance from the lens to the water surface was about 0.8 m. The frame width and height for the videos were 1280 and 720, respectively, and the frame rate was 29 frames/second. Before the start of ejecting the discharges, the rotating table was run for several minutes to establish a steady ambient environment, and the undulation of the surface was not very obvious for the current experimental conditions. The discharges were tap water mixed with Regal food dye (approximately 10%), which contains citric acid, FD&C #1, and sodium benzoate. The solution was neutrally buoyant within detectable limits. The same dye has been previously used and reported by Schreiner et al. [57]. These experiments were conducted to determine the reasonable ranges for Q₀ and T (rotating period) for the current experimental conditions. The tests with Q₀ and T out of the ranges were either found to be significantly influenced by the lateral confinements, or the corresponding Fr and Ro were not usually seen. The experimental parameters for the considered cases are summarized in

Table 1. Parameters of the considered cases.

Case	Q0 (cm3/s)	T (s)	ω	Fr	Ro
C01	13.5	17.5	537	5.42	4700
C02	24.1	17.5	959	9.68	8390
C03	12.6	20.5	501	5.06	5139
C04	20.2	20.5	804	8.11	8238
C05	14.8	24.0	589	5.95	7066
C06	15.3	24.0	609	6.15	7305
C07	13.1	32.0	521	5.26	8340
C08	11.7	40.0	466	4.70	9311
C09	14.7	40.0	585	5.91	11,698
C10	9.7	60.0	386	3.90	11,579

.

2.3. Governing Equations

The standard continuity and momentum equations for incompressible multiple fluids can be written as [32]:

$$\frac{\mathit{\partial }\rho }{\mathit{\partial }t}+\nabla \cdot \rho \mathbf{U}=0$$

(4)

$$\rho \cdot 2\omega U+\frac{\mathit{\partial }\rho \mathbf{U}}{\mathit{\partial }t}+\nabla \cdot \left(\rho \mathrm{UU}\right)=-\nabla ·\left(p_{rgh}\right)-gh\nabla \rho +\nabla \cdot \left(\rho \mathbf{T}\right)$$

(5)

with

$$\rho ={\alpha }_1{\rho }_1+{\alpha }_2{\rho }_2={\alpha }_1{\rho }_1+\left(1-{\alpha }_1\right){\rho }_2$$

(6)

$$\mathbf{T}=-\frac{2}{3}{\overline{\mu }}_{eff}\nabla \cdot \mathrm{UI}+{\overline{\mu }}_{eff}\nabla \mathbf{U}+{\overline{\mu }}_{eff}{\left(\nabla \mathbf{U}\right)}^T$$

(7)

$${\overline{\mu }}_{eff}={\alpha }_1{\left({\mu }_{eff}\right)}_1+{\alpha }_2{\left({\mu }_{eff}\right)}_2$$

(8)

$${\left({\mu }_{eff}\right)}_i={\left(\mu -{\mu }_t\right)}_i$$

(9)

where t = time, U = velocity, ρ = density, p = pressure, g = gravitational acceleration, α = volume fraction (normalized concentration field), μ = dynamic viscosity, and μ_t = turbulent viscosity. The terms $\nabla \cdot \left(p_{rgh}\right)$ and $gh\nabla \rho$ are obtained by $P=p_{rgh}+\rho gh$. The subscript i denotes either fluid 1 (effluent) or 2 (ambient water).

To incorporate the effects of rotation, the Coriolis term can be added at the left-hand side of the momentum equations, and the term can be written as $2\rho \mathbf{\omega }\times \mathbf{U}$, where $\mathbf{\omega }$ represents the angular velocity vector.

The α field, which is actually the normalized concentration field, can be modeled using a transport equation, as

$$\frac{\mathit{\partial }{\alpha }_1}{\mathit{\partial }t}+\nabla \cdot \left(\mathbf{U}{\alpha }_1\right)=\nabla \cdot \left(\left(D_{ab}+\frac{{\nu }_t}{S_C}\right)\nabla {\alpha }_1\right)$$

where D_ab is the molecular diffusivity, ν_t is the turbulent eddy viscosity, and S_C is the turbulent Schmidt number.

As for turbulence modeling, the standard k-ε model [58], which is one of the most widely used turbulence closures, was used. The model can be expressed as:

$$\frac{\mathit{\partial }k}{\mathit{\partial }t}+\nabla \cdot \left(\mathbf{U}k\right)-\nabla \left[(\frac{{\nu }_t}{{\sigma }_k}+\nu )\nabla k\right]=G-\epsilon$$

$$\frac{\mathit{\partial }\epsilon }{\mathit{\partial }t}+\nabla \cdot \left(\mathbf{U}\epsilon \right)-\nabla \left[(\frac{{\nu }_t}{{\sigma }_{\epsilon }}+\nu )\nabla \epsilon \right]=c_{1\epsilon }G\frac{\epsilon }{k}-c_{2\epsilon }\frac{{\epsilon }^2}{k}$$

$${\nu }_t=c_{\mu }\frac{k^2}{\epsilon }$$

where $k$ is the turbulent kinetic energy; $\epsilon$ is the turbulent energy dissipation rate; ${\nu }_t$ is the turbulent kinematic viscosity; $\nu$ is the kinematic viscosity; $G$ is the production of turbulence due to shear; ${\sigma }_k$, ${\sigma }_{\epsilon }$, $c_{1\epsilon }$, $c_{2\epsilon }$, and $c_{\mu }$ are model constants equal to 1.0, 1.3, 1.44, 1.92, and 0.09, respectively.

2.4. Numerical Models

The current work used two different numerical models for distinct purposes. The first numerical model was only used to better visualize the experimental images. The recorded images do not contain any velocity data, and it is impossible to observe the effluent development from the images. Thus, the OpenTELEMAC-Mascaret software was used to create models to generate vector fields for the images. In this model, the effluent boundaries were imported into the model, and the models were run to generate flow vectors. It should be emphasized that these vectors were only for visualization purposes and can only roughly represent the actual flow directions, so it cannot be used to provide reliable data for development of data-driven models.

The second model was newly implemented in the framework of OpenFOAM. It solves the governing equations for mass and momentum conservation with the Coriolis term and was employed to conduct synthetic numerical experiments. Since it was developed based upon the standard TwoLiquidMixingFoam solver available in OpenFOAM and considered the Coriolis force, it is named as TwoLiquidMixingCoriolisFoam in this study. The implemented solver was used to simulate the cases considered in the preliminary laboratory tests. The horizontal computational domain was kept consistent with the actual tank, that is, 0.74 m × 0.74 m. The vertical computational domain was set to be 0.15 m, which is consistent with the actual water depth. In terms of boundary conditions, the velocity inlet was imposed at the nozzle surface. The imposed velocity inlet at the nozzle surface can be calculated with $\mathrm{U}=\frac{{\mathrm{Q}}_0}{\mathrm{A}}$, $\mathrm{A}=\mathsf{\pi }{\mathrm{R}}^2$. The value of Q₀ can be read in Table 1. The slip wall boundary conditions were assigned to the bottom and side walls of the tank, and the inlet–outlet boundary condition was assigned to the top patch (atmosphere). The grid resolution was determined based on a mesh sensitivity analysis, and the number of cells in the final mesh was 656,992.

In terms of numerical schemes, the temporal terms were discretized using the Euler scheme. The gradient terms and viscous stresses tensor terms were discretized using the Gauss linear scheme. The α divergence term was discretized using the Gauss van-Leer scheme. The Laplacian terms were discretized using the corrected Gauss linear scheme. The k and ε divergence terms were discretized by the Gauss limited linear and Gauss upwind schemes, respectively, and the linear approach was used as interpolation schemes. The PIMPLE algorithms were used to couple pressure and momentum. The geometric agglomerated algebraic multigrid (GAMG) preconditioner was used for the pressure field. The Gauss Seidel solver was employed as the smooth solver for the α and velocity fields. The schemes or algorithms were selected because of their good performance and suitability in the current cases. More details about these algorithms have been well documented elsewhere in the literature [33,59,60].

The default time step was set as 1 s, but the actual transient time step was calculated by the solver based on the Courant number, the maximum value of which was set to be 0.5. The simulations were run up to 30 s, and the time interval for output writing was set as 1 s. The numerical outputs were written in the ASCII format with a precision of 6.

2.5. Parameter-Based Field Reconstruction Convolutional Neural Network (PFR-CNN)

CNNs, or Convolutional Neural Networks, are specifically designed to extract spatial features in image-like data. Compared to traditional neural networks, which process input data as one-dimensional vectors, CNNs use a series of convolutional layers that apply filters to the input data, allowing them to efficiently recognize patterns and relationships within images. The neurons in traditional ANNs are fully connected to all neurons in the previous layers but are not connected to other neurons in the same layers, so they are less capable of extracting local spatial features and contain many redundant connections. CNNs use convolutional layers to create a feature map and highlight local patterns and thus can better predict the spatial distribution of target variables. They also typically contain down-sampling layers (e.g., pooling layers) to reduce the data resolution and thus can provide higher efficiency. The characteristic length was set the same as the diameter of the port.

The flow and mixing properties of effluent-driven flows in rotating fluids are mainly governed by Fr and Ro, and thus, these two dimensionless parameters were used as the primary input variables. To model the transient processes, the time t was also used as an additional input variable. These three variables were divided by their maximum values before being fed into CNNs. The spatial distribution of the normalized concentration field at the central plane (z = 0.105 m) was used as the output variable.

Figure 3 shows the structure of the PFR-CNN designed in this study. In this CNN, the three dimensionless input variables were fed into the network and converted into input feature maps via a set of layers (input, dense, batch-normalization, etc.). It then performed up-sampling to include more information and down-sampling to reduce the data resolution. In order to overcome the issues of internal covariate shift and overfitting, the batch-normalization approach [61] was used for regularization, which alters the input distribution in the process of training and normalizes the output in the hidden layers. In terms of down-sampling, convolutional layers were utilized in this work. These convolutional layers for down-sampling purposes can be seen as trainable pooling layers and were employed in this study because the network was large enough for the current dataset. In terms of activation function, the ReLU approach was utilized because of its widespread usage and satisfactory efficiency. The PFR-CNN algorithm determined the parameters by minimizing the errors, and the errors were measured using the loss function based on the mean squared error.

A single PFR-CNN was trained in this work, which can predict the spatial distribution of normalized concentration for effluent-driven solute transport in rotating fluids at different time instants using Fr, Ro, and t. The network was constructed using Keras in TensorFlow 2. The Adam optimizer was used as the optimization algorithm. The learning rate, epoch, and batch size were set as 0.001, 1000, and 4, respectively.

2.6. Multigene Genetic Programming (MGGP)

MGGP is a relatively new machine learning technology that evolved from genetic programming (GP). GP is derived from Darwin’s evolutionary theory and imitates the natural selection and genetic mechanism of the biological world [62]. It is an extension of the genetic algorithm (GA). MGGP is a variant of GP, which combines the modeling ability of GP and statistical regression method. In traditional genetic programming, the evolutionary model is composed of one tree, while in MGGP, each model is composed of the weighted linear combination of several trees. In the MGGP technique, a population is composed of many individuals, and each individual represents a feasible solution [63]. The technique searches for the optimal solution through certain genetic operations, and it iterates step by step according to the optimal fitness. Some common genetic operations include selection, replication, exchange, and mutation n [64]. An advantage of MGGP is that it does not need to assume the model structures in advance. This can eliminate the errors due to models’ improper assumptions, figure out some deeply hidden relationships between variables, and lower the requirement of expertise. MGGP is an additive summation model, in which some trees can be weighted and linearly combined by multigene symbolic regression. Compared with some other machine learning algorithms, such as the GP approach, the complexity of the model is reduced because the required depth of trees is lowered, and thus a relatively compact model can be generated [65]. At the comparable level of complexity, the MGGP technique is typically more accurate, as an MGGP chromosome allows for multiple genes.

The workflow of the MGGP algorithm is shown in Figure 4. In this study, the input features included Fr, Ro, t, and the indices for the rows (i) and columns (j) of the raster. The output feature was the normalized concentration. The MGGP algorithm was performed using a modified MATLAB script GPTIPS2 [46,66]. The population was set as 500. The tournament size was set as 100, and the Pareto tournament and elite fraction were both set as 0.3. The number of genes and the tree depth were both set as 10. The algorithm was terminated based on one of the following conditions: the number of generations reaches 1000, or the training time reaches 5 h.

2.7. Pre- and Post-Processing of the Data

The outputs of the numerical data were written from 1 s to 30 s, and the numerical data were exported into CSV (Comma-Separated Values) files. The data extraction was performed using the open-source software ParaView: we first loaded the numerical outputs, set a slice at the central plane, and exported the data at the slice. A matrix containing the coordinates and normalized alpha values (x, y, and α) corresponding to each time instant was prepared for each case using MATLAB. For CNNs, the matrices with multiple arrays were converted into raster-like matrices for α. The matrices were then converted into grayscale images, which were finally served as synthetic data. For MGGP, additional arrays comprising the normalized Fr, Ro, t, i, and j were added into the matrices, which were directly used as the corresponding synthetic data. In terms of post-processing for the PFR-CNN outputs, the predicted images were resized and converted back to matrices. The matrices were used to generate pseudo-color images for visualization and reshaped as arrays of α for error analyses. With respect to the post-processing for the MGGP predictions, the predicted data were re-organized to generate the matrices similar to the converted matrices for PFR-CNN outputs, which can subsequently be used for pseudo-color image generation and error analyses.

2.8. Performance Metrices

As for the evaluation of model performance, two indicators, RMSE (root-mean-squared error) and R² (coefficient of correlation), were used:

$$RMSE=\sqrt{\frac{1}{N}{\Sigma }_{i=1}^N{({\alpha }_a-{\alpha }_p)}^2}$$

$$R^2={\left|\left.\frac{N{\displaystyle \sum _{i=1}^N({\alpha }_p{\alpha }_a)-{\displaystyle \sum _{i=1}^N\left({\alpha }_p\right){\displaystyle \sum _{i=1}^N\left({\alpha }_a\right)}}}}{\sqrt{\left[{\displaystyle \sum _{i=1}^N\left({\alpha }_p{}^2\right)-{\left({\displaystyle \sum _{i=1}^N{\alpha }_p}\right)}^2}\right]\left[{\displaystyle \sum _{i=1}^N\left({\alpha }_a{}^2\right)-{\left({\displaystyle \sum _{i=1}^N{\alpha }_a}\right)}^2}\right]}}\right|\right.}^2$$

where α_a indicates the actual values of α, α_p indicates the predictions, and N denotes the number of data pairs.

3. Results

3.1. General Observations and Numerical Experiments

Figure 5 is a typical instantaneous figure that shows the evolution of the effluent-induced flow observed in the preliminary laboratory experiments. The preliminary experiments were conducted to provide some fundamental qualitative observations and select suitable cases and thus did not capture the flow vectors. After each experiment, the recorded data were transferred to the computer for analysis. Reference points were marked on the tank, and the perspective distortions of the images were corrected using MATLAB and the open-source image editor GIMP (GNU Image Manipulation Program), with the reference points being used for the corrections. The snapshots at different time instants were extracted, and effluent boundaries were detected based on the grayscale values of the pixels. The boundaries of the effluents were extracted and imported into the software BlueKenue, and the geometry and boundary condition files were prepared for the open-source CFD package OpenTELEMAC-Mascaret, which can provide the flow fields based on the flow outlines. The flow vectors, roughly estimated with the CFD code, were sketched on the same figure. The flow vectors are not accurate but are helpful in visualizing the evolution of the effluents.

In this case, (C05), the nozzle diameter was 0.4 cm. The flowrate was 14.8 cm³/s, and the rotation period was 24.0 s. As can be seen in the figure, at the very early stage (e.g., t = 2 s), the jet trajectory was close to a straight line, similar to a free jet without the Coriolis effect, partially because the flow properties were primarily governed by the initial momentum. Gradually, the Coriolis effects became more obvious, and the jet started bending. For example, at about t = 6 s, there was a clear shift of the jet centerline towards the right side. At about t = 8 s, the jet became perpendicular to the initial jet direction. After that, the jet continuously developed, and the trajectory appeared to be spiral-shaped. The flow patterns in the other tests were quite similar to this case.

To validate the numerical model, the experimental case was reproduced by the numerical model developed in OpenFOAM. Figure 6 presents the sample snapshots of the normalized concentration field at different time sequences for the same case as Figure 5. It should be noted that Figure 5 was adapted based on pictures taken above the tank instead of measurements at the central plane while the plane in Figure 6 was exactly the central plane, so there are some minor differences between the results in the two figures. However, it can be seen that the numerical model reproduced the above general observations very well and the results were consistent with general flow theories, demonstrating that the Coriolis term has been correctly incorporated in the numerical modeling system. The numerical model provides predictions based on strict principles of physics, which are reliable, and the outputs can better show the spatial distribution of the concentration field compared to the preliminary experimental images. Overall, the good match validated the satisfactory performance of the CFD method. Therefore, the numerical model was employed to conduct numerical experiments to generate a comprehensive dataset.

The numerical experiments considered different scenarios with different Fr and Ro values. The numerical data at the central plane were exported and processed to create grayscale maps for the spatial distribution of the normalized solute concentration. For a better visualization, pseudo-color was imposed on these maps, and sample maps are shown in Figure 7. The figure clearly shows that the effluents were significantly influenced by Fr and Ro. Generally, the effluents extended farther in the initial flow direction (the y direction) in the cases with greater values of Fr and Ro. This can be explained by the relative importance of the initial momentum and the Coriolis force. A free jet in a non-rotating system generally has two regimes in space: zone of flow establishment (ZFE) and zone of established flow (ZEF). ZFE extends from the port surface to the point where the centerline velocity is affected by the water entrainment at the jet edges, and ZEF extends from the end point of zone of flow establishment to infinity [67]. The experimental results implied the existence of this flow structure, as the mixing properties near the port surface showed obviously different behavior than those in the far field. However, the ZFE-ZEF theory is very difficult to apply to the current problem because the fluid mechanisms became quite complicated when the Coriolis force was considered, and thus this study explained the observations directly from the viewpoint of the balance between momentum and Coriolis force. It is acknowledged that conducting more numerical experiments may be useful in further enriching the dataset. However, the current synthetic experiments have already covered broad data ranges, and it is meaningful to evaluate the performance of ML algorithms with limited amount of data. Therefore, the number of numerical tests in the current study is believed to be adequate.

3.2. Assessment of the PFR-CNN Algorithm

To determine the optimal value of epoch, the convergence history of the training and validation process for the CNN based models was recorded, which was shown in Figure 8. The mean absolute error (MAE) at the early stage was high, but it rapidly dropped to values below 0.005 and became statistically steady after about 400 epochs. At the early stage, the oscillations in the errors for the validating data were quite evident, and after about 400 epochs, the differences between the errors for the training and validating datasets became negligible, indicating that the issue of overfitting was well controlled. To be conservative, the number of epochs was set as 1000, which is sufficient based on the convergence history shown in the figure. To further evaluate the performance of the trained and validated PFR-CNN models, two additional cases were considered: C04 and C08. These two cases were not used in the processes of model training and validation and were thus unseen data for model testing.

Figure 9 presents the concentration fields at various time instants for case C04 obtained by the ground truth dataset and the developed PFR-CNN-based model. Overall, the spatial distributions of solute concentrations predicted by the PFR-CNN-based model showed a good agreement with those obtained by the numerical model. There was some noise in the PFR-CNN predictions, which can be attributed to the relatively small size of dataset. However, the key characteristic parameters, including the distance at which the turning of the effluent becomes obvious, the terminal location, and the curvature of the effluent path, can be clearly identified and matched the ground truths very well. The ground truths and PFR-CNN predictions for the case C08 were compared in Figure 10, and the comparisons reconfirmed the good performance of the developed PFR-CNN-based models.

The RMSE and R² values were calculated for each map, and the results are presented as heatmaps in Figure 11 and Figure 12, respectively. It can be seen that the RMSE values were well below 0.01, and most of the R² values exceeded 0.9. The matrices showed that the errors for the cases near the edges of the heatmaps were relatively larger, primarily because there were comparably fewer samples for these extreme values. In general, the RMSE values were very low, and the R² values were very high, confirming that the established model is a promising tool for predicting effluent-driven solute transport in rotating fluids.

4. Discussion

Modeling the mixing and transport patterns in large water bodies, which are typically subjected to the effects of Earth’s rotation, has many practical applications in the field of hydrodynamic, river, and coastal engineering. For instance, wastewater or cooling water effluents are often discharged into large lakes or coastal regions [6], which are influenced by the earth’s rotation. The interactions between riverine outflows and oceans often induce river plumes, which are subjected to the Coriolis force [7]. River intrusions with sediment, nutrients, and contaminants can also be common in stratified and rotational lacustrine and coastal systems [5]. It is also of significant importance in inland water research. For example, many large reservoirs concentrate their incoming water into jets to break up the stratification and increase the circulation of the water in the reservoirs, and the earth’s rotation has evident effects on the flow and mixing patterns in these reservoirs [12]. These applications require accurate predictions of effluent-driven solute transport in rotating fluids for proper assessment of the environment and ecosystem and sound design of engineering projects and management measures.

Mathematical modeling of effluent-driven solute transport in rotationally influenced systems has received less attention compared to the modeling of non-rotational systems. Thus, as introduced in Section 1 of this paper, the existing models have not adequately handled the relevant phenomena. A key contribution of this study is the implementation of the new numerical solver TwoLiquidMixingCoriolisFoam within the framework of OpenFOAM. The solver was implemented to solve the modified governing equations with the Coriolis term incorporated. Figure 13 compares the concentration fields at various time instants for Case C06 with and without the Coriolis force considered, and the results showed that the solute transport was significantly altered by the Coriolis force. The results in Figure 13 along with those in Section 2 demonstrated that the solver can effectively capture the flow and mixing patterns in a rotating environment. The solver is a new powerful tool for simulating effluent-driven solute transport. It can combine the merits of both near-field and far-field models; specifically, it can provide detailed predictions near the effluents and consider the Coriolis effect. It can also be readily modified and extended thanks to the open-source nature of OpenFOAM and can be easily coupled with different turbulence closures, numerical algorithms, and other modules available in OpenFOAM. Furthermore, the numerical solver has been employed to conduct numerical experiments, which provided a new synthetic dataset that is useful for developing data-driven models.

A main contribution of this study is the development of the PFR-CNN for rapidly predicting the concentration field in rotating fluids. To the best of the authors’ knowledge, such a CNN has not been previously reported. The PFR-CNN-based model can quickly predict the normalized concentration field using Fr and Ro. Compared to numerical models, the most important merit of the PFR-CNN-based model is that it can provide predictions in a few seconds. The average computational cost of the numerical simulations was approximately 5 h. This is much faster than numerical models, which typically require a computational time of several hours or even days. This merit can provide many benefits to the community of researchers and engineers involved in ocean and offshore engineering. For example, the developed PFR-CNN-based model enables quick predictions of emergent water pollution events and fast design of relevant engineering projects, such as diffusers and outfalls. In addition, compared to numerical models, the developed PFR-CNN-based model requires less expertise to use, as it only requires the modeler to replace the data in the input files to make new predictions. It also requires less disk storage space than numerical models. Due to the fact that the network can make arbitrary new predictions in a few seconds, it is not necessary to install the CFD software and store the outputs. The storage size of the file for the trained network was 149 MB, whereas that of the raw numerical outputs for the 10 cases was around 30 GB. The trained model can be executed without CFD software, so it can be easily coupled with other software or embedded into other platforms. Furthermore, the current model was trained using numerical data, so its prediction accuracy may not be higher than the numerical model, but the PFR-CNN model can be easily and continuously improved with more high-fidelity data. It can be expected that a PFR-CNN model would outperform numerical models in the future, when spatially and temporally dense measurements are available.

To further evaluate the merits of the proposed PFR-CNN, the MGGP algorithm was also utilized to predict the same concentration fields. The MGGP algorithm was selected to act as a reference algorithm because it is an advanced version of traditional ML algorithms and has been successfully applied to predict the mixing properties of effluents. The ANFIS (Adaptive-Network-Based Fuzzy Inference System) was also attempted, but its data processing was too slow, and thus it was not further considered. Figure 14 presents the RMSE and R² values for the MGGP predictions corresponding to the training cases, respectively. The PFR-CNN counterparts are also shown in the same plots. The results in the figures show that the PFR-CNN algorithm significantly outperformed the MGGP algorithm. It is acknowledged that the prediction accuracy of the MGGP algorithm could be substantially improved. However, the performance of different algorithms should be evaluated based on the trade-off between accuracy and efficiency, and thus two termination conditions were set for the MGGP algorithm: the number of generations and the termination time. The MGGP algorithm in this study was actually terminated by the latter condition. The results showed that the MGGP predictions were substantially worse than PFR-CNN at comparable computational costs. As previously introduced, the good performance of CNN in dealing with image-like data can be mainly attributed to the convolutional layers and down-sampling layers, which can better extract spatial features and reduce computational costs. These merits of the CNN algorithm imply that the proposed PFR-CNN can potentially be employed to predict the spatial distribution of any physical variable in the lentic, ocean, and earth system.

Despite the contributions of this work, there are also some limitations in this study. The current work focuses on the concentration fields at the central plane, and it is possible in future studies to extend the PFR-CNN model to reconstruct 3D concentration fields. In addition to the concentration field, the flow and turbulence fields can also be predicted using the network through transfer studies. As an explanatory example, the current study used laboratory-scale experiments, but experiments in larger-scale basins are preferred in further research. This work focuses on neutrally buoyant effluents discharged horizontally into the receiving water body, and positively or negatively buoyant effluents discharged at certain angles can be investigated in future studies.

5. Conclusions

It is important to develop effective and efficient tools to rapidly predict the effluent mixing and solute transport processes in large water bodies for a sound assessment of the relevant environmental and ecological effects. This work implemented a new numerical solver TwoLiquidMixingCoriolisFoam that can predict multiple mixing fluids under the effects of rotation and employed the solver to conduct numerical experiments. The data from the numerical experiments were subsequently utilized to established PFR-CNN-based models. The network designed in this work can effectively predict the spatial distribution of the target variable using the governing parameters. More specifically, the developed PFR-CNN-based models can predict the concentration fields of effluent-driven solute transport in rotationally influenced fluids based on the Froude and Rossby numbers. The results showed that the developed PFR-CNN-based models can accurately predict the transient variations in the spatial distribution of the normalized concentrations. As a reference algorithm, the MGGP algorithm was also used to develop ML-based models. However, the performance comparisons showed that the proposed PFR-CNN approach can significantly outperform the MGGP algorithm at comparable computational costs. The PFR-CNN-based model can substantially reduce the requirements of computational resources and disk storage space of numerical models, is easy to use, and can be continuously improved. It has many practical applications in the field of ocean and water engineering regarding the wastewater or cooling water effluents discharged into large lakes or coastal regions, the interactions between riverine outflows and oceans, incoming water in large reservoirs, and river intrusions with sediment, nutrients, and contaminants in stratified and rotational lacustrine and ocean systems. The advantages of the proposed network imply that it can potentially be employed to predict the spatial distribution of any physical variable in the lentic, ocean, and earth system and as such can be used in broader applications. This study demonstrated that the proposed approach was a promising tool and it can facilitate accurate estimation of effluent-driven solute transport in rotationally influenced fluids. The topic has many practical applications. For instance, wastewater or cooling water effluents are often discharged into large lakes or coastal regions, which are influenced by the earth’s rotation. The interactions between riverine outflows and oceans often induce river plumes, which are subjected to the Coriolis force. River intrusions with sediment, nutrients, and contaminants can also be common in stratified and rotational lacustrine and coastal systems. The proposed approach is a promising tool in these applications, which are very important applications in the field of water resources and ocean and offshore engineering. The current work focuses on the concentration fields at the central plane, and it is possible in future studies to extend the PFR-CNN model to reconstruct 3D concentration fields. As an explanatory example, the current study used laboratory-scale experiments, but experiments in larger-scale basins are preferred in further research. Future studies are recommended to extend the PFR-CNN model to reconstruct 3D concentration fields, to consider ablation study, to predict flow and turbulence fields, to perform experiments in larger-scale basins, and to investigate positively or negatively buoyant effluents discharged at certain angles.