Densimetric Froude Number Research Articles

Density effects on the hydrodynamics and mixing at a large confluence with a compound channel tributary

Abstract Confluences are marked by converging streamlines, complex hydrodynamics features and mixing processes. Understanding such intricate flow patterns and their effects on the mixing at natural river junctions is difficult, particularly for large confluences with a compound channel tributary, due to the scarcity of field data. This research aims to assess the effects of density differences on hydrodynamics and mixing at the large river confluence between the Yangtze River and its tributary, the Poyang Lake, whose outflow channel has a large inner-side floodplain. Field data were collected during three surveys under various flow conditions to analyse how the alternate flow mechanism (compound channel vs single channel) of the Poyang Lake outflow channel influences the confluence mixing process. In high flow and relatively high flow conditions, the Poyang Lake outflow channel functioned as a compound channel and switched to a single channel in low flow. Acoustic Doppler current profiling (ADCP) was used in all three surveys to explore the features of hydrodynamics and the mixing process at the confluence, supplemented by water quality sampling to characterise the mixing patterns. Secondary flows observed during the field surveys were found to be affected by alterations in the flow mechanism (compound channel vs single channel) of the Poyang Lake outflow channel and the density effects, which were characterised using the densimetric Froude number Frd. Ultimately, the findings obtained in these field surveys confirm the role of density differences between the tributaries in significantly affecting hydrodynamics features and the mixing process at river confluences.

Trajectory and spreading of falling circular dense jets in shallow stagnant ambient water

This study investigates the trajectory and spreading of a horizontal circular dense jet flow discharged above the water surface, the flow of which falls into shallow stagnant ambient water. For this purpose, 27 experiments were performed using three diameters, flow rates, and falling heights to investigate jet trajectory. In addition, nine experiments with constant falling height were conducted to study jet spreading. The densimetric Froude number of the jet flow at the nozzle outlet and water surface of the present study ranged from 0.72 to 4.22 and 36.96 to 86.10, respectively. The data pertaining to this study were extracted and analyzed through image processing. The results of the experiments showed that by increasing both the momentum and falling height, the trajectory of the jet flow reached a farther distance from the discharge nozzle. The spreading of the outer boundaries of the dense circular falling jet flow tended to extend downstream of the impact point and had an elliptical motion on the bed. The relationship between radial distance from the impingement point to the outer boundary of flow and time was determined to have a power of 0.67 for flow along the flume and 0.59 for flow along the flume's width.

Modeling flow resistance and geometry of dunes bed form in alluvial channels using hybrid RANN–AHA and GEP models

Dunes formation in sandy rivers significantly impacts flow resistance, subsequently affecting water levels, flow velocity, river navigation, and hydraulic structures performance. Accurate prediction of flow resistance and dune geometry (length and height) is essential for environmental engineering and river management. The current paper introduces two models to evaluate the flow resistance and geometry of dunes formed in sand-bed channels. The first model, RANN–AHA is a hybrid artificial intelligence model using the recurrent artificial neural network (RANN) linked with the artificial hummingbird optimization algorithm (AHA) to optimize the biases and weights of the neural network model. The second model uses gene expression programming (GEP) as a nonlinear approach based on a genetic algorithm (GA) and genetic programming (GP) to explicitly determine dune characteristics. For both models, the input parameters include flow and sediment characteristics, while Manning's roughness coefficient (nM), and relative dune height, h/H or h/L, were used as output parameters where h is the dune height, H is the flow depth above the dune crest, and L is the dune length. Five different published flume data sets were compiled for the analysis. Sensitivity analysis was done using different combinations of input parameters. It was found that the combination of hydraulic radius divided by median diameter (RH/d50), Reynolds number (Re), Particle densimetric Froude number (F∗), and grain Froude number (FG) yielded the best prediction accuracy for estimating Manning nM and relative height, h/H or h/L, with a root mean square error (RMSE) = 0.00027, 0.0504, and 0.0078 and a correlation coefficient (R) = 0.9989, 0.942, and 0.9272, respectively. Model verification proved that the RANN–AHA model outperformed the GEP model and most of the previous studies available in the literature when predicting the roughness coefficient and dune geometry in sand bed channels.

Modeling of scour hole characteristics under turbulent wall jets using machine learning

The novelty of the present study is to investigate the parameters that depict the scour hole characteristics caused by turbulent wall jets and develop new mathematical relationships for them. Four significant parameters i.e., depth of scouring, location of scour depth, height of the dune and location of dune crest are identified to represent a complete phenomenon of scour hole formation. From the gamma test, densimetric Froude number, apron length, tailwater level, and median sediment size are found to be the key parameters that affect these four dependent parameters. Utilizing the previous data sets, Multi Regression Analysis (linear and non-linear) has been performed to establish the relationships between the dependent parameters and influencing independent parameters. Further, artificial neural network-particle swarm optimisation (ANN-PSO) and gene expression programming (GEP) based models are developed using the available data. In addition, results obtained from these models are compared with proposed regression equations and the best models are identified employing statistical performance parameters. The performance of the ANN-PSO model (RMSE = 1.512, R2 = 0.605), (RMSE = 6.644, R2 = 0.681), (RMSE = 6.386, R2 = 0.727) and (RMSE = 1.754, R2 = 0.636) for predicting four significant parameters are more satisfactory than that of regression and other soft computing techniques. Overall, by analysing all the statistical parameters, uncertainty analysis and reliability index, ANN-PSO model shows good accuracy and predicts well as compared to other presented models.

Benchmarking the performance and uncertainty of machine learning models in estimating scour depth at sluice outlets

ABSTRACT This study investigates the performance of six machine learning (ML) models – Random Forest (RF), Adaptive Boosting (ADA), CatBoost (CAT), Support Vector Machine (SVM), Lasso Regression (LAS), and Artificial Neural Network (ANN) – against traditional empirical formulas for estimating maximum scour depth after sluice gates. Our findings indicate that ML models generally outperform empirical formulas, with correlation coefficients (CORR) ranging from 0.882 to 0.944 for ML models compared with 0.835–0.847 for empirical methods. Notably, ANN exhibited the highest performance, followed closely by CAT, with a CORR of 0.936. RF, ADA, and SVM performed competitive metrics around 0.928. Variable importance assessments highlighted the dimensionless densimetric Froude number (Fd) as significantly influential, particularly in RF, CAT, and LAS models. Furthermore, SHAP value analysis provided insights into each predictor's impact on model outputs. Uncertainty assessment through Monte Carlo (MC) and Bootstrap (BS) methods, with 1,000 iterations, indicated ML's capability to produce reliable uncertainty maps. ANN leads in performance with higher mean values and lower standard deviations, followed by CAT. MC results trend towards optimistic predictions compared with BS, as reflected in median values and interquartile ranges. This analysis underscores the efficacy of ML models in providing precise and reliable scour depth predictions.

Advancements in predicting scour depth induced by turbulent wall jets: A comparative analysis of mathematical formulations and machine learning models

This study examines the scour depth induced by turbulent wall jets and proposes novel mathematical formulations to predict the depth of scouring. Through a comprehensive gamma test, key parameters influencing the scour depth are identified, including the apron length, densimetric Froude number, median sediment size, tailwater level, Reynolds number, and Froude number of the jet. Regression analysis is subsequently conducted to establish relationships between the dependent parameter and the aforementioned independent variables. A comparative analysis is then undertaken between the measured scour depths and those predicted by existing equations from previous studies. Furthermore, predictive models leveraging the support vector machine, artificial neural network with particle swarm optimization, M5 tree algorithm, gene expression programming, and adaptive neuro-fuzzy inference system (ANFIS) are developed using the collected data. Statistical metrics are employed to evaluate the performance of each model and the regression equation. The effectiveness of each model in predicting scour depth is demonstrated. Notably, ANFIS yields a coefficient of determination of 0.809 and a root mean square error (RMSE) of 1.585. Multi-nonlinear regression analysis exhibits a coefficient of determination of 0.752 and an RMSE of 0.421, while the M5 tree achieves a coefficient of determination of 0.739 and an RMSE of 1.874, demonstrating superior performance compared to other machine learning techniques and regression equations employed in this study.

Water renewal and stratification modelling in small estuaries

Water renewal and flushing in small, intermittently open or closed estuaries is receiving increasing attention particularly in light of the climate change induced alterations in run-off, wave and sediment transport conditions along coasts. The challenges of predicting the stratification-circulation state and the balance between tidal or freshwater flushing in response to the mouth dynamics of small, wave-dominated estuaries is the focus of the paper. Such predictions are required for determining estuary freshwater requirements or establishing an estuary's capacity to maintain sound water quality under pollutant discharges. Advances in simulating changes in stratification-circulation over long time scales are limited. Instead attention has focused on generating indices of stratification or water quality state using heuristic methods. In this paper, systems dynamics modelling is applied to simulate the non-linear response of the estuary to changes in river and marine water fluxes. The estuary is modelled as a basin with a specified water volume to water level relationship, connected to the sea by a channel with variable sill height, but fixed width. The direction and magnitude of the flow through the mouth determines whether the sill height erodes or accretes and hence the mouth dynamics (see Slinger, 2017). The tidal flux through the mouth co-determines the volumetric exchange of salt, influencing both the stratification state of the estuary and the degree of tidal or freshwater flushing. This is also influenced by run-off. The resulting dynamic balance is captured in two bulk indices, the Estuarine Richardson number and the bulk densimetric Froude number. Using measured data from the Great Brak Estuary, South Africa, the model is calibrated. Model simulations demonstrate the importance of tidal flushing and concomitant mouth breaching for water renewal as freshwater flushing declines under scenarios of increased water abstraction. Although the estuary remains partially mixed, there is increased average salinity and a more uniform the water column. Water releases and mouth breaching bring about a more natural stratification-circulation state, but these effects are short-lived.

Investigation of dredging pattern due to changes in jet cross-section

Sediment management is a significant part of the decision-making process that hydraulic engineers need to undertake as it has a direct bearing on the environment. Sediment dredging from around intake structures and reservoirs impose a high operating cost. Jets can efficiently remove large quantities of sediment at low operational costs. This study assesses the scouring pattern development by changing nozzle parameters to achieve maximum scouring conditions. Toward this end, scour holes induced by jets generated using the nozzles with four inner angles (30°, 45°, 60°, and 90°) are tested on a cohesionless sediment bed. The experimental results show that the inner nozzle angle, α, and the densimetric Froude number, F0, affect the scour pattern. This work illustrates that the scour hole dimensions grow with increasing α and F0. Based on the velocity measurements, for the range of jet discharges considered herein, the relative velocity Um/ U0 is increased by 20%–25% for the nozzle with α = 90° compared to the nozzle with α = 30°.

Hydro‐Sedimentary Processes of a Plunging Hyperpycnal River Plume Revealed by Synchronized Remote Imagery and Gridded Current Measurements

AbstractThe present knowledge of plunging hyperpycnal river plumes is mainly based on two‐dimensional (confined) laboratory experiments. Several hypotheses on three‐dimensional (unconfined) flow processes have been made, but not tested in situ. In this field study, the dominant three‐dimensional hydro‐sedimentary processes related to unconfined plunging were elucidated by synchronizing autonomous time‐lapse camera images with boat‐towed acoustic Doppler current profiler measurements. It was found that the flow field complies with two‐dimensional conceptualizations along the plume centerline. Perpendicular to the centerline, the plume slumped laterally due to its density excess and simultaneously converged laterally due to its vertical divergence. This combination led to a narrowing of the plume near the surface, resulting in a sediment‐rich triangle‐shaped pattern on the surface near the inflow, and a rather stable width near the bed. The formation of secondary flow cells transporting riverine water and suspended particulate matter (SPM) away from the plume near the bed, up toward the surface and back toward the plume near the surface, was revealed. An increase of the average SPM concentration and the SPM flux in the main flow direction indicates net sediment erosion under the investigated conditions of high discharge and sediment load. This suggests transient storage of sediment during conditions of lower discharge and sediment load, and high morphological activity in the plunging area. These findings allowed extending a classical conceptual plunging model to laterally unconfined sediment‐laden plunging inflows for conditions of well‐mixed ambient water in the plunging region and inflow densimetric Froude numbers exceeding 1, common in nature.

CFD analysis of the full-scale resistance of an oil tanker in presence of a mud–water interface

The presence of mud layers on the bottom of ports and waterways can have negative effects on the hydrodynamic behaviour of marine vessels. This numerical study investigates the effect of muddy seabeds on the full-scale resistance of an oil tanker sailing straight ahead. The objective is to determine the influence of factors such as the densimetric Froude number, UKC and mud rheology at speeds between 3 and 9 knots. The numerical study is conducted using a finite-volume Reynolds-Averaged Navier–Stokes (RANS) flow solver combined with the Volume-Of-Fluid (VOF) method to capture the mud–water interface. At certain critical speeds, the presence of mud increased the ship’s total resistance by up to 15 times compared to the case with solid bottoms. The non-Newtonian rheology of mud was found to influence the ship’s resistance mainly at low speeds and when sailing through the mud layer. This article also shows that, when sailing through mud, the computed resistance at high speeds may be underestimated because of two effects, namely ‘water lubrication’ and ‘numerical ventilation’.

Apron and macro roughness as scour countermeasures downstream of block ramps

ABSTRACT A block ramp is a grade-control structure commonly used to stabilize the river bed with a high efficiency of energy dissipation. This study investigates the effects of using macro roughness and a downstream apron as scour countermeasures on the maximum scour depth downstream of block ramps in straight rivers. All the experiments have been carried out in a horizontal channel and in clear water conditions (no sediment transport). A wide range of hydraulic conditions including densimetric Froude numbers, water drop heights, and tailwater depths for three different rough ramp slopes (1 V:3 H, 1 V:5 H, and 1 V:7 H) was tested. The results have been compared with the results from a smooth ramp (no roughness), which has been considered as a reference. Results show that both macro roughness on the ramp and downstream aprons work well as scour countermeasures while as a comparison, using macro roughness (big-size stones) on the ramp is more effective than downstream aprons to reduce the scour depth. The other finding of this study shows that increasing the downstream apron decreases the length of the scour hole, consequently decreasing the volume of eroded material.

Buoyant miscible viscoplastic displacements in vertical pipes: Flow regimes and their characterizations

We experimentally study miscible displacement flows of a light Newtonian fluid by a heavy viscoplastic fluid, in a vertical pipe with a large aspect ratio (δ−1≫1). We use camera imaging, laser-induced fluorescence, and ultrasound Doppler velocimetry techniques, to capture and process data. Four dimensionless parameters, namely, the Reynolds (Re), Bingham (B), viscosity ratio (M), and densimetric Froude (Fr) numbers (or their combinations), mainly govern the flow dynamics. We identify and characterize three distinct flow regimes, including plug, separation, and mixing regimes, while we describe each regime's dynamics in detail, particularly in terms of the velocity and concentration fields as well as the displacement front velocity. In addition, we analyze the plug regime concerning the residual wall layers, the separation regime in terms of the separation dynamics, spatiotemporal separation zone, and viscoplastic layer thinning, and the mixing regime regarding the mixing index and macroscopic diffusion. Finally, we develop a simplified model to help delineate the flow regime classification, in the plane of Re/Fr2 and M.

Predicting maximum scour depth at sluice outlet: a comparative study of machine learning models and empirical equations

Estimating the maximum scour depth of sluice outlets is pivotal in hydrological engineering, directly influencing the safety and efficiency of water infrastructure. This research compared traditional empirical formulas with advanced machine learning (ML) algorithms, including RID, SVM, CAT, and XGB, utilizing experimental datasets from prior studies. Performance statistics highlighted the efficacy of the ML algorithms over empirical formulas, with CAT and XGB leading the way. Specifically, XGB demonstrated superiority with a correlation coefficient (CORR) of 0.944 and a root mean square error (RMSE) of 0.439. Following closely, the CAT model achieved a CORR of 0.940, and SVM achieved 0.898. For empirical formulas, although CORR values up to 0.816 and RMSE values of 0.799 can be obtained, these numbers are still lower than most ML algorithms. Furthermore, a sensitivity analysis underscored the densimetric Froude number (Fd) as the most crucial factor in ML models, with influences ranging from 0.839 in RID to 0.627 in SVM. Uncertainty in ML model estimates was further quantified using the Monte Carlo technique with 1,000 simulations on testing datasets. CAT and XGB have shown more stability than the other models in providing estimates with mean CORRs of 0.937 and 0.946, respectively. Their 95% confidence intervals (CIs) are [0.929–0.944] for CAT and [0.933–0.954] for XGB. These results demonstrated the potential of ML algorithms, particularly CAT and XGB, in predicting the maximum scour depth. Although these models offer high accuracy and higher 95% CI than others, the empirical formulas retain their relevance due to their simplicity and quick computation, which may still make them favored in certain scenarios.

A Novel Efficient Method of Estimating Suspended‐To‐Total Sediment Load Fraction in Natural Rivers

AbstractSediment transport load monitoring is important in civil and environmental engineering fields. Monitoring the total load is difficult, especially because of the cost of the bed load transport measurement. This study proposes estimation models for the suspended‐to‐total load fraction using dimensionless hydro‐morphological variables. Two prominent variable combinations were identified using the recursive feature elimination for support vector regression (SVR): (a) width‐to‐depth ratio, dimensionless particle size, flow Reynolds number, densimetric Froude number, and falling particle Reynolds number, and (b) flow Reynolds number, Froude number, and densimetric Froude number. The explicit relations between the suspended‐to‐total load fraction and the two combinations were revealed by two modern symbolic regression methods: multi‐gene genetic programming and Operon. The five‐variable SVR model showed the best performance. Clustering analyses using a self‐organizing map and Gaussian mixture model, respectively, identified the underlying relationships between dimensionless variables. Subsequently, the one‐at‐a‐time sensitivity of the input variables of the empirical models was investigated. The suspended‐to‐total load fraction is positively related to the flow Reynolds number and is inversely related to the densimetric Froude number. The models developed in this study are practical and easy to implement in other suspended sediment monitoring methods because they only require basic measurable hydro‐morphological variables, such as velocity, depth, width, and median bed material size.

Novel wall-attached multidirectional jets: Mathematical formulation, CFD modeling and experimental validation

Achieving a healthy and comfortable indoor air environment has long been an objective of human endeavors. For a novel ventilation system, the wall-attached multidirectional jet pattern, the mathematical model describing the jet trajectory, temperature and velocity changes was first proposed to predict airflow attachment and separation moments from walls, covering a densimetric Froude number range from 0.5 to 4.0. Subsequently, this mathematical model was solved using a comprehensive numerical methodology and validated by full-scale experiments, which demonstrated that this established model could accurately depict the wall-attached multidirectional jet flow, with controllable errors — 16.4% for jet trajectory and 8.2% errors for jet axis temperature — when maintaining a supply velocity of 0.2 m/s. Also, the flow characteristics within a room featuring internal heat sources were analyzed, varying air supply velocities from 0.1 to 0.8 m/s and jet angles from 0° to 60°. Finally, this flow pattern of the wall-attached multidirectional jet was further confirmed and reproduced through CFD modelling which illustrated temperature stratification ranging from 18.1 °C to 25.2 °C, similar to the pattern observed in displacement ventilation. Present research could be valuable for the development of wall-attached multidirectional jet and the establishment of a healthy indoor air environment in practical applications.

Local scour around submerged angled spur dikes under ice cover

Local scour is a phenomenon leading to the localized lowering of the channel bed due to the imbalance of sediment transport. As spur dikes protrude into the natural channels, local scour could be triggered. Accurate estimation of local scour around spur dikes is crucial for the effectiveness of erosion control and prevention and habitat enhancement measures. In the current study, the correlations between the maximum scour depth and the overtopping ratio, spur dike dimensions, ice cover roughness, and grain size of the bed material are investigated. Under both open channel and ice-covered flow conditions, a variety of experiments were done in a large-scale outdoor flume with different experimental setups. The results revealed that the scour depths around submerged spur dikes increased with increases in the densimetric Froude number and the decreases in the overtopping ratio and alignment angle. The maximum scour depth around a submerged angled vertical wall spur dike is significantly affected by the presence of an ice cover on the water surface, namely, the rougher the cover, the deeper the scour hole. Based on data collected from the laboratory experiments, an existing maximum scour depth estimation equation has been modified to consider the influence of the cover condition and the submergence level. The calculated results showed high accuracy in estimation of the measured data.

Insights into Wall-Jet Scouring Characterization with the Kinect Device: Benefits and Limitations

This study explores the use of the Kinect device for measuring Three-Dimensional (3D) Digital Elevation Models (DEMs) of wall-jet scour holes and its benefits and limitations. The Kinect device can accurately measure wall-jet scour holes in dry conditions with an accuracy of ±0.05 cm, but significant errors occur in the presence of water due to water surface refraction. A code based on Snell’s law was developed to correct these errors, resulting in good agreement with point gauge measurements with an accuracy of ±0.12 cm. The method is not recommended for tail-water depth ratios smaller than three and densimetric Froude numbers larger than 7.75 due to water surface fluctuations. The findings have important implications for improving the efficiency and accuracy of scour hole measurements in hydraulic engineering, particularly for submerged wall-jet scouring, and the developed code can also be applied to other optical measurement devices.

Immiscible displacement flows in axially rotating pipes

We experimentally study buoyant immiscible displacement flows in an axially rotating pipe, with varying flow parameters, such as the mean imposed flow velocity, density difference, pipe rotation speed, and pipe inclination angle. Via employing image processing and ultrasound Doppler velocimetry techniques, we analyze key flow features, including displacement regimes, interfacial instabilities, interfacial front velocities, and velocity and concentration fields. We find that immiscible displacement flows are distinguished by the emergence of one or two heavy fluid fronts, particularly depending on the rotation speed. Furthermore, our dimensional analysis reveals that the displacement flow is governed by four dimensionless parameters, including the Reynolds, densimetric Froude (or Archimedes), and Rossby numbers, as well as the pipe inclination angle. Using these dimensionless groups, we succeed in categorizing the main flow regimes as efficient and inefficient displacements. Moreover, we classify the interfacial regimes as stable, intermittently unstable, kinks, and separating interfacial patterns. Our analysis shows that the interfacial instabilities observed are indeed characterized by the Kelvin–Helmholtz instability. Our analysis of the velocity fields suggests remarkable differences between displacements in stationary and rotating pipes, especially in terms of the absence and presence of a countercurrent flow, respectively. Finally, our assessment of concentration fields using a Fourier transform approach provides a preliminary fundamental understanding of the characteristics of concentration waves and their corresponding amplitudes.

Local Scour around Side-by-Side Double Piers in Channel Bends under Ice-Covered Conditions—An Experimental Study

The pier scour process is normally intensified in the presence of an ice cover, which poses risks to the longevity and safety of bridges. In the present study, the impact of the densimetric Froude number, locations, and pier spacing of side-by-side piers on the local scour depth under ice-covered flow conditions were investigated based on clear water scour experiments in an S-shaped laboratory flume. The results demonstrated that the local scour at piers along the convex bank was more substantial than that along the concave bank when other factors stayed identical. The densimetric Froude number clearly has more impact on local scour at piers along the convex bank than that along the concave bank. Different from the mechanism of the pier scour in a straight channel, the scour depth around a pier along the convex bank in the S-shaped flume increases as the distance between two piers (or pier spacing) increases, while it decreases around the piers along the concave bank. Similar scour patterns were observed when the side-by-side piers were installed at different bend apex cross-sections. The maximum local scour depths at piers along the convex bank measured at different bend apex cross-sections were relatively unchanged when other influencing factors were held constant. However, the maximum scour depth around piers along the concave bank decreased as the bends increased toward downstream.

Investigation of heavy gas dispersion characteristics in a static environment: Spatial distribution and volume flux prediction

The unexpected release of hazardous gases due to natural/industrial accidents or man-made disasters can have tragic consequences. Compared with other gaseous pollutants, heavy gases easily accumulate in personnel working areas and are more difficult to eliminate through existing mechanical ventilation systems. In this paper, to improve air quality in enclosed environments, the spatial characteristics of heavy gas dispersion were obtained based on an analysis of the basic properties of heavy gas dispersion. Schlieren experiments and image processing were conducted with shadow correction and noise reduction. The computational fluid dynamics (CFD) was used for predicting the dispersion of heavy gas. The numerical simulations cover a wide range of the densimetric Froude number (Fr) from 6.45 to 25.79, and a best-fit relation of Fr and the height of heavy gas fountain was provided. The radial velocity, radial density and axial density at cross-section of heavy gas dispersion were normalized using the unified scaling law. The axial density reached a minimum value at a height of 10 times the inlet diameter. Further decreases in the inertial force or decreases in the buoyant force can reduce the dilution of heavy gas by ambient fluids, which means that heavy gases diffuse faster and farther. An alternative method for estimating volume flux during gas dispersion using Fr was also proposed.