Velocity Vector Field Research Articles

An airflow velocity field reconstruction method with sparse or incomplete data using physics-informed neural network

Velocity field is critical for creating a healthy and comfortable indoor environment but it is difficult to be measured experimentally. Due to the limitations of measurement instruments and site conditions, the captured data are usually sparse, incomplete, and low-dimensional, which limits further research on indoor velocity field. This study proposes a practical reconstruction and super-resolution method based on physics-informed learning that seamlessly integrates imperfect measured data and physical laws. The ability to reconstruct the airflow velocity field with imperfect data as the training dataset was validated using a three-dimensional and three-component airflow field generated with an axial fan. This method successfully recovered the airflow field from sparse or insufficient data (even when the amount of data was only 5 % of the original vector fields) and incomplete data. This indicates that the proposed method can reconstruct and super-resolve the airflow velocity vector fields from sparse, incomplete, and low-dimensional data.

Experimental heat transfer analysis of the effects of thermophysical properties on PCM at freezing conditions employing Particle Image Velocimetry

Cold chain logistics of fresh produce distribution are facing new challenges owing to the increasing consumer demand, often traveling long distances, and facing temperature variations during storage and distribution stages. Nowadays, solutions for improving food preservation by searching effective energy storage strategies, such as the use of PCM, are capturing more interest. Nevertheless, most studies are focused on organic PCM with ambient temperature melting points analyses. This leaves an important research gap in the experimentation of other types of eutectic materials with melting points below 0 °C, something of great interest for food preservation in the refrigeration industry. Velocity vector field and heat transfer via natural convection in a small cavity filled with two different PCM at below 0 °C conditions were experimentally investigated employing temperature measurements and Particle Image Velocimetry. In this regard, brine (18.8 wt%) and ethylene glycol-water (EGW) blend (30.5 wt%) PCM with equal melting temperature, but different thermophysical properties, were selected to compare their thermal behavior. It was investigated the effects of thermophysical properties such as density, specific heat capacity, thermal conductivity, and viscosity on melting front, heat transfer and flow patterns through the evaluation of liquid fraction, heat storage, Nu, Gr, Ra, and velocity field. The maximum Nu value for brine was almost the double of that of EGW. Pr values of EGW were in average 1.88 times higher than those of brine. Besides, Gr and Ra values of brine were in average 7.23 and 3.97 higher than those of EGW, respectively.

Velocity estimation of thunderstorm movement and dealiasing of single Doppler radar during convective events

The use of meteorological radars in monitoring current weather conditions is crucial regarding the observation of the evolution and dissipation of thunderstorms. Thus, Doppler velocities being measured in each radar scan and velocity vectors derived from Numerical Weather Prediction (NWP) models—that are usually not as highly resolving as radar scans—are combined, as a monitoring utility during the evolution of a convective weather event. The objective of this study is to develop a new method that allows the implementation of a thunderstorm movement velocity estimation technique combining block matching and optical flow techniques. This new method constitutes a nowcasting (NWC) application that enables the use of a single Doppler radar without the need of using NWPs. The method relies on the estimation of the thunderstorm movement vector velocities (Doppler velocity) for each constant altitude plan position indicator (CAPPI) and through correction for aliasing errors to obtain 3D vector velocity fields for convective systems. The performance of the method is evaluated for selected case studies of convective thunderstorms under different synoptic scale conditions over Greece, a geographical area with challenges in forecasting due to its sharp relief and the need for optimization of the use of radar products.

Characterizations of η–ρ-Einstein solitons in spacetimes and f(ℛ)-gravity

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then the integral curves generated by the velocity vector field [Formula: see text] are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient [Formula: see text]–[Formula: see text]-Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient [Formula: see text]–[Formula: see text]-Einstein soliton in [Formula: see text]-gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function [Formula: see text] remains invariant under the velocity vector [Formula: see text]. Also, for two models [Formula: see text] and [Formula: see text] ([Formula: see text] = constant and [Formula: see text] is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition.

Three dimensional analysis of the exhalation flow in the proximity of the mouth

The human exhalation flow is characterized in this work from the three-dimensional velocimetry results obtained by using the stereo particle image velocimetry (SPIV) measurement technique on the flow emitted from a realistic airway model. For this purpose, the transient exhalation flow through the mouth of a person performing two different breaths corresponding to two metabolic rates, standing relaxed (SR) and walking active (WA), is emulated and studied. To reproduce the flow realistically, a detailed three-dimensional model obtained from computed tomography measurements on real subjects is used. To cope with the variability of the experimental data, a subsequent analysis of the results is performed using the TR-PIV (time resolved particle image velocimetry) technique. Exhalation produces a transient jet that becomes a puff when flow emission ends. Three-dimensional vector fields of the jet velocity are obtained in five equally spaced transverse planes up to a distance of ▪ from the mouth at equally spaced time instants ▪ which will be referred to as phases (φ), from the beginning to the end of exhalation. The time evolution during exhalation of the jet area of influence, the velocity field and the jet air entrainment have been characterized for each of the jet cross sections. The importance of the use of realistic airway models for the study of this type of flow and the influence of the metabolic rate on its development are also analyzed. The results obtained contribute to the characterization of the human exhalation as a pathway of the transmission of pathogens such as SARS-CoV-2 virus.

Numerical simulation on hydrodynamics loads during tail rudder deployment procedure of supercavitating vehicle

In the state of navigating covered with liquid fluid, the hydrodynamics loads of the supercavitating vehicle during the deployment process of the tail rudder have an important impact on the stability analysis of the vehicle and the performance design of the tail rudder. In this study, a numerical simulation method for the deployment process of the rudder based on the overlapping grids technology and the concept of relative motion between grid domains is established. Combined with the design requirements of the supercavitating vehicle, the change law of hydrodynamics loads under different deployment rates, attack angles during the deployment process, before and after deployment of the tail rudder are researched in detail. Through the analysis of the velocity vector field, pressure field near the rudder and pressure coefficients of the rudder surface, the reasons for the changes of the hydrodynamic loads in different states are revealed. The results have important reference value for the research on the flow field characteristics of the deployment process of the tail rudder for the supercavitating vehicle, and provide an important design basis for the performance design of the tail rudder.

Error analysis of the linearized Crank-Nicolson FEM for the incompressible vector potential magnetohydrodynamic system

A linearized fully discrete Crank-Nicolson finite element scheme is proposed for solving the three-dimensional incompressible magnetohydrodynamic equations based on a magnetic vector potential formulation, where the magnetic induction is written to a rotation of magnetic vector potential. By using the MINI element and lowest order Nédélec edge element to approximate the velocity field and pressure of fluid and magnetic vector potential, respectively, the numerical solution of magnetic induction can preserve the exactly divergence-free condition in fully discrete level. Error estimates for the velocity field and magnetic vector potential are rigorously analyzed under some reasonable regularity assumptions of exact solution. Finally, numerical results are given to support the theoretical analysis.

Toward accurate and fast velocity quantification with 3D ultrashort TE phase-contrast imaging.

Traditional phase-contrast MRI is affected by displacement artifacts caused by non-synchronized spatial- and velocity-encoding time points. The resulting inaccurate velocity maps can affect the accuracy of derived hemodynamic parameters. This study proposes and characterizes a 3D radial phase-contrast UTE(PC-UTE) sequence to reduce displacement artifacts. Furthermore, it investigates the displacement of a standard Cartesian flow sequence by utilizing a displacement-free synchronized-single-point-imaging MR sequence (SYNC-SPI) that requires clinically prohibitively long acquisition times. 3D flow data was acquired at 3T at three different constant flow rates and varying spatial resolutions in a stenotic aorta phantom using the proposed PC-UTE, a Cartesian flow sequence, and a SYNC-SPI sequence as reference. Expected displacement artifacts were calculated from gradient timing waveforms and compared to displacement values measured in the in vitro flow experiments. The PC-UTE sequence reduces displacement and intravoxel dephasing, leading to decreased geometric distortions and signal cancellations in magnitude images, and more spatially accurate velocity quantification compared to the Cartesian flow acquisitions; errors increase with velocity and higher spatial resolution. PC-UTE MRI can measure velocity vector fields with greater accuracy than Cartesian acquisitions (although pulsatile fields were not studied) and shorter scan times than SYNC-SPI. As such, this approach is superior to traditional Cartesian 3D and 4D flow MRI when spatial misrepresentations cannot be tolerated, for example, when computational fluid dynamics simulations are compared to or combined with in vitro or in vivo measurements, or regional parameters such as wall shear stress are of interest.

Time-independent, paraxial and time-dependent Madelung trajectories near zeros

The Madelung trajectories associated with a wavefunction are the integral curves (streamlines) of its phase gradient, interpretable in terms of the local velocity (momentum) vector field. The pattern of trajectories provides an immediately visualisable representation of the wave. The patterns can be completely different when the same wave equation describes different physical contexts. For the time-independent Schrödinger or Helmholtz equation, trajectories circulate around the phase singularities (zeros) of the wavefunction; and in the paraxially approximate wave, streamlines spiral slowly in or out of the zeros as well as circulating. But if the paraxial wave equation is reinterpreted as the time-dependent Schrödinger equation, its Madelung trajectories do not circulate around the zeros in spacetime: they undulate while avoiding them, except for isolated trajectories that encounter each zero in a cusp singularity. The different local trajectory geometries are illustrated with two examples; a local model explains the spacetime cusps.

Influences of longitudinal ventilation and two same of obstacles placed upstream and downstream of the fire source on gas temperature distribution in a tunnel

In case of tunnel fire, vehicles can be blocked both upstream and downstream of the fire source. This paper presents a numerical study on influences of longitudinal ventilation and two same of obstacles placed upstream and downstream of the fire source on gas temperature distribution in a tunnel. The heat release rate, ventilation velocity and blockage ratio are varied. Results show that for a given ventilation velocity, as the blockage ratio increases, the inclination of fire plume gradually changes from the downstream to the upstream and the downstream smoke spread distance below the ceiling decreases. The Coanda effect (which refers to the adhesion of fluid on a convex surface) and the velocity vector field are introduced to explain the attachment of fire plume to the upstream obstacle. As the blockage ratio increases, the gas temperatures in the upstream of the fire source increase, while the gas temperatures in the downstream of the fire source decrease. The maximum temperature tends to decrease with increasing ventilation velocity. The effect of blockage ratio on maximum temperature is obvious for small ventilation velocities, while it becomes insignificant for large velocities. Then the maximum temperature model concerning the block effect and ventilation velocity are proposed.

Robust and fast stochastic 4D Flow Vector-field signature technique for quantifying composite flow dynamics from 4D Flow MRI: Application to left atrial flow in atrial fibrillation

4D flow MRI is an emerging imaging modality that maps voxel-wise blood flow information as velocity vector fields that is acquired in 7-dimensional image volumes (3 spatial dimensions + 3 velocity directions + time). Blood flow in the cardiovascular system is often complex and composite involving multiple flow dynamics and patterns (e.g., vortex flow, jets, stagnating flow) that occur and interact simultaneously. The spectrum of such complex flow dynamics is embedded in the velocity vector field dynamics derived from 4D Flow MRI. However, current flow metrics cannot fully measure high-dimensional vector-field data and embedded complex composite flow data. Instead, these methods need to break down the vector-field data into secondary scalar fields of individual flow components using fluid dynamics operators. These methods are gradient-based and sensitive to data uncertainties, and only focus on individual flow components of the overall composite flow, therefore potentially underestimating the severity of overall flow changes associated with cardiovascular diseases. To address these limitations, in MICCAI 2021, we introduced a novel comprehensive stochastic 4D Flow vector-field signature technique that works directly on the entire spatiotemporal velocity vector field. This technique uses efficient stochastic gradient-free interrogation of multi-million flow vector-pairs per patient to derive the patient's unique flow profile of the complex composite flow alterations and in real-time processing. The signature technique's probabilistic gradient-free formulation should allow for highly robust quantification despite inherent errors in 4D flow MRI acquisitions. Here, we extend the application of the 4D flow vector-field signature technique to the left atrium to analyze complex composite flow changes in patients with atrial fibrillation. In 128 subjects, we performed extensive sensitivity testing and determined that the vector-field signature technique is highly robust to typical sources of data uncertainties in 4D flow MRI: degradation in spatiotemporal resolution, added gaussian noise, and segmentation errors. We demonstrate the excellent generalizability of the stochastic convergence from the aorta to the left atrium and between different 4D Flow MRI acquisition protocols. We compare the robustness of our technique to existing advanced flow quantification metrics of kinetic energy, vorticity, and energy loss demonstrating a superior performance of up-to 14-fold. Our results show the potential diagnostic and clinical utility of our signature technique in identifying distinctly altered composite flow signatures in atrial fibrillation patients independent of existing flow metrics.

Numerical simulation of the combustion chamber and performance analysis of microalgae-based biodiesel/diesel-powered CI engine: An axisymmetric flow approach

This study presents a comprehensive analysis of the combustion chamber characteristics of a variable compression ratio compression ignition engine powered with microalgae biodiesel/diesel and their impact on engine performance and emissions. Both numerical and experimental methods have been employed to study the problem. A computational fluid dynamics method is used to study the axisymmetric flow in the cylindrical combustion chamber. Numerical simulations are obtained by solving time-independent steady, incompressible Navier-Stokes equation in a two-dimensional cylindrical coordinate system. Fluid flow characteristics, such as velocity vector field, streamlines, and pressure distribution are presented for different biodiesel blends. The velocity vector field analysis shows that biodiesel blends exhibit high velocities near the axisymmetric wall region. Swirl motion generated near the outer periphery of the combustion chamber influences the flow patterns. The experimental study involves the engine performance and its emissions. Due to the increase of biodiesel blend ratios in B20 (20% biodiesel and 80% diesel), B40 (40% biodiesel and 60% diesel), B50 (50% biodiesel and 50% diesel), and B100 (100% biodiesel), the brake thermal efficiency (BTE) decreases for a given brake power. B20, B100, and diesel exhibit 32.87%, 29.15%, and 34.45% BTE respectively. Higher brake-specific fuel consumption is obtained at no-load engine operations due to incomplete combustion of fuel. B20, B40, B50, and B100 show reductions of 16.04%, 20.85%, 25.66%, and 32.08% CO, respectively, as compared to diesel. HC emissions are very low for all biodiesel blends, while B20 exhibits a decrease of 7.35% HC compared to diesel. However, NOx emissions increase with the biodiesel blend ratio, with B100 exhibiting a 22.56% increase in NOx compared to diesel. Smoke emissions decrease with an increase in the biodiesel blend ratio, while B20 shows a 4.64% reduction. Results show that B20 offers a favorable balance between emissions reduction and engine performance. This study determines the biodiesel combustion characteristics and insights for the practical implementation of biodiesel blends in the transportation and energy sectors.

The influence of the lobed ejector on the horizontal axial tidal turbine and flow characteristic analysis

The power coefficient of the horizontal axis tidal turbine (HATT) is limited to 0.593 by the Betz limitation, however, the turbine's efficiency can be improved by incorporating a diffuser or lobe ejector. In this study. In this paper, water tunnel experiments were conducted to obtain the power and thrust coefficients of the original turbine and the turbine with a lobe ejector at different tip speed ratios (TSR). The accuracy of the numerical simulation was then verified by comparing it with the experimental results. The results reveal that the power coefficient of the turbine with a lobe ejector is up to 623% higher than that of the original turbine; however, it also leads to a negative thrust coefficient higher than the original turbine by 94.64%. To determine the optimal position of the lobe ejector, a numerical simulation of the turbine with a diffuser, which consists of a single inner lobe channel, was conducted under the same conditions. The findings demonstrate that the power coefficient of the diffuser is 123% higher than that of the original turbine, and the negative thrust is higher at 26.64%, significantly less than that of the lobe ejector. This indicates that the effect of the lobe ejector largely depends on the ejector effect of the outflow channel. Subsequently, to investigate the ejection effect of the lobe ejector, various aspects, including the vortex structure of the flow field, pressure coefficient of the blade section, velocity vector field of the rotor's center plane, and wake velocity curve, were analyzed. Moreover, the changes in the position of flow separation points along different sections of the blade from the blade root to the blade tip, the variation of the spiral structure of the blade root vortex, the evolution of the velocity vector field in the low-velocity region of the turbine's wake flow field and the alteration of the velocity curve of the wake flow field were further examined.

Pumping the brakes on RNA velocity by understanding and interpreting RNA velocity estimates.

RNA velocity analysis of single cells offers the potential to predict temporal dynamics from gene expression. In many systems, RNA velocity has been observed to produce a vector field that qualitatively reflects known features of the system. However, the limitations of RNA velocity estimates are still not well understood. We analyze the impact of different steps in the RNA velocity workflow on direction and speed. We consider both high-dimensional velocity estimates and low-dimensional velocity vector fields mapped onto an embedding. We conclude the transition probability method for mapping velocity estimates onto an embedding is effectively interpolating in the embedding space. Our findings reveal a significant dependence of the RNA velocity workflow on smoothing via the k-nearest-neighbors (k-NN) graph of the observed data. This reliance results in considerable estimation errors for both direction and speed in both high- and low-dimensional settings when the k-NN graph fails to accurately represent the true data structure; this is an unknown feature of real data. RNA velocity performs poorly at estimating speed in both low- and high-dimensional spaces, except in very low noise settings. We introduce a novel quality measure that can identify when RNA velocity should not be used. Our findings emphasize the importance of choices in the RNA velocity workflow and highlight critical limitations of data analysis. We advise against over-interpreting expression dynamics using RNA velocity, particularly in terms of speed. Finally, we emphasize that the use of RNA velocity in assessing the correctness of a low-dimensional embedding is circular.

Machine Learning Approach for Flow Fields Over a Circular Cylinder Based on Particle Image Velocimetry Measurements

Flow around a circular cylinder has been experimentally studied at Reynolds numbers of Re = 4 × 103 and Re = 8 × 103 using Particle Image Velocimetry (PIV). The Artificial Neural Network (ANN) method, a subset of machine learning, has been implemented to capture complex flow patterns and relationships within data to estimate the experimental results. Experimental and estimated results regarding instantaneous contour graphics for streamwise (u) and cross-stream (v) velocity components with velocity vector fields have been compared. Furthermore, drag coefficient (CD) and Strouhal number (St) have been validated. In flow features, instantaneous streamwise velocity components indicated a lower pressure zone in the wake region of the cylinder owing to flow separation. The ANN method has predicted the positions of the minimum velocity clusters with 3.2 % for Re = 8 × 103 as a mean relative error compared to the experimental results. Moreover, the average relative difference of 10.82 % between the experimental and the estimated results for the clusters with the minimum streamwise velocity components has been attained at Re = 4 × 103 and Re = 8 × 103. Regarding the cross-stream velocity components, the mean relative difference values have been observed for the clusters with maximum values of 6.83 % and the clusters with minimum values of 3.66 %. Drag coefficients have also been obtained, and these values are very close as CD = 1.090 and CD = 1.084 for PIV and ANN methods at Re = 8 × 103, respectively. Furthermore, St values of the circular cylinder are St = 0.216 at Re = 4 × 103 and St = 0.211 at Re = 8 × 103, which agree with the results of the literature.

Analysis of Ozone Production Reaction Rate and Partial Discharge Power in a Dielectric-Barrier Acrylic Chamber with 60 Hz High-Voltage Electrodes: CFD and Experimental Investigations

We present a synergistic investigation into ozone production dynamics within a high-voltage reaction chamber operating from 1.75 kV to 10 kV, at 60 Hz. The output ozone concentration dependence on a 60 Hz root mean square high-voltage for an air input pressure of 0.1 bar was measured using an ultraviolet-based sensor. A finite-element numerical model of the reaction chamber was developed and used for obtaining the reaction rate for the generated ozone for each level of input voltage and the internal distribution of the gas vector velocity field and ozone concentration. An expression to estimate the ozone reaction rates from the measured ozone concentrations was also derived. It was then compared and validated using the developed computational model. Therefore, in this work, a formula is proposed to estimate the ozone production by providing the experimental reaction rates. The other objective is to analyze the hysteresis phenomena observed in the partial discharges’ levels in the reaction chamber and in the output ozone concentration. Finally, we establish a connection between the total power of partial discharges and the consequent ozone production, clarifying the role of cumulative partial discharges’ power levels in governing the resultant output ozone concentration.

Predicting virtual acoustic source properties in a stereophonic sound-field using the coherence properties of the acoustic velocity vector field

It is widely understood that the mutual coherence properties, including the relative phase, of the channel signals in a stereophonic sound reproduction system significantly influence a listener’s perception of the resulting virtual acoustic sources. This talk presents a quantitative model that combines wavefield coherence theory, the acoustic velocity vector, and known perceptual cues to make quantitative predictions of the perceived location and spatial spread of horizontally placed stereophonic virtual sources as a function of the mutual coherence properties of the stereo channel signals. It will be seen that a small number of parameters, including the intrinsic coherence time of the source signal, the relative phase of the channels, and perceptual averaging times, account for most of the variation in the cues that determine virtual acoustic source localization. The output of the coherence model will be compared to the results of earlier experiments, including ones investigating the “phaseyness” problem and precedence effect. Additionally, the potential of this model to account for reverberation in the listening space will be demonstrated.

Experimental and numerical study of flow structure in a model of distal anastomosis of femoral artery

Physical and numerical simulation of the steady blood flow was carried out focusing on the distal end-to-side anastomosis of a graft to the femoral artery. The angle between the graft and the artery was 60°. Two Reynolds numbers were considered that corresponded to the physiological range when estimated using the average (Re = 240) and maximum (Re = 1640) blood flow through the human femoral artery over a single cardiac cycle. The experiment included flow visualization and measurements of instantaneous vector fields of velocity using an optical SIV (Smoke Image Velocimetry) technique. Direct numerical simulation (DNS) was employed for numerical study. The distribution of blood flows through the branches of the main artery was 80 % in the antegrade and 20 % in the retrograde direction. Main patterns in the distribution of velocity and its root-mean-square fluctuations in both branches of the main artery were revealed. Flow separation regions were observed in the branching area. Additionally, secondary flows (Prandtl’s vortices of the first kind) were revealed. The flow in the branching area was still laminar Re = 240, while the signs of transition to turbulence were documented within separation regions at Re = 1640. Distributions of streamwise and circumferential components of skin friction vector were estimated. In some regions of flow, these components were shown to be comparable in size. The location of regions was revealed in which the friction magnitude in the considered flow regimes was lower than the one in an unimpaired artery (developed laminar flow in a circular pipe) at appropriate Reynolds numbers.

Experimental and numerical study on explosion behavior of hydrogen-air mixture in an obstructed closed chamber

The flame evolution, flame front speed and pressure dynamics are investigated through experimental and simulation analysis. The effects of the obstacle on velocity vector field and vorticity field are analyzed. The results indicate that a rise in blockage ratio causes a rapid rise in flame front speed. The turbulent combustion, flame backflow, and overpressure are significantly enhanced as obstacle distance and blockage ratio increase. The jet flame results in a surge in flame front speed, and a momentary increase in pressure rise. The flame-vortex interaction causes strong turbulent combustion and a rapid overpressure increase. The overpressure displays regular periodic fluctuations and high-frequency oscillations with high amplitudes, which can be attributed to the dynamic and impulsive responses. The vortex results in the formation of a curled flame with high localized vorticity. The backflow flame, exhibiting an extremely high vorticity magnitude, is induced by the overall pressure gradient and flow field recirculation.

Analytical and HYDRUS solutions for exfiltration through inclined seepage faces

Steady, phreatic flows in wedge-shaped aquifers subtended by a tilted bedrock and discharged through a hillslope seepage face are studied by 2-D and 1-D analytical models for fully saturated and a FEM numerical model for saturated-unsaturated flows. The curvilinear triangle in the hodograph domain is conformally mapped onto a reference half-plane where the Hilbert problem is solved for two holomorphic functions. Free boundaries and sizes of seepage face segments are found for given flow rates of the incident flows seeping downhills. The approximate (Dupuit-Forchheimer) water table is conjugated with a saturated triangle where the exact 1-D solution gives a constant Darcian velocity as a function of the angles of the soil wedge. HYDRUS-2D model gives the distributions of the pressure head, volumetric moisture content, isotachs, vector-fields of velocity and other characteristics of seepage described by the Richards-Richardson equation. Applications to stability of exfiltrating zones near tailwaters of earth dams are discussed.